diff --git "a/B9FKT4oBgHgl3EQfXi52/content/tmp_files/load_file.txt" "b/B9FKT4oBgHgl3EQfXi52/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/B9FKT4oBgHgl3EQfXi52/content/tmp_files/load_file.txt" @@ -0,0 +1,869 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf,len=868 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='11795v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='AP] 27 Jan 2023 Higher regularity for weak solutions to degenerate parabolic problems Andrea Gentile - Antonia Passarelli di Napoli∗ Dipartimento di Matematica e Applicazioni “R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Caccioppoli” Universitá di Napoli “Federico II”, via Cintia - 80126 Napoli e-mail: andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='gentile@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='it,antpassa@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='it January 30, 2023 Abstract In this paper, we study the regularity of weak solutions to the following strongly degen- erate parabolic equation ut − div � (|Du| − 1)p−1 + Du |Du| � = f in ΩT , where Ω is a bounded domain in Rn for n ≥ 2, p ≥ 2 and ( · )+ stands for the positive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that f ∈ L2 loc (ΩT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Widely degenerate problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Second order regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Higher integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' AMS Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 35B45, 35B65, 35D30, 35K10, 35K65 1 Introduction In this paper, we study the regularity properties of weak solutions u : ΩT → R to the following parabolic equation ut − div � (|Du| − 1)p−1 + Du |Du| � = f in ΩT = Ω × (0, T ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) which appears in gas filtration problems taking into account the initial pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For a precise description of this motivation we refer to [1] and [3, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The main feature of this equation is that it possesses a wide degeneracy, coming from the fact that its modulus of ellipticity vanishes at all points where |Du| ≤ 1 and hence its principal part of behaves like a p-Laplacian operator only at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In this paper we address two interrelated aspects of the regularity theory for solutions to parabolic problems, namely the higher differentiability and the higher integrability of the weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1), with the main aim of weakening the assumption on the datum f with respect to the available literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' ∗Aknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The work of the authors is supported by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The authors have been also supported by the Universitá degli Studi di Napoli “Federico II” through the project FRA-000022-ALTRI-CDA-752021-FRA-PASSARELLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 1 2 These questions have been exploited in case of non degenerate parabolic problems with quadratic growth by Campanato in [9], by Duzaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' in [13] in case of superquadratic growth, while Scheven in [17] faced the subquadratic growth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In the above mentioned papers, the problem have been faced or in case of homogeneous equations or considering sufficiently regular datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' It is worth mentioning that the higher integrability of the gradient of the solution is achieved through an interpolation argument, once its higher differentiability is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' This strategy has revealed to be successful also for degenerate equations as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Indeed the higher integrability of the spatial gradient of weak solutions to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1), has been proven in [3] , under suitable assumptions on the datum f in the scale of Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We’d like to recall that a common feature for nonlinear problems with growth rate p > 2 is that the higher differentiability is proven for a nonlinear expression of the gradient which takes into account the growth of the principal part of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Indeed, already for the non degenerate p-Laplace equation, the higher differentiability refers to the function Vp (Du) = � 1 + |Du|2� p−2 4 Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In case of widely degenerate problems, this phenomenon persists, and higher differentiability results, both for the elliptic and the parabolic problems, hold true for the function H p 2 (Du) = (|Du| − 1) p 2 + Du |Du|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' It is worth noticing that, as it can be expected, this function of the gradient doesn’t give information on the second regularity of the solutions in the set where the equation degenerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Actually, since every 1-Lipschitz continuous function is a solution to the elliptic equation div (Hp−1 (Du)) = 0, where Hp−1 (Du) = (|Du| − 1)p−1 + Du |Du|, no more than Lipschitz regularity can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover, it is well known that in case of degenerate problems (already for the degenerate p-Laplace equation, with p > 2) a Sobolev regularity is required for the datum f in order to get the higher differentiability of the solutions (see, for example [8] for elliptic and [3] for parabolic equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Actually, the sharp assumption for the datum in the elliptic setting has been determined in [8] as a fractional Sobolev regularity suitably related to the growth exponent p and the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The main aim of this paper is to show that without assuming any kind of Sobolev regularity for the datum, but assuming only f ∈ L2, we are still able to obtain higher differentiability for the weak solutions but outside a set larger than the degeneracy set of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' It is worth mentioning that, while for the p-Laplace equation the degeneracy appears for p > 2, here, even in case p = 2, under a L2 integrability assumption on the datum f, the local W 2,2 regularity of the solutions cannot be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Actually, we shall prove the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let n ≥ 2, p ≥ 2 and f ∈ L2 loc (ΩT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover, let us assume that u ∈ C0 � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' L2 (Ω) � ∩ Lp loc � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' W 1,p loc (Ω) � is a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then, for any δ ∈ (0, 1), we have Gδ � (|Du| − 1 − δ)+ � ∈ L2 loc � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' W 1,2 loc (Ω) � , where Gδ(t) := ˆ t 0 s(s + δ) p−2 2 √ 1 + δ + s2 ds, for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover the following estimate ˆ Q R 16 ��D � Gδ � (|Du| − δ − 1)+ ����2 dz ≤ c (n, p) R2δ2 �ˆ QR (|Du|p + 1) dz + 1 δp ˆ QR |f|2 dz � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2) holds for any R > 0 such that QR = QR (z0) ⋐ ΩT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 3 As already mentioned, the weak solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) are not twice differentiable, and hence it is not possible in general to differentiate the equation to estimate the second derivative of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We overcome this difficulty by introducing a suitable family of approximating problems whose solutions are regular enough by the standard theory ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The major effort in the proof of previous Theorem is to establish suitable estimates for the solutions of the regularized problems that are uniform with respect to the approximation’s parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Next, we take advantage from these uniform estimates in the use of a comparison argument aimed to bound the difference quotient of a suitable nonlinear function of the gradient of the solution that vanishes in the set { |Du| ≤ 1 + δ }, with δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Roughly speaking, due to the weakness of our assumption on the datum, we only get the higher differentiability of a nonlinear function of the gradient of the solutions that vanishes in a set which is larger with respect to that of the degeneracy of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' This is quite predictable, since the same kind of phenomenon occurs in the setting of widely degenerate elliptic problems (see, for example [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Anyway, as a consequence of the higher differentiability result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1, we establish a higher integrability result for the spatial gradient of the solution to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1), which is the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Under the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1, we have Du ∈ L p+ 4 n loc (ΩT ) with the following estimate ˆ Q ρ 2 |Du|p + 4 n dz ≤ c (n, p) ρ 2(n+2) n �ˆ Q2ρ � 1 + |Du|p + |f|2� dz � 2 n +1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3) for every parabolic cylinder Q2ρ (z0) ⋐ ΩT , with a constant c = c(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The proof of previous Theorem consists in using an interpolation argument with the aim of establishing an estimate for the Lp+ 4 n norm of the gradient of the solutions to the approximating problems that is preserved in the passage to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We conclude mentioning that the elliptic version of our equation naturally arises in optimal transport problems with congestion effects, and the regularity properties of its weak solutions have been widely investigated (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' [2, 4, 6, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover, we’d like to stress that, for sake of clarity, we confine ourselves to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1), but we believe that our techniques apply as well to a general class of equations with a widely degenerate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 2 Notations and preliminaries In this paper we shall denote by C or c a general positive constant that may vary on different occasions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Relevant dependencies on parameters will be properly stressed using parentheses or sub- scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The norm we use on Rn will be the standard Euclidean one and it will be denoted by | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In particular, for the vectors ξ, η ∈ Rn, we write ⟨ξ, η⟩ for the usual inner product and |ξ| := ⟨ξ, ξ⟩ 1 2 for the corresponding Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For points in space-time, we will use abbreviations like z = (x, t) or z0 = (x0, t0), for spatial variables x, x0 ∈ Rn and times t, t0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We also denote by B (x0, ρ) = Bρ (x0) = { x ∈ Rn : |x − x0| < ρ } the open ball with radius ρ > 0 and center x0 ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' when not important, or clear from the context, we shall omit to indicate the center, denoting: Bρ ≡ B (x0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Unless otherwise stated, different balls in the same context will have the same center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover, we use the notation Qρ (z0) := Bρ (x0) × ��� t0 − ρ2, t0 � , z0 = (x0, t0) ∈ Rn × R, ρ > 0, for the backward parabolic cylinder with vertex (x0, t0) and width ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We shall sometimes omit the dependence on the vertex when the cylinders occurring share the same vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Finally, for a cylinder Q = A × (t1, t2), where A ⊂ Rn and t1 < t2, we denote by ∂parQ := (A × { t1 }) ∪ (∂A × [t1, t2]) the usual parabolic boundary of Q, which is nothing but the standard topological boundary without the upper cap A × { t2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 4 We now recall some tools that will be useful to prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For the auxiliary function Hλ : Rn → Rn defined as Hλ(ξ) := \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (|ξ| − 1)λ + ξ |ξ| if ξ ∈ Rn \\ {0} , 0 if ξ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) where λ > 0 is a parameter, we record the following estimates (see [7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1]): Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' If 2 ≤ p < ∞, then for every ξ, η ∈ Rn it holds ⟨Hp−1(ξ) − Hp−1(η), ξ − η⟩ ≥ 4 p2 ���H p 2 (ξ) − H p 2 (η) ��� 2 , |Hp−1(ξ) − Hp−1(η)| ≤ (p − 1) ����H p 2 (ξ) ��� p−2 p + ���H p 2 (η) ��� p−2 p � ���H p 2 (ξ) − H p 2 (η) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we record the following estimates (see [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8]) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let ξ, η ∈ Rk with |ξ| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then, we have |Hp−1(ξ) − Hp−1(η)| ≤ c(p) � (|ξ| − 1) + (|η| − 1)+ �p−1 |ξ| − 1 |ξ − η| and ⟨Hp−1(η) − Hp−1(ξ), ·η − ξ⟩ ≥ min { 1, p − 1 } 2p+1 (|ξ| − 1)p |ξ| (|ξ| + |η|) |η − ξ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' With the use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1), a function u ∈ C0 � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' L2 (Ω) � ∩Lp � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' W 1,p (Ω) � is a weak solution of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) if ˆ ΩT (u · ∂tϕ − ⟨Hp−1 (Du) , Dϕ⟩) dz = − ˆ ΩT fϕ dz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2) for every ϕ ∈ C∞ 0 (ΩT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In the following, we shall also use the well known auxiliary function Vp : Rn → Rn defined as Vp(ξ) := � 1 + |ξ|2� p−2 4 ξ, where p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For every ξ, η ∈ Rn there hold 1 c1 (p) |Vp(ξ) − Vp(η)|2 ≤ � 1 + |ξ|2 + |η|2� p−2 2 |ξ − η|2 ≤ c1(p) �� 1 + |ξ|2� p−2 2 ξ − � 1 + |η|2� p−2 2 η, ξ − η � , We refer to [16, Chapter 12] or to [15, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2] for a proof of these fundamental inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For further needs, we also record the following interpolation inequality whose proof can be found in [12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1] Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Assume that the function v : Qr(z0) ∪ ∂parQr(z0) → R satisfies v ∈ L∞ � t0 − r2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lq (Br (x0)) � ∩ Lp � t0 − r2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' W 1,p 0 (Br (x0)) � for some exponents 1 ≤ p, q < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then the following estimate ˆ Qr(z0) |v|p+ pq n dz ≤ c � sup s∈(t0−r2,t0) ˆ Br(x0) |v(x, s)|q dx � p n ˆ Qr(z0) |Dv|p dz holds true for a positive constant c depending at most on n, p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 Difference quotients We recall here the definition and some elementary properties of the difference quotients (see, for ex- ample, [15, Chapter 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For every function F : Rn → RN the finite difference operator in the direction xs is defined by τs,hF(x) = F (x + hes) − F(x), where h ∈ R, es is the unit vector in the direction xs and s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The difference quotient of F with respect to xs is defined for h ∈ R \\ {0} as ∆s,hF(x) = τs,hF(x) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We shall omit the index s when it is not necessary, and simply write τhF(x) = F(x + h) − F(x) and |∆hF(x)| = |τhF(x)| |h| for h ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let F ∈ W 1,p (Ω), with p ≥ 1, and let us set Ω|h| := { x ∈ Ω : dist (x, ∂Ω) > |h| } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then: (a) ∆hF ∈ W 1,p � Ω|h| � and Di(∆hF) = ∆h(DiF), for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' , n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (b) If at least one of the functions F or G has support contained in Ω|h|, then ˆ Ω F ∆hG dx = − ˆ Ω G ∆−hF dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (c) We have ∆h (FG) (x) = F (x + hes) ∆hG(x) + G(x)∆hF(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The next result about the finite difference operator is a kind of integral version of Lagrange Theorem (see [15, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' If 0 < ρ < R, |h| < R − ρ 2 , 1 < p < +∞, and F ∈ W 1,p � BR, RN� , then ˆ Bρ |τhF(x)|p dx ≤ cp(n) |h|p ˆ BR |DF(x)|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover ˆ Bρ |F(x + hes)|p dx ≤ ˆ BR |F(x)|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We conclude this section with the following fundamental result, whose proof can be found in [15, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2]: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let F : Rn → RN, F ∈ Lp � BR, RN� with 1 < p < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Suppose that there exist ρ ∈ (0, R) and a constant M > 0 such that n � s=1 ˆ Bρ |τs,hF(x)|p dx ≤ M p |h|p for every h, with |h| < R − ρ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then F ∈ W 1,p � Bρ, RN� and ∥DF∥Lp(Bρ) ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover ∆s,hF → DsF strongly in Lp loc (BR) , as h → 0, for each s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 Some auxiliary functions and related algebraic inequalities In this section we introduce some auxiliary functions and we list some of their properties, that will be used in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For any k > 1 and for s ∈ [0, +∞), let us consider the function gk(s) = s2 k + s2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3) for which we record the following Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let k > 1, and let gk be the function defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then for every A, B ≥ 0 the following Young’s type inequality A · B[s · g′ k ((s − k)+)] ≤ 2 √ 2k � αA2gk ((s − k)+) + ασA2 + cαB2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4) holds for every parameters α, σ > 0 with a constant cα independent of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover, there exists a constant ck > 0, depending on k, such that sg′ k �� s2 − k � + � ≤ ck, ∀s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Since g′ k(s) = 2ks (k + s2)2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6) both the conclusions trivially hold for s ≤ √ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Now assume that s > ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k and note that Young’s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='inequality implies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='A · B [s · g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k ((s − k)+)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='A · B · s · g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k ((s − k)+) [σ + (s − k)+] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='[σ + (s − k)+] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='αA2s · g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k ((s − k)+) [σ + (s − k)+] + cα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='B2s · g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k ((s − k)+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='[σ + (s − k)+] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='αA2s · g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k ((s − k)+) (s − k)+ + ασA2s · g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k ((s − k)+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+cα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='B2s · g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k ((s − k)+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='[σ + (s − k)+] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='αA2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2ks(s − k)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k + (s − k)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='�2 + ασA2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2ks(s − k)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k + (s − k)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+cαB2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2ks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='k + (s − k)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='(s − k)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='[σ + (s − k)+],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7) where we used the explicit expression of g′ k(s) at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3) and since t k + t2 ≤ 1, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7) we deduce A · B [s · g′ k ((s − k)+)] ≤ αA2 2ks k + (s − k)2 + gk ((s − k)+) +ασA2 2ks k + (s − k)2 + + cαB2 2ks k + (s − k)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8) Setting h(s) = s k + (s − k)2 + , we can easily check that h(k) = 1, lim s→+∞ h(s) = 0, max s∈[k,+∞) h(s) = h �� k2 + k � = 1 2 � 1 + � 1 + 1 k � < √ 2 7 and so 2ks k + (s − k)2 + ≤ 2 √ 2k ∀s > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Inserting this in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8), we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In order to prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5), let us notice that, recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6), we have sg′ �� s2 − k � + � = 2ks � s2 − k � + � k + (s2 − k)2 + �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' So, since the function sg′ �� s2 − k � + � is continuous in the interval � s ≥ 0 �� s2 > k � = �√ k, +∞ � and lim s→+∞ 2ks � s2 − k � + � k + (s2 − k)2 + �2 = 0, then there exists a constant ck > 0 such that sg′ �� s2 − k � + � ≤ ck for every s ≥ 0, which is the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For any δ > 0, let us define Gδ(t) := ˆ t 0 s(s + δ) p−2 2 √ 1 + δ + s2 ds, for t ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9) and observe that G′ δ(t) = t(t + δ) p−2 2 √ 1 + δ + t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='10) Next Lemma relates the function Gδ (|ξ|) with H p 2 (ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let Gδ be the function defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9) and H p 2 be the one defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) with λ = p 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then we have ��Gδ � (|ξ| − δ − 1)+ � − Gδ � (|η| − δ − 1)+ ���2 ≤ cp ���H p 2 (ξ) − H p 2 (η) ��� 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='11) for any ξ, η ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' If |ξ| < 1 + δ and |η| < 1 + δ there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' So will assume that |ξ| > 1 + δ, and without loss of generality we may suppose that |η| ≤ |ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Since Gδ(t) is increasing, we have ��Gδ (|ξ| − 1 − δ) − Gδ � (|η| − 1 − δ)+ ��� = Gδ (|ξ| − 1 − δ) − Gδ � (|η| − 1 − δ)+ � = ˆ |ξ|−1−δ (|η|−1−δ)+ s(s + δ) p−2 2 √ 1 + δ + s2 ds ≤ ˆ |ξ|−1−δ (|η|−1−δ)+ (s + δ) p−2 2 ds = 2 p � (|ξ| − 1) p 2 − � (|η| − δ − 1)+ + δ � p 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Now, it can be easily checked that (|ξ| − 1) p 2 − � (|η| − δ − 1)+ + δ � p 2 8 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (|ξ| − 1) p 2 − δ p 2 if |ξ| > δ + 1 and |η| ≤ δ + 1 (|ξ| − 1) p 2 − (|η| − 1) p 2 if |ξ| > δ + 1 and |η| > δ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In the first case, we have ���(|ξ| − 1) p 2 − δ p 2 ��� = (|ξ| − 1) p 2 − δ p 2 ≤ (|ξ| − 1) p 2 − (|η| − 1) p 2 + = ���H p 2 (ξ) ��� − ���H p 2 (η) ��� ≤ ���H p 2 (η) − H p 2 (ξ) ��� , while, in the second, (|ξ| − 1) p 2 − � (|η| − δ − 1)+ + δ � p 2 = ���H p 2 (ξ) ��� − ���H p 2 (η) ��� ≤ ���H p 2 (η) − H p 2 (ξ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Therefore, ��Gδ � (|ξ| − δ − 1)+ � − Gδ � (|η| − δ − 1)+ ���2 ≤ cp ���H p 2 (ξ) − H p 2 (η) ��� 2 for every ξ, η ∈ Rn, which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Arguing as in [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1], we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then the following inequalities hold cp,δ(t + δ) p 2 − ˜cp,δ ≤ Gδ(t) ≤ 2 p(t + δ) p 2 with constants ˜cp,δ and cp,δ < 2 p depending on p and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' If p = 2, one can easily calculate Gδ(t) = ˆ t 0 s √ 1 + δ + s2 ds = �� 1 + δ + s2 �t 0 = � 1 + δ + t2 − √ 1 + δ, from which immediately follows 1 2 (t + δ) − 1 2 �√ 1 + δ + δ � ≤ Gδ(t) ≤ t + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The right inequality is a simple consequence of the trivial bound s √ 1+δ+s2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For the left inequality we start observing that � 1 + δ + s2 ≤ √ 1 + δ + s =⇒ Gδ(t) ≥ ˆ t 0 s (s + δ) p−2 2 √ 1 + δ + s ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Now, we calculate the integral in previous formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' By the change of variable r = √ 1 + δ + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we get ˆ t 0 s (s + δ) p−2 2 √ 1 + δ + s ds = ˆ t+ √ 1+δ √ 1+δ � r − √ 1 + δ � � r − √ 1 + δ + δ � p−2 2 r ds = ˆ t+ √ 1+δ √ 1+δ � r − √ 1 + δ + δ � p−2 2 ds − √ 1 + δ ˆ t+ √ 1+δ √ 1+δ � r − √ 1 + δ + δ � p−2 2 r ds ≥ 2 p �� r − √ 1 + δ + δ � p 2 �t+√1+δ √1+δ − √ 1 + δ ˆ t+ √ 1+δ √ 1+δ � r − √ 1 + δ + δ � p 2 −2 ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' since 0 < δ ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we have δ ≤ √ 1 + δ and therefore r − √ 1 + δ + δ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Calculating the last integral in previous formula, we get ˆ t 0 s(s + δ) p−2 2 √ 1 + δ + s ds 9 ≥ 2 p �� r − √ 1 + δ + δ � p 2 �t+√1+δ √1+δ − 2 √ 1 + δ p − 2 �� r − √ 1 + δ + δ � p 2 −1�t+√1+δ √1+δ = 2 p � (t + δ) p 2 − δ p 2 � − 2 √ 1 + δ p − 2 � (t + δ) p 2 −1 − δ p 2 −1� = 2 p(t + δ) p 2 − 2 √ 1 + δ p − 2 (t + δ) p 2 −1 + 2 √ 1 + δ p − 2 δ p 2 −1 − 2 pδ p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Therefore the lemma will be proven if there exists a constant cp,δ < 2 p such that cp,δ(t + δ) p 2 ≤ 2 p(t + δ) p 2 − 2 √ 1 + δ p − 2 (t + δ) p 2 −1 + 2 √ 1 + δ p − 2 δ p 2 −1 − 2 pδ p 2 which, setting h(t) = 2 √ 1 + δ p − 2 (t + δ) p 2 −1 + � cp,δ − 2 p � (t + δ) p 2 , is equivalent to prove that there exists cp,δ such that h(t) ≤ 2 √ 1 + δ p − 2 δ p 2 −1 − 2 pδ p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' It is easy to check that h(t) attains his maximum for t + δ = 2 √ 1 + δ 2 − pcp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ and so h(t) ≤ h � 2 √ 1 + δ 2 − pcp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ − δ � = � 2 √ 1 + δ � p 2 � 1 2 − pcp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ � p−2 2 2 p (p − 2) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' to complete the proof it’s enough to solve the following equation � 2 √ 1 + δ � p 2 � 1 2 − pcp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ � p−2 2 2 p (p − 2) = 2 √ 1 + δ p − 2 δ p 2 −1 − 2 pδ p 2 which is equivalent to 1 2 − pcp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ = � δ 2 √ 1 + δ � p p−2 � p �√ 1 + δ − δ � δ + 2 � 2 p−2 that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' for 0 < δ < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' admits a unique solution cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ < 2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 3 The regularization For ε > 0, we introduce the sequence of operators Aε(ξ) := (|ξ| − 1)p−1 + ξ |ξ| + ε � 1 + |ξ|2� p−2 2 ξ and by uε ∈ C0 � t0 − R2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' L2 (BR) � ∩ Lp � t0 − R2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' u + W 1,p 0 (BR) � we denote the unique solution to the corresponding problems \uf8f1 \uf8f2 \uf8f3 uε t − div (Aε (Duε)) = f ε in QR (z0) uε = u in ∂parQR (z0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) where QR (z0) ⋐ ΩT with R < 1, f ε = f ∗ ρε with ρε the usual sequence of mollifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' One can easily check that the operator Aε satisfies p-growth and p-ellipticity assumptions with constants depending 10 on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Therefore, by the results in [13], we have Vp (Duε) ∈ L2 loc � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' W 1,2 loc (BR (x0) , Rn) � and |Duε| ∈ L p+ 4 n loc (QR) and, by the definition of Vp(ξ), this yields DVp (Duε) ≈ � 1 + |Duε|2� p−2 4 D2uε ∈ L2 loc � QR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Rn×n� =⇒ ��D2u���� ∈ L2 loc (QR) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2) By virtue of [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1], we also have H p 2 (Duε) ∈ L2 loc � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' W 1,2 loc (Ω, Rn) � and, by the definition of H p 2 (ξ), it follows ���DH p 2 (Du) ��� ≤ cp (|Duε| − 1) p−2 2 + |D2uε| ∈ L2 loc � QR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Rn×n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 Uniform a priori estimates The first step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 is the following estimate for solutions to the regularized problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let uε ∈ C0 � t0 − R2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' L2 (BR) � ∩ Lp � t0 − R2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' u + W 1,p 0 (BR) � be the unique solu- tion to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then the following estimate sup τ∈(t0−4ρ2,t0) ˆ Bρ � |Duε(x, τ)|2 − 1 − δ � + dx + ˆ Qρ ��D � Gδ � (|Duε| − δ − 1)+ ����2 dz ≤ c ρ2 �ˆ Q2ρ (1 + |Duε|p) dz + δ2−p ˆ Q2ρ |f ε|2 dz � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4) holds for any ε ∈ (0, 1] and for every Qρ ⋐ Q2ρ ⋐ QR, with a constant c = c(n, p) independent of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The weak formulation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) reads as ˆ QR (uε · ∂tϕ − ⟨Aε (Duε) , Dϕ⟩) dz = − ˆ QR f ε · ϕ dz for any test function ϕ ∈ C∞ 0 (QR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Recalling the notation used in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2), and replacing ϕ with ∆−hϕ = τ−hϕ h for a sufficiently small h ∈ R \\ { 0 }, by virtue of the properties of difference quotients, we have ˆ QR � ∆huε · ∂tϕ − ⟨∆hHp−1 (Duε) , Dϕ⟩ − ε � ∆h �� 1 + |Duε|2� p−2 2 Duε � , Dϕ �� dz = − ˆ QR f ε · ∆−hϕ dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5) Arguing as in [13, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1], from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5) we get ˆ QR ∂t∆huε · �� dz + ˆ QR ⟨∆hHp−1 (Duε) , Dϕ⟩ dz +ε ˆ QR � ∆h �� 1 + |Duε|2� p−2 2 Duε � , Dϕ � dz = ˆ QR f ε · ∆−hϕ dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For Φ ∈ W 1,∞ 0 (QR) non negative and g ∈ W 1,∞ (R) non negative and non decreasing, we choose ϕ = Φ · ∆huε · g � |∆huε|2� in previous identity, thus getting ˆ QR ∂t (∆huε) ∆huε · g � |∆huε|2� Φ dz 11 + ˆ QR � ∆hHp−1 (Duε) , D � Φ∆huεg � |∆huε|2��� dz +ε ˆ QR � ∆h �� 1 + |Duε|2� p−2 2 Duε � , D � Φ∆hug � |∆huε|2��� dz = ˆ QR f ε · ∆−h � Φ∆huε · g � |∆huε|2�� dz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' ˆ QR ∂t (∆huε) ∆huε · g � |∆huε|2� Φ dz + ˆ QR Φ � ∆hHp−1 (Duε) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' ∆hDuε · g � |∆huε|2�� dz +ε ˆ QR Φ � ∆h �� 1 + |Duε|2� p−2 2 Duε � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' ∆hDuε · g � |∆huε|2�� dz +2 ˆ QR Φ � ∆hHp−1 (Duε) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' |∆huε|2 ∆hDuε · g′ � |∆huε|2�� dz +2ε ˆ QR Φ � ∆h �� 1 + |Duε|2� p−2 2 Duε � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' |∆huε|2 ∆hDuε · g′ � |∆huε|2�� dz = − ˆ QR � ∆hHp−1 (Duε) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' DΦ · ∆huε · g � |∆huε|2�� dz −ε ˆ QR � ∆h �� 1 + |Duε|2� p−2 2 Duε � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' DΦ · ∆huε · g � |∆huε|2�� dz + ˆ QR f ε · ∆−h � Φ∆huε · g � |∆huε|2�� dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6) that we rewrite as follows Jh,1 + Jh,2 + Jh,3 + Jh,4 + Jh,5 = −Jh,6 − Jh,7 + Jh,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Arguing as in [5],the first integral in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6) can be expressed as follows Jh,1 = ˆ QR ∂t (∆huε) ∆huε · g � |∆huε|2� Φ dz = 1 2 ˆ QR ∂t � |∆huε|2� g � |∆huε|2� Φ dz = 1 2 ˆ QR ∂t �ˆ |∆huε|2 0 g(s) ds � Φ dz = −1 2 ˆ QR �ˆ |∆huε|2 0 g(s) ds � ∂tΦ dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2, since Φ, g are non negative, we have Jh,2 ≥ ˆ QR Φ · g � |∆huε|2� |∆hDuε|2 (|Duε| − 1)p |Duε| (|Duε| + |Duε(x + h)|) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' The right inequality in the assertion of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4 yields Jh,3 ≥ εcp ˆ QR Φ · g � |∆huε|2� |∆hVp (Duε)|2 dz Moreover, again by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4 and the fact that g′(s) ≥ 0, we infer Jh,4 + Jh,5 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Therefore (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6) implies −1 2 ˆ QR �ˆ |∆huε|2 0 g(s) ds � ∂tΦ dz + ˆ QR Φ · g � |∆huε|2� |∆hDuε|2 (|Duε| − 1)p |Duε| (|Duε| + |Duε(x + h)|) dz 12 +cpε ˆ QR Φ · g � |∆huε|2� |∆hVp (Duε)|2 dz ≤ ˆ QR |DΦ| |∆hHp−1 (Duε)| |∆huε| · g � |∆huε|2� dz +ε ˆ QR |DΦ| ����∆h �� 1 + |Duε|2� p−2 2 Duε ����� |∆huε| · g � |∆huε|2� dz + ˆ QR |f ε| ���∆−h � Φ∆huε · g � |∆huε|2����� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7) Now let us consider a parabolic cylinder Qρ (z0) ⋐ Q2ρ (z0) ⋐ QR (z0) with ρ < 2ρ < R and t0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For a fixed time τ ∈ � t0 − 4ρ2, t0 � and θ ∈ (0, t0 − τ), we choose Φ(x, t) = η2(x)χ(t)˜χ(t) with η ∈ C∞ 0 (B2ρ (x0)), 0 ≤ η ≤ 1, χ ∈ W 1,∞ ([0, T ]) with ∂tχ ≥ 0 and ˜χ a Lipschitz continuous function defined, for 0 < τ < τ + θ < T , as follows ˜χ(t) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 if t ≤ τ 1 − t − τ θ if τ < t ≤ τ + θ 0 if τ + θ < t ≤ T so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7) yields Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 + Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 + Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3 := 1 2 ˆ B2ρ η2χ(τ) �ˆ |∆huε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='τ)|2 0 g(s) ds � dx +cp ˆ Qτ η2χ(t) · g � |∆huε|2� |∆hDuε|2 (|Duε| − 1)p |Duε| (|Duε| + |Duε(x + h)|) dz +cpε ˆ Qτ η2χ(t)g � |∆huε|2� |∆hVp (Duε)|2 dz ≤ 2 ˆ Qτ ηχ(t) |Dη| |∆hHp−1 (Duε)| |∆huε| · g � |∆huε|2� dz +2ε ˆ Qτ ηχ(t) |Dη| ����∆h �� 1 + |Duε|2� p−2 2 Duε ����� |∆huε| · g � |∆huε|2� dz + ˆ Qτ χ(t) |f ε| ���∆−h � η2∆huε · g � |∆huε|2����� dz +1 2 ˆ Qτ η2∂tχ(t) �ˆ |∆huε|2 0 g(s) ds � dz =: Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4 + Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5 + Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6 + Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8) where we used the notation Qτ = B2ρ (x0) × � t0 − 4ρ2, τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Since g ∈ W 1,∞ ([0, ∞)), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2), by the last assertion of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9 and by Fatou’s Lemma, we have lim inf h→0 (Ih,1 + Ih,2 + Ih,3) ≤ 1 2 ˆ B2ρ η2χ(τ) �ˆ |Duε(x,τ)|2 0 g(s) ds � dx +cp ˆ Qτ η2χ(t) · g � |Duε|2� ��D2uε��2 (|Duε| − 1)p |Duε|2 dz +cpε ˆ Qτ η2χ(t)g � |Duε|2� |DVp (Duε)|2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9) and lim h→0 Ih,7 = 1 2 ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='10) Now let us observe that |DHp−1 (Duε)| ≤ cp (|Duε| − 1)p−2 + ��D2uε�� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='11) 13 and, using Hölder’s inequality with exponents � 2(p−1) p−2 , 2(p−1) p � , we have ˆ BR |DHp−1 (Duε)| p p−1 dx ≤ cp ˆ BR � (|Duε| − 1)p−2 + ��D2uε�� � p p−1 dx ≤ cp �ˆ BR (|Duε| − 1)p + dx � p−2 2(p−1) �ˆ BR � (|Duε| − 1) p−2 2 + ��D2uε�� �2 dx � p 2(p−1) , and since, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3), the right hand side of previous inequality is finite again by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9, we have ∆hHp−1 (Duε) → DHp−1 (Duε) strongly in L2 � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' L p p−1 (BR) � as h → 0, which, since ∆huε → Duε strongly in L2 (0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lp (BR)) as h → 0, implies lim h→0 Ih,4 = 2 ˆ Qτ ηχ(t) |Dη| |DHp−1 (Duε)| |Duε| g � |Duε|2� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='12) Using similar arguments, we can check that lim h→0 Ih,5 = 2ε ˆ Qτ ηχ(t) |Dη| ����D �� 1 + |Duε|2� p−2 2 Duε ����� |Duε| · g � |Duε|2� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='13) Now, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7(c), it holds ���∆−h � η2∆huε · g � |∆huε|2����� ≤ c∥Dη∥∞ |∆huε| ���g � |∆huε|2���� +c |∆−h (∆huε)| ���g � |∆huε|2���� +c |∆huε|2 ���g′ � |∆huε|2���� |∆hDuε| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' and choosing g such that sg′ � s2� ≤ M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='14) for a positive constant M, we have ���∆−h � η2∆huε · g � |∆huε|2����� ≤ c∥Dη∥∞ |∆huε| ���g � |∆huε|2���� +c |∆−h (∆huε)| ���g � |∆huε|2���� +cM |∆huε| |∆−hDuε| (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='15) Since ∆huε → Duε, ∆−h (∆huε) → D2uε, ∆−hDuε → D2uε strongly in L2 � 0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' L2 loc (Ω) � as h → 0, and f ε ∈ C∞ (ΩT ), thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='15), we have lim h→0 Ih,6 = ˆ Qτ χ(t) |f ε| ���D � η2Duε · g � |Duε|2����� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='16) So, collecting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='16), we can pass to the limit as h → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8), thus getting 1 2 ˆ B2ρ η2χ(τ) �ˆ |Duε(x,τ)|2 0 g(s) ds � dx +cp ˆ Qτ η2χ(t) · g � |Duε|2� ��D2uε��2 (|Duε| − 1)p |Duε|2 dz +cpε ˆ Qτ η2χ(t)g � |Duε|2� |DVp (Duε)|2 dz 14 ≤ 2 ˆ Qτ ηχ(t) |Dη| |DHp−1 (Duε)| |Duε| · g � |Duε|2� dz +2ε ˆ Qτ ηχ(t) |Dη| ����D �� 1 + |Duε|2� p−2 2 Duε ����� |Duε| · g � |Duε|2� dz + ˆ Qτ χ(t) |f ε| ���D � η2Duε · g � |Duε|2����� dz +1 2 ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz =: ˜I1 + ˜I2 + ˜I3 + ˜I4, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='17) for every g ∈ W 1,∞(0, +∞) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='14) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Now, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='11) and by Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we have ˜I1 + ˜I2 ≤ cp ˆ Qτ ηχ(t) |Dη| (|Duε| − 1)p−2 + ��D2uε�� |Duε| · g � |Duε|2� dz +cp · ε ˆ Qτ ηχ(t) |Dη| � 1 + |Duε|2� p−1 2 ��D2uε�� · g � |Duε|2� dz ≤ σ ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 · g � |Duε|2� dz +σε ˆ Qτ η2χ(t) � 1 + |Duε|2� p−2 2 ��D2uε��2 · g � |Duε|2� dz +cσ ˆ Qτ χ(t) |Dη|2 (|Duε| − 1)p−4 + |Duε|4 · g � |Duε|2� dz +cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='σ · ε ˆ Qτ χ(t) |Dη|2 � 1 + |Duε|2� p 2 · g � |Duε|2� dz ≤ σ ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 · g � |Duε|2� dz +σε ˆ Qτ η2χ(t) |DVp (Duε)|2 · g � |Duε|2� dz +cσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ∥Dη∥2 L∞ ∥g∥L∞ ˆ Qτ χ(t) (1 + |Duε|)p dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='18) where we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2), and where σ > 0 is a parameter that will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Now, using Young’s Inequality, we estimate the term ˜I3, as follows ˜I3 ≤ c ˆ Qτ χ(t) |f ε| η |Dη| |Duε| · g � |Duε|2� dz +c ˆ Qτ χ(t) |f ε| η2 ��D2uε�� · g � |Duε|2� dz +c ˆ Qτ χ(t) |f ε| η2 |Duε|2 ��D2uε�� · g′ � |Duε|2� dz ≤ c ∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) |f ε|2 dz +c ∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) |Duε|2 dz +c ˆ Qτ η2χ(t) |f ε| ��D2uε�� · g � |Duε|2� dz +c ˆ Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � |Duε|2� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='19) Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='19) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we get 1 2 ˆ B2ρ η2χ(τ) �ˆ |Duε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='τ)|2 0 g(s) ds � dx 15 +cp ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz +cpε ˆ Qτ η2χ(t)g � |Duε|2� |DVp (Duε)|2 dz ≤ σ ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 · g � |Duε|2� dz +σε ˆ Qτ η2χ(t) |DVp (Duε)|2 · g � |Duε|2� dz +cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='σ ∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) |f ε|2 dz +cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='σ∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) (1 + |Duε|)p dz +c ˆ Qτ η2χ(t) |f ε| ��D2uε�� · g � |Duε|2� dz +c ˆ Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � |Duε|2� dz +1 2 ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' for a sufficiently small σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' gives 1 2 ˆ B2ρ η2χ(τ) �ˆ |Duε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='τ)|2 0 g(s) ds � dx +cp ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz +cpε ˆ Qτ η2χ(t)g � |Duε|2� |DVp (Duε)|2 dz ≤ cp∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) |f ε|2 dz +cp∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) (1 + |Duε|)p dz +c ˆ Qτ η2χ(t) |f ε| ��D2uε�� · g � |Duε|2� dz +c ˆ Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � |Duε|2� dz +1 2 ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' neglecting the third integral in the left hand side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' implies 1 2 ˆ B2ρ η2χ(τ) �ˆ |Duε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='τ)|2 0 g(s) ds � dx +cp ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ cp∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) |f ε|2 dz +cp∥Dη∥∞ ∥g∥L∞ ˆ Qτ ηχ(t) (1 + |Duε|)p dz +c ˆ Qτ η2χ(t) |f ε| ��D2uε�� · g � |Duε|2� dz +c ˆ Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � |Duε|2� dz +1 2 ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='20) 16 Now, for δ ∈ (0, 1), recalling the notation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3), we choose g(s) = g1+δ � (s − 1 − δ)+ � that is g(s) = (s − 1 − δ)2 + 1 + δ + (s − 1 − δ)2 + , that is legitimate since g ∈ W 1,∞([0, +∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Moreover, with this choice, we have g(s) ∈ [0, 1], for every s ≥ 0, and thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5), there exists a constant cδ > 0 such that sg′ � s2� ≤ cδ for every s ≥ 0, so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Therefore, since g(s) vanishes on the set where s ≤ 1 + δ and g(s) ≤ 1 for every s, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='20) becomes 1 2 ˆ B2ρ η2χ(τ) �ˆ |Duε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='τ)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='g(s) ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+cp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ η2χ(t) · g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε|2� (|Duε| − 1)p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='��D2uε��2 dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ∩{|Duε|2>1+δ} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='η2χ(t) |f ε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='��D2uε�� (|Duε| − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='(|Duε| − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ ∩{|Duε|2>1+δ} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='η2χ(t) |f ε| |Duε|2 (|Duε| − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='(|Duε| − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='��D2uε�� g′ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+cp∥Dη∥∞ ∥χ∥L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 + |Duε|p + |f ε|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ η2∂tχ(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='�ˆ |Duε|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='g(s) ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='cp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ η2χ(t) |f ε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='��D2uε�� (|Duε| − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ cp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='��D2uε�� g′ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='|Duε|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+cp∥Dη∥∞ ∥χ∥L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 + |Duε|p + |f ε|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='Qτ η2∂tχ(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='�ˆ |Duε|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='g(s) ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' where we used that sup x∈( √ 1+δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='+∞) x (x − 1) p 2 = √ 1 + δ �√ 1 + δ − 1 � p 2 = √ 1 + δ �√ 1 + δ + 1 � p 2 δ p 2 ≤ cp δ p 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' since δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Using Young’s inequality in the first integral in the right hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' previous estimate yields 1 2 ˆ B2ρ η2χ(τ) �ˆ |Duε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='τ)|2 0 g(s) ds � dx +cp ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ cp(β) δp ˆ Qτ η2χ(t) |f ε|2 · g � |Duε|2� dz +β ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 · g � |Duε|2� dz + cp δ p 2 ˆ Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) p 2 + |Duε| ��D2uε�� g′ � |Duε|2� dz +cp∥Dη∥∞ ∥χ∥L∞ ˆ Qτ � 1 + |Duε|p + |f ε|2� dz 17 + ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Choosing β sufficiently small, reabsorbing the second integral in the right hand side by the left hand side and using that g(s) ≤ 1, we get ˆ B2ρ η2χ(τ) �ˆ |Duε(x,τ)|2 0 g(s) ds � dx + ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ c cp δ p 2 ˆ Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) p 2 + |Duε| ��D2uε�� g′ � |Duε|2� dz + ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz c ∥Dη∥2 ∞ ∥χ∥∞ ˆ Qτ (1 + |Duε|)p dz +c ∥χ∥L∞ �cp δp + ∥Dη∥L∞ � ˆ Qτ |f ε|2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='21) We now estimate the first integral in the right side of previous inequality with the use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4) with s = |Duε|2, A = (|Duε| − 1) p 2 + |Duε| ��D2uε��, B = cp δ p 2 |f ε| and k = 1 + δ, thus getting cp δ p 2 ˆ Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) p 2 + |Duε| ��D2uε�� g′ � |Duε|2� dz ≤ 2α ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 g � |Duε|2� dz +2ασ ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 dz +cα,p δp ˆ Qτ η2χ(t) |f ε|2 dz, with constants c, cα both independent of σ and where we used that δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' By virtue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3), taking the limit as σ → 0 in previous inequality, we have cp δ p 2 ˆ Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) p 2 + |Duε| ��D2uε�� g′ � |Duε|2� dz ≤ 2α ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 g � |Duε|2� dz +cα,p δp ˆ Qτ η2χ(t) |f ε|2 dz, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='22) Inserting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='22) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='21), we find ˆ B2ρ η2χ(τ) �ˆ |Duε(x,τ)|2 0 g(s) ds � dx + ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ 2α ˆ Qτ η2χ(t)(|Duε| − 1)p + |Duε|2 ��D2uε��2 g � |Duε|2� dz +cα,p δp ˆ Qτ η2χ(t)|f ε|2 dz 18 + ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz c ∥Dη∥2 ∞ ∥χ∥∞ ˆ Qτ (1 + |Duε|)p dz +c ∥χ∥L∞ �cp δp + ∥Dη∥L∞ � ˆ Qτ |f ε|2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Choosing α = 1 4 , we can reabsorb the first integral in the right hand side by the left hand side, thus obtaining ˆ B2ρ η2χ(τ) �ˆ |Duε(x,τ)|2 0 g(s) ds � dx + ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ c ∥Dη∥2 ∞ ∥χ∥∞ ˆ Qτ (1 + |Duε|)p dz + c δp ∥χ∥L∞ (1 + ∥Dη∥L∞) ˆ Qτ |f ε|2 dz +c ˆ Qτ η2∂tχ(t) �ˆ |Duε|2 0 g(s) ds � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='23) By the definition of g, we have ˆ ζ 0 g(s) ds = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 0 if 0 < ζ ≤ 1 + δ ˆ ζ 1+δ (s − 1 − δ)2 1 + δ + (s − 1 − δ)2 ds if ζ > 1 + δ, and so it is easy to check that ˆ ζ 0 g(s) ds = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 0 if 0 < ζ ≤ 1 + δ ζ − 1 − δ − √ 1 + δ arctan �ζ − 1 − δ √ 1 + δ � if ζ > 1 + δ, that is ˆ ζ 0 g(s) ds = (ζ − 1 − δ)+ − √ 1 + δ arctan �(ζ − 1 − δ)+ √ 1 + δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Therefore, by previous equality and the properties of χ and η, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='23) implies ˆ B2ρ η2χ(τ) � |Duε(x, τ)|2 − 1 − δ � + dx + ˆ Qτ η2χ(t) · g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ c ∥Dη∥2 ∞ ∥χ∥∞ ˆ Qτ (1 + |Duε|)p dz + c δp ∥χ∥L∞ (1 + ∥Dη∥L∞) ˆ Qτ |f ε|2 dz +c ˆ Qτ η2∂tχ(t) � |Duε|2 − 1 − δ � + dz +c ∥∂tχ∥∞ |Qτ| + c ∥χ∥∞ |BR| , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='24) which holds for almost every τ ∈ � t0 − 4ρ2, t0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We now choose a cut-off function η ∈ C∞ (B2ρ (x0)) with η ≡ 1 on Bρ (x0) such that 0 ≤ η ≤ 1 and 19 |Dη| ≤ c ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' For the cut-off function in time, we choose χ ∈ W 1,∞ � t0 − R2, t0, [0, 1] � such that χ ≡ 0 on � t0 − R2, t0 − 4ρ2� , χ ≡ 1 on � t0 − ρ2, t0 � and ∂tχ ≤ c ρ2 on � t0 − 4ρ2, t0 − ρ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' With these choices, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='24) gives sup τ∈(t0−4ρ2,t0) ˆ Bρ χ(τ) � |Duε(x, τ)|2 − 1 − δ � + dx + ˆ Qρ g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ c ρ2 ˆ Q2ρ (1 + |Duε|p) dz + c ρ2δp ˆ Q2ρ |f ε|2 dz +c |Q2ρ| ρ2 + c |B2ρ| , and since ρ < 2ρ < R < 1, and Q2ρ = Bρ × � t0 − 4ρ2, t0 � , we have sup τ∈(t0−4ρ2,t0) ˆ Bρ � |Duε(x, τ)|2 − 1 − δ � + dx + ˆ Qρ g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 dz ≤ c ρ2 ˆ Q2ρ (1 + |Duε|p) dz + c ρ2δp ˆ Q2ρ |f ε|2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='25) Now, with Gδ(t) defined at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9), recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='10), we have ��D � Gδ � (|Duε| − δ − 1)+ ����2 ≤ (|Duε| − δ − 1)2 + 1 + δ + (|Duε| − δ − 1)2 + � (|Duε| − δ − 1)+ + δ �p−2 ��D2uε��2 = g (|Duε|) � (|Duε| − δ − 1)+ + δ �p−2 ��D2uε��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Since g(s) is nondecreasing, we have g(s) ≤ g � s2� , and therefore ��D � Gδ � (|Duε| − δ − 1)+ ����2 ≤ g � |Duε|2� (|Duε| − 1)p−2 + ��D2uε��2 ≤ cp δ2 g � |Duε|2� (|Duε| − 1)p + |Duε|2 ��D2uε��2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='26) where we also used that g(s) = 0, for 0 < s ≤ 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='26) in the left hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='25), we obtain sup τ∈(t0−4ρ2,t0) ˆ Bρ � |Duε(x, τ)|2 − 1 − δ � + dx + ˆ Qρ ����D � Gδ � (|Duε| − δ − 1)+ ����2 dz ≤ c ρ2δ2 �ˆ Q2ρ (1 + |Duε|p) dz + 1 δp ˆ Q2ρ |f ε|2 dz � , which is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let uε ∈ C0 � t0 − R2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' L2 (BR) � ∩ Lp � t0 − R2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' u + W 1,p 0 (BR) � be the unique solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Then the following estimate ˆ Q ρ 2 ��τh � Gδ � (|Duε| − δ − 1)+ ����2 dz 20 ≤ c|h|2 ρ2δ2 �ˆ Q2ρ (1 + |Duε|p) dz + 1 δp ˆ Q2ρ |f ε|2 dz � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='27) holds for |h| < ρ 4, for any parabolic cylinder Q2ρ ⋐ QR (z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 4 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 This section is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1, that will be divided in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In the first one we shall establish an estimate that will allow us to measure the L2-distance between H p 2 (Du) and H p 2 (Duε) in terms of the L2-distance between f and f ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' In the second one, we conclude combining this comparison estimate with the one obtained for the difference quotient of the solution to the regularized problem at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Step 1: the comparison estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We formally proceed by testing equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) with the map ϕ = k(t) (uε − u), where k ∈ W 1,∞ (R) is chosen such that k(t) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 if t ≤ t2, − 1 ω (t − t2 − ω) if t2 < t < t2 + ω, 0 if t ≥ t2 + ω, with t0 − R2 < t2 < t2 + ω < t0, and then letting ω → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We observe that, at this stage, it is important that uε and u agree on the parabolic boundary ∂parQR (z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Proceeding in a standard way (see for example [13]), for almost every t2 ∈ � t0 − R2, t0 � , we find 1 2 ˆ BR(x0) |uε (x, t2) − u (x, t2)|2 dx + ˆ QR,t2 ⟨Hp−1 (Duε) − Hp−1 (Du) , Duε − Du⟩ dz +ε ˆ QR,t2 �� 1 + |Duε|2� p−2 2 Duε, Duε − Du � dz = ˆ QR,t2 (f − f ε) (uε − u) dz, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) where we used the abbreviation QR,t2 = BR (x0) × � t0 − R2, t2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1, the Cauchy- Schwarz inequality as well as Young’s inequality, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) we infer λp sup t∈(t0−R2,t0) ∥uε(·, t) − u(·, t)∥2 L2(BR(x0)) +λp ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz + ε ˆ QR(z0) |Duε|p dz ≤ ˆ QR |f − f ε| |uε − u| dz + ε ˆ QR |Duε|p−1 |Du| dz ≤ ˆ QR |f − f ε| |uε − u| dz + ε · cp ˆ QR |Du|p dz +1 2 · ε ˆ QR |Duε|p dz, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2) where we set λp = min � 1 2, 4 p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Reabsorbing the last integral in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2) by the left-hand side, we arrive at sup t∈(t0−R2,t0) ∥uε(·, t) − u(·, t)∥2 L2(BR(x0)) 21 + ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz + ε 2λp ˆ QR |Duε|p dz ≤ ε cp ˆ QR |Du|p dz + cp ˆ QR |f − f ε| |uε − u| dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3) Using in turn Hölder’s inequality and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5, we get ˜I := ˆ QR |f − f ε| |uε − u| dz ≤ C (R, n, p) ∥f − f ε∥L2(QR) · �ˆ QR |uε − u|p+ 2p n dz � n p(n+2) ≤ c (n, p, R) ∥f − f ε∥L2(QR) · �ˆ QR |Duε − Du|p dz � n p(n+2) � sup t∈(t0−R2,t0) ∥uε(·, t) − u(·, t)∥2 L2(BR(x0)) � 1 n+2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4) Now, let us notice that ˆ QR |Duε − Du|p dz = ˆ QR∩{|Duε|≥1} (|Duε| − 1 + 1)p dz + ˆ QR∩{|Duε|<1} |Duε|p dz + ˆ QR |Du|p dz ≤ cp ˆ QR � (|Duε| − 1)p + � dz + ˆ QR (|Du|p + 1) dz ≤ cp ˆ QR ����H p 2 (Duε) − H p 2 (Du) + H p 2 (Du) ��� 2� dz + cp ˆ QR (|Du|p + 1) dz ≤ cp ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz + cp ˆ QR (|Du|p + 1) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we get ˜I ≤ c (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' R) ∥f − f ε∥L2(QR(z0)) �ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz + ˆ QR (|Du|p + 1) dz � n p(n+2) � sup t∈(t0−R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='t0) ∥uε(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t) − u(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t)∥2 L2(BR(x0)) � 1 n+2 ≤ c (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' R) ∥f − f ε∥L2(QR) · �ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz � n p(n+2) � sup t∈(t0−R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='t0) ∥uε(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t) − u(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t)∥2 L2(BR(x0)) � 1 n+2 +c (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' R) ∥f − f ε∥L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+2) � sup t∈(t0−R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='t0) ∥uε(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t) − u(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t)∥2 L2(BR(x0)) � 1 n+2 and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' by Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we get ˜I ≤ β ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz + β sup t∈(t0−R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='t0) ∥uε(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t) − u(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t)∥2 L2(BR(x0)) +c (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' β) ∥f − f ε∥ n+2 n+1 L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+1) 22 +c (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' β) ∥f − f ε∥ p(n+2) n(p−1)+p L2(QR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='6) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3), we obtain sup t∈(t0−R2,t0) ∥uε(·, t) − u(·, t)∥2 L2(BR(x0)) + ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz + ε 2λp ˆ QR |Duε|p dz ≤ β ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz + β sup t∈(t0−R2,t0) ∥uε(·, t) − u(·, t)∥2 L2(BR(x0)) +c (n, p, R, β) ∥f − f ε∥ n+2 n+1 L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+1) +c (n, p, R, β) ∥f − f ε∥ p(n+2) n(p−1)+p L2(QR) + ε cp ˆ QR |Du|p dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7) Choosing β = 1 2 and neglecting the third non negative term in the left hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='7), we get sup t∈(t0−R2,t0) ∥uε(·, t) − u(·, t)∥2 L2(BR(x0)) + ˆ QR ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz ≤ c (n, p, R) ∥f − f ε∥ n+2 n+1 L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+1) +c (n, p, R) ∥f − f ε∥ p(n+2) n(p−1)+p L2(QR) + ε cp ˆ QR |Du|p dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8) For further needs, we also record that, combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8), we have ˆ QR |Duε|p dz ≤ c (n, p, R) ∥f − f ε∥ n+2 n+1 L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+1) +c (n, p, R) ∥f − f ε∥ p(n+2) n(p−1)+p L2(QR) + ε cp ˆ QR |Du|p dz +cp ˆ QR (|Du|p + 1) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9) Step 2: The conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let us fix ρ > 0 such that Q2ρ ⊂ QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We start observing that ˆ Q ρ 2 ��τh � Gδ � (|Du| − δ − 1)+ ����2 dz ≤ c ˆ Q ρ 2 ��τh � Gδ � (|Duε| − δ − 1)+ ����2 dz +c ˆ Qρ ��Gδ � (|Duε| − δ − 1)+ � − Gδ � (|Du| − δ − 1)+ ���2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We estimate the right hand side of previous inequality using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='27) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='11), as follows ˆ Q ρ 2 ��τh � Gδ � (|Du| − δ − 1)+ ����2 dz ≤ c|h|2 ρ2 �ˆ Q2ρ (1 + |Duε|p) dz + δ2−p ˆ Q2ρ |f ε|2 dz � +cp ˆ Q2ρ ���H p 2 (Duε) − H p 2 (Du) ��� 2 dz that, thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='8), implies ˆ Q ρ 2 ��τh � Gδ � (|Du| − δ − 1)+ ����2 dz 23 ≤ c|h|2 ρ2 �ˆ Q2ρ (1 + |Duε|p) dz + δ2−p ˆ Q2ρ |f ε|2 dz � +c (n, p, R) ∥f − f ε∥ n+2 n+1 L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+1) +c (n, p, R) ∥f − f ε∥ p(n+2) n(p−1)+p L2(QR) + ε cp ˆ QR |Du|p dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='10) Now, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9), we get ˆ Q2ρ (1 + |Duε|p) dz ≤ c (n, p, R) ∥f − f ε∥ n+2 n+1 L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+1) +c (n, p, R) ∥f − f ε∥ p(n+2) n(p−1)+p L2(QR) + ε cp ˆ QR |Du|p dz +cp ˆ QR (|Du|p + 1) dz which, combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='10), implies ˆ Q ρ 2 ��τh � Gδ � (|Du| − δ − 1)+ ����2 dz ≤ c (n, p) |h|2 ρ2 � c(R) ∥f − f ε∥ n+2 n+1 L2(QR) · �ˆ QR (|Du|p + 1) dz � n p(n+1) +c(R) ∥f − f ε∥ p(n+2) n(p−1)+p L2(QR) + ε ˆ QR |Du|p dz + ˆ QR (|Du|p + 1) dz + δ2−p ˆ QR |f ε|2 dz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Taking the limit as ε → 0, and since f ε → f strongly in L2 (BR), we obtain ˆ Q ρ 2 ��τh � Gδ � (|Du| − δ − 1)+ ����2 dz ≤ c (n, p) |h|2 ρ2 �ˆ QR (|Du|p + 1) dz + δ2−p ˆ QR |f|2 dz �� , and thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9, we have Gδ � (|Du| − δ − 1)+ � ∈ L2 � t0 − ρ2, t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' W 1,2 (Bρ) � with the follow- ing estimate ˆ Q ρ 2 ��D � Gδ � (|Du| − δ − 1)+ ����2 dz ≤ c (n, p) ρ2 �ˆ QR (|Du|p + 1) dz + δ2−p ˆ QR |f|2 dz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Since previous estimate holds true for any ρ > 0 such that 4ρ < R, we may choose ρ = R 8 thus getting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' 5 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2 The higher differentiability result of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1 allows us to argue as in [13, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='3] and [17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2] to obtain the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We start observing that ���D �� Gδ � (|Duε| − 1 − δ)+ �� 4 np + 1���� 24 ≤ c ��Gδ � (|Duε| − 1 − δ)+ ��� 4 np ��D � Gδ � (|Duε| − 1 − δ)+ ���� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1) where c ≡ c(n, p) > 0 and Gδ(t) is the function defined at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' With the notation we used in the previous sections, for B2ρ (x0) ⋐ BR (x0), let ϕ ∈ C∞ 0 (Bρ (x0)) and χ ∈ W 1,∞ ((0, T )) be two non-negative cut-off functions with χ(0) = 0 and ∂tχ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we fix a time t0 ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' T ) and apply the Sobolev embedding theorem on the time slices Σt := Bρ(x0) × {t} for almost every t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' t0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' to infer that ˆ Σt ϕ2 �� Gδ � (|Duε| − 1 − δ)+ �� 4 np + 1�2 dx ≤ c �ˆ Σt ���D � ϕ � Gδ � (|Duε| − 1 − δ)+ �� 4 np + 1���� 2n n+2 dx � n+2 n ≤ c �ˆ Σt ���ϕ D �� Gδ � (|Duε| − 1 − δ)+ �� 4 np + 1���� 2n n+2 dx � n+2 n +c �ˆ Σt ��� ��Gδ � (|Duε| − 1 − δ)+ ��� 4 np + 1 Dϕ ��� 2n n+2 dx � n+2 n =: c I1(t) + c I2(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' in the second to last line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' we have applied Minkowski’s and Young’s inequalities one after the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We estimate I1(t) and I2(t) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Let us first consider I1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='1), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='12 and Hölder’s inequality with exponents �n + 2 n , n + 2 2 � , we deduce I1(t) ≤ c �ˆ Σt ϕ 2n n+2 � (|Duε| − 1) 2 n + ��DGδ � (|Duε| − 1 − δ)+ ��� � 2n n+2 dx � n+2 n ≤ c ˆ Σt ϕ2 ��DGδ � (|Duε| − 1 − δ)+ ���2 dx �ˆ supp(ϕ) (|Duε| − 1)2 + dx � 2 n ≤ c ˆ Σt ϕ2 ��DGδ � (|Duε| − 1 − δ)+ ���2 dx �ˆ supp(ϕ) |Duε|2 dx � 2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' We now turn our attention to I2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='12 and Hölder’s inequality yield I2(t) ≤ c �ˆ Σt (|Duε| − 1) np + 4 n+2 + |Dϕ| 2n n+2 dx � n+2 n ≤ c �ˆ Σt � |Dϕ|2 |Duε|p� n n+2 |Du| 4 n+2 dx � n+2 n ≤ c ˆ Σt |Dϕ|2 |Duε|p dx �ˆ supp(ϕ) |Duε|2 dx � 2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content=' Putting together the last three estimates, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FKT4oBgHgl3EQfXi52/content/2301.11795v1.pdf'} +page_content='12 in the left hand side, and integrating with respect to time, we obtain ˆ Qt0 χϕ2 (|Duε| − 1) p + 4 n + dz ≤ c ˆ t0 0 χ �ˆ supp(ϕ) |Duε(x, t)|2 dx � 2 n �ˆ Σt � ϕ2 ��DGδ � (|Duε| − 1 − δ)+ ���2 + |Dϕ|2 |Du|p� dx � dt ≤ c ˆ Qt0 χ � ϕ2 ��DGδ � (|Duε| − 1 − δ)+ ���2 + |Dϕ|2 |Du|p� dz 25 � sup 0