diff --git "a/59A0T4oBgHgl3EQfN_9T/content/tmp_files/load_file.txt" "b/59A0T4oBgHgl3EQfN_9T/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/59A0T4oBgHgl3EQfN_9T/content/tmp_files/load_file.txt" @@ -0,0 +1,1546 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf,len=1545 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='02154v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='AP] 5 Jan 2023 Generalised Young Measures and characterisation of gradient Young Measures Tommaso Seneci Abstract Given a function f ∈ C(Rd) of linear growth, we give a new way of representing accumulation points of ˆ Ω f(vi(z))dµ(z), where µ ∈ M+(Ω), and (vi)i∈N ⊂ L1(Ω, µ) is norm bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We call such representa- tions "generalised Young Measures".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' With the help of the new representations, we then characterise these limits when they are generated by gradients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' when vi = Dui for ui ∈ W 1,1(Ω, Rm), via a set of integral inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Contents 1 Intro 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Terminology and symbols .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 5 2 Generalised Young Measures on separable compactifications 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Generalised Young Measures as generalised objects .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Parametrized measures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': 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+page_content='1 Non-separability of the space of quasi-convex functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Characterisation of Gradient Young Measures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Inhomogenization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 40 A Appendix 48 2 1 Intro 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Terminology and symbols For a vector v ∈ Rm, we write |v| = ��m i=1 v2 i otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Given a function f : X → Z and W an arbitrary set, we call graph(f) ≡graphX(f) = {(x, f(x)) ∈ X × Z such that x ∈ X}, graphX×W (f) = {(x, w, f(x)) ∈ X × W × Z such that x ∈ X, w ∈ W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We say that a function f : Rd → R has p growth if there is C > 0 such that |f(z)| ≤ C(1 + |z|p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The identity matrix is indicated by 1, or 1d ∈ Rd×d if we need to specify the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The Lebesgue measure is indicated by dx, or Ln depending on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The set of finite Borel d-vector measures is indicated by M(Ω, Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For E ⊂ M(Ω), E+ is the set of positive Borel measures that belong to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For a measure µ ∈ M(Ω)+, we write Lp(Ω, µ, Rd) to mean the space of µ-measurable functions f : Ω → Rm such that ˆ Ω |f(x)|pdµ(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For µ ∈ M(Ω, Rd), we write its restriction to a µ-measurable set E ⊂ Ω µ E : A Borel set �→ µ(E ∩ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If µ ∈ M(Ω) and f ∈ L(Ω, µ, Rd), we call fdµ the measure in M(Ω, Rd) defined by U �→ ˆ U fdµ, where U runs through all µ-measurable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If X is any space, the Dirac delta is indicated, for x ∈ X, by ˆ X f(y)dδx(y) = f(x), where f : X → Rd is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let X be a metric space, µ ∈ M+(Ω) and (fj)j∈N ⊂ Lp(Ω, µ, Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We say that the se- quence (fj)j is p-equi-integrable (or simply equi-integrable if p is clear from the context) if it is norm bounded and lim k↑∞ sup j∈N ˆ |fj|p>k |fj|pdµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 3 The set of test functions is D(Ω, Rd) = C∞ c (Ω, Rd) = {f : Ω → Rd : f is infinitely differentiable and has compact support}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We do not insist on the topology this space is endowed with, as it is standard and nowhere used in the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For the derivative of a function u ∈ L1(Ω, Rm) we mean the matrix-valued distribution Du = [∂jui]i,j ∈ (D(Ω, Rm)∗)n such that ˆ Ω ui∂jφdx = −⟨∂jui, φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The Sobolev space of functions with integrable derivatives is W 1,1(Ω, Rm) = {u ∈ L1(Ω, Rm) : Du ∈ L1(Ω, Rm×n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The set of functions of bounded variation is BV (Ω, Rm) = {u ∈ L1(Ω, Rm) : Du ∈ M(Ω, Rm×n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For a function u ∈ BV (Ω, Rm) we can write Du = ∇udLn Ω + Dsu, Dsu = Dju Ju + Dcu where ∇udLn is the absolutely continuous part, Dju is the jump part concentrated on a n − 1 rectifiable set Ju, and Ds is the Cantor part, which is absolutely continuous with respect to Hn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For a function U ∈ BV (Ω, Rm), we call BVU(Ω, Rm) = � u ∈ BV (Ω, Rm) : there is a sequence uj ∈ D(Ω, Rm) such that uj weak* in BV −−−−−−−−→ u − U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The set of special functions of bounded variation is SBV (Ω, Rm) = {u ∈ BV (Ω, Rm) : Dsu = Dju, or equivalently Dcu = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Introduction Young Measures were first introduced by Young in [You37] to study the minima of integral energies of the form inf �ˆ 1 0 f(u(t), u′(t))dt : u ∈ C1([0, 1]), u(0) = a, u(1) = b, ∥u′∥∞ ≤ K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The author wanted to understand what conditions on f would guarantee the existence of a minimizing curve u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Young had the intuition that, for an extremely general class of functions f, minimising sequences always converge to a "generalised" curve t �→ (u(t), νt) where ν is a probability measure on the image of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This translates to the following equality inf u(0)=1,u(1)=b ˆ 1 0 f(u(t), u′(t))dt = lim j ˆ 1 0 f(uj(t), u′ j(t))dt = ˆ 1 0 ˆ R f(u(t), y)dνt(y)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' So the question of the existence of a minimiser can be reformulated as to whether such objects are gradients of a curve or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' νt might fail to be a gradient when the minimizing sequence oscillates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Young’s original work focused on the case n = 1 and was carried out via functional analytic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This approach was later extended in [Bal89, BL73] to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We call these generalised functions "oscillation Young Measures".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The method developed by Young is not powerful enough to tackle problems arising in modern mathematics, as it can only handle sequences (vj)j∈N that are bounded in L∞ rather than in some Lebesgue space Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The first attempt to well represent generalised limits of integrable functions is due to DiPerna and Majda, [DM87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Functions vj : Ω → Rd are seen as Dirac deltas on the product space Ω × Rd, which is subsequently compactified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' An accumulation point, in the sense of these new generalised functions, is then found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Such an accumulation point is a measure defined on an abstract compactification of Ω × Rd, and as such, it is not clear how to represent it in the original, non-compact, space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In [AB97], an explicit formula for such accumulation points was obtained for a class of integrands that grow "nicely" infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In what follows, we give a general formula for describing Young Measures for a large class of integrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The construction of Young Measures follows mainly the work by DiPerna and Majda [DM87] and lecture notes taken from a class given by Kristensen [Kri15], see also [Rin18], chapter 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This generalisation is based on the canonical way of constructing Haus- dorff compactifications starting from continuous functions, see [CC76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' A small reduction lemma gives a clearer, and somehow geometrical interpretation of such limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This formula captures oscillations at infinity, which are now let occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We also prove a few structure theo- rems that relate different compactifications and Young Measures representations to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This generalisation of Young Measures is then applied to study extensions and variations - within the class of functions of bounded variations BV (Ω, Rm) - of energies that depend on gradients u �→ ˆ Ω f(Du(x))dx, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16) where u ∈ D(Ω, Rd), Ω ⊂ Rn is a bounded domain, and f ∈ C(Rm×n) has linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Given any such f, there is no way to extend (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16) to the class BV so that such extension is 5 continuous with respect to sequential weak* convergence in C0(Ω, Rm×n)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We can however find an extension which is lower semi-continuous for certain fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In [Mor52], Morrey established the equivalence of lower semi-continuity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16) to a condition named "quasi-convexity", which can be written as a Jensen-type inequality ˆ Ω f(z + Dφ(x))dx ≥ |Ω|f(z) ∀φ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='17) The original result by Morrey works in the setting of weak* convergence in W 1,∞(Ω, Rm), and it was subsequently extended to the case W 1,p(Ω, Rm), 1 ≤ p < ∞ and weak convergence in [AF84], for positive integrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' As for signed integrands, the same result was proven in [BZ90] and it is one of the first examples where Young Measures are employed for proving lower semi-continuity in the space of gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To be more specific, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='17) can be rephrased as a Jensen-type inequality for measures of the form {νx : νx = Dφ(x)#dLn Ω, φ ∈ C∞ c (Ω, Rm)}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='18) where νx acts on f in the following way: ˆ Ω f(z + Dφ(x))dx = ˆ Ω ˆ Rm×n f(z + w)dνx(w)dx ≡ ˆ Ω ⟨νx, f⟩dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The lower semi-continuity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16) becomes a functional analytic inequality of the form ˆ Ω ���νx, f⟩dx ≥ ˆ Ω f(Du(x))dx and Du(x) = ˆ Rm×n zdνx, and in this case we call x �→ νx a "Gradient Young Measure".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This class can be seen as the closure of the set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='18) in the weak* topology of measures over the graph of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The opposite is also true and was proven for the first time in [KP91, KP94], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' every measure-valued function x �→ νx, for which a Jensen’s type inequality holds against quasi-convex functions of suitable growth, is the limit of a sequence of gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The aforementioned results hold in the setting of weak convergence in W 1,p, 1 ≤ p < ∞ and weak* convergence in W 1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This is a natural condition to assume when p > 1, but not when p = 1, as the Lebesgue space L1(Ω, Ln) is not reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, a bounded sequence in L1(Ω, Ln) can concentrate and converge to measures that are singular with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In terms of gradients, the closure of W 1,1(Ω, Rm) so that its unit ball is weak* compact is the set of functions of bounded variations BV (Ω, Rm), precisely the set of functions whose derivatives are measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This concentration phenomenon is exclusive of the case p = 1, and so regards integrands that have linear growth at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It turns out, as proven in [ADM92], that when f has linear growth and it’s quasi-convex, the integral functional u �→ ´ Ω f(∇u)dx, f ≥ 0 is still lower semi-continuous in BV (Ω, Rm) with respect to the weak* topology, but there is a deficit of mass when gradients concentrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Letting f be so that f ∞(z) = lim t→∞,zn→z f(tzn) t (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='21) 6 exists for all zn → z, t → ∞, the lower semi-continuous envelope of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16) in the space BV (Ω, Rm), with respect to sequential weak* convergence, is, for f non-negative, u �→ ˆ Ω f(∇u(x))dx + f ∞ � Dsu |Dsu|(x) � d|Dsu|(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In this case, a Young Measure formulation of the Jensen’s-type inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='17) has to take into account the singular part of Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In the spirit of the previous results, one is tempted to prove a duality-type characterisation of Young Measures with concentrations and quasi-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Differently from the case without concentration, here we assumed f ∞ to exist as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='21), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' However, as shown in [Mül92], quasi-convex functions can oscillate at infinity, meaning that f ∞(z) may not exist for some z ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This suggests that to obtain a Jensen-type inequality and characterisation result for gradient Young Measure in the case p = 1, it is necessary to specify a compactification at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The characterisation for gradient Young Measures when p = 1 has been already obtained on the so-called "sphere compactification" - functions for which f ∞(z) exists for all z - see [KR10a, KR10b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' After showing that the class of quasi-convex functions of linear growth is too big to be included within any separable compactification, we reprove the characterisation result for gradient Young Measures on separable compactifications of quasi-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This restricts the number of quasi-convex functions to be considered at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' However, it is also inevitable because a compactification containing all quasi-convex functions would be so big that its topology would fail to be metrisable and separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 7 2 Generalised Young Measures on separable compactifications In this section, we construct generalised Young Measures and provide a new geometric rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Concentration is let "oscillate with different amplitudes at infinity".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To do so, we embed a space of functions into a bigger compact set and subsequently use the theory of Hausdorff compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Generalised Young Measures as generalised objects Generalised Young Measures are objects that were known to exist since Majda and Di Perna [DM87], and have been used in a few instances, see for example [FK10, KR96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' However, their existence per se does not give enough clarity on their properties, and so makes it hard to work with such objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We give a new interpretation and geometric representation that better captures oscillation and concentration effects that occur in limits of the form lim j ˆ Ω f(vj(x))dµ(x), where vj ∈ Lp(Ω, µ, Rd) is a norm-bounded sequence and f ∈ C(Rd) has p-growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Under these assumptions, it is easy to see that, up to a subsequence, f ◦ vj converges to a measure ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' mathematically this means that ˆ Ω f(vj(x))φ(x)dµ(x) → ˆ Ω φ(x)dν(x) for all φ ∈ C0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It’s clear that ν = ν(f) is a linear function of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' What is not clear is how such dependence can be represented in terms of µ and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Without a clear representation, it is not possible to set up a system of calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This section is dedicated to working out a geometric interpretation of the relation between ν and f, which we will then call Young Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We will mainly concentrate on the more interesting and harder case of p = 1 and vj = Duj gradients, where concentration effects create rather complicated structures, and cannot in general be separated from oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 A function f : Rd → R is said to have p-growth if there is a constant C ≥ 0 such that |f(z)| ≤ C(1 + |z|p) ∀z ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' When p = 1, we say that such functions have linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Before proceeding with formal definitions, we give a heuristic interpretation of Young Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' When vj → v strongly in L1(µ) then the limit Young Measure is trivial, meaning that ˆ Ω f(vj(x))φ(x)dµ(x) → ˆ Ω f(v(x))φ(x)dµ(x) for all f ∈ C(Rd) of linear growth and φ ∈ C0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This is a simple consequence of the Vitali convergence Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 48 (or the generalised dominated convergence theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' When strong L1 convergence fails, only two things can go wrong: 8 oscillation - vj oscillates around µ-almost every point x ∈ E ⊂ Ω with µ(E) > 0, and generates a probability distribution on the target space f(vj(x)) ⇝ ⟨νx, f⟩ = ˆ Rd f(z)dνx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' concentration - |vj| concentrates to a measure 0 ̸= λ ∈ M+(Ω) - equivalently (x, vj(x)) concentrates to the boundary of some compactification of Ω×Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Around λ-almost every point, vj(x) goes to infinity and its "support" collapse to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' That is to say, f(vj(x)) ⇝ �f(z) |z| with |z| ≫ 0 � ∼ ˆ ∂K f ∞(w)dν∞ x (w), where K is some compactification containing Rd that extends f to f ∞ on the remainder of Rd within K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Parametrized measures In order to construct Young Measures, we regard functions as maps from a domain Ω into the set of probability measures over a target space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Ordinary functions f : Ω → Rd, x �→ f(x) are embedded into maps Ω → M+ 1 (Rd), x �→ δf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Preliminary to the construction, we introduce two basic concepts that are at the core of this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Let X and Z be locally compact, separable metric spaces and λ ∈ M+(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' A map ν : X → M+(Z) is said to be λ-measurable if for each φ ∈ C0(Z) the function x �→ ⟨ν(x), φ⟩ is λ-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We shall often write the measure-valued map ν as a parametrized measure (νx)x∈X, where νx : = ν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Given a measure ν on a product space X × Z, it can always be decomposed as a product of its projection onto X and its cross section on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='3 (Disintegration of measures) Let X and Z be compact metric spaces and denote by π: X × Y → Z the projection mapping onto the first coordinate π(x, y) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For u ∈ M+(X ×Z) and λ = π#ν ∈ M+(X) (the pushforward of ν via π) there exists a unique λ- measurable parametrized measure (ηx)x∈X, ηx ∈ M+ 1 (Z) such that for all φ ∈ C(X), ψ ∈ C(Z) we have ⟨ν, φ ⊗ ψ⟩ = ˆ X ⟨ηx, ψ⟩φ(x)dλ(x) = ˆ X ˆ Z ψ(z)dηx(z)φ(x)dλ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For a proof of this result see [AFP00], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In this case we write ν = ηxdλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Generalized Young measures In what follows, we show how to obtain a good representation of Young Measures on general compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The procedure is adapted from some lecture notes taken from a homonym course given by Jan Kristensen at the University of Oxford, [Kri15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Some of the results can also be found in [Rin18], chapter 12, where they are only proven on the sphere compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 9 Throughout this section, Ω ⊂ Rn is open and bounded, µ ∈ M+(Ω) and (vj)j ⊂ Lp(Ω, µ, Rd) is a bounded sequence, 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Assume vj ⇀ v in Lp when 1 < p < ∞ or vj ⇀∗ v in C0(Ω, Rd)∗ when p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Given a continuous integrand Φ: Ω × Rd → R satisfying the p-growth condition |Φ(x, z)| ≤ C(1 + |z|)p ∀(x, z) ∈ Ω × Rd, we seek to represent limits of �´ Ω Φ(x, vj(x))dx � j as j → ∞, possibly passing through suitable subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='4 For each j, the map Φ acts on the graph of vj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Φ(x, vj(x)) = Φ ◦ (x, vj(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Therefore, we look for the limiting distribution of (x, vj(x)) as j → ∞, and more precisely the Φ-moment of this limiting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Morally speaking, since (Φ(·, vj))j is bounded in L1(Ω, µ), (Φ(·, vj)dµ)j is bounded in M(Ω) ≂ C(Ω)∗, so by the abstract compactness principle Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 48, there exists a limit measure which depends on the integrand Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='3 Functional analytic setup Let z ∈ Rd �→ ˆz = z 1+|z| ∈ Bd be a homeomorphism Rd �→ Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Define the class of functions of p-growth in the z variable to be Gp = Gp(Ω, Rd) := � Φ ∈ C(Ω × Rd) : sup (x,z) |Φ(x, z)| (1 + |z|)p < ∞ � , and for Φ ∈ Gp put (TΦ)(x, ˆz) := (1 − |ˆz|)pΦ � x, ˆz 1 − |ˆz| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then T : Gp → BC(Ω × Bd) is an isometric isomorphism provided Gp is normed by ∥TΦ∥∞ and BC(Ω × Bd) by ∥ · ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The inverse operator is (T −1Ψ)(x, z) = (1 + |z|)pΨ � x, z 1 + |z| � , where Ψ ∈ BC(Ω × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The dual operator T ∗ : BC(Ω×Bd)∗ → G∗ p is again an isometric isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We are interested in the limits of ´ Ω Φ(x, vj(x))dx for Φ ∈ Gp and may define ξvj ∈ G∗ p by ξvj(Φ) := ˆ Ω Φ(x, vj(x))dx, Φ ∈ Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Note ∥ξvj∥ = sup ∥Φ∥Gp ξvj(Φ) = ˆ Ω (1 + |vj|)pdx so (ξvj) is a bounded sequence in G∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' But G∗ p ≂ BC(Ω × Bd)∗, and because BC (hence Gp) is not separable we do not necessarily have sequential compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We must restrict the integrands Φ to a separable subspace of Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Hausdorff compactification In this subsection, we present how to construct a compactification of the space X = Ω × Rd from a family of bounded and continuous functions F ⊂ BC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Roughly speaking, such compactification is a compact set eF X that contains X as a dense subset and on which all f ∈ F admit a continuous extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The idea behind such construction is to look at the graph of each f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because of the boundedness assumption, each function has its image contained in a closed bounded interval of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Therefore, the graph is embedded into a closed subset of an (infinite-dimensional) hypercube, which is compact in the product topology by Tychonoff theorem, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Preliminaries on Hausdorff compactifications Most of the results will be stated without proof, which can be found in chapters 1 and 2 of [CC76] and in chapter 4 of [Fol99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We first define three classes of functions that are rich enough to determine the topological structure of their domains: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='5 Consider a family of functions F ⊂ BC(X), we say that F separates points from closed sets if for each C ⊂ X closed and x ∈ X \\ C there exists f ∈ C(X) such that f(x) ̸∈ f(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, we define what a compactification of a topological space is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='6 A compactification of X is a compact Hausdorff space αX and an embedding α: X → αX (continuous and so that α−1 : αX → X exists and is continuous) such that α(X) is dense in αX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It is useful to remark that because α is continuous, any function f ∈ C(αX) can be restricted to a continuous function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Indeed, f ◦ α is the composition of a bounded continuous function with continuous function, and thus it belongs to BC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On the other hand, because α(X) is dense in αX, each f ∈ C(αX) is uniquely recovered from f ◦ α ∈ C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Given a family F ⊂ BC(X) that separates points from closed sets, there is a canonical way of generating a compactification αX on which every f ∈ F has a continuous extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='7 To each family F ⊂ BC(X) that separates points from closed sets, we associate a canonical embedding eF : X → Πf∈F � inf f, sup f � , x �→ {f(x)}f∈F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' eF X := eF(X) is a compactification of X The above theorem is a direct implication of Tychonoff’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' When F ⊂ BC(X) is a family that separates points from closed sets, then eF : X �→ Πf∈F � inf f, sup f � is open and continuous, and so it is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' However, the map eF makes sense even if F does not separate points from closed sets, and eF X is always a compact subset of Πf∈F � inf f, sup f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='8 Let F ⊂ BC(X) be a family that separates points from closed sets and eF X its induced compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Each f ∈ F embeds into C(eF (X)) in an obvious way and admits a unique extension f ∈ C(eF X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Every y ∈ eFX is an accumulation point of eF (X), so we can find a net yλ = Πf∈F f(xλ) such that yλ → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' A way of extending f ∈ F is by setting f : eF X → R, y � = lim λ Πf∈F f(xλ) � �→ f(y) = lim λ f(xλ), which does not depend on the choice of xγ as far as limγ f(xγ) = limλ f(xλ) for each f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Suppose that we have given a family F and its associated compactification eF X, and we consider the compactification of F ∪ {f}, where f ∈ BC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We expect the latter compacti- fication to be bigger than the former, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' to be a space where all the previous extensions can be further extended to continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='9 Given two compactifications αX and γX of X, we say that αX ≥ γX if there exists a continuous function f : αX → γX such that f ◦ α = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, we write αX ≂ γX if αX ≥ γX and γX ≥ αX, or equivalently if f : αX → γX is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In the following paper, we will sometimes refer to a generic compactification K without specify- ing the underlying family generating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The reason why is stated by the following astonishing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='10 Given a compactification αX of X, there exists a family F ⊂ BC(X) that separates points from closed sets such that eF X ≂ αX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Representation of compactifications Consider a family F ⊂ BC(X) that separates points from closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' According to the Hausdorff compactification theory (see above subsections), its induced compactification can be written as a subset of the hypercube Πf∈F � inf f, sup f � , where the sides of this cube are as many as the functions f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because F can be uncountable, its compactification could be hard to deal with from an analytical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We seek a better representation of such space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The idea behind the following result is that if we know the limits of functions f, g ∈ BC(Ω, R), we also know the limits of f n + gm, n, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='11 Let F be a family of functions f : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We call A(F) the algebra generated by F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' A(F) = {f n + gm : f, g ∈ F, n, m ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='12 (Representation theorem) Let F ⊂ BC(X) be a closed sub-algebra that separates points from closed sets and let F ′ ⊂ F be such that A(F ′) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then eF ′X is a compactification of X and eF ′X ≂ eF X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let eF ′X be the (formal) compactification of F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Clearly eF X ≥ eF ′X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove the opposite inclusion, we must find a continuous function T : eF ′X → eF X such that T ◦eF ′ = eF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fix y = limλ Πf∈F ′f(xλ) ∈ eF X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If g ∈ A(F ′), limλ g(xλ) exists and coincides on all nets xγ such that y = limγ Πf∈F ′f(xγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, let g ∈ A(F ′) and find a sequence {fn}n∈N ⊂ A(F ′) such that fn → g uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because ∥fn∥∞ is bounded, so is {limλ fn(xλ)}n∈N, and so we can extract a subsequence {fnk}k∈N such that limk limλ fnk(xλ) = L ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fix ε > 0 and N ∈ N such that ∥fnN − g∥∞ < ε 3 and find ˜λ ∈ Λ such that |fnN(xλ) − limλ fnN(xλ)| < ε 3 for all λ ≥ ˜λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Finally |g(xλ) − L| ≤ |g(xλ) − fnN(xλ)| + |fnN (xλ) − lim λ fnN(xλ)| + | lim λ fnN(xλ) − L| < ε for all λ ≥ ˜λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' the net g(xλ) converges to L ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, by the uniqueness of limλ g(xλ), we conclude that the original sequence {limλ fn(xλ)}n∈N converges to L, which also proves that the limit does not depend on the particular net xγ as far as limλ f(xλ) = limγ f(xγ) for all f ∈ F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This shows that the map T : eF ′X → eF X, y = lim λ Πf∈F ′f(xλ) �→ Ty = lim λ Πf∈F f(xλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' is a well-defined isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because its inverse is the projection map πF ′|T(eF ′X), which is continuous and open, then T ◦ eF ′ : x �→ Πf∈F ′f(x) �→ Πf∈F f(x) = eF (x) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove that eF ′X is a compactification of X, we notice that F separates points, as so does F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If U ⊂ X is open, so is eF ′(U) = T −1(eF (U)), and eF ′ is an injective continuous open map, thus it is an embedding onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='3 Restriction on non-linearity For the sake of this work, it is important that the set of functions we work with is separable (has a countable dense set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let F ⊂ BC(Ω×Bd) be a closed separable algebra that separates points from closed sets and let eF Ω × Bd be its compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because eF Ω × Bd is a compact Hausdorff space we have the following isometric isomorphism of its dual C(eF Ω × Bd)∗ ≂ M(eF Ω × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='13 If F ⊂ BC(X) is separable, so is C(eFX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let {fn}n∈N be dense in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By the Stone-Weierstrass theorem, the algebra generated by {1} ∪ {fn}n∈N ⊂ C(eF X) is dense in C(eF X), and so C(eF X) is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Because C(eF Ω × Bd) is separable we also have the abstract sequential compactness prin- ciple Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 48 on its dual M(eF Ω × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let T −1F ⊂ Gp the corresponding 13 algebra (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' ×p) on the set of continuous functions with p-growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' There is an isometric isomorphism T −1F ˜T≂ C(eF Ω × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By the Riesz representation theorem, we can write its adjoint as ˜T ∗ : M(eF Ω × Bd) → (T −1F)∗, ν �→ � Φ �→ ( ˜T ∗ν, Φ) = (ν, ˜TΦ) = ˆ eF Ω×Bd ˜TΦdν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' � Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='14 Let X, Z be completely regular Hausdorff spaces and let F ⊂ BC(X) and G ⊂ BC(Z) be closed sub-algebras that separate points from closed sets and contain the constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The following spaces are all isometrically isomorphic to each other C(eF ∪GX × Z) ≂ C(eF X × eGZ) ≂ A(C(eF X) × C(eGZ)) ≂ A(F ∪ G), where each f ∈ F and g ∈ G is extended to a function on the product space by keeping constant the other variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, F ∪ G ⊂ BC(X × Z) separates points from closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The proof of the above lemma is a straightforward application of the Stone-Weierstrass the- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We underline that it’s important to take families F, G defined exclusively on each respective space, and the theorem is false if we instead add f = f(x, y) that is not of the form above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Morally speaking, what the previous lemma says is that on product spaces it is enough to work with the compactifications in each coordinate separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, their dual elements (measures) can be tested against tensor products of functions that depend on each variable independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='4 Representation of Young measures Let Ω ⊂ Rn be open and bounded and G′ ⊂ BC(Bd) and F ′ ⊂ BC(Ω) be closed separable sub- algebras that separate points from closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let T −1F = A ⊂ Gp the corresponding algebra, with respect to the ×p product, in the set of functions having p-growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' F is isometrically isomorphic to C(eFΩ × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' With abuse of notation, we are going to call T the isomorphism between A and C(eF Ω × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To each function u ∈ Lp(Ω, Rd) we associate an elementary Young measure ξu ∈ A∗ by setting ξu : A → R, Φ �→ ˆ Ω Φ(x, u(x))dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, consider a bounded sequence {un}n∈N ⊂ Lp(Ω, Rd), supn ∥un∥p ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' As Φ ∈ Gp, the sequence of elementary Young measures is also bounded ∥ξun∥ = sup ∥Φ∥≤1 ���� ˆ Ω Φ(x, un(x))dµ(x) ���� = ˆ Ω (1 + |un|)pdµ ≤ µ(Ω) + Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because of the isomorphism A∗ ≂ C(eF Ω × Bd)∗ ≂ M(eF Ω × Bd), there exists a subsequence, relabelled in the same way, and ν ∈ A∗ such that ξun ⇀∗ ν in A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Set L := (T ∗)−1ν ∈ M(eF Ω × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 14 We now study the measure L to find a better representation for ν ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let ψ ∈ C(eF Ω×Bd), we immediately notice that L ∈ M+(eF Ω × Bd) as a consequence of the following equality ≪ ν, T −1ψ ≫= lim n ˆ Ω (T −1ψ)(x, un(x))dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because the constant functions belong to G′, we can plug T −1ψ = φ(x)(1+ |z|)p, φ ∈ C(eF ′Ω) into the previous equation and obtain the identity ˆ eF Ω×Bd φ(x)dL(x, z) = lim n ˆ Ω φ(x)(1 + |un(x)|)pdµ(x) = ˆ eF ′ φ(x)dλ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 14, the projection π: eF Ω × Bd → eF ′Ω is well-defined, and we can write ˜λ = π#L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Note that hereby ˜λ ∈ M+(eF ′(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Find the unique ˜λ-measurable parametrized family {˜νx}x∈eF ′Ω such that νx ∈ M+ 1 (eG′Bd) ˜λ-almost every x and ⟨L, Φ⟩ = ˆ eF ′Ω ⟨˜νx, Φ(x, ·)⟩d˜λ(x) ∀ Φ ∈ C(eF Ω × Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For any φ ∈ C0(Ω) take Φ = φ(1 − | · |)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We compute ˆ eF ′Ω φ(x)⟨˜νx, (1 − | · |)p⟩d˜λ(x) = ˆ eF Ω×Bd φ(x)(1 − |z|)pdL(x, z) = lim n ˆ Ω (φ1Rd)(x, un(x))dµ(x) = ˆ Ω φ(x)dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because µ ∈ C0(Ω)∗ we immediately conclude that µ = ⟨˜νx, (1 − | · |)p⟩˜λ Ω, where µ is extended on eF ′Ω by µ(E) ≡ µ(e−1 F ′ (E)), E ⊂ eF ′Ω Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Apply the Radon- Nikodym theorem and write ˜λ = ˜λ µdµ + ˜λs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' From the previous identification, we get � ⟨˜νx, (1 − | · |)p⟩ ˜λ µ = 1 µ − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' ⟨˜νx, (1 − | · |)p⟩ = 0 ˜λs − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=', where the second condition implies that ˜νx(eG′(Bd)) = 0 ˜λs-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' the measures are concen- trated on the boundary ∂eG′(Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On eG′(Bd) we have 0 < (1 − |z|)p ≤ 1, whereas |z| = 1 on ∂eG′(Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular � ˜λ µ = 1 ⟨˜νx,(1−|·|)p⟩ ≥ 1 µ − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' ˜νx(∂eG′(Bd)) = 1 ˜λs − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 15 Now let φ ∈ BC(Rd) and define ⟨νx, φ⟩ = ˜λ µ(x) ˆ eG′(Bd) (1 − |z|)pφ � z 1 − |z| � d˜νx(z) = ˜λ µ(x) ˆ Bd(1 − |z|)pφ � z 1 − |z| � d(eG′)#˜νx(z) In particular νx ∈ M+ 1 (Rd) and {νx}x∈Ω is µ-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let λ = ˜νx(∂eG′(Bd))˜λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then λ ∈ M+(eF ′Ω) and it decomposes into λ =˜νx(∂eG′(Bd)) ˜λ µµ + ˜νx(∂eG′(Bd))˜λs =˜νx(∂eG′(Bd)) ˜λ µµ + ˜λs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For λ-almost every x ∈ eF ′Ω and for ψ ∈ C(∂eG′Bd) set ⟨ν∞ x , ψ⟩ = 1 ˜νx(∂eG′(Bd)) ˆ ∂eG′(Bd) ψ(z)d˜νx(z), hereby ν∞ x ∈ M+ 1 (∂eG′(Bd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For each Φ ∈ A, its recession function is defined to be the restriction φ∞ = φ|eF ′Ω×∂eG′(Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Finally, we obtain the formula ⟨L, TΦ⟩ = ˆ eF ′(Ω) ⟨˜νx, TΦ(x, ·)⟩d˜λ = ˆ eF ′Ω ˆ eG′(Bd) TΦd˜νx \uf8eb \uf8ec \uf8ed ˜λ µdµ + =0 ���� d˜λs \uf8f6 \uf8f7 \uf8f8 + ˆ eF ′Ω ∂eG′(Bd) TΦd˜νx (˜νx(∂eG′(Bd))d˜λ) = ˆ Ω ⟨νx, Φ(x, ·)⟩dµ + ˆ eF ′Ω ⟨ν∞ x , Φ∞(x, ·)⟩dλ and the representation of the Young Measure as the triple ν = � {νx}x∈Ω, λ, {ν∞ x }x∈eF ′Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' where νx ∈ M+ 1 (Rd) for µ-almost every x ∈ Ω, λ ∈ M+(eF ′Ω), and ν∞ x ∈ M+ 1 (∂eG′(Bd)) for λ-almost every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 16 We say that un converges in the sense of Young measures to ν, and write un Y p(µ,eA) −−−−−−→ ν or just un Y p(µ,A) −−−−−→ ν, where µ is the measure that "regulates and weights" oscillation and concentration of un, and eA is the compactification at infinity, generated by the family A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' From this point onwards the family F ′ in Ω will always be the set of functions C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='5 Properties of generalised Young Measures and connection to Young Measures on the sphere Here we show how the above construction generalises the more classical setting of Young Measures on the sphere, see [Res68] for the original idea behind their representation, and [AB97] for their modern implementation in the calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We then study how the new representation for generalised Young Measures behaves geometrically, and its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' As a reminder, we state here the definition of integrands with a regular recession at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='15 The set of integrands admitting a regular recession at infinity is Ep(Ω, Rd) = � Φ ∈ C(Ω × Rd) : lim t→∞ Φ(x, tz) tp ∈ R locally uniformly in (x, z) ∈ Ω × Rd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because we intend to generalise the theory of Young Measures on functions with a regular recession, we need to extend the above class and at the same time preserve good topological properties of such a larger class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To do so, consider countably many functions gi ∈ BC(Bd) and their representations as integrands of p-growth gi( z 1+|z|)(1 + |z|)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We are interested in understanding how to represent, in a simple way, Young Measures relative to the compactifi- cation generated by Ep ∪ {gi( z 1+|z|)(1 + |z|)p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In the language of Hausdorff compactifications, set G′ = C(Bd) ∪ {gi, i ∈ N} and F ′ = C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because C(Bd) ⊂ G′, the closure of the algebra generated by either family is separable, separates points from closed sets and contains the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Call F = G′ ∪ F ′, and without loss of generality, we can assume that ∥gi∥ ≤ 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16 C(eF Ω×Bd) is isometrically isomorphic to C(Ω×graph(gi)), where (gi): Bd → [−1, 1]N, z �→ (gi(z))i∈N and the topology on the target space is the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It is metrised by d(z, w) = |z − w| + � i 2−i|gi(z) − gi(w)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 14 it is enough to prove that C(egi,i∈NBd) ≂ C(graph(gi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 12 provides the isomorphism A(C(Bd) ∪ {gi, i ∈ N}) = A(1, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' , zd, gi, i ∈ N), and we conclude by noticing that (1, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' , zd, gi, i ∈ N)(Bd) is homeomorphic to graph(gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The topological equivalence between such metrics and the product topology is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 17 When gi ∈ C(Bd), then C(eF Ω × Bd) ≂ C(Ω × Bd), and therefore we recover the usual sphere representation for the compactification induced by Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This means the obvious, that we can add functions that have a regular recession and we still obtain the same space (up to homeomorphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By definition, the compactification of Bd can be represented by the space Γ of sequences {zn}n∈N ⊂ Bd such that zn → z ∈ Bd and gi(zn) converges for all i ∈ N, and two such sequences {zn}n∈N and {wn}n∈N are identified provided lim n |zn − wn| + � i 2−i|gi(zn) − gi(wn)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='17 We call egi,i∈N the compactification, and ∂egi,i∈N = egi,i∈N\\graph(gi, i ∈ N), we can write the triple Young measure as ν = � {νx}x∈Ω, λ, {ν∞ x }x∈Ω � , where νx ∈ M+ 1 (Rd) for µ-almost every x ∈ Ω, λ ∈ M+(Ω), and ν∞ x ∈ M+ 1 (∂egi,i∈N) for λ-almost every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that ∂egi,i∈N is an abuse of notation and refers to the boundary of the embedded space within the compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' So far we have constructed compactifications by "glueing" gis on top of the functions z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' , zd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' that is to say on top of the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It is sometimes useful to iterate this argument, to stack another countable family {fi, i ∈ N} on top of the compactification egi,i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This process gives the same compactification as if we were considering the two families at once, as the following lemma shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='18 Let egi,i∈N be a compactification of Bd and fi ∈ BC(Bd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then egi,fi,i∈N ≂ graphgraph(gi)fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This is a trivial consequence of the fact that {(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' , zd, gi(z), fi(z)), z ∈ Bd} ={(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' , zd, w, z) : w = gi(z), y = fi(z), z ∈ Bd} (extending fi to constant in the variable w) ={(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' , zd, w, z) : y = fi(z, w), w = gi(z), z ∈ Bd} =graphgraph(gi)(fi), z ∈ Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ A standard application of the disintegration lemma yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='19 Consider a compactification egi,fi,i∈N and ν∞ ∈ M(∂egi,fi,i∈N), then ν∞ = P(zn)n∈Nd˜ν∞ where ˜ν∞ ∈ M(∂egi), (zn)n ∈ ∂egi, and P(zn)n is a probability measure defined on the space of subsequences (zni)i of (zn)n so that fi(zni) converges for all i ∈ N (with sequences being equivalents if all the limits are).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 18 For the case of oscillating functions fi, we also write the compactification as efiX ≡ efi and the convergence as vj Y p(µ,fi) −−−−−→ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' When working with the sphere compactification, we will simply write vj Y p(µ,Bd) −−−−−−→ ν, or just vj Y p(µ) −−−−→ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also, because here we mainly consider the case p = 1, we omit the superscript p in Y p and write vj Y (µ,efi) −−−−−→ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We now study the relation of Young Measures with respect to different compactifications and different underlying measures µ ∈ M+(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Using Chacon Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 48, we can prove the following structure theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='20 Let vj Y (µ,efi,i∈N) −−−−−−−→ (νx, λ, ν∞ x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then for all ψ ∈ C0(Rd), we have ψ(vj) ⇀ ⟨νx, ψ⟩ weakly in L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because ψ(vj) ∈ L∞ then is the sequence is equi-integrable and there is a subsequence that converges weakly in L1 to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because ψ∞ = 0, testing against φ ∈ D(Ω) we get ˆ Ω φψ(uj)dµ → ˆ Ω φvdµ = ˆ Ω φ⟨νx, ψ⟩dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ We can improve the above weak convergence result to show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='21 Let uj Y (µ) −−−→ � νx, 0, N/A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For every a ∈ L1(Ω, µ) such that a > 0 µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' and for all ψ ∈ C0(Rd) we have ψ �uj a � a ⇀ ⟨νx, ψ � a(x) � ⟩a(x) in L1(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' As expected, this implies that oscillations do not depend on the particular compactification chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Before proving the above results we show the following uniform approximation result: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='22 Let µ ∈ M+(Ω) and a ∈ L1(Ω, µ), a > 0 µ-almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' There exists an ∈ L1(Ω), 0 < an < a, so that an(x) ∈ Q for all x ∈ Ω and ∥an − a∥∞ + ���� a an − 1 ���� ∞ n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let an = � k∈N,k≥1 χa−1� [ k n, k+1 n ) � k n, where N is the set of strictly positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because an ≤ a then an ∈ L1 and it also assumes countably many values at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also |an(x) − a(x)| ≤ 1 n so it converges uniformly to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Furthermore 1 = k n k n ≤ a(x) an(x) ≤ k+1 n k n = k + 1 k and so a an converges uniformly to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Now we can prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Using the previous approximation, we write, for 1-Lipschitz ψ : Rd → R, ˆ Ω ψ �uj a � a = ˆ Ω =I � �� � ψ �uj a � a − ψ �uj an � a + =II � �� � ψ �uj an � a − ψ �uj an � an +ψ � uj an � an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The first two terms are bounded by |I| ≤ ˆ Ω a ���� uj a − uj an ���� = ˆ Ω |uj| ����1 − a an ���� ≤ sup j ∥uj∥ ����1 − a an ���� ∞ |II| ≤ ˆ Ω � 1 + |uj| an � |a − an| ≤ (1 + sup j |uj|) ����1 − a an ���� ∞ , which goes to 0 as n → ∞ uniformly in j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' As for the third term, calling Ek = a−1� [ k n, k+1 n ) � , we can use dominated convergence theorem to pass to the limit lim j ˆ Ω ψ �uj an � an = lim j � k ˆ Ek ψ � uj k n � k n = � k ˆ Ek ⟨νx, ψ � k n � ⟩k n = ˆ Ω ⟨νx, ψ � · an � ⟩an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Another application of the dominated convergence theorem will let us conclude the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ We can finally conclude with a structure theorem regarding concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='23 Consider two separable algebras (that separate points from closed sets) A and B of G1 and let a ∈ L1(Ω, µ), a > 0 µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let vj ∈ L1(Ω, µ) be a sequence so that vj Y (µ,A) −−−−→ � νx, λν, ν∞ x � and vj a Y (a dµ,B) −−−−−−→ � ηx, λη, η∞ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then νx = � a(x) � # ηx, λν = λη = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, decomposing ν∞ x = P ν (zn)nd˜ν∞ x and η∞ x = P η (zn)nd˜η∞ x , where ˜ν∞ x and ˜η∞ x are the projections on the sphere according to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='19, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 18, then ˜ν∞ x = ˜η∞ x λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' with vj Y (µ,Bd) −−−−−→ � νx, λ, ˜ν∞ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For all ψ ∈ C0(Rd) we have that ψ(vj) is equi-integrable so that ψ(vj) ⇀ ⟨ηx, ψ⟩ in L1(µ) and ψ �vj a � ⇀ ⟨νx, ψ⟩ in L1(adµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, identify vj with its subsequence and find Ek so that vj ⇀ v in L1(Ek, µ) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By inner approximation, we can assume that all such E′ ks are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consider now the sequence vjχEk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then vjχEk Y (µ,A) −−−−→ � ηxχEk + δ0χEc k, 0, N/A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 19 we then have, for φ ∈ C0(Ω) and ψ ∈ C0(Rd), because Ω \\ Ek is open, ˆ Ω φ⟨νx, ψ⟩a(x)dµ = lim j ˆ Ω φψ �vj a � adµ = lim j ˆ Ω\\Ek φψ �vj a � adµ + ˆ Ek φψ �vj a � adµ = ˆ Ω\\Ek φ⟨νx, ψ⟩adµ + ˆ Ek φ⟨ηx, ψ � · a � ⟩adµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, let φ ∈ C(Ω), then lim j ˆ Ω φ|vj|dµ = ˆ Ω ⟨νx, | · |⟩φdµ + ˆ Ω φdλν = lim j ˆ Ω φ ���vj a ��� adµ = ˆ Ω ⟨ηx, | · | a(x)⟩φa(x)dµ + ˆ Ω φdλη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Using the previous part we conclude that λν = λη = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Finally, let f ∈ C(∂Bd) and extending by 1-homogeneity we obtain that lim j ˆ Ω φf(vj)dµ = ˆ Ω φ⟨νx, f⟩dµ + ˆ Ω φ⟨ν∞ x , f ∞⟩dλ = ˆ Ω φ⟨νx, f⟩dµ + ˆ Ω φ ˆ ˆ f ∞dP ν (zn)d˜ν∞ x dλν = ˆ Ω φ⟨νx, f⟩dµ + ˆ Ω φ ˆ f ∞d˜ν∞ x dλν = ˆ Ω φ⟨νx, f � a(x) � ⟩a(x)dµ + ˆ Ω φ⟨η∞ x , f ∞⟩dλ = ˆ Ω φ⟨νx, f⟩dµ + ˆ Ω φ ˆ f ∞dP η (zn)d˜η∞ x dλη = ˆ Ω φ⟨νx, f⟩dµ + ˆ Ω φ ˆ f ∞d˜η∞ x dλη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Notice that the previous identification with the concentration angle measure fails if we only consider Γ = A ∩ B which does not necessarily generate the sphere compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This is so because sequences (uj)j can concentrate around values of a that are measure-discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' However, equality holds true if a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 21 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='24 Following the assumptions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 20, if a = 1, Γ = A ∩ B and writing ν∞ x = P ν (zn)nd(γν)∞ x and η∞ x = P η (zn)nd(γη)∞ x , where γη and γν are the projections onto the compactification generated by Γ, then (γν)∞ x = (γη)∞ x λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This is proven similarly at the end of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 20 and testing against functions belonging in A(Γ) and using the decomposition of angle Young Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Next, we show that the lack of concentration is equivalent to the equi-integrability of the generating sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='25 Let (vj)j ∈ L1(Ω, µ, Rd) be so that vj Y (µ,efi,i∈N) −−−−−−−→ � νx, λ, ν∞ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then the sequence (vj)j∈N is equi-integrable if and only if λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover vj → v strongly in L1(Ω, µ, Rd) if and only if λ = 0 and νx = δv(x) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because λ does not depend on the compactification (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 20), we can apply the same theorem from the sphere compactification, [Rin18], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 347, lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='14 and [Rin18], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 348, corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='15, to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Before stating the next two structure results, we prove that T −1 is a bounded operator from Lip(efi,i∈N) to Lip(Rd), provided the compactification is generated by Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In this case, by Lip(Rd) we mean the weighted norm ∥f∥Lip(Rd) := ∥Tf∥∞ + sup x̸=y |f(x) − f(y)| |x − y| = ���� f 1 + | · | ���� ∞ + sup x̸=y |f(x) − f(y)| |x − y| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' As for the compactification, the metric is always intended as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='26 Let efi,i∈N be a separable compactification metrised by the usual metric, where fi ∈ Lip(Rd) are normalised so that ∥f∥Lip(Rd) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then sup x̸=y |g(x) − g(y)| |x − y| ≤ 5Lip(Tg, efi,i∈N) for all maps g: Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Without loss of generality assume that ∥Tg∥Lip ≤ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' for all |x|, |y| < 1, ����g � x 1 − |x| � (1 − |x|) − g � y 1 − |y| � (1 − |y|) ���� ≤ |x − y| + � i 2−i|Tfi(x) − Tfi(y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 22 Then |g(x) − g(y)| = ���� g(x) 1 + |x|(1 + |x|) − g(x) 1 + |x|(1 + |y|) + g(x) 1 + |x|(1 + |y|) − g(y) 1 + |y|(1 + |y|) ���� = |g(x)| 1 + |x| ���1 + |x| − 1 − |y| ��� + (1 + |y|) ���� g(x) 1 + |x| − g(y) 1 + |y| ���� ≤|x − y| + (1 + |y|) ����� x 1 + |x| − y 1 + |y| ���� + � i 2−i ����Tfi( x 1 + |x|) − Tfi( y 1 + |y|) ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For all i we have that ������Tfi � x 1 + |x| � − Tfi � y 1 + |y| ����� = ���� fi(x) 1 + |x| − fi(y) 1 + |y| ���� , and therefore after multiplying by 1 + |y| we obtain ���� fi(x) 1 + |x| � 1 + |x| + (|y| − |x|) � − fi(y) ���� = ����fi(x) − fi(y) + fi(x) 1 + |x|(|y| − |x|) ���� ≤|fi(x) − fi(y)| + |fi(x)| 1 + |x||x − y| ≤ 2|x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ We now show that the above lemma allows us to test Young Measures on Lipschitz compact- ifications against Lipschitz functions of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='27 (Kantorovich semi-norm) Let X be a metric space and µ ∈ M(X), then the (formal) Kantorovich norm of µ is ∥µ∥K = sup ∥φ∥Lip≤1 ˆ X φdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The above formula induces a pseudo-distance between measures by setting d(µ, η)K = ∥µ − η∥K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It turns out that this is indeed a metric on the positive cone of non-negative Measures, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='In particular, by taking Ψ ∈ Lip(efi,i∈N) and the pull-back T −1 we deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='28 Let efi,i∈N be a separable compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then every ν ∈ Y (efi,i∈N, µ) is defined by testing it against Lipschitz functions of the form φ ⊗ ψ, where ∥φ∥Lip(Ω) ≤ 1, ∥ψ∥Lip(Rd) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For this reason, we remind once again of the norm we will be using on the space efi,i∈N throughout this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='29 We say that efi,i∈N is a Lipschitz compactification if each fi is Lipschitz continuous, and renormalised so that Lip(Tf) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The norm on Lip(efi,i∈N) will always be ∥g∥Lip(efi,i∈N) := sup x∈efi,i∈N |g(x)| + sup x̸=y |g(x) − g(y)| defi,i∈N(x, y) 23 where defi,i∈N(x, y) = |x − y| + � i 2−i|Tfi(x) − Tfi(y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To conclude this subsection, we state decomposition results for Young Measures regarding oscillation and concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Originally proven in the context of the sphere compactification, [KR19], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 29, we here extend them to general compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='30 Let vj ∈ L1(Ω, µ) so that vj Y (efi,i∈N,µ) −−−−−−−→ � νx, λ, ν∞ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We can write vj = oj + cj, where oj ∈ L1(Ω, µ) is equi-integrable, oj Y (efi,i∈N,µ) −−−−−−−→ � νx, 0, N/A � and cj ∈ L1(Ω, µ) so that cj Y (efi,i∈N,µ) −−−−−−−→ � δ0, λ, ν∞ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The converse is also true, for each such sequence oj, cj as above, their sum converges to the former Young Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' A standard diagonal argument gives us kj ↑ ∞ so that oj = vjχ|vj| ≤ kj is equi- integrable and generates oj Y (efi,i∈N,µ) −−−−−−−→ � νx, 0, N/A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then, letting cj = vj − oj, for η ∈ C(Ω), Tψ ∈ C(efi,i∈N) we have ˆ Ω η(ψ(cj) − ψ(vj)) = ˆ |vj|≤kj η(ψ(0) − ψ(oj)) + ˆ |vj|>kj η(ψ(vj) − ψ(vj)) = ˆ Ω η(ψ(0) − ψ(oj)) → ˆ Ω η(ψ(0) − ⟨νx, ψ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Writing ψ(cj) = � ψ(cj) − ψ(vj) � + ψ(vj) and letting j → ∞ we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ We remark here that, when considering certain subsets of Y (for example Young Measures generated by gradients, see next section), oj and cj might generate different types of Young Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The next lemma is an extension of the previous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='31 Let vj ∈ L1(µ) and wj ∈ L1(µ) generate vj Y (µ,efi,i∈N) −−−−−−−→ � δv(x), λη, η∞ x � and wj Y (µ,efi,i∈N) −−−−−−−→ � νx, λν, ν∞ x � with λη ⊥ λν, for some v ∈ L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then the sum of the sequence generates vj + wj Y (µ,efi,i∈N) −−−−−−−→ � δv(x) ∗ νx, λν + λη, k∞ x � , where k∞ x = � ν∞ x λν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' η∞ x λη-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Write wj = oj + cj as in the previous lemma and put bj = vj − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We claim that bj + cj Y (µ,efi,i∈N) −−−−−−−→ � δ0, λν + λη, k∞ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' No oscillation is a consequence of the fact that bj + cj → 0 in µ-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, let φ ∈ C(Ω), ∥φ∥Lip ≤ 1 and Ψ ∈ efi,i∈N, ∥TΦ∥Lip ≤ 1 with Ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let Eν and Eη be sets where λν and λη are concentrated, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For ε > 0 find Cν ⊂ Eν, Cη ⊂ Eη compact sets and Oν ⊃ Cν, Oη ⊃ Cη open sets such that λη(Ω \\ Cη) + λν(Oη) + λν(Ω \\ Cν) + λη(Oν) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consider a function ρ ∈ C(Rn) with χCη ≤ ρ ≤ χOη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We write ˆ Ω φ � Ψ(bj + cj) − Ψ(bj) − Ψ(cj) � ( =I ���� ρ + II � �� � 1 − ρ)dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We estimate the first guy by lim sup j |I| ≤ lim sup j 2 ˆ Ω ρ|bj| = 2 ˆ Ω ρdλν ≤ 2λν(Ω ∩ Oη) ≤ 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Similarly for the second term lim sup j |II| ≤ lim sup j 2 ˆ Ω (1 − ρ)|cj| ≤ 2λη(Ω \\ Cη) ≤ 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' But the first term converges to 0 and therefore we conclude for the representation of Young Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='6 Terminology We dedicate this part to clarifying the terminology of Young Measures adopted throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='32 Given a separable algebra A of Gp that separates points from closed sets, a p-Young measure is a triple ν = � (νx)x∈Ω, λ, (ν∞ x )x∈Ω � where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' (νx)x∈Ω is µ-measurable and νx ∈ M+ 1 (Rd) µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We call it the oscillation Young measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' λ ∈ M+(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We call it the concentration measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' (ν∞ x )x∈Ω is λ-measurable and ν∞ x ∈ M+ 1 (∂eA) λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We call it the concentration angle Young measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' the moment condition ˆ Ω ˆ Rd |z|pdνx(z) < ∞ must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The collection of all such triples is denoted by Y p = Y p(Ω, µ, eA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Where obvious from the context, we will not specify the domain Ω, the measure µ, the family A or the target space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also, when λ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' when there is no concentration, there is no point in specifying the compactification we are working with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' From every triple ν = � νx, λ, ν∞ x � one can construct the measure L ∈ C(eF Ω × Rd) and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='33 The following equality holds: Y p = T ∗ � L ∈ M+(eF Ω × Bd)+ : ˆ eF Ω×Bd φ(x)(1 − |ˆz|)pdL = ˆ Ω φ(x)dµ(x) ∀φ ∈ C(eGΩ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular Y p is a weak* closed and convex subset of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let L ∈ M+(eF Ω × Rd)+ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It was already shown that T ∗L ∈ Y p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On the other side, if ν = � νx, λ, ν∞ x � , we let L = νxdµ + ν∞ x dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Testing against a test function φ = φ(x) that only depends on x, ˆ eF Ω×Bd TφdL = ˆ eF Ω×Bd φ(x)(1 − |ˆz|)pdL = ˆ Ω φdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Conclude by noticing that the above characterisation amounts to Y p = T ∗ \uf8eb \uf8ed � φ∈C(eF Ω) � L ∈ M+(eF Ω × Bd)+ : ˆ eF Ω×Bd φ(x)(1 − ˆz)pdL = ˆ Ω φdµ �\uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='34 With a straightforward adaptation of a classical argument for the sphere com- pactification, one can prove that given any Young Measure of the above form ν ∈ Y (Ω, µ, efi,i∈N) with supp(µ Ω) = Ω and µ non-atomic, then there is a sequence of smooth functions uj ∈ D(Ω, Rd) such that uj Y (Ω,µ,efi,i∈N) −−−−−−−−−→ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We do not transcribe the proof here because it won’t be used at any point in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 26 We now give a formal definition of what elementary Young Measures are, as a way to embed functions and measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='35 Let µ ∈ M+(Ω) and v ∈ Lp(Ω, µ, Rd), 1 ≤ p ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The corresponding elementary p-Young measure is ξv := � (δv(x))x∈Ω, 0, N/A � ∈ Y (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' When p = 1, we extend the definition to l ∈ M(Ω, Rd) by setting, for l = l µdµ + ls,µ, ξl := �� δ l(x) µ(x) � x∈Ω , |ls,µ|, � δ ls,µ |ls,µ| � x∈Ω � ∈ Y (µ, ∂Bd) Note that for Φ ∈ Ep we have ≪ ξv, Φ ≫= ˆ Ω Φ(x, v(x))dν(x) and ≪ ξl, Φ ≫= ˆ Ω Φ � x, l µ(x) � dµ(x) + ˆ Ω Φ∞ � x, ls,µ |ls,µ|(x) � d|ls,µ|(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In general, there is no clear way of defining elementary Young Measures on compactifications that are larger than the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We will see later, however, that this can be done in very specific cases when we have more structure on A and more information on the measure l ∈ M(Ω, Rd) we are trying to embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='36 (Barycentre of a p-Young measure) Let ν = � (νx)x∈Ω, λ, (ν∞ x )x∈Ω � ∈ Y p(µ, A), where A is so that eA ≥ Bd (in the sense of compactifications, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We call its barycentre ν = � νx 1 < p < ∞ νxµ + ν∞ x λ p = 1, which is the following quantity νx = ˆ Rd zdνx(z) ν∞ x = ˆ ∂eA zdν∞ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In the above definition, "≥" is the ordering over the set of Hausdorff compactifications of a topological space (see the subsection on Hausdorff compactifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, the barycen- tre does not depend on the compactification, as far as eA ≥ Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Indeed z = [zj]j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=',d extended to eA coordinate-wise, and so ˆ ∂eA zdν∞ x = ˆ ∂Bd ˆ {(wn)n} zdPz((wn)n)dπ∂Bdν∞ x = ˆ ∂Bd zdπ∂Bdν∞ x , as the coordinate map z �→ zj is constant on sequences (wn)n that converge to the same value w ∈ ∂Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that x �→ νx is µ-measurable and x �→ ν∞ x is λ-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, ν ∈ Lp(Ω, µ, Rd) for 1 < p < ∞ and ν ∈ M(Ω, Rd) for p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It is easy to see that when 1 < p < ∞, vj ⇀ v ∈ Lp(Ω, µ, Rd) we have v = νv, and when p = 1, ρj ⇀∗ ρ ∈ C0(Ω, Rd)∗, then ρ = ξρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='7 Stronger notions of convergence To conclude the discussion about generalised Young Measures, we mention some stronger notions of convergence such as strict convergence and µ-strict convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' These modes of convergence explain why we chose such canonical embedding for measures in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, we give a simple example of why such canonical embedding has no meaning when the compactification is larger than the sphere one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='37 Let ηj, η ∈ M(Ω, Rd), we say that ηj s−→ η (ηj converges strictly to η) if ηj → η weakly* in C0(Ω, Rd)∗ and |ηj|(Ω) → |η|(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It is easy to see that if ηj → η strictly, then |ηj| ⇀∗ |η| in C0(Ω, Rd)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The above convergence prevents small-scale cancellations and concentration on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' However, it does not prevent oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prevent oscillation, we must choose a "weight" µ ∈ M+(Ω) and ask for convergence of ηj on the graph (µ, ηj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We thus obtain a notion of µ-strict convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='38 We say that ηj µ−s −−→ η (ηj converges µ-strictly to η) if ηj → η weakly* in C0(Ω, Rd)∗ and (µ, ηj) s−→ (µ, η) in C0(Ω, R × Rd)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Similarly to what was observed in the case of strict convergence, such a notion implies that |(µ, ηj)| ⇀∗ |(µ, ηj)| in C0(Ω)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, µ-strict convergence of ηj to η simply amounts to weak* convergence and additional convergence of the following quantity: writing ηj = ηj µ dµ + ηs,µ j , ηs,µ j ⊥ µ, |(ηj, µ)|(Ω) = ���� �ηj µ dµ, µ � + (ηs,µ j , 0) ���� = ˆ Ω � 1 + ���� ηj µ ���� 2 dµ + |ηs,µ j |(Ω) → ˆ Ω � 1 + ���� η µ ���� 2 dµ + |ηs,µ|(Ω) = |(η, µ)|(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' When µ = Ln, we refer to such convergence as area-strict convergence, in analogy with the area formula for smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Reshetnyak continuity theorem (see [Res68] for the original) shows that strict convergence is equivalent to the convergence of 1-homogeneous functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='39 Let f(x, z) ∈ C(Ω × Rd) be 1-homogeneous in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If ηj → η strictly in the sense of measures, then ˆ Ω f � x, ηj |ηj| � d|ηj|(x) → ˆ Ω f � x, η |η| � d|η|(x) In case f is not 1-homogeneous but has an extension on the sphere compactification, we can obtain the following auto-convergence result by requiring µ-strict convergence instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='40 Let f ∈ E(Ω × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If ηj µ−s −−→ η in C0(Ω, Rd)∗ then ˆ Ω f � x, ηj µ � dµ + f ∞ � x, ηs,µ j |ηs,µ j | � d|ηs,µ j | → ˆ Ω f � x, η µ � dµ + f ∞ � x, ηs,µ |ηs,µ| � d|ηs,µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 28 These are well-known results, but we write down a proof of the latter one because it gives an insight into how to move from one type of convergence to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consider the so-called perspective functional ˜f(x, z, t) = � f(x, z t )|t| t ̸= 0 f ∞(x, z) t = 0 which is positively 1-homogeneous in the last variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By the Reshetnyak continuity theorem, we know that, for η ∈ M(Ω, Rd), ˆ Ω ˜f(x, (η, µ)) = ˆ Ω f � x, η dµ � dµ + f ∞ � x, ηs,µ |ηs,µ| � d|ηs,µ| is sequentially continuous in the µ-strict topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Upon taking f(x, z) = |z| we get that µ-strict convergence implies strict convergence, for every µ ∈ M+(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In light of these results, for a fixed measure µ ∈ M+(Ω), the canonical embedding of measures η ∈ M(Ω, Rd) into the set of Young Measures on the sphere η ∈ M(Ω, Rd) �→ ξη = � δ η µ , |ηs,µ|, δ ηs,µ |ηs,µ| � is sequentially µ-strictly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This is a solid justification for this choice of embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For the same reason, we can show why on larger compactifications we don’t have, in general, a canonical choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='41 Let µ ∈ M+(Ω) with the property that there is x ∈ supp(µ Ω) with δx ⊥ µ, and let f ∈ C(Rd) of linear growth, f ̸∈ E1(Rd) (oscillate at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' There is (uj)j∈N ⊂ D(Ω, Rd), uj µ−strictly −−−−−−→ zδx in M(Ω, Rd) for some z ∈ ∂Bd, but ˆ Ω f(uj(x))dµ(x) does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The above theorem implies that for all efi,i∈N ≥ ef (in the sense of compactifications), ξuj does not converge in Y (µ, efi,i∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because of the assumptions on µ, we can find x ∈ supp(µ Ω) so that δx ⊥ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that the result is unchanged if we instead consider f(x) + C1|z| + C2, so taking C1, C2 > 0 big enough we can assume that f ≥ 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Find z ∈ ∂Bd and zj, wj → z so that lim n Tf(zj) = M and lim j Tf(wj) = m exist, and M > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, because x ∈ supp(µ) then µ(Br(x)) > 0 for all r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' There are two possible scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' First, x ∈ Ω, in which case we consider only those balls Br(x) so that B2r(x) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If x ∈ ∂Ω then we can δr ↓ 0 so that µ(Br(x))∩Ω−δ(r)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Either way, we call Br(x) or Br(x)∩Ω−δ(r) simply Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Furthermore, we can find ε = ε(r) > 0 so that Bε r = (Br)ε = {x ∈ Rn : d(x, Br) < ε} ⊂ Ω 29 and lim r↓0 µ(Bε r) µ(Br) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For each such r find φr ∈ D(Bε r), 0 ≤ φr ≤ 1 so that φr(Br) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Put ur = φr ´ φrdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Clearly ur ⇀∗ δx in C(Ω)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, refine the sequences (zj)j and (wj)j so that there is rj ↓ 0 so that ˆ Br2j φ2jdµ = 1 − |zj| and ˆ Bεr2j+1 φ2j+1dµ = 1 − |wj|, and put uj = \uf8f1 \uf8f2 \uf8f3 urjz j 2 if j is even, urjw j−1 2 if j is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' First we show that uj µ−strictly −−−−−−→ zδx in M(Ω, Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because zj, wj → z, it is enough to show that ur µ−strictly −−−−−−→ δx in M(Ω) as r ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because ur ⇀∗ δx in C(Ω)∗ then lim inf r→0 |(urdµ, µ)|(Ω) ≥ |(δx, µ)|(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To achieve the opposite inequality, we calculate |(urdµ, µ)|(Ω) =|(0, µ)|(Ω \\ Bε r) + |(ur, 1)dµ|(Ω ∩ Bε r) = µ(Ω \\ Bε r) + ˆ Bεr � 1 + usrdµ =µ(Ω \\ Bε r) + ´ Bεr ��´ φrdµ �2 + φsrdµ ´ φrdµ ≤µ(Ω \\ Bε r) + ´ Bεr � µ(Bεr)2 + 1dµ µ(Br) =µ(Ω \\ Bε r) + µ(Bε r) � µ(Bεr)2 + 1 µ(Br) → µ(Ω) + 1 = |(δx, µ)|(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next, we study how the integral behaves on alternating integers of the sequence (uj)j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If j = 2i then lim inf i ˆ Ω f(u2i(x))dµ(x) ≥ lim inf i ˆ Br2i f � zi 1 − |zi| � dµ = lim i Tf(zi) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On the other side, if j = 2i + 1 we get the upper bound lim sup i ˆ Ω f(u2i+1(x))dµ(x) ≤ lim sup i ˆ Bεr2i+1 f � wi 1 − |wi| � dµ = lim i Tf(wi) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 30 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='42 The assumption on µ is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If for all x ∈ supp(µ Ω) we have δx ̸⊥ µ, then all such x’s belong to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consider the atomic decomposition of µ: µ = µa + µn−a = � n µ(xn)δxn + µn−a, see Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If µ(Ω \\ {xn, n ∈ N}) > 0 then we could find x ∈ Ω \\ {xn, n ∈ N} and δx ⊥ µ, so that µn−a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then µ = � n µ(xn)δxn, and the set {xn, n ∈ N} contains its accumulation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular {xn, n ∈ N} = supp(µ) is a compact subset of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It is easy to see that, in this setting, if (φj)j∈N ⊂ L1(µ) is bounded in norm and φj Y (µ,efi,i∈N) −−−−−−−→ � νx, λ, ν∞ x � , then λ ≪ µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' λ = � n λ(xn)δxn (because the space is countable and compact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also φj ⇀∗ φ = � νx + ν∞ x λ µ � dµ in C0(X)∗, X = {xn, n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Assume also that φj → φ µ-strictly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This amounts to the following ˆ X f(φj)dµ → ˆ X ⟨νx, f⟩ + ⟨ν∞ x , f ∞⟩λ µdµ = ˆ X f(φ)dµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='157) where f(z) = � 1 + |z|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' f is a strictly convex function, therefore the inequality f(x + y) ≤ f(x) + f ∞(y) is strict unless y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We have ⟨νx, f⟩ + ⟨ν∞ x , f ∞⟩λ µ ≥f(νx) + f ∞ � ν∞ x λ µ � > f � νx + ν∞ x λ µ � = f(φ) unless ν∞ x = 0 λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' So φ = νx µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=', and using convexity once again in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='157), and the fact that f ∞ = 1, we get f(φ) =⟨νx, f⟩ + ⟨ν∞ x , f ∞⟩λ µ ≥ f(νx) + λ µ = f(φ) + λ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' So λ = 0, which implies that the sequence φj does not concentrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Moreover, φ(x) = νx, which means that φj → φ in measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then φj → φ strongly in L1(µ), and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='41, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 29 is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 31 3 Characterisation of gradient Young Measure on general compactifications In this section, we show that generalised gradient Young Measures are characterised by a set of integral inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' A characterisation result was previously obtained in the context of the sphere compactification, see [KR10a] and [KR19] for the result on general differential operators, and it’s here extended to general compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Non-separability of the space of quasi-convex functions We start by showing that the set of quasi-convex functions having linear growth is non- separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lack of separability prevents sequential compactness and other essential properties that were used to develop the theory of generalised Young Measures (see the section above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Therefore, we are forced to consider only smaller countable collections of quasi-convex functions at the time, and cannot work with the entire class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To show that the class is non-separable, we modify the example by Muller [Mül92] and generate quasi-convex functions that oscillate at different amplitudes in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 The space of quasi-convex functions f : R2×2 → R having linear growth is not separable with respect to ∥T · ∥∞,B2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The idea behind the proof is to construct an uncountable family {fΛ}Λ so that ∥TfΛ−TfΓ∥∞ ≥ c for some universal constant c > 0 and all Λ ̸= Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We will split the proof of the above result into two different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' First, we show that we have quasi-convex functions that oscillate along every possible sequence of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 There is c > 0 such that for every Λ ⊂ N infinite so that Λc is also infinite there exists fΛ : R2×2 → R quasi-convex and having linear growth such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' fΛ(3j 1) = 0 for all j ∈ Λ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' fΛ(3j 1) 3j ≥ c for all j ∈ Λc sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove this theorem we first need a preliminary lemma, whose proof can be found in [Mül92], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 299, lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='3 For k ∈ R+ we let gk : R2×2 → R, F �→ |F1,1 − F2,2| + |F1,2 + F2,1| + (2k − |F1,1 + F2,2|)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then there exists c1 > 0 such that Qgk(0) ≥ c1k for all sufficiently large ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We can then prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 32 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let Λ ⊂ N as in the proposition above and set fΛ = QgΛ where gΛ(F) = |F1,1 − F2,2| + |F1,2 + F2,1| + inf{|F1,1 + F2,2 − 2 · 3i|, i ∈ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We have that gΛ(3j 1) = 0 for all j ∈ Λ and so fΛ(3j 1) = 0 as well given that gΛ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We first derive the following lower bound: compute gΛ(3j 1 + G) = |G1,1 − G2,2| + |G1,2 + G2,1| + inf{|G1,1 + G2,2 + 2(3j − 3i)| : i ∈ Λ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' because 3j is increasing we estimate, for arbitrary β ∈ R, inf i∈Λ |2(3j − 3i) + β| ≥ � inf i̸=j |2(3j − 3i)| − |β| �+ = � 2(3j − 3j−1) − |β| �+, and so gΛ(3j 1 + G) ≥ gk(G) for all matrices G and k = 3j − 3j−1, and gk given by gk(F) = |F1,1 − F2,2| + |F1,2 + F2,1| + (2k − |F1,1 + F2,2|)+ does not depend on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 32 to find c > 0 (independent of Λ) such that fΛ(3j 1) = QgΛ(3j 1) ≥ Qgk(0) ≥ c1(3j − 3j−1) for j big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Dividing everything by 3j we get fΛ(3j 1) 3j ≥ 2 3c1 ≡ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ It’s not a priori clear if these functions are "far away from each other at infinity", as subsets of natural numbers could intersect infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To show that there is a wide variety of sequences that differ at infinity, we define the following relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='4 Given Γ, Λ any two sequences (not necessarily subsets of N), we say that Γ ≤ Λ provided Γ is eventually a subset of Λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Λ = (ai),i∈N, Γ = (bi)i∈N, Λ ≤ Γ ⇐⇒ there exists k > 0 : (ai)i≥k is a subsequence of (bi)i≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let [Λ] be the equivalence class of Λ with respect to ≤, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Γ ∈ [Λ] if Γ ≤ Λ ≤ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If Λ′ ∈ [Λ] and Γ′ ∈ [Γ] then Λ ≤ Γ if and only if Λ′ ≤ Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We use � F, ≤ � to indicate the set of equivalence classes with the inherited order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The above ordering is needed because, to show that we have uncountably many sequences that are independent of each other at infinity, we will use Zorn’s lemma and find a maximal set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' One could also reason that the Stone-Cech compactification of natural numbers is not metrisable and reason by contradiction using a suitably adapted version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 12, but we decided to not pursue this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='5 There exists an uncountable family G ⊂ F such that for every different pair Λ, Γ ∈ G, Λ is not comparable to either Γ nor Γc with respect to ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 33 The above means that we can find a set G such that given any two sequences of natural numbers in Λ, Γ ∈ G, one is frequently in the other sequence and its complement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Λ ∩ Γ and Λ ∩ Γc are both infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consider the set of subsets G = {G ⊂ F : no pair within G is comparable according to ≤}, ordered by inclusion ⊂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The above set is non-empty, which can be seen by taking Λ = 2N and Γ = 4N ∪ (4N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Every chain in G has an upper limit given by its union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By Zorn’s lemma, there exists a maximal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' I claim that the maximal element has uncountably many elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove the claim we first assume that the maximal element G ⊂ G is countable or finite and show that it is always possible to extract an extra incomparable element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To do so, I will show that given any countable or finite collection of infinite natural numbers {cj i, i ∈ N}j∈N there is {ci, i ∈ N} such that {ci, i ∈ N} ∩ {cj i, i ∈ N} is infinite for all j and {ci, i ∈ N} ∩ {cj i , i ∈ N} < {cj i, i ∈ N} according to the order previously established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consider the isomorphism N : F → {0, 1}N, {ci, i ∈ N} �→ � Nci = � 1 if i ∈ {ck, k ∈ N} 0 otherwise � i∈N Practically speaking, we are replacing subsets of natural numbers to sequences that take values 1 when the i-th number is in the set, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Set initially Nci = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' At i1 so that Nc1 i1 = 1 for the first time put Nci1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We then iterate "diagonally" in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' At the n-th iteration find in+1 so that Ncj kj = 1 for all 1 ≤ j ≤ n and some in < kj−1 < kj and Ncj tj = 1 for all 1 ≤ j ≤ n + 1 and kn < tj−1 < tj < tn+1 = in+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Set Nctj = 1 for all 1 ≤ j ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This procedure stops if the maximal set is finite, otherwise can be iterated countably many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This way we guarantee that Nckj = 0 for in < kj and Ncj kj = 1, which means that Nci skips infinitely many 1s from each sequence (Ncj i)i∈N, for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On the other side Nctj = 1 = Ncj tj, tj ≤ in+1, so Nci is also frequently in every sequence (Ncj i)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Going back to our countable maximum element G = � {aj i, i ∈ N}, j ∈ N � , we can apply the previous construction to find {ci, i ∈ N} ∈ F generated by the countable family {cj i , i ∈ N}j∈N = � {aj i, i ∈ N} , N \\ {aj i, i ∈ N} � j∈N Because {ci, i ∈ N} is frequently and properly in {aj i, i ∈ N} and its complement N\\{aj i, i ∈ N} for all j, then {ci, i ∈ N} is not comparable to any member of the family and this contradicts the maximality of our set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ We are now ready to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By the lemma we can find an uncountable set of uncomparable subsequences of N, call it G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' I claim that if Λ, Γ ∈ G, Λ ̸= Γ then fΛ and fΓ have different recessions at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Find {xn} ∈ efΛ with {xn} ≥ {3j 1, j ∈ Λ}, where the inequality could be strict given that there might be more zero points at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Given our construction, we immediately have that 34 {xn} ̸≥ {3j 1, j ∈ N \\ Λ} as fΛ(3j 1) ≥ c3j for all j ∈ N \\ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also, Γ intersects N \\ Λ and Λ infinitely many times, and vice versa, so that lim sup n fΓ(xn) |xn| ≥ lim sup j∈Λ∩Γ fΓ(3j 1) 3j ≥ c, and lim inf n fΓ(xn) |xn| ≤ lim inf j∈Λ∩Γ fΓ(3j 1) 3j = 0, which shows that {xn} ̸∈ efΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In terms of the non-separability, by the definition of xn, we have lim n fΛ(xn) |xn| = 0, thus ∥TfΛ − TfΓ∥∞ ≥ lim sup n ���� fΓ(xn) |xn| − fΛ(xn) |xn| ���� = lim sup n ���� fΓ(xn) |xn| ���� ≥ c, where c is independent of Λ or Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Given that the space is metric, having such a property prevents separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We can end this section with the following corollary that incorporates higher dimensions: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='6 The set of quasi-convex functions having linear growth f : Rm×n → R is sep- arable in the topology induced by ∥T(·)∥∞ if and only if min(m, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If m or n is 1, quasi-convex functions are convex and therefore the space is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This is because convex functions admit a limit at infinity in every direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' in this case, we actually recover the sphere compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On the other side, for a function g: R2×2 → R we let gP ≡ g ◦ P : Rn×m → R, P : Rm×n → R2×2, M �→ �M1,1, M1,2 M2,1, M2,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If g is quasi-convex and locally bounded we let φ ∈ D(Q, Rm) and compute ˆ Q gP(Dφ + z) = ˆ Q⊂Rn−2 dLn−2 ˆ [0,1]2 dx1x2g(PDφ + Pz) ≥ ˆ Q⊂Rn−2 dLn−2gP(z) = gP(z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' gP is quasi-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then the set fΛP is uncountable and ∥T(fΛP − fΓP)∥∞,Bm×n = ∥T(fΛ − fΓ)∥∞,B2×2 ≥ c if Γ ̸= Λ, so the space cannot be separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Notice that separability is important to achieve both the Young Measure representation and for sequential compactness in the inherited weak star topology of Young Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 Characterisation of Gradient Young Measures In this section, we characterise Gradient Young Measures (on separable compactification) via certain Jensen-like integral inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' It is worth mentioning that the result for the sphere compactification, achieved in [KR10a], can be easily improved in consideration of the fact that lim supt→∞ f(tz)t−1 is 1-homogeneous rank one convex, and so convex at points rank(z) = 1 (see [KK16], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 528)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In what follows, we cannot use this type of auto-convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In our context, f ∞ lives on a general compactification and convexity at points of rank one, as a Jensen’s type inequality, is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove our result, we will adopt the same strategy as in [KR19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Preliminarily to stating the theorem, we define the upper recession of a function relative to a general compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='7 Let efi,i∈N be a separable compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For any f having linear growth, we define f ♯,efi,i∈N((zn)) = sup (wn)n∈[(zn)n] lim sup n Tf(wn), where (wn)n are sequences belonging to the equivalence class of (zn)n within efi,i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The reason why we introduce this notion is that the strategy for proving the characterisation theorem makes use of the trivial fact that f ≥ f qc, f qc being the quasi-convex envelope of f (see, for example, [Dac07]) f qc(z) = inf φ∈D(Q) ˆ Q f(z + Dφ(x))dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' However, f qc does not need to live in the same class of separable quasi-convex functions, so the implication lim n f(zn) |zn| exists ⇒ lim n f qc(zn) |zn| exists could be false for some functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fix any compactification efi,i∈N and a function Tg ∈ efi,i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If g ≥ f then g∞((zn)) = lim n Tg(zn) ≥ sup (zn)∈[(zn)] lim sup n Tf(zn) = f ♯,efi,i∈N((zn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' f ♯,efi,i∈N does not need to be continuous on efi,i∈N with respect to its product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' How- ever, we can show that it is still upper semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='8 Let efi,i∈N be a separable compactification and f a function having linear growth, then f ♯,efi,i∈N is upper semi-continuous on ∂efi,i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that ∂efi,i∈N is metrisable, so it is enough to show sequential upper semi- continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let (zj n)n ∈ ∂efi,i∈N so that (zj n)n j−→ (zn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By the very definition of g♯,efi,i∈N((zj n)n), for fixed ε > 0 we can find nj ≥ kj, where kj is a natural number to be selected, so that g♯,efi,i∈N((zj n)n) ≤ ε + Tg((zj nj)), where (zj nj) is a constant sequence and belongs to efi,i∈N \\ 36 ∂efi,i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We want to show that we can select kj so that (zj nj)j belongs in the equivalence class of (zn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By applying the dominated convergence theorem we get lim j � i lim n 2−i|fi(zj n) − fi(zn)| = 0 = lim j lim n � i 2−i|fi(zj n) − fi(zn)|, and so can find kj so that � i 2−i|fi(zj n) − fi(zn)| ≤ εj ∀n ≥ kj, where 0 ≤ εj ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This shows that the above sequence (zj nj)j ∈ [(zn)n] (the equivalence class), thus lim sup j g♯,efi,i∈N((zj n)n) ≤ ε + lim sup j Tg((zj nj)) ≤ ε + g♯,efi,i∈N((zn)n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By the arbitrariness of ε > 0 we conclude upper semi-continuity of g♯,efi,i∈N □ In particular, g♯,efi,i∈N is Borel measurable on efi,i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The above statement can be generalised to extensions of functions over more general compact metric spaces, but this version suffices for our scopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We are interested in studying those Young Measures that are generated by gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' So we define the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='9 We say that ν ∈ Y (efi,i∈N) is a (generalised) gradient Young Measure if there exists a sequence uj ∈ BV (Ω, Rm) such that Duj Y (efi,i∈N) −−−−−−→ � νx, λ, ν∞ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We use GY (efi,i∈N) to refer to these subsets of Young Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The convergence of measure derivatives has not been fully comprehended yet and it is still the subject of active research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This means that it is not so clear how rich the above class is, and with which frequency gradients oscillate - or at least within the setting of weak* convergence of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='10 We can use the characterisation lemma for Young Measure on the sphere to show that the class is still quite vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Indeed, by [KR10a], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 541 Theorem 1, fix any z = a ⊗ b and ν∞ ∈ P(∂B) with ν∞ = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then gradient Young Measure on the sphere � δ0, Hn−1 (B ∩ b⊥), ν∞� satisfies the characterisation theorem from [KR10a], with u = aχx·b≥0 and so it is generated by a sequence of gradients Duj ∈ BV (B1(0)), B1(0) ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because Duj is bounded in BV , we can find a subsequence (ujk)k∈N such that Dujk Y (efi,i∈N), as k→∞ −−−−−−−−−−−−→ � δ0, Hn−1 (B ∩ b⊥), η∞� , with clearly π∂Bη∞ = ν∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='19, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 18 to write η∞ = Pzdν∞, z ∈ ∂Bd, it remains an open question to understand how many probabilities Pz over subsequences zn → z can be generated by gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 37 We now state the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='11 Let Ω ⊂ Rn be a bounded Lipschitz domain and efi,i∈N be a separable compact- ification of quasi-convex functions and consider a generalised Young Measure ν ∈ Y (efi,i∈N) that satisfies λ(∂Ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then ν ∈ GY (efi,i∈N) is a Young Measure generated by a sequence (φj ⋆ (Du Ω) + Duj) Y (efi,i∈N) −−−−−−→ � νx, λ, ν∞ x � , where u ∈ BV (Ω, Rm), uj ∈ D(Ω, Rm) and ∥uj∥1 → 0, and φj is any sequence of mollifiers with φj ⇀∗ δ0, if and only if there is u ∈ BV (Ω, Rm) such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' ≪ 1 ⊗ | · |, ν ≫< +∞, and for all f quasi-convex and having linear growth, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' f(∇u(x))dx ≤ ⟨νx, f⟩dx + ⟨ν∞ x , f ♯,efi,i∈N⟩ λ Ln dx and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' f ∞(Dsu) ≤ ⟨ν∞ x , f ♯,efi,i∈N⟩dλs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We can adjust the above theorem to fix the boundary of the converging sequence so that it’s always equal to u in the sense of trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='12 If ν ∈ Y (efi,i∈N) is generated by a sequence φj ⋆ (Du Ω) + Duj as above, then there exists another sequence vj ∈ C∞(Ω) ∩ W 1,1 u (Ω) such that D(vj + uj) Y (efi,i∈N) −−−−−−→ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, ν ∈ GY (efi,i∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We can find uj → u strictly in BV (Ω) with uj ∈ C∞(Ω) ∩ W 1,1 u (Ω), see for example [KR10a] Lemma 1 for a proof of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In the construction of the ujs just mentioned, it is possible to select φj ⋆ (Du Ω) on Ω−ε for j big enough, φj as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also, without loss of generality, we can assume that |Du|(∂Ω−ε) = |Duj|(∂Ω−ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because the fis are all Lipschitz, it is enough to test against f ∈ Lip(Rm×n) with Lip(f) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We then compute ˆ Ω |f(φj ⋆ (Du Ω) + Dvj) − f(Duj + Dvj)|dx ≤ ˆ Ω |φj ⋆ (Du Ω) − Duj| ≤ ˆ Ω\\Ω−ε |φj ⋆ (Du Ω)| + |Duj| = Ij + IIj By strict convergence of both integrands, we have that lim sup j Ij + IIj ≤ 2|Du|(Ω \\ Ω−ε), and so use a diagonal argument to conclude the existence and equality of the limit Young Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 38 It’s implicit in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 38 that Du = ν = ⟨νx, ·⟩dx + ⟨ν∞ x , ·⟩dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also, the above inequalities can be written in the sense of distribution, in the form ˆ Ω φ(x)⟨νx, f⟩dx + ˆ Ω φ(x) ˆ ∂efi,i∈N f ♯,efi,i∈Ndν∞ x dλ ≥ ˆ Ω φ(x)f(∇u(x))dx + φ(x)f ∞ � Dsu |Dsu| � d|Dsu| for all φ ∈ D(Ω), φ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove the characterisation result we will follow the same strategy as in [KR19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We initially prove the result for homogeneous gradient Young Measures and then extend the theorem to the inhomogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that Young Measures that act on functions f = f(z) that only depend on z can be represented by ˆ Ω ⟨νx, f⟩dx + ˆ Ω ⟨ν∞ x , f ∞⟩dλ = ˆ Ω fdν0 + ˆ ∂efi,i∈N f ∞dν∞, where for efi,i∈N the separable compactification that extends f, ν0 = νxdLn ∈ M+(Ω) and ν∞ = ν∞ x dλ ∈ M+(∂efi,i∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The (push-forward) Kantorovich metric is then ∥(ν0, ν∞)∥K = sup Φ∈H,∥TΦ∥Lip(efi,i∈N)≤1 ����� ˆ Rd Φdν0 + ˆ efi,i∈N Φ∞dν∞ ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For z ∈ Rd we let Y be the set of pairs � ν0, ν∞� ∈ M+ 1 (Rd) × M+(efi,i∈N) such that there is a sequence uj ∈ D(Q, Rm), where Q is the unit cube, so that z + Duj Y (efi,i∈N) −−−−−−→ � ν0, ν∞� and ∥uj∥1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The following proposition follows from obvious variations of the proofs contained in [KR19], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 8, lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='7,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='8,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The proofs are essentially the same as they only use the separability of the compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='13 The family {εz+Du : u ∈ D(Q, Rm)} is weakly* dense in Y, and Y is a weak* closed and convex subset of homogeneous Young Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We can now prove the main theorem in case � ν0, ν∞� is a homogeneous gradient Young Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='14 Let ν = � ν0, ν∞� ∈ M+ 1 (Ω) × M+(∂efi,i∈N) and z ∈ Rm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then ν ∈ Y if and only if there is z ∈ Rm×n such that ˆ Rm×n fdν0 + ˆ ∂efi,i∈N f ♯,efi,i∈Ndν∞ ≥ f(z) for all f : Rm×n → R quasi-convex and having linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 39 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Suppose that ν ∈ Y and let z + Duj, uj ∈ D(Q, Rm) be the generating sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' for all Φ ∈ T −1efi,i∈N, ˆ Q Φ(z + Duj)dx → ⟨ν, Φ⟩ = ˆ Rm×n Φdν0 + ˆ ∂efi,i∈N Φdν∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fix an arbitrary f having linear growth and quasi-convex and let ef,fi,i∈N the bigger compact- ification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Upon extracting a subsequence we have that z + Duj Y (ef,fi,i∈N) −−−−−−−→ � ν0, ˜ν∞� , where we identify the gradient Young Measure with its tensor products as f = f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By quasi- convexity, we have ˆ Rm×n fdν0 + ˆ ∂ef,fi,i∈N f ∞d˜ν∞ = lim sup j ˆ Q f(z + Duj)dx ≥ f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On the other side, using the decomposition of angle concentration Young Measure Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='19, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 18 we also obtain that ˆ ∂ef,fi,i∈N f ∞d˜ν∞ = ˆ ∂efi,i∈N ˆ f ∞dP(zn)ndν∞ ≤ ˆ ∂efi,i∈N f ♯,efi,i∈Ndν∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For the other implication, because Y is weakly* closed and convex, we can write Y = ∩H where H are half-spaces containing Y, which can be written as H = {l ∈ H∗ : l(Φ) ≥ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, we can test the above inequality against εz+Du and get t ≤ εz+Du(Φ) ≤ ˆ Q Φ(z + Du)dx for all u ∈ D(Q, Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Passing to the infimum over all such us we deduce t ≤ Φqc(z) and so ⟨ν, Φ⟩ = ˆ Rm×n Φdν0 + ˆ ∂efi,i∈N Φ∞dν∞ ≥ ˆ Rm×n Φqcdν0 + ˆ ∂efi,i∈N (Φqc)♯,efi,i∈Ndν∞ ≥ Φqc(z) ≥ t which shows that ν ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 Inhomogenization In what follows, we will prove a semi-approximation result for the absolutely continuous and singular parts separately and then put them together via Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In each case, we will use a covering argument to boil it down to the homogeneous case, which was solved in the above section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consider a standard mollifier φt(x) = tn−1φ(x t ), where φ ∈ D(Q) and let M = ∥Dφ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Also, unless otherwise specified, the norm on Rn is the maximum norm ∥x∥ = maxi |xi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 40 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='15 Given ε > 0 there is tε > 0 and a family ϕt ∈ D(Ω, Rm) with ∥ϕt∥1 ≤ ε so that ���� ˆ Ω ηΦ(0) + η⟨Φ∞, ν∞ x ⟩dλs − ˆ Ω ηΦ(φt ⋆ (ν∞dλs) + Dϕt)dx ���� < ε for all t ∈ (0, tε) uniformly in ∥η∥Lip ≤ 1 and ∥TΦ∥Lip,graph(f) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The idea behind this approximation result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The singular part of the centre of mass (which is just Du ∈ M(Ω, Rd)) is approximated by mollification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Such a procedure generates area-strictly convergent smooth approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' At the same time, we generate angle concentration and oscillation via compactly supported functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because the first type of convergence is very strong, and the latter doesn’t concentrate, the two modes of convergence don’t interfere with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that this strategy would not be possible using the bare notion of weak* convergence because of the lack of quantifiability, whereas the (equivalent in this case) Kantorovich metric gives us an "exact" quantity to approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Before proving the above statement we remind that, according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='26, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 22, T pulls back bounded sets of Lipschitz functions on efi,i∈N to bounded sets of Lipschitz functions on Rm×n (provided fi are Lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Therefore, all the functions in the following theorem can be taken to be, after renormalisation, 1-Lipschitz in both spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' From now on, after fixing a compactification, we will always identify ∥TΦ∥Lip = ∥TΦ∥Lip(efi,i∈N) = ∥TΦ∥∞ + sup x̸=y |TΦ(x) − TΦ(y)| |x − y| + � i 2−i|Tfi(x) − Tfi(y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fix ε > 0 and apply Luzin’s theorem to the λs map x ∈ Ω → (δ0, ν∞ x ) ∈ M+ 1 (Rd) × M+(∂efi,i∈N) ֒→ � (T −1efi,i∈N)∗�+ to find a compact set C = Cε ⊂ Ω with λs(Ω \\ C) < λs(Ω)ε restricted to which the above map is uniformly continuous, with modulus of continuity ω = ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Without loss of generality, assume that Ln(Cs) = 0 and because λs(∂Ω) = 0 then ∆ = ∆ε = d(Cs, ∂Ω) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For the moment, fix two integers a, b ∈ N and put t = 2−a, so that φt > 0 if and only if ∥x∥ < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let a be so large that 2t ≤ ∆ and a ≥ log2 � 2 ∆ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Denote by F the collection of a+b-th generation dyadic cubes Q in Rn so that d(Q, ∂Ω) > 2−a, and for each such Q ∈ F we define rQ = Q φ ⋆ (λs Cs)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that rQ > 0 means that dist(Q, Cs) < t, and so for each such Q we can find xQ ∈ Cs so that d(xQ, Q) < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Denote by Fs the set of those Q ∈ F for which rQ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, if Q ∈ Fs we can find xQ ∈ Cs so that supQ ∥x − xQ∥ < 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 41 For every quasi-convex function having linear growth we have f(z + w) ≤ f(z) + f ∞(w) for all z ∈ Rm×n and rank(w) = 1, see [KK16], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 536, lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='5 (we don’t need regular recession for this result to hold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then by assumption, we have f(rQν∞ xQ) ≤ f(0) + rQf ∞(ν∞ xQ) ≤ f(0) + rQ ˆ ∂efi,i∈N f ♯,efi,i∈Ndν∞ xQ for all f quasi-convex and having linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Going back to the homogeneous case Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 39, we can select ϕQ ∈ D(Q, Rm) with ∥ϕQ∥1 < ελs(Q) such that ∥(δ0, ν∞ xQrQ) − εrQν∞ xQ+DϕQ∥K < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Define ϕ = � Q∈Fd ϕQ ∈ D(Ω, Rm) and ∥ϕ∥1 ≤ ελs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The sought-after map is then ξs = φ ⋆ (ν∞ x dλs + Dϕ) ∈ D(Rn, Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove that this function is the desired one, we fix ∥η∥Lip ≤ 1, ∥Ψ∥Lip(efi,i∈N) ≤ 1 as in the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We have ˆ Ω η⟨Φ∞, φ ⋆ (ν∞dλs)⟩dx = ˆ Ω η⟨Φ∞, φ ⋆ (ν∞dλs Cs)⟩dx + =E1 � �� � ˆ Ω η⟨Φ∞, φ ⋆ (ν∞dλs Ω \\ Cs)⟩dx, and |E1| ≤ ελs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Notice that here ˆ Ω η⟨Φ∞, φ ⋆ (ν∞dλs U)⟩dx = ˆ Ω η(x) ˆ U φ(x − y) ˆ ∂efi,i∈N Φ∞dν∞ y dλs(y)dx where U = Ω or Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Since for each Q ∈ F with rQ = 0 ˆ Q η⟨Φ∞, φ ⋆ (ν∞dλs Cs)⟩dx = 0 and dist(∪F, ∂Ω) > 2t, we get ˆ Ω ⟨Φ∞, φ ⋆ (ν∞dλs Cs)⟩dx = � Q∈Fs ˆ Q η⟨Φ∞, φ ⋆ (ν∞dλs Cs)⟩dx + E2 = � Q∈Fs �ˆ Q ηdx⟨Φ∞, ν∞ xQ⟩rQ + EQ 3 � + E2, 42 where |E2| ≤ λs(Cs ∩ (∂Ω)2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The third error is estimated in the following way: |EQ 3 | ≤ ���� ˆ Q � η − ˆ Q η � ⟨Φ∞, φ ⋆ (ν∞dλs Cs)⟩dx ���� + ���� ˆ Q η �ˆ Q ⟨Φ∞, φ ⋆ (ν∞dλs Cs)⟩dx − ⟨Φ∞, ν∞ xQ⟩rQ ����� ≤∥η∥Lip Ln(Q) 1 n ∥Φ∞∥ ˆ Q φ ⋆ λsdx + ∥η∥Lip ˆ Q ˆ Cs φ(x − y)⟨Φ∞, ν∞ y − ν∞ xQ⟩dλs(y)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, we obtain |EQ 3 | ≤t ˆ Q φ ⋆ λsdx + ˆ Q ˆ Cs φ(x − y)ω(∥y − xQ∥)dλs(y)dx ≤(t + ω(3t)) ˆ Q φ ⋆ λsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' From each Q ∈ Fs we get Φ(0) + ⟨Φ∞, ν∞ xQ⟩rQ = ˆ Q Φ(rQν∞ xQ + DφQ)dx + |·|≤ε ���� EQ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Further computations show that ˆ Q |Φ(rQν∞ xQ + DφQ) − Φ(0)|dx ≤∥εrQν∞ xQ+DφQ∥K ≤∥(δ0, ν∞ xQrQ)∥K + ε ≤ 1 + rQ + ε and consequently ˆ Q η ˆ Q Φ(rQν∞ xQ + DφQ)dx = ˆ Q ηΦ(rQν∞ xQ + DφQ)dx + EQ 5 , where the error term is upper bounded by |EQ 5 | ≤ sup Q ����η − ˆ Q η ���� Ln(Q)(1 + rQ + ε) ≤ Ln(Q) 1 n ˆ Q (2 + φ ⋆ λs)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Finally, we estimate the last term with ˆ Q ηΦ(rQν∞ xQ + DφQ)dx = ˆ Q ηΦ(φ ⋆ ν∞dλs) + DφQ)dx + EQ 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To bound the 6th error term we introduce an extra quantity ���� ˆ Q η � Φ(φ ⋆ ν∞dλs) + DφQ)dx − Φ(φ ⋆ ν∞dλs Cs) + DφQ) � dx ���� �� ˆ Q φ ⋆ (λs Ω \\ Cs)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 43 and ���� ˆ Q η � Φ(rQν∞ xQ + DφQ)dx − Φ(φ ⋆ ν∞dλs Cs) + DφQ) � dx ���� ≤ ˆ Q |rQν∞ xQ − φ ⋆ (ν∞dλs Cs)|dx ≤ ���� ˆ Q φ ⋆ � (ν∞ xQ − ν∞� dλs Cs)dx ���� + ˆ Q ���� Q φ ⋆ (ν∞dλs Cs)dx′ − φ ⋆ (ν∞dλs Cs) ���� dx ≤ ˆ Q ˆ Cs φ(x − y)|ν∞ xQ − ν∞ y |dλsdx + ˆ Q ���� Q φ ⋆ (ν∞dλs Cs)dx′ − φ ⋆ (ν∞dλs Cs) ���� dx ≤ω(3t) ˆ Q φ ⋆ λsdx + EQ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The 7th error term is estimated by EQ 7 ≤ ˆ Q Q ˆ Cs ��φ(x′ − y) − φ(x − y) ��|ν∞ y |dλs(y)dx′dx ≤√n ˆ Q Q ˆ Q+tQ ∥x − x′∥ ˆ 1 0 ��Dφ(x + (x′ − x)τ)(x′ − x) ��dτdλs(y)dx′dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Given the scaling of φ = φt with respect to t, we have |Dφ| ≤ t−1M(χQ)t, so the above can be bounded by EQ 7 ≤ √nM Ln(Q) 1 n t ˆ Q (χ2Q)t ⋆ λsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For our choice of a and b, we have that Ln(Q) = 2−n(a+b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' With ξs defined above we obtain that ˆ Ω η⟨Φ∞, φ ⋆ (ν∞dλs)⟩dx = ˆ ⋒Fs ηΦ(ξs) + E, where |E| ≤ελs(Ω) + (t + ω(3t)) ˆ ∪Fs � φ ⋆ λs + 2−a−b(2 + φ ⋆ λs) + φ ⋆ (λs Ω \\ Cs) + ω(3t) φ ⋆ λsdx + √nM2−b(χ2Q)t ⋆ λs� dx ≤ � 2ε + 2−a + 2ω(32−a) + 2−a−b + cnM2−b� λs(Ω) + 21−a−bLn(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To conclude we add ˆ Ω\\∪Fs ηΦ(ξs)dx = ˆ Ω\\∪Fs ηdxΦ(0) to both sides, and obtain ˆ Ω η � Φ(0) + ⟨Φ∞, φ ⋆ (ν∞dλs Ω)⟩ � dx = ˆ Ω Φ(ξs)dx + E + ˆ ∪Fs ηdxΦ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 44 Because ∪Fs ⊂ (Cs)2t and since Ln(Cs) = 0 we can find aε ≥ a, bε ≥ b such that |E| + ���� ˆ ∪Fs ηdxΦ(0) ���� ≤ 3ε(Ln + λs)(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The left-hand side tends to ˆ Ω ηdxΦ(0) + ˆ Ω η⟨Φ∞, ν∞ x ⟩dλs(x) as a → ∞, uniformly in η and Φ, and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ We now move on to the absolutely continuous part, which is proven similar and is a bit easier to construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='16 Let ε > 0, there is tε > 0 and ψt ∈ D(Ω, Rm) with ∥ψt∥1 ≤ ε so that ���� ˆ Ω η(⟨Φ, νx⟩ + λa(x)⟨Φ∞, ν∞ x ⟩)dx − ˆ Ω ηΦ � φt ⋆ (ν + ν∞λa(x))dx Ω + Dψt � dx ���� < ε holds for t ∈ (0, tε), uniformly in ∥η∥Lip ≤ 1 and ∥TΦ∥Lip(efi,i∈N) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fix ε ∈ (0, 1) and apply Luzin’s theorem to the Ln-measurable map Ω ∋ x �→ (νx, λa(x)ν∞ x ) ∈ M+ 1 (Rd) × M+(∂efi,i∈N) ֒→ � (T −1efi,i∈N)∗�+ to find a compact set Ca ⊂ Ω such that ˆ Ω\\Ca M(x)dx < ε, M(x) = ⟨νx, | · |⟩ + λa(x), and ω be the modulus of continuity over Ca, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' ∥(νx, λa(x)ν∞ x ) − (νy, λa(y)ν∞ y )∥K ≤ ω(∥x − y∥) for all x, y ∈ Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fix d ∈ N and s ∈ (0, 1) and let Fa be the family of dyadic cubes in Rn of side length t = 2−d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Fa = {Q ∈ Dd : d(Q, ∂Ω) > t, Ln(Q ∩ Ca) > sLn(Q)}, where the distance is induced by ∥ · ∥∞ over vectors in Rn, and d and s will be selected later in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For every Q ∈ Fa select xQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 39 we have ψQ ∈ D(Q, Rn×m) with ∥ψQ∥1 < εLn(Q) Ln(Ω) and ∥(νxQ, λa(xQ)ν∞ xQ) − ενxQ+λa(xQ)ν∞ xQ+Dψ∞∥K < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Let ψ = � Q ψQ ∈ D(Ω, Rm×n) and ∥ψ∥1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then ˆ Ω η(⟨Ψ, νx⟩ + λa(x)⟨Ψ∞, ν∞ x ⟩)dx = � Q∈Fa ˆ Q η(⟨Ψ, νx⟩ + λa(x)⟨Ψ∞, ν∞ x ⟩)dx + E1 with |E1| ≤ ˆ Ω\\∪Fa M(x)dx ≤ ε 45 for large enough d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Next � Q∈Fa ˆ Q η(⟨Ψ, νx⟩ + λa(x)⟨Ψ∞, ν∞ x ⟩)dx = � Q∈Fa Q η ˆ Q (⟨Ψ, νx⟩ + λa(x)⟨Ψ∞, ν∞ x ⟩)dx + E2, where |E2| ≤ t � Q∈Fa ˆ Q |⟨Φ, νx⟩ + λa(x)⟨Φ∞, ν∞ x ⟩|dx ≤ t ˆ Ω M(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' We further estimate, on every set Q ∈ Fa, ���� ˆ Q ⟨Φ, νx⟩ + λa(x)⟨Φ∞, ν∞ x ⟩ − Ln(Q) � ⟨Φ, νxQ⟩ + λa(xQ)⟨Φ∞, ν∞ xQ⟩ ����� ≤ω(t) s Ln(Q) + 1 − s s ˆ Q |⟨Φ, νx⟩ + λa(x)⟨Φ∞, ν∞ x ⟩|dx + ˆ Q\\Ca |⟨Φ, νx⟩ + λa(x)⟨Φ∞, ν∞ x ⟩|dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' By the linear growth of f, we get that � Q∈Fa Q η ˆ Q ⟨Φ, νx⟩ + λa(x)⟨Φ∞, ν∞ x ⟩dx = � Q∈Fa � ⟨Φ, νxQ⟩ + λa(xQ)⟨Φ∞, ν∞ xQ⟩ � ˆ Q η + E3 where |E3| ≤ ω(t) s Ln(Q) + 1 − s s ˆ Q M(x)dx + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' For every Q ∈ Fa we have that f(xQ) = Q Φ(νxQ + λa(xQ)ν∞ xQ + DφQ) + |·|≤ε ���� EQ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Set va(x) = νx + λa(x)ν∞ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Letting Φ = z · ei, (ei) canonical basis of Rm×n, we obtain, from continuity over Ca, |va − va(xQ)| ≤ ω(t) on Q ∩ Ca for each Q ∈ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Consequently, ˆ Q |va − va(xQ)|dx ≤ ω(t) s Ln(Q) + ˆ Q\\Ca |va|dx + 1 − s s ˆ Q |va|dx for all Q ∈ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because |va| ≤ M(x) and Lip(Φ) ≤ 5, then � Q∈Fa Q ηdx ˆ Q Φ(va(xQ) + DψQ)dx = � Q∈Fa Q ηdx ˆ Q Φ(va + DψQ)dx + E5 with |E5| ≤ 5ω(t) s Ln(Q) + 5ε + 51 − s s ˆ Ω M(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 46 Combining some of the previous estimates, we get � Q∈Fa Q ηdx ˆ Q Φ(va + DψQ)dx = � Q∈Fa ˆ Q ηΦ(va + DψQ)dx + E6, and |E6| ≤t � Q∈Fa ˆ Q |Φ(va + DψQ)|dx ≤ t|E5| + t � Q∈Fa ˆ Q |Φ(va(xQ) + DψQ)|dx ≤t|E5| + tεLn(Ω) + t � Q∈Fa Ln(Q)(⟨|Φ|, νxQ⟩ + λa(xQ)⟨|Φ|∞, ν∞ xQ⟩) ≤t|E5| + tεLn(Ω) + tω(t) s Ln(|) + �ˆ Q\\Ca +1 − s s ˆ Ω � M(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Finally, if φt is a standard mollifier, then φt ⋆ va Ω L1(Ω) −−−−→ va as t → 0, and so ˆ Ω ηΦ(va + Dψ)dx = ˆ Ω ηΦ(φt ⋆ va Ω + Dψ)dx + E7, where using again that Φ is Lipschitz over Rm×n, |E7| ≤ Lip(Φ) ˆ Ω |φt ⋆ va Ω − va|dx ≤ 5 ˆ Ω |φt ⋆ va Ω − va|dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 47 A Appendix Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='1 (Vitali convergence theorem) Let µ ∈ M+(Ω) and fn, f ∈ L1(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then fn → f in L1(Ω, µ) if and only if fn → f in measure and fn is uniformly integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' See [BR07], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 268, theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='2 (Stone-Weierstrass) Let X be a compact Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' If A is a closed subalgebra of C(X) that separates points, then either A = C(X) or there is x0 ∈ X so that A = {f ∈ C(X) : f(x0) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In particular, A = C(X) if and only if A contains the constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' See [Fol99], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 139, theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='3 (Tychonoff) If {Xα}α∈A is a family of compact spaces, Πα∈AXα is compact in the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='4 (Banach-Alaoglu sequential version) Let X be a separable Banach space and B ⊂ X∗ the closed unit ball of the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then B is weakly* sequentially compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' See [Lax14], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 107, theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='5 (Chacon biting lemma) Let µ ∈ M+(Ω) and vj ∈ L1(Ω, µ) be a sequence such that supj ∥vj∥ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' There are sets Ek ⊂ Ek+1, µ(Ω \\ Ek) → 0 and a subsequence (vji)i of (vj)j and v ∈ L1(µ) so that vji ⇀ v in L1(Ek, µ) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' See [BM89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='6 (Kantorovich metric) Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' The Kantorich metric on M+(X) generates the same topology as the weak* topology of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' See [KR19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='7 Let η ∈ M(Ω, Rd), Ω ⊂ Rn open bounded set, and (φε)0<ε≤1 be a family of standard mollifiers, supp(φ1) ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then η ⋆ φε area-strictly −−−−−−−→ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Because (Ln Ω, ηε) ⇀∗ (Ln Ω, η), we immediately obtain the lower bound lim inf ε→0 |(Ln, ρε)|(Ω) ≥ |(Ln, ρ)|(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' To prove the upper semi-continuity of the above quantity, notice the following equality: (Ln Ω, ρε) = (Ln Ω, ρ) ⋆ φε − (Ln Ω−ε ⋆ (δ0 − φε), 0), where Ω−ε = {x ∈ Ω : d(x, ∂Ω) > ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 48 Using then Jensen’s inequality lim sup ε→0 |(Ln, ρε)|(Ω) ≤ lim sup ε→0 |(Ln Ω, ρ) ⋆ φε|(Ω) + |(Ln Ω−ε ⋆ (δ0 − φε), 0)|(Ω) ≤|(Ln Ω, ρ)|(Ω) + lim sup ε→0 2Ln(Ω \\ Ω−ε) = |(Ln, ρ)|(Ω) □ Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content='8 (Atomic decomposition) Let µ ∈ M(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Then there exists a purely atomic measure µa and a non-atomic measure µn−a such that µ = µa + µn−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' See [FL07], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' □ 49 References [AB97] Jean-Jacques Alibert and Guy Bouchitté.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Non-uniform integrability and generalized young measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Journal of Convex Analysis, 4:129–148, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' [ADM92] Luigi Ambrosio and Gianni Dal Maso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' On the relaxation in bv (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' rm) of quasi- convex integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Journal of functional analysis, 109(1):76–97, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' [AF84] Emilio Acerbi and Nicola Fusco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Semicontinuity problems in the calculus of varia- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Archive for Rational Mechanics and Analysis, 86(2):125–145, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' [AFP00] Luigi Ambrosio, Nicola Fusco, and Diego Pallara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Functions of bounded variation and free discontinuity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Courier Corporation, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' [Bal89] John M Ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' A version of the fundamental theorem for young measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' In PDEs and continuum models of phase transitions, pages 207–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Springer, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' [BL73] Henri Berliocchi and Jean-Michel Lasry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59A0T4oBgHgl3EQfN_9T/content/2301.02154v1.pdf'} +page_content=' Intégrandes normales et mesures paramétrées en calcul des variations.' metadata={'source': 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