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+arXiv:2301.02154v1  [math.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. We call such representa-
+tions "generalised Young Measures". With the help of the new representations, we then
+characterise these limits when they are generated by gradients, i.e. when vi = Dui for
+ui ∈ W 1,1(Ω, Rm), via a set of integral inequalities.
+
+Contents
+1
+Intro
+3
+1.1
+Terminology and symbols
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.2
+Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Generalised Young Measures on separable compactifications
+8
+2.1
+Generalised Young Measures as generalised objects . . . . . . . . . . . . . . . .
+8
+2.1.1
+Parametrized measures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.1.2
+Generalized Young measures . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.1.3
+Functional analytic setup
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2.2
+Hausdorff compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+2.2.1
+Preliminaries on Hausdorff compactifications
+. . . . . . . . . . . . . . .
+11
+2.2.2
+Representation of compactifications . . . . . . . . . . . . . . . . . . . . .
+12
+2.3
+Restriction on non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+2.4
+Representation of Young measures
+. . . . . . . . . . . . . . . . . . . . . . . . .
+14
+2.5
+Properties of generalised Young Measures and connection to Young Measures
+on the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+2.6
+Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+2.7
+Stronger notions of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+3
+Characterisation of gradient Young Measure
+on general compactifications
+32
+3.1
+Non-separability of the space of quasi-convex functions . . . . . . . . . . . . . .
+32
+3.2
+Characterisation of Gradient Young Measures . . . . . . . . . . . . . . . . . . .
+36
+3.2.1
+Inhomogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+40
+A Appendix
+48
+2
+
+1
+Intro
+1.1
+Terminology and symbols
+• For a vector v ∈ Rm, we write |v| =
+��m
+i=1 v2
+i otherwise specified.
+• 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}.
+• We say that a function f : Rd → R has p growth if there is C > 0 such that
+|f(z)| ≤ C(1 + |z|p).
+• The identity matrix is indicated by
+1, or
+1d ∈ Rd×d if we need to specify the dimension.
+• The Lebesgue measure is indicated by dx, or Ln depending on the context. The set of
+finite Borel d-vector measures is indicated by M(Ω, Rd). For E ⊂ M(Ω), E+ is the set
+of positive Borel measures that belong to E.
+• 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) < +∞.
+• For µ ∈ M(Ω, Rd), we write its restriction to a µ-measurable set E ⊂ Ω
+µ
+E : A Borel set �→ µ(E ∩ A).
+• 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.
+• 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.
+• Let X be a metric space, µ ∈ M+(Ω) and (fj)j∈N ⊂ Lp(Ω, µ, Rd). 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.
+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}.
+We do not insist on the topology this space is endowed with, as it is standard and
+nowhere used in the work.
+• 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⟩.
+• The Sobolev space of functions with integrable derivatives is
+W 1,1(Ω, Rm) = {u ∈ L1(Ω, Rm) : Du ∈ L1(Ω, Rm×n)}.
+• The set of functions of bounded variation is
+BV (Ω, Rm) = {u ∈ L1(Ω, Rm) : Du ∈ M(Ω, Rm×n)}.
+• 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.
+• 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
+�
+.
+• The set of special functions of bounded variation is
+SBV (Ω, Rm) = {u ∈ BV (Ω, Rm) : Dsu = Dju, or equivalently Dcu = 0}.
+4
+
+1.2
+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
+�
+.
+The author wanted to understand what conditions on f would guarantee the existence of a
+minimizing curve u(t). 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. 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.
+So the question of the existence of a minimiser can be reformulated as to whether such objects
+are gradients of a curve or not. νt might fail to be a gradient when the minimizing sequence
+oscillates.
+Young’s original work focused on the case n = 1 and was carried out via functional analytic
+methods. This approach was later extended in [Bal89, BL73] to higher dimensions. We call
+these generalised functions "oscillation Young Measures". 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.
+The first attempt to well represent generalised limits of integrable functions is due to DiPerna
+and Majda, [DM87]. Functions vj : Ω → Rd are seen as Dirac deltas on the product space
+Ω × Rd, which is subsequently compactified. An accumulation point, in the sense of these new
+generalised functions, is then found. 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. In [AB97], an explicit formula for such accumulation points was
+obtained for a class of integrands that grow "nicely" infinity.
+In what follows, we give a general formula for describing Young Measures for a large class
+of integrands. 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. This generalisation is based on the canonical way of constructing Haus-
+dorff compactifications starting from continuous functions, see [CC76].
+A small reduction
+lemma gives a clearer, and somehow geometrical interpretation of such limits. This formula
+captures oscillations at infinity, which are now let occur. We also prove a few structure theo-
+rems that relate different compactifications and Young Measures representations to each other.
+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,
+(1.16)
+where u ∈ D(Ω, Rd), Ω ⊂ Rn is a bounded domain, and f ∈ C(Rm×n) has linear growth.
+Given any such f, there is no way to extend (1.16) to the class BV so that such extension is
+5
+
+continuous with respect to sequential weak* convergence in C0(Ω, Rm×n)∗. We can however
+find an extension which is lower semi-continuous for certain fs. In [Mor52], Morrey established
+the equivalence of lower semi-continuity of (1.16) 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).
+(1.17)
+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. 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. To be more specific, (1.17) can be rephrased
+as a Jensen-type inequality for measures of the form
+{νx : νx = Dφ(x)#dLn
+Ω, φ ∈ C∞
+c (Ω, Rm)},
+(1.18)
+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.
+The lower semi-continuity of (1.16) 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". This class can be seen as the
+closure of the set (1.18) in the weak* topology of measures over the graph of f. The opposite
+is also true and was proven for the first time in [KP91, KP94], i.e. 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.
+The aforementioned results hold in the setting of weak convergence in W 1,p, 1 ≤ p < ∞ and
+weak* convergence in W 1,∞. 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. In particular, a bounded sequence
+in L1(Ω, Ln) can concentrate and converge to measures that are singular with respect to the
+Lebesgue measure. 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. This concentration phenomenon is exclusive of
+the case p = 1, and so regards integrands that have linear growth at infinity. 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. Letting f be
+so that
+f ∞(z) =
+lim
+t→∞,zn→z
+f(tzn)
+t
+(1.21)
+6
+
+exists for all zn → z, t → ∞, the lower semi-continuous envelope of (1.16) 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).
+In this case, a Young Measure formulation of the Jensen’s-type inequality (1.17) has to take
+into account the singular part of Du. 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. Differently from the case without concentration, here we assumed f ∞ to exist as
+in eq. (1.21), p. 6. However, as shown in [Mül92], quasi-convex functions can oscillate at
+infinity, meaning that f ∞(z) may not exist for some z ∈ Rm. 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.
+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]. 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.
+This restricts the number of quasi-convex functions to be considered at once. 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.
+7
+
+2
+Generalised Young Measures on separable compactifications
+In this section, we construct generalised Young Measures and provide a new geometric rep-
+resentation. Concentration is let "oscillate with different amplitudes at infinity". To do so,
+we embed a space of functions into a bigger compact set and subsequently use the theory of
+Hausdorff compactifications.
+2.1
+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]. However, their
+existence per se does not give enough clarity on their properties, and so makes it hard to work
+with such objects.
+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. Under
+these assumptions, it is easy to see that, up to a subsequence, f ◦ vj converges to a measure
+ν; mathematically this means that
+ˆ
+Ω
+f(vj(x))φ(x)dµ(x) →
+ˆ
+Ω
+φ(x)dν(x)
+for all φ ∈ C0(Ω). It’s clear that ν = ν(f) is a linear function of f. What is not clear is
+how such dependence can be represented in terms of µ and f. Without a clear representation,
+it is not possible to set up a system of calculus. This section is dedicated to working out
+a geometric interpretation of the relation between ν and f, which we will then call Young
+Measure. 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.
+Definition 2.1 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.
+When p = 1, we say that such functions have linear growth.
+Before proceeding with formal definitions, we give a heuristic interpretation of Young Measures.
+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). This is a simple consequence of the Vitali
+convergence Theorem A.1, p. 48 (or the generalised dominated convergence theorem).
+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;
+• concentration - |vj| concentrates to a measure 0 ̸= λ ∈ M+(Ω) - equivalently (x, vj(x))
+concentrates to the boundary of some compactification of Ω×Rd. Around λ-almost every
+point, vj(x) goes to infinity and its "support" collapse to 0. 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.
+2.1.1
+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. Ordinary functions f : Ω → Rd, x �→ f(x)
+are embedded into maps Ω → M+
+1 (Rd), x �→ δf(x).
+Preliminary to the construction, we
+introduce two basic concepts that are at the core of this theory.
+Definition 2.2 Let X and Z be locally compact, separable metric spaces and λ ∈ M+(X).
+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.
+We shall often write the measure-valued map ν as a parametrized measure (νx)x∈X, where
+νx : = ν(x). 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.
+Theorem 2.3 (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. 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).
+For a proof of this result see [AFP00], p. 57. In this case we write
+ν = ηxdλ.
+2.1.2
+Generalized Young measures
+In what follows, we show how to obtain a good representation of Young Measures on general
+compactifications. The procedure is adapted from some lecture notes taken from a homonym
+course given by Jan Kristensen at the University of Oxford, [Kri15]. Some of the results can
+also be found in [Rin18], chapter 12, where they are only proven on the sphere compactification.
+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 < ∞. Assume
+vj ⇀ v in Lp when 1 < p < ∞
+or
+vj ⇀∗ v in C0(Ω, Rd)∗ when p = 1.
+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.
+Remark 2.4 For each j, the map Φ acts on the graph of vj, i.e. Φ(x, vj(x)) = Φ ◦ (x, vj(x)).
+Therefore, we look for the limiting distribution of (x, vj(x)) as j → ∞, and more precisely the
+Φ-moment of this limiting distribution.
+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.4, p. 48, there exists a limit
+measure which depends on the integrand Φ.
+2.1.3
+Functional analytic setup
+Let z ∈ Rd �→ ˆz =
+z
+1+|z| ∈ Bd be a homeomorphism Rd �→ Bd. 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|
+�
+.
+Then T : Gp → BC(Ω × Bd) is an isometric isomorphism provided Gp is normed by ∥TΦ∥∞
+and BC(Ω × Bd) by ∥ · ∥∞. The inverse operator is
+(T −1Ψ)(x, z) = (1 + |z|)pΨ
+�
+x,
+z
+1 + |z|
+�
+,
+where Ψ ∈ BC(Ω × Bd).
+The dual operator T ∗ : BC(Ω×Bd)∗ → G∗
+p is again an isometric isomorphism. 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.
+Note
+∥ξvj∥ = sup
+∥Φ∥Gp
+ξvj(Φ) =
+ˆ
+Ω
+(1 + |vj|)pdx
+so (ξvj) is a bounded sequence in G∗
+p. But G∗
+p ≂ BC(Ω × Bd)∗, and because BC (hence Gp)
+is not separable we do not necessarily have sequential compactness.
+We must restrict the
+integrands Φ to a separable subspace of Gp.
+10
+
+2.2
+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). 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. The idea behind such construction is to look at the
+graph of each f ∈ F. Because of the boundedness assumption, each function has its image
+contained in a closed bounded interval of R. 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.3, p. 48.
+2.2.1
+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].
+We first define three classes of functions that are rich enough to determine the topological
+structure of their domains:
+Definition 2.5 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).
+Next, we define what a compactification of a topological space is.
+Definition 2.6 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.
+It is useful to remark that because α is continuous, any function f ∈ C(αX) can be restricted
+to a continuous function on X. Indeed, f ◦ α is the composition of a bounded continuous
+function with continuous function, and thus it belongs to BC(X). On the other hand, because
+α(X) is dense in αX, each f ∈ C(αX) is uniquely recovered from f ◦ α ∈ C(X).
+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.
+Theorem 2.7 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 .
+eF X := eF(X) is a compactification of X
+The above theorem is a direct implication of Tychonoff’s theorem. 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.
+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
+�
+.
+11
+
+Lemma 2.8 Let F ⊂ BC(X) be a family that separates points from closed sets and eF X its
+induced compactification. Each f ∈ F embeds into C(eF (X)) in an obvious way and admits a
+unique extension f ∈ C(eF X).
+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. 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.
+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). We expect the latter compacti-
+fication to be bigger than the former, i.e. to be a space where all the previous extensions can
+be further extended to continuous functions.
+Definition 2.9 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. Moreover, we write αX ≂ γX
+if αX ≥ γX and γX ≥ αX, or equivalently if f : αX → γX is a homeomorphism.
+In the following paper, we will sometimes refer to a generic compactification K without specify-
+ing the underlying family generating it. The reason why is stated by the following astonishing
+result.
+Theorem 2.10 Given a compactification αX of X, there exists a family F ⊂ BC(X) that
+separates points from closed sets such that eF X ≂ αX.
+2.2.2
+Representation of compactifications
+Consider a family F ⊂ BC(X) that separates points from closed sets.
+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. Because F can be uncountable, its compactification could be
+hard to deal with from an analytical point of view. We seek a better representation of such
+space.
+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.
+Definition 2.11 Let F be a family of functions f : X → R.
+We call A(F) the algebra
+generated by F, i.e.
+A(F) = {f n + gm : f, g ∈ F, n, m ∈ N}.
+Follows.
+Theorem 2.12 (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. Then eF ′X is a
+compactification of X and eF ′X ≂ eF X.
+12
+
+Proof. Let eF ′X be the (formal) compactification of F ′. Clearly eF X ≥ eF ′X. To prove the
+opposite inclusion, we must find a continuous function T : eF ′X → eF X such that T ◦eF ′ = eF .
+Fix y = limλ Πf∈F ′f(xλ) ∈ eF X. If g ∈ A(F ′), limλ g(xλ) exists and coincides on all nets
+xγ such that y = limγ Πf∈F ′f(xγ). Next, let g ∈ A(F ′) and find a sequence {fn}n∈N ⊂ A(F ′)
+such that fn → g uniformly. 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. 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. 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.e. the net g(xλ) converges to L ∈ R. 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 ′. This shows that the map
+T : eF ′X → eF X, y = lim
+λ Πf∈F ′f(xλ) �→ Ty = lim
+λ Πf∈F f(xλ).
+is a well-defined isomorphism. 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. To prove that eF ′X is a compactification of X, we notice that F separates
+points, as so does F ′. 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.
+□
+2.3
+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). Let F ⊂ BC(Ω×Bd) be a closed separable algebra that separates
+points from closed sets and let eF Ω × Bd be its compactification.
+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).
+Lemma 2.13 If F ⊂ BC(X) is separable, so is C(eFX).
+Proof. Let {fn}n∈N be dense in F. 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.
+□
+Because C(eF Ω × Bd) is separable we also have the abstract sequential compactness prin-
+ciple Theorem A.4, p.
+48 on its dual M(eF Ω × Bd).
+Let T −1F ⊂ Gp the corresponding
+13
+
+algebra (w.r.t. ×p) on the set of continuous functions with p-growth. There is an isometric
+isomorphism
+T −1F
+˜T≂ C(eF Ω × Bd).
+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.
+�
+Lemma 2.14 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. 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. Moreover, F ∪ G ⊂ BC(X × Z) separates points from closed sets.
+The proof of the above lemma is a straightforward application of the Stone-Weierstrass the-
+orem.
+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.
+Morally speaking, what the previous lemma says is that on product spaces it is enough to
+work with the compactifications in each coordinate separately. Moreover, their dual elements
+(measures) can be tested against tensor products of functions that depend on each variable
+independently.
+2.4
+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. Let T −1F = A ⊂ Gp the corresponding algebra,
+with respect to the ×p product, in the set of functions having p-growth. F is isometrically
+isomorphic to C(eFΩ × Bd).
+With abuse of notation, we are going to call T the isomorphism between A and C(eF Ω × Bd).
+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).
+Next, consider a bounded sequence {un}n∈N ⊂ Lp(Ω, Rd), supn ∥un∥p ≤ C. 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.
+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∗. Set
+L := (T ∗)−1ν ∈ M(eF Ω × Bd).
+14
+
+We now study the measure L to find a better representation for ν ∈ A∗. 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).
+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).
+By Lemma 2.14, p. 14, the projection
+π: eF Ω × Bd → eF ′Ω
+is well-defined, and we can write ˜λ = π#L. Note that hereby ˜λ ∈ M+(eF ′(Ω)). 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).
+For any φ ∈ C0(Ω) take Φ = φ(1 − | · |)p. 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).
+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.
+Apply the Radon-
+Nikodym theorem and write
+˜λ =
+˜λ
+µdµ + ˜λs.
+From the previous identification, we get
+�
+⟨˜νx, (1 − | · |)p⟩ ˜λ
+µ = 1
+µ − a.e.
+⟨˜νx, (1 − | · |)p⟩ = 0
+˜λs − a.e.,
+where the second condition implies that ˜νx(eG′(Bd)) = 0 ˜λs-a.e., i.e. the measures are concen-
+trated on the boundary ∂eG′(Bd). On eG′(Bd) we have 0 < (1 − |z|)p ≤ 1, whereas |z| = 1 on
+∂eG′(Bd). In particular
+� ˜λ
+µ =
+1
+⟨˜νx,(1−|·|)p⟩ ≥ 1
+µ − a.e.
+˜νx(∂eG′(Bd)) = 1
+˜λs − a.e.
+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. Let λ = ˜νx(∂eG′(Bd))˜λ. 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.
+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)).
+For each Φ ∈ A, its recession function is defined to be the restriction
+φ∞ = φ|eF ′Ω×∂eG′(Bd).
+Finally, we obtain the formula
+⟨L, TΦ⟩ =
+ˆ
+eF ′(Ω)
+⟨˜νx, TΦ(x, ·)⟩d˜λ
+=
+ˆ
+eF ′Ω
+ˆ
+eG′(Bd)
+TΦd˜νx
+
+
+
+˜λ
+µdµ +
+=0
+����
+d˜λs
+
+
+ +
+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
+�
+.
+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.
+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.
+From this point onwards the family F ′ in Ω will always be the set of functions C(Ω).
+2.5
+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. We then study how the
+new representation for generalised Young Measures behaves geometrically, and its properties.
+As a reminder, we state here the definition of integrands with a regular recession at infinity.
+Definition 2.15 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
+�
+.
+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. 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. 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}. In the language of Hausdorff compactifications,
+set G′ = C(Bd) ∪ {gi, i ∈ N} and F ′ = C(Ω). Because C(Bd) ⊂ G′, the closure of the algebra
+generated by either family is separable, separates points from closed sets and contains the
+constants. Call F = G′ ∪ F ′, and without loss of generality, we can assume that ∥gi∥ ≤ 1 for
+all i.
+Lemma 2.16 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.
+It is
+metrised by
+d(z, w) = |z − w| +
+�
+i
+2−i|gi(z) − gi(w)|.
+Proof. By Lemma 2.14, p.
+14 it is enough to prove that C(egi,i∈NBd) ≂ C(graph(gi)).
+Theorem 2.12, p. 12 provides the isomorphism
+A(C(Bd) ∪ {gi, i ∈ N}) = A(1, z1, . . . , zd, gi, i ∈ N),
+and we conclude by noticing that (1, z1, . . . , zd, gi, i ∈ N)(Bd) is homeomorphic to graph(gi).
+The topological equivalence between such metrics and the product topology is standard.
+□
+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. This means the obvious, that
+we can add functions that have a regular recession and we still obtain the same space (up to
+homeomorphisms).
+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.
+Definition 2.17 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.
+Notice that ∂egi,i∈N is an abuse of notation and refers to the boundary of the embedded space
+within the compactification.
+So far we have constructed compactifications by "glueing" gis on top of the functions z1, . . . , zd;
+that is to say on top of the unit ball. It is sometimes useful to iterate this argument, to stack
+another countable family {fi, i ∈ N} on top of the compactification egi,i∈N. This process gives
+the same compactification as if we were considering the two families at once, as the following
+lemma shows.
+Lemma 2.18 Let egi,i∈N be a compactification of Bd and fi ∈ BC(Bd). Then
+egi,fi,i∈N ≂ graphgraph(gi)fi.
+Proof. This is a trivial consequence of the fact that
+{(z1, . . . , zd, gi(z), fi(z)), z ∈ Bd} ={(z1, . . . , zd, w, z) : w = gi(z), y = fi(z), z ∈ Bd}
+(extending fi to constant in the variable w) ={(z1, . . . , zd, w, z) : y = fi(z, w), w = gi(z), z ∈ Bd}
+=graphgraph(gi)(fi), z ∈ Bd.
+□
+A standard application of the disintegration lemma yields the following.
+Corollary 2.19 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).
+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.
+When working with the sphere compactification, we will simply write
+vj
+Y p(µ,Bd)
+−−−−−−→ ν, or just vj
+Y p(µ)
+−−−−→ ν.
+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.
+We now study the relation of Young Measures with respect to different compactifications and
+different underlying measures µ ∈ M+(Ω). Using Chacon Lemma A.5, p. 48, we can prove
+the following structure theorems.
+Lemma 2.20 Let vj
+Y (µ,efi,i∈N)
+−−−−−−−→ (νx, λ, ν∞
+x ). Then for all ψ ∈ C0(Rd), we have
+ψ(vj) ⇀ ⟨νx, ψ⟩ weakly in L1(µ).
+Proof. Because ψ(vj) ∈ L∞ then is the sequence is equi-integrable and there is a subsequence
+that converges weakly in L1 to v. Because ψ∞ = 0, testing against φ ∈ D(Ω) we get
+ˆ
+Ω
+φψ(uj)dµ →
+ˆ
+Ω
+φvdµ =
+ˆ
+Ω
+φ⟨νx, ψ⟩dµ.
+□
+We can improve the above weak convergence result to show the following.
+Lemma 2.21 Let uj
+Y (µ)
+−−−→
+�
+νx, 0, N/A
+�
+. For every a ∈ L1(Ω, µ) such that a > 0 µ-a.e. and
+for all ψ ∈ C0(Rd) we have
+ψ
+�uj
+a
+�
+a ⇀ ⟨νx, ψ
+�
+·
+a(x)
+�
+⟩a(x) in L1(Ω, µ).
+As expected, this implies that oscillations do not depend on the particular compactification
+chosen.
+Before proving the above results we show the following uniform approximation result:
+Lemma 2.22 Let µ ∈ M+(Ω) and a ∈ L1(Ω, µ), a > 0 µ-almost everywhere. 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.
+19
+
+Proof. Let
+an =
+�
+k∈N,k≥1
+χa−1�
+[ k
+n, k+1
+n )
+� k
+n,
+where N is the set of strictly positive integers. Because an ≤ a then an ∈ L1 and it also
+assumes countably many values at a time. Also |an(x) − a(x)| ≤ 1
+n so it converges uniformly
+to a. 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.
+□
+Now we can prove Lemma 2.21, p. 19
+Proof. 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.
+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. 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.
+Another application of the dominated convergence theorem will let us conclude the result.
+□
+We can finally conclude with a structure theorem regarding concentration.
+Proposition 2.23 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.e. 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
+�
+.
+Then νx =
+�
+·
+a(x)
+�
+# ηx, λν = λη = λ. 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 2.19, p. 18, then
+˜ν∞
+x = ˜η∞
+x λ-a.e. with
+vj
+Y (µ,Bd)
+−−−−−→
+�
+νx, λ, ˜ν∞
+x
+�
+.
+20
+
+Proof. 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).
+Next, identify vj with its subsequence and find Ek so that vj ⇀ v in L1(Ek, µ) for all k. By
+inner approximation, we can assume that all such E′
+ks are compact. Consider now the sequence
+vjχEk. Then
+vjχEk
+Y (µ,A)
+−−−−→
+�
+ηxχEk + δ0χEc
+k, 0, N/A
+�
+.
+By Lemma 2.21, p. 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.
+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.
+Using the previous part we conclude that λν = λη = λ.
+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.
+□
+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. This is
+so because sequences (uj)j can concentrate around values of a that are measure-discontinuous.
+However, equality holds true if a = 1.
+21
+
+Lemma 2.24 Following the assumptions of Proposition 2.23, p. 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.e.
+Proof. This is proven similarly at the end of Proposition 2.23, p.
+20 and testing against
+functions belonging in A(Γ) and using the decomposition of angle Young Measures.
+□
+Next, we show that the lack of concentration is equivalent to the equi-integrability of the
+generating sequence.
+Theorem 2.25 Let (vj)j ∈ L1(Ω, µ, Rd) be so that
+vj
+Y (µ,efi,i∈N)
+−−−−−−−→
+�
+νx, λ, ν∞
+x
+�
+.
+Then the sequence (vj)j∈N is equi-integrable if and only if λ = 0.
+Moreover vj → v strongly in L1(Ω, µ, Rd) if and only if λ = 0 and νx = δv(x) for µ-a.e. x ∈ Ω.
+Proof. Because λ does not depend on the compactification (see Proposition 2.23, p. 20), we
+can apply the same theorem from the sphere compactification, [Rin18], p. 347, lemma 12.14
+and [Rin18], p. 348, corollary 12.15, to conclude.
+□
+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. 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|
+.
+As for the compactification, the metric is always intended as in Lemma 2.16, p. 17.
+Lemma 2.26 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. Then
+sup
+x̸=y
+|g(x) − g(y)|
+|x − y|
+≤ 5Lip(Tg, efi,i∈N)
+for all maps g: Rd → R.
+Proof. Without loss of generality assume that ∥Tg∥Lip ≤ 1, i.e. 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)|.
+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|)
+����
+�
+.
+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|.
+□
+We now show that the above lemma allows us to test Young Measures on Lipschitz compact-
+ifications against Lipschitz functions of Rd.
+Definition 2.27 (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.
+The above formula induces a pseudo-distance between measures by setting d(µ, η)K = ∥µ −
+η∥K. It turns out that this is indeed a metric on the positive cone of non-negative Measures,
+Lemma A.6, p. 48.In particular, by taking Ψ ∈ Lip(efi,i∈N) and the pull-back T −1 we deduce
+the following.
+Lemma 2.28 Let efi,i∈N be a separable compactification.
+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.
+For this reason, we remind once again of the norm we will be using on the space efi,i∈N
+throughout this thesis.
+Definition 2.29 We say that efi,i∈N is a Lipschitz compactification if each fi is Lipschitz
+continuous, and renormalised so that Lip(Tf) ≤ 1. 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)|.
+To conclude this subsection, we state decomposition results for Young Measures regarding
+oscillation and concentration. Originally proven in the context of the sphere compactification,
+[KR19], p. 29, we here extend them to general compactifications.
+Lemma 2.30 Let vj ∈ L1(Ω, µ) so that
+vj
+Y (efi,i∈N,µ)
+−−−−−−−→
+�
+νx, λ, ν∞
+x
+�
+.
+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
+�
+.
+The converse is also true, for each such sequence oj, cj as above, their sum converges to the
+former Young Measure.
+Proof. 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
+�
+. 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⟩).
+Writing ψ(cj) =
+�
+ψ(cj) − ψ(vj)
+�
++ ψ(vj) and letting j → ∞ we conclude.
+□
+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.
+The next lemma is an extension of the previous result.
+Lemma 2.31 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). 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.e.
+η∞
+x
+λη-a.e.
+24
+
+Proof. Write wj = oj + cj as in the previous lemma and put bj = vj − v. We claim that
+bj + cj
+Y (µ,efi,i∈N)
+−−−−−−−→
+�
+δ0, λν + λη, k∞
+x
+�
+.
+No oscillation is a consequence of the fact that bj + cj → 0 in µ-measure.
+Next, let φ ∈
+C(Ω), ∥φ∥Lip ≤ 1 and Ψ ∈ efi,i∈N, ∥TΦ∥Lip ≤ 1 with Ψ(0) = 0. Let Eν and Eη be sets where
+λν and λη are concentrated, respectively. 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.
+Consider a function ρ ∈ C(Rn) with χCη ≤ ρ ≤ χOη. We write
+ˆ
+Ω
+φ
+�
+Ψ(bj + cj) − Ψ(bj) − Ψ(cj)
+�
+(
+=I
+����
+ρ
++
+II
+� �� �
+1 − ρ)dµ.
+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.
+Similarly for the second term
+lim sup
+j
+|II| ≤ lim sup
+j
+2
+ˆ
+Ω
+(1 − ρ)|cj| ≤ 2λη(Ω \ Cη) ≤ 2ε.
+But the first term converges to 0 and therefore we conclude for the representation of Young
+Measures.
+□
+2.6
+Terminology
+We dedicate this part to clarifying the terminology of Young Measures adopted throughout
+this paper.
+Definition 2.32 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. (νx)x∈Ω is µ-measurable and νx ∈ M+
+1 (Rd) µ-a.e. x ∈ Ω.
+We call it the oscillation Young measure;
+2. λ ∈ M+(Ω).
+We call it the concentration measure;
+3. (ν∞
+x )x∈Ω is λ-measurable and ν∞
+x ∈ M+
+1 (∂eA) λ-a.e. x ∈ Ω. We call it the concentration
+angle Young measure;
+25
+
+4. the moment condition
+ˆ
+Ω
+ˆ
+Rd |z|pdνx(z) < ∞
+must hold.
+The collection of all such triples is denoted by Y p = Y p(Ω, µ, eA).
+Where obvious from the context, we will not specify the domain Ω, the measure µ, the family
+A or the target space Rd. Also, when λ = 0, i.e. when there is no concentration, there is no
+point in specifying the compactification we are working with.
+From every triple ν =
+�
+νx, λ, ν∞
+x
+�
+one can construct the measure L ∈ C(eF Ω × Rd) and
+vice-versa.
+Lemma 2.33 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)
+�
+.
+In particular Y p is a weak* closed and convex subset of A∗.
+Proof. Let L ∈ M+(eF Ω × Rd)+ as above. It was already shown that T ∗L ∈ Y p. On the
+other side, if ν =
+�
+νx, λ, ν∞
+x
+�
+, we let
+L = νxdµ + ν∞
+x dλ.
+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.
+Conclude by noticing that the above characterisation amounts to
+Y p = T ∗
+
+
+�
+φ∈C(eF Ω)
+�
+L ∈ M+(eF Ω × Bd)+ :
+ˆ
+eF Ω×Bd φ(x)(1 − ˆz)pdL =
+ˆ
+Ω
+φdµ
+�
+ .
+□
+Remark 2.34 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.
+We do not transcribe the proof here because it won’t be used at any point in this work.
+26
+
+We now give a formal definition of what elementary Young Measures are, as a way to embed
+functions and measures.
+Definition 2.35 Let µ ∈ M+(Ω) and v ∈ Lp(Ω, µ, Rd), 1 ≤ p ≤ ∞.
+The corresponding
+elementary p-Young measure is
+ξv :=
+�
+(δv(x))x∈Ω, 0, N/A
+�
+∈ Y (µ).
+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).
+In general, there is no clear way of defining elementary Young Measures on compactifications
+that are larger than the sphere. 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.
+Definition 2.36 (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 2.9, p.
+12). 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 .
+In the above definition, "≥" is the ordering over the set of Hausdorff compactifications of a
+topological space (see the subsection on Hausdorff compactifications). Moreover, the barycen-
+tre does not depend on the compactification, as far as eA ≥ Bd.
+Indeed z = [zj]j=1,...,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.
+Notice that x �→ νx is µ-measurable and x �→ ν∞
+x is λ-measurable. In particular,
+ν ∈ Lp(Ω, µ, Rd) for 1 < p < ∞
+and
+ν ∈ M(Ω, Rd) for p = 1.
+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.
+27
+
+2.7
+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. These modes of
+convergence explain why we chose such canonical embedding for measures in the previous
+paragraph. Moreover, we give a simple example of why such canonical embedding has no
+meaning when the compactification is larger than the sphere one.
+Definition 2.37 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).
+It is easy to see that if ηj → η strictly, then |ηj| ⇀∗ |η| in C0(Ω, Rd)∗. The above convergence
+prevents small-scale cancellations and concentration on the boundary. However, it does not
+prevent oscillation. To prevent oscillation, we must choose a "weight" µ ∈ M+(Ω) and ask
+for convergence of ηj on the graph (µ, ηj). We thus obtain a notion of µ-strict convergence.
+Definition 2.38 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)∗.
+Similarly to what was observed in the case of strict convergence, such a notion implies that
+|(µ, ηj)| ⇀∗ |(µ, ηj)| in C0(Ω)∗. 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).
+When µ = Ln, we refer to such convergence as area-strict convergence, in analogy with the
+area formula for smooth functions.
+Reshetnyak continuity theorem (see [Res68] for the original) shows that strict convergence is
+equivalent to the convergence of 1-homogeneous functionals.
+Theorem 2.39 Let f(x, z) ∈ C(Ω × Rd) be 1-homogeneous in z. 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.
+Corollary 2.40 Let f ∈ E(Ω × Rd). 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|.
+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.
+Proof. 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. 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.
+□
+Upon taking f(x, z) = |z| we get that µ-strict convergence implies strict convergence, for every
+µ ∈ M+(Ω).
+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. This is a solid justification for this choice of embedding.
+For the same reason, we can show why on larger compactifications we don’t have, in general,
+a canonical choice.
+Theorem 2.41 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). 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.
+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).
+Proof. Because of the assumptions on µ, we can find x ∈ supp(µ
+Ω) so that δx ⊥ µ. 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. Find z ∈ ∂Bd and zj, wj → z so that
+lim
+n Tf(zj) = M and lim
+j Tf(wj) = m exist, and M > m.
+Next, because x ∈ supp(µ) then µ(Br(x)) > 0 for all r. There are two possible scenarios.
+First, x ∈ Ω, in which case we consider only those balls Br(x) so that B2r(x) ⊂ Ω. If x ∈ ∂Ω
+then we can δr ↓ 0 so that µ(Br(x))∩Ω−δ(r)) > 0. Either way, we call Br(x) or Br(x)∩Ω−δ(r)
+simply Br. 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.
+For each such r find φr ∈ D(Bε
+r), 0 ≤ φr ≤ 1 so that φr(Br) = 1. Put ur =
+φr
+´ φrdµ. Clearly
+ur ⇀∗ δx in C(Ω)∗. 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 =
+
+
+
+urjz j
+2
+if j is even,
+urjw j−1
+2
+if j is odd.
+First we show that uj
+µ−strictly
+−−−−−−→ zδx in M(Ω, Rd). Because zj, wj → z, it is enough to show
+that ur
+µ−strictly
+−−−−−−→ δx in M(Ω) as r ↓ 0. Because ur ⇀∗ δx in C(Ω)∗ then
+lim inf
+r→0 |(urdµ, µ)|(Ω) ≥ |(δx, µ)|(Ω).
+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).
+Next, we study how the integral behaves on alternating integers of the sequence (uj)j∈N. 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.
+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.
+□
+30
+
+Remark 2.42 The assumption on µ is sharp. If for all x ∈ supp(µ
+Ω) we have δx ̸⊥ µ, then
+all such x’s belong to Ω. Consider the atomic decomposition of µ:
+µ = µa + µn−a =
+�
+n
+µ(xn)δxn + µn−a,
+see Theorem A.8, p. 49. If µ(Ω \ {xn, n ∈ N}) > 0 then we could find x ∈ Ω \ {xn, n ∈ N}
+and δx ⊥ µ, so that µn−a = 0. Then µ = �
+n µ(xn)δxn, and the set {xn, n ∈ N} contains its
+accumulation points. In particular {xn, n ∈ N} = supp(µ) is a compact subset of Ω. 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.e. λ = �
+n λ(xn)δxn (because the space is countable and compact). Also
+φj ⇀∗ φ =
+�
+νx + ν∞
+x
+λ
+µ
+�
+dµ
+in C0(X)∗, X = {xn, n ∈ N}. Assume also that φj → φ µ-strictly. This amounts to the
+following
+ˆ
+X
+f(φj)dµ →
+ˆ
+X
+⟨νx, f⟩ + ⟨ν∞
+x , f ∞⟩λ
+µdµ =
+ˆ
+X
+f(φ)dµ,
+(2.157)
+where f(z) =
+�
+1 + |z|2. f is a strictly convex function, therefore the inequality f(x + y) ≤
+f(x) + f ∞(y) is strict unless y = 0. 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.e. So φ = νx µ-a.e., and using convexity once again in (2.157), 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.
+So λ = 0, which implies that the sequence φj does not concentrate. Moreover, φ(x) = νx,
+which means that φj → φ in measure. Then φj → φ strongly in L1(µ), and Theorem 2.41, p.
+29 is false.
+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. 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.
+3.1
+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. 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).
+Therefore, we are forced to consider only smaller countable collections of quasi-convex functions
+at the time, and cannot work with the entire class. 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.
+Theorem 3.1 The space of quasi-convex functions f : R2×2 → R having linear growth is not
+separable with respect to ∥T · ∥∞,B2×2.
+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. We will split the proof of the above result
+into two different parts. First, we show that we have quasi-convex functions that oscillate
+along every possible sequence of natural numbers.
+Proposition 3.2 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. fΛ(3j
+1) = 0 for all j ∈ Λ and
+2.
+fΛ(3j
+1)
+3j
+≥ c for all j ∈ Λc sufficiently large.
+To prove this theorem we first need a preliminary lemma, whose proof can be found in [Mül92],
+p. 299, lemma 4.
+Lemma 3.3 For k ∈ R+ we let
+gk : R2×2 → R, F �→ |F1,1 − F2,2| + |F1,2 + F2,1| + (2k − |F1,1 + F2,2|)+.
+Then there exists c1 > 0 such that
+Qgk(0) ≥ c1k
+for all sufficiently large ks.
+We can then prove Proposition 3.2, p. 32.
+32
+
+Proof. 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}.
+We have that gΛ(3j
+1) = 0 for all j ∈ Λ and so fΛ(3j
+1) = 0 as well given that gΛ ≥ 0.
+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};
+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. Apply Lemma 3.3, p. 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.
+Dividing everything by 3j we get
+fΛ(3j
+1)
+3j
+≥ 2
+3c1 ≡ c.
+□
+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. To show that there is a wide variety
+of sequences that differ at infinity, we define the following relation.
+Definition 3.4 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.e.
+Λ = (ai),i∈N, Γ = (bi)i∈N,
+Λ ≤ Γ
+⇐⇒
+there exists k > 0 : (ai)i≥k is a subsequence of (bi)i≥0.
+Let [Λ] be the equivalence class of Λ with respect to ≤, i.e. Γ ∈ [Λ] if Γ ≤ Λ ≤ Γ.
+If Λ′ ∈ [Λ] and Γ′ ∈ [Γ] then Λ ≤ Γ if and only if Λ′ ≤ Γ′.
+We use
+�
+F, ≤
+�
+to indicate the set of equivalence classes with the inherited order.
+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. 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 2.12, p.
+12, but we decided to not pursue this path.
+Lemma 3.5 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 ≤.
+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.e. �� ∩ Γ
+and Λ ∩ Γc are both infinite.
+Proof. Consider the set of subsets
+G = {G ⊂ F : no pair within G is comparable according to ≤},
+ordered by inclusion ⊂. The above set is non-empty, which can be seen by taking Λ = 2N and
+Γ = 4N ∪ (4N + 1). Every chain in G has an upper limit given by its union. By Zorn’s lemma,
+there exists a maximal element.
+I claim that the maximal element has uncountably many
+elements. 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.
+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. 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.
+Set initially Nci = 0 for all i. At i1 so that Nc1
+i1 = 1 for the first time put Nci1 = 1. We then
+iterate "diagonally" in the following way. 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. Set Nctj = 1 for all 1 ≤ j ≤ n + 1. This procedure stops if the
+maximal set is finite, otherwise can be iterated countably many times.
+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. On the other side Nctj = 1 =
+Ncj
+tj, tj ≤ in+1, so Nci is also frequently in every sequence (Ncj
+i)i∈N.
+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.
+□
+We are now ready to prove Proposition 3.2, p. 32.
+Proof. By the lemma we can find an uncountable set of uncomparable subsequences of N, call
+it G. I claim that if Λ, Γ ∈ G, Λ ̸= Γ then fΛ and fΓ have different recessions at infinity.
+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. Given our construction, we immediately have that
+34
+
+{xn} ̸≥ {3j
+1, j ∈ N \ Λ} as fΛ(3j
+1) ≥ c3j for all j ∈ N \ Λ. 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. 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.
+□
+Given that the space is metric, having such a property prevents separability. We can end this
+section with the following corollary that incorporates higher dimensions:
+Corollary 3.6 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.
+Proof. If m or n is 1, quasi-convex functions are convex and therefore the space is separable.
+This is because convex functions admit a limit at infinity in every direction; in this case, we
+actually recover the sphere compactification.
+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
+�
+.
+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.e. gP is quasi-convex. 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.
+□
+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.
+35
+
+3.2
+Characterisation of Gradient Young Measures
+In this section, we characterise Gradient Young Measures (on separable compactification) via
+certain Jensen-like integral inequalities.
+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. 528)). In what follows, we
+cannot use this type of auto-convexity. 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.
+To prove our result, we will adopt the same strategy as in [KR19].
+Preliminarily to stating the theorem, we define the upper recession of a function relative to a
+general compactification.
+Definition 3.7 Let efi,i∈N be a separable compactification. 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.
+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.
+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.
+Fix any compactification efi,i∈N and a function Tg ∈ efi,i∈N. If g ≥ f then
+g∞((zn)) = lim
+n Tg(zn) ≥
+sup
+(zn)∈[(zn)]
+lim sup
+n
+Tf(zn) = f ♯,efi,i∈N((zn)).
+f ♯,efi,i∈N does not need to be continuous on efi,i∈N with respect to its product topology. How-
+ever, we can show that it is still upper semi-continuous.
+Lemma 3.8 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.
+Proof. Notice that ∂efi,i∈N is metrisable, so it is enough to show sequential upper semi-
+continuity. Let (zj
+n)n ∈ ∂efi,i∈N so that (zj
+n)n
+j−→ (zn)n. 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. We want to show that we can select kj so that (zj
+nj)j belongs in the equivalence class
+of (zn)n. 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. 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).
+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. The above statement can be generalised
+to extensions of functions over more general compact metric spaces, but this version suffices
+for our scopes.
+We are interested in studying those Young Measures that are generated by gradients. So we
+define the following.
+Definition 3.9 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
+�
+.
+We use GY (efi,i∈N) to refer to these subsets of Young Measures.
+The convergence of measure derivatives has not been fully comprehended yet and it is still the
+subject of active research. 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.
+Remark 3.10 We can use the characterisation lemma for Young Measure on the sphere to
+show that the class is still quite vast. Indeed, by [KR10a], p. 541 Theorem 1, fix any z = a ⊗ b
+and ν∞ ∈ P(∂B) with ν∞ = z. 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. 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∞. Using Corollary 2.19, p. 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.
+37
+
+We now state the main theorem of this section.
+Theorem 3.11 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.
+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. ≪ 1 ⊗ | · |, ν ≫< +∞, and for all f quasi-convex and having linear growth,
+2. f(∇u(x))dx ≤ ⟨νx, f⟩dx + ⟨ν∞
+x , f ♯,efi,i∈N⟩ λ
+Ln dx and
+3. f ∞(Dsu) ≤ ⟨ν∞
+x , f ♯,efi,i∈N⟩dλs.
+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.
+Lemma 3.12 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.
+In particular, ν ∈ GY (efi,i∈N).
+Proof. 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. 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 3.11, p. 38. Also,
+without loss of generality, we can assume that |Du|(∂Ω−ε) = |Duj|(∂Ω−ε) = 0. Because the
+fis are all Lipschitz, it is enough to test against f ∈ Lip(Rm×n) with Lip(f) ≤ 1. 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.
+□
+38
+
+It’s implicit in Theorem 3.11, p. 38 that
+Du = ν = ⟨νx, ·⟩dx + ⟨ν∞
+x , ·⟩dλ.
+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.
+To prove the characterisation result we will follow the same strategy as in [KR19]. We initially
+prove the result for homogeneous gradient Young Measures and then extend the theorem to
+the inhomogeneous case. 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).
+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∞
+����� .
+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. The following proposition follows from obvious variations of the proofs contained
+in [KR19], p. 8, lemmas 3.7,3.8,3.9. The proofs are essentially the same as they only use the
+separability of the compactification.
+Lemma 3.13 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.
+We can now prove the main theorem in case
+�
+ν0, ν∞�
+is a homogeneous gradient Young
+Measure.
+Proposition 3.14 Let ν =
+�
+ν0, ν∞�
+∈ M+
+1 (Ω) × M+(∂efi,i∈N) and z ∈ Rm×n. 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.
+39
+
+Proof. Suppose that ν ∈ Y and let z + Duj, uj ∈ D(Q, Rm) be the generating sequence, i.e.
+for all Φ ∈ T −1efi,i∈N,
+ˆ
+Q
+Φ(z + Duj)dx → ⟨ν, Φ⟩ =
+ˆ
+Rm×n Φdν0 +
+ˆ
+∂efi,i∈N
+Φdν∞.
+Fix an arbitrary f having linear growth and quasi-convex and let ef,fi,i∈N the bigger compact-
+ification. 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). 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).
+On the other side, using the decomposition of angle concentration Young Measure Corollary 2.19,
+p. 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∞.
+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}.
+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). 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.
+□
+3.2.1
+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 2.31, p. 24. In each case,
+we will use a covering argument to boil it down to the homogeneous case, which was solved in
+the above section.
+Consider a standard mollifier φt(x) = tn−1φ(x
+t ), where φ ∈ D(Q) and let M = ∥Dφ∥∞. Also,
+unless otherwise specified, the norm on Rn is the maximum norm ∥x∥ = maxi |xi|.
+40
+
+Lemma 3.15 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.
+The idea behind this approximation result is the following. The singular part of the centre
+of mass (which is just Du ∈ M(Ω, Rd)) is approximated by mollification. Such a procedure
+generates area-strictly convergent smooth approximations.
+At the same time, we generate
+angle concentration and oscillation via compactly supported functions. 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. 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.
+Before proving the above statement we remind that, according to Lemma 2.26, p. 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). Therefore, all the functions in the following theorem can
+be taken to be, after renormalisation, 1-Lipschitz in both spaces.
+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)|.
+Proof. 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. Without loss of generality,
+assume that Ln(Cs) = 0 and because λs(∂Ω) = 0 then
+∆ = ∆ε = d(Cs, ∂Ω) > 0.
+For the moment, fix two integers a, b ∈ N and put t = 2−a, so that φt > 0 if and only if
+∥x∥ < t. Let a be so large that
+2t ≤ ∆
+and
+a ≥ log2
+� 2
+∆
+�
+.
+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.
+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. Denote by Fs the set of those Q ∈ F for which rQ > 0. In particular, if
+Q ∈ Fs we can find xQ ∈ Cs so that supQ ∥x − xQ∥ < 2t.
+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. 536, lemma 2.5 (we don’t need regular
+recession for this result to hold). 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.
+Going back to the homogeneous case Proposition 3.14, p. 39, we can select ϕQ ∈ D(Q, Rm)
+with ∥ϕQ∥1 < ελs(Q) such that
+∥(δ0, ν∞
+xQrQ) − εrQν∞
+xQ+DϕQ∥K < ε.
+Define ϕ = �
+Q∈Fd ϕQ ∈ D(Ω, Rm) and ∥ϕ∥1 ≤ ελs(Ω). The sought-after map is then
+ξs = φ ⋆ (ν∞
+x dλs + Dϕ) ∈ D(Rn, Rd).
+To prove that this function is the desired one, we fix ∥η∥Lip ≤ 1, ∥Ψ∥Lip(efi,i∈N) ≤ 1 as in the
+assumptions. 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). 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. 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). 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.
+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.
+From each Q ∈ Fs we get
+Φ(0) + ⟨Φ∞, ν∞
+xQ⟩rQ =
+ˆ
+Q
+Φ(rQν∞
+xQ + DφQ)dx +
+|·|≤ε
+����
+EQ
+4 .
+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.
+Finally, we estimate the last term with
+ˆ
+Q
+ηΦ(rQν∞
+xQ + DφQ)dx =
+ˆ
+Q
+ηΦ(φ ⋆ ν∞dλs) + DφQ)dx + EQ
+6 .
+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,
+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 .
+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.
+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.
+For our choice of a and b, we have that Ln(Q) = 2−n(a+b). 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).
+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).
+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).
+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.
+□
+We now move on to the absolutely continuous part, which is proven similar and is a bit easier
+to construct.
+Lemma 3.16 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.
+Proof. 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.e.
+∥(νx, λa(x)ν∞
+x ) − (νy, λa(y)ν∞
+y )∥K ≤ ω(∥x − y∥)
+for all x, y ∈ Ca.
+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.e.
+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. For every Q ∈ Fa select xQ. By Proposition 3.14, p. 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.
+Let ψ = �
+Q ψQ ∈ D(Ω, Rm×n) and ∥ψ∥1 ≤ ε. 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. 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.
+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.
+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.
+For every Q ∈ Fa we have that
+f(xQ) =
+ 
+Q
+Φ(νxQ + λa(xQ)ν∞
+xQ + DφQ) +
+|·|≤ε
+����
+EQ
+4 .
+Set va(x) = νx + λa(x)ν∞
+x . 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. 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. 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.
+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.
+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.
+This concludes the proof.
+□
+47
+
+A
+Appendix
+Theorem A.1 (Vitali convergence theorem) Let µ ∈ M+(Ω) and fn, f ∈ L1(Ω, µ). Then
+fn → f in L1(Ω, µ) if and only if fn → f in measure and fn is uniformly integrable.
+Proof. See [BR07], p. 268, theorem 4.5.4.
+□
+Theorem A.2 (Stone-Weierstrass) Let X be a compact Hausdorff space. 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}. In particular, A = C(X) if and only if A contains the constant
+functions.
+Proof. See [Fol99], p. 139, theorem 4.45.
+□
+Theorem A.3 (Tychonoff) If {Xα}α∈A is a family of compact spaces, Πα∈AXα is compact
+in the product topology.
+Theorem A.4 (Banach-Alaoglu sequential version) Let X be a separable Banach space
+and B ⊂ X∗ the closed unit ball of the dual. Then B is weakly* sequentially compact.
+Proof. See [Lax14], p. 107, theorem 12.
+□
+Lemma A.5 (Chacon biting lemma) Let µ ∈ M+(Ω) and vj ∈ L1(Ω, µ) be a sequence
+such that supj ∥vj∥ < ∞. 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.
+Proof. See [BM89].
+□
+Lemma A.6 (Kantorovich metric) Let (X, d) be a metric space. The Kantorich metric
+on M+(X) generates the same topology as the weak* topology of measures.
+Proof. See [KR19].
+□
+Lemma A.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. Then
+η ⋆ φε
+area-strictly
+−−−−−−−→ η.
+Proof. Because (Ln
+Ω, ηε) ⇀∗ (Ln
+Ω, η), we immediately obtain the lower bound
+lim inf
+ε→0 |(Ln, ρε)|(Ω) ≥ |(Ln, ρ)|(Ω).
+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}.
+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.8 (Atomic decomposition) Let µ ∈ M(Ω). Then there exists a purely atomic
+measure µa and a non-atomic measure µn−a such that µ = µa + µn−a.
+Proof. See [FL07], p. 13.
+□
+49
+
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