A Variational Theory of Convolution-Type Functionals 9819906849, 9789819906840

Table of contents :
Preface
Contents
1 Introduction
References
2 Convolution-Type Energies
2.1 Notation
2.2 Setting of the Problem and Comments
2.3 Assumptions
Reference
3 The -Limit of a Class of Reference Energies
3.1 The -Limit of G[a]
References
4 Asymptotic Embedding and Compactness Results
4.1 An Extension Result
4.2 Control of Long-Range Interactions with Short-Range Interactions
4.3 Compactness in Lp Spaces
4.4 Poincaré Inequalities
References
5 A Compactness and Integral-Representation Result
5.1 The Integral-Representation Theorem
5.2 Truncated-Range Functionals
5.3 Fundamental Estimates
5.4 Proof of the Integral-Representation Theorem
5.5 Convergence of Minimum Problems
5.6 Euler-Lagrange Equations
5.6.1 Regularity of Functionals F
5.6.2 Relations with Minimum Problems
References
6 Periodic Homogenization
6.1 A Homogenization Theorem
6.2 The Convex Case
6.3 Relaxation of Convolution-Type Energies
6.4 An Extension Lemma from Periodic Lipschitz Domains
6.5 Homogenization on Perforated Domains
References
7 A Generalization and Applications to Point Clouds
7.1 Perturbed Convolution-Type Functionals
7.2 Application to Functionals Defined on Point Clouds
References
8 Stochastic Homogenization
References
9 Application to Convex Gradient Flows
9.1 The Minimizing-Movement Approach to Gradient Flows
9.2 Homogenized Flows for Convex Energies
References
Index

Citation preview

SpringerBriefs on PDEs and Data Science Roberto Alicandro · Nadia Ansini · Andrea Braides Andrey Piatnitski · Antonio Tribuzio

A Variational Theory of ConvolutionType Functionals

SpringerBriefs on PDEs and Data Science Editor-in-Chief Enrique Zuazua, Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Bayern, Germany

Series Editors Irene Fonseca, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA Franca Hoffmann, Hausdorff Center for Mathematics, University of Bonn, Bonn, Germany Shi Jin, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, Shanghai, China Juan J. Manfredi, Department of Mathematics, University Pittsburgh, Pittsburgh, PA, USA Emmanuel Trélat, CNRS, Laboratoire Jacques-Louis Lions, Sorbonne University, PARIS CEDEX 05, Paris, France Xu Zhang, School of Mathematics, Sichuan University, Chengdu, Sichuan, China

SpringerBriefs on PDEs and Data Science targets contributions that will impact the understanding of partial differential equations (PDEs), and the emerging research of the mathematical treatment of data science. The series will accept high-quality original research and survey manuscripts covering a broad range of topics including analytical methods for PDEs, numerical and algorithmic developments, control, optimization, calculus of variations, optimal design, data driven modelling, and machine learning. Submissions addressing relevant contemporary applications such as industrial processes, signal and image processing, mathematical biology, materials science, and computer vision will also be considered. The series is the continuation of a former editorial cooperation with BCAM, which resulted in the publication of 28 titles as listed here: https://www.springer. com/gp/mathematics/bcam-springerbriefs

Roberto Alicandro • Nadia Ansini • Andrea Braides • Andrey Piatnitski • Antonio Tribuzio

A Variational Theory of Convolution-Type Functionals

Roberto Alicandro Department of Electrical and Information Engineering University of Cassino and Southern Lazio Cassino, Frosinone, Italy

Nadia Ansini Department of Mathematics Sapienza University of Rome Rome, Italy

Andrea Braides SISSA Trieste, Italy

Andrey Piatnitski Department of Technology UiT The Arctic University of Norway Narvik, Norway

Antonio Tribuzio Institute for Applied Mathematics University of Heidelberg Heidelberg, Baden-Württemberg, Germany

ISSN 2731-7595 ISSN 2731-7609 (electronic) SpringerBriefs on PDEs and Data Science ISBN 978-981-99-0684-0 ISBN 978-981-99-0685-7 (eBook) https://doi.org/10.1007/978-981-99-0685-7 Mathematics Subject Classification: 49J45, 49J55, 74Q05, 35B27, 35B40, 45E10 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

These notes stem from a group activity in Rome stimulated by a visit of Andrey Piatnitski at the University of Rome Tor Vergata, where he was also giving a course on the homogenization of convolution operators. Noting the timely nature of that subject and the potentiality of applications in different directions, from evolution phenomena with long-range diffusion to Data Science, the working group focussed on providing an abstract framework for a comprehensive theory of convolutiontype functionals. In that project, we had in mind on one hand the analogy with theories regarding the passage discrete-to-continuum developed in the last 20 years, and on the other hand a number of previous homogenization results for convolution operators both in the static and dynamic framework. The result was a quite complete formalization based on a very general compactness theorem in the spirit of the De Giorgi localization methods for .-convergence, and a series of preliminary abstract results, such as embeddings, Poincarè inequalities, etc., with a number of applications, in particular to stochastic homogenization, to energies on point clouds and to gradient flows, which are just some of the potential directions of the theory. The results presented here are intended to provide an environment, together with the related technical tools, in which to frame both static and dynamic problems related to multiple-scale variational models where non-local interactions at a small scale are involved. As such, the notes are addressed to a readership of Mathematical Analysts and Applied Mathematicians. Beside the material elaborated in the working group, the present notes are complemented by references to peridynamics, in whose terminology our analysis gives a general framework for the limit as the horizon tends to zero, and a revisitation of some recent works on perforated domains in the light of the compactness theorem. We are very grateful to Enrique Zuazua for having proposed us to contribute to this book series. We also acknowledge the fruitful interaction with Valeria Chiadò Piat, Lorenza D’Elia, Carolin Kreisbeck and Elena Zhizhina on issues connected with the content of these notes.

v

vi

Preface

The authors acknowledge the support of the Italian MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Antonio Tribuzio was partially supported by the Deutsche Forschungsgemeinschaft through SPP 2256, project ID 441068247. Cassino, Italy Rome, Italy Trieste, Italy Narvik, Norway Heidelberg, Germany October 2022

Roberto Alicandro Nadia Ansini Andrea Braides Andrey Piatnitski Antonio Tribuzio

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 7

2

Convolution-Type Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Setting of the Problem and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 12 17

3

The .-Limit of a Class of Reference Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The -Limit of Gε [aε ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 27

4

Asymptotic Embedding and Compactness Results. . . . . . . . . . . . . . . . . . . . . . . 4.1 An Extension Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Control of Long-Range Interactions with Short-Range Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Compactness in Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Poincaré Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29

A Compactness and Integral-Representation Result . . . . . . . . . . . . . . . . . . . . 5.1 The Integral-Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Truncated-Range Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fundamental Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of the Integral-Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Convergence of Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Regularity of Functionals Fε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Relations with Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 44 49 51 54 54 55 58

5

32 34 37 40

vii

viii

Contents

6

Periodic Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A Homogenization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Convex Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Relaxation of Convolution-Type Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 An Extension Lemma from Periodic Lipschitz Domains . . . . . . . . . . . . . 6.5 Homogenization on Perforated Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 67 69 78 83 89

7

A Generalization and Applications to Point Clouds . . . . . . . . . . . . . . . . . . . . . 91 7.1 Perturbed Convolution-Type Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 Application to Functionals Defined on Point Clouds. . . . . . . . . . . . . . . . . . 96 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8

Stochastic Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9

Application to Convex Gradient Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Minimizing-Movement Approach to Gradient Flows . . . . . . . . . . . . 9.2 Homogenized Flows for Convex Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 111 114

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 1

Introduction

Abstract We provide a general treatment of a class of functionals modeled on convolution energies with kernel having finite p-moments. Such model energies approximate the p-th norm of the gradient as the kernel is scaled by letting a small parameter .ε tend to 0. We first provide the necessary functional-analytic tools to show coerciveness of families of such functionals with respect to strong .Lp convergence. The main result is a compactness and integral-representation theorem which shows that limits of convolution-type energies are local integral functionals with p-growth defined on a Sobolev space. This result is applied to obtain periodic homogenization results, to study applications to functionals defined on point-clouds, to stochastic homogenization and to the study of limits of the related gradient flows. Keywords Nonlocal energies · Peridynamics · Population dynamics · Compactness · Integral representation · Periodic homogenization · Stochastic homogenization · Point clouds · Gradient flows

Scope of these notes is a general asymptotic analysis of families of non-local functionals of the form  . Wε (x, y, u(y) − u(x)) dx dy, (1.1) ×

with . a Lipschitz domain in .Rd , as the parameter .ε tends to 0, and the energies ‘concentrate’ on the diagonal .x = y. Energies of this form are customary in the theory of peridynamics (see e.g. [9, 22, 23, 36–39]), where .ε represents the range of interaction between points, so that indeed the energies above are of the form  .

{(x,y)∈×:|x−y|≤ε}

Wε (x, y, u(y) − u(x)) dx dy.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_1

(1.2)

1

2

1 Introduction

A prototypical assumption on .Wε is that Wε (x, y, z) =

.

1  y − x z W x, , . εd ε ε

(1.3)

If .u ∈ C 1 (; Rm ) then we may approximate .u(x) − u(y) with .∇u(x)(x − y), so that, after a change of variables .y = x + εξ , formally we can write the energies above approximatively as  .

    u(x + εξ ) − u(x)  dxdξ ∼ W x, ξ, W x, ξ, ∇u(x) ξ dxdξ . ε ×B1 (0) ×B1 (0) (1.4)

This formula suggests that the limit may be a local functional representable as an integral. Under some growth and convexity hypothesis on W , indeed it has been proved that the .-limit is defined on some Sobolev space .W 1,p (; Rm ) and has the local form     .F (u) = W0 (x, ∇u) dx, where W0 (x, M) = W x, ξ, Mξ dξ (1.5) 

B1 (0)

for every .m × d matrix M (see [5, 6, 31]). This argument will be made rigorous as a very particular case of the results in the following. In order to build up a general theory for functionals (1.1), we start with considering convolution functionals; i.e., functionals of the form 

1 .

εd+p

a ×

y − x  |u(y) − u(x)|p dx dy, ε

(1.6)

as a simpler class of prototype energies. In formula (1.6) . is a Lipschitz domain in Rd and a is a sufficiently integrable positive kernel; namely, a satisfies

.

 .

Rd

a(ξ )|ξ |p dξ < +∞.

(1.7)

This choice corresponds to choosing .W (x, ξ, z) = a(ξ )|z|p in (1.3). Functionals as in (1.6) can be seen as an approximation and a generalization of an .Lp -norm of the gradient of u, and as such have been used e.g. in non-local approaches to phasetransition problem (see Alberti and Bellettini [1]). Again, if u is of class .C 1 (), we may approximate .u(y) − u(x) with .∇u(x), y − x, so that, up to an error which can be neglected as .ε → 0, energies (1.6) can be rewritten as .

1 εd

 a ×

 y − x  y − x p   ∇u(x),  dx dy. ε ε

(1.8)

1 Introduction

3

After the change of variables .y = x + εξ and letting .ε → 0, we obtain 

p

p

∇u a dx, where z a =

.



 Rd

a(ξ )|z, ξ |p dξ.

(1.9)

In the particular case when a is radially symmetric we obtain a constant times the p-th power. Limits of energies similar to (1.6), of the form .

1 εd

 a

 y − x  |u(y) − u(x)|p ε

×

|y − x|p

dy dx,

(1.10)

have also been studied by Bourgain et al. [10] as an alternative definition of the Lp -norm of the gradient of a Sobolev function (see e.g. also [35]), as part of a number of works stemming from a general interest towards non-local variational problems arisen in the last twenty years (see e.g. the survey paper [21] or the book [18]). It is worth recalling that a non-linear version of functionals (1.6) with truncated quadratic potentials had been previously proposed as an approximation of the Mumford-Shah energy by De Giorgi and subsequently studied by Gobbino [26] (see also [27], and corresponding energies in peridynamics in [30]). Furthermore, discrete energies of the form

.



1 .

εd+p

aij |uj − ui |p ,

(1.11)

i,j ∈L

where .L is a d-dimensional lattice, .u : εL → Rm and .ui = u(εi), have been widely investigated as a discrete approximation of integral functionals of p-growth (see e.g. [2, 11, 12, 33]; see also [4, 16] for static and dynamic lattice problems with interfaces). Such energies can be seen as a discrete version of functionals (1.6). In the case .p = 2, some energies of the form (1.6) derive from models in population dynamics where macroscopic properties can be reduced to studying the evolution of the first-correlation functions describing the population density u in the system [24, 28]. With that interpretation in mind, in the simplest formulation one can consider their perturbations 1 .

εd+2

 bε (x, y) a ×

y − x  |u(y) − u(x)|2 dx dy, ε

(1.12)

that may take into account inhomogeneities of the environment encoded in the (nonnegative) coefficient .bε . Functionals modeled on these energies, with .bε obtained by scaling a given periodic function b or with a random coefficient .bε = bεω depending on the realization of a random variable, have been considered in the context of

4

1 Introduction

homogenization for u scalar in [14, 15], producing a limit elliptic homogeneous functional  . Ahom ∇u(x), ∇u(x) dx . (1.13) 

This limit can be expressed in terms of .-convergence and implies the convergence of related minimum problems. Another type of perturbations of functionals (1.6) is encountered in a different application to Data Science, studied by García Trillos and Slepcev [25], who examine energies approximating the total variation of u of the form (1.6) 1 .

εd+p

 a ×

 T (y) − T (x)  ε ε |u(y) − u(x)|p dx dy, ε

(1.14)

when .p = 1, with .Tε :  →  and discuss its stability in terms of the convergence of .Tε to the identity. This result has been extended to the case of energies with p-growth for .p > 1 in [20] (see also [19] in the context of free-discontinuity problems). In this book, we provide a general treatment of ‘convolution-type’ energies (1.1) modeled on (1.6) and (1.12) (and including also the case (1.14) with .p > 1) that allow for a general non-linear dependence on .u(x) − u(y) and inhomogeneity on x and y. More precisely, the functionals that we will consider are of the form 1 .

εd+p

 fε (x, y, u(x) − u(y)) dx dy,

(1.15)

×

with .p > 1; i.e., we consider the scaled function .fε = εd+p Wε in (1.1). The functions .fε :  ×  × Rm → R are quite general. In order to compare functionals (1.15) with the energies defined in (1.6) we assume that . is a bounded domain and that there exist two kernels .a1 and .a2 such that a1

.

y − x  y − x  (|z|p − εp ) ≤ fε (x, y, z) ≤ a2 (|z|p + εp ). ε ε

(1.16)

A non-degeneracy condition for the limit is ensured e.g. by assuming that a1 (ξ ) ≥ c0

.

if |ξ | ≤ r0

for some .c0 , r0 > 0, while a decay condition in .a2 provides that the limit be finite exactly on .W 1,p (; Rm ). Note that, for a wider applicability of this analysis, considering a dependence on .ε of the kernel .a2 = a2ε is also necessary. Since this general form of the kernels may cause the limit energy to be non-local, some uniform conditions on the decay of the .a2ε must be required to ensure that the limit be a local integral energy. These assumptions will be stated precisely in Sect. 2.2. The

1 Introduction

5

central result of this book is that, up to subsequences, the energies above converge to an energy of the form  f0 (x, ∇u) dx,

.

(1.17)



with domain .W 1,p (; Rm ). This convergence is expressed as a .-limit with respect to the .Lp -topology in .. This is justified by a Compactness Theorem, which states that, if a is the characteristic function of a ball, any sequence .{uε } bounded in .Lp () with equibounded energies (1.6) admits a subsequence converging to some .u ∈ W 1,p (; Rm ) with respect to the .Lp (; Rm )-topology. This result is complemented by the validity of suitable Poincaré inequalities, which allow to prove the equi-coerciveness of the functionals subjected to boundary data and the application of the direct methods of .-convergence to the asymptotic description of minimum problems. It is worth noting that even in the ‘homogeneous’ case of functionals of the form (1.1) with .Wε as in (1.3) and .W = W (ξ, z) not convex in z the functionals in (1.1) are not lower semicontinuous with respect to the weak p .L -convergence, and their lower-semicontinuous envelope is not even an integral functional [29, 32]. In particular, it is then not possible to justify a simple expansion argument as in (1.4). We also include various applications. First, to the homogenization of non-local functionals of the form   x y 1 , , u(x) − u(y) dx dy, . f (1.18) d+p ε ε ε × with f periodic in the first variable. In this case the limit integrand .f0 in (1.17) is independent of x and can be characterized by a non-local asymptotic formula, which can be further simplified to a non-local cell-problem formula if f is convex in the last variable. A second application is to a class of non-local functionals of the form 1 .

εd+p

 fε (Tε (x), Tε (y), u(y) − u(x))ρ(x)ρ(y) dx dy,

(1.19)

×

which generalize (1.14). If the image of .Tε is discrete these energies can be interpreted as a continuum interpolation of discrete energies. In particular, following [25] we can use these functionals to describe the behavior of energies defined on point clouds. A third application is a stochastic homogenization theorem; i.e., the characterization of the limit of functionals in (1.18) when the integrand .f = f (ω) is a statistically homogeneous (in the first variable) random function defined through a measure-preserving ergodic dynamical system. We characterize the limit using an asymptotic non-local homogenization formula, and prove that the limit is deterministic under ergodicity assumptions. Related results in a discrete setting can be found e.g. in [3, 7, 8, 13]. Finally, we also treat some evolutionary problems using the methods of minimizing movements if the functions .fε are convex in the

6

1 Introduction

last variable. In particular, we consider the homogenization case (1.18), and show that the solutions of gradient flows for those energies, which take the form ∂t uε (t, x) = −

.

1 εd+1

 ∇z f 

 y x u (t, x) − u (t, y)  ε ε , , dy ε ε ε   x y u (t, y) − u (t, x)  1 ε ε dy, + d+1 ∇z f , , ε ε ε ε 

converge to the solution of the corresponding gradient flow for the limit homogenized energy. Previously, homogenization problems for parabolic equations with linear periodic not necessarily symmetric convolution type operators have been studied in [34], where homogenization results were obtained by means of two-scale expansions technique. The plan of the book is as follows. In Chap. 2 we introduce the necessary notation and introduce the class of convolution-type energies under examination. In particular, we change the formal appearance of our energies so as to highlight the range of the interactions. We compare our hypotheses with the corresponding ones for integral functionals, commenting on analogies and differences. We state a weaker version of the coerciveness assumption that may be of use when dealing e.g. with perforated domains (see [17]). We also introduce the special convolution energies .Gε [a] of type (1.6) and in particular .Grε when .a = χBr , which are used as comparison energies throughout the notes (in particular, since they are a lower bound for the energies we consider, it is sufficient to state compactness results for families of functions .{uε } with .Grε (uε ) bounded). Chapter 4 contains some general results, that mirror the analog results in Sobolev spaces, of extension from Lipschitz sets, compactness with respect the .Lp convergence and Poincaré inequalities, where the role of the p-th norm of the gradient is played by .Grε as .ε → 0. The fundamental Lemma 4.1 allows to control long-range interactions with short-range interactions, while Compactness Theorem 4.2 guarantees both a compact embedding in .Lp for sequences with equibounded .Grε -energies, and that their limits belong to .W 1,p . In Chap. 3 we consider the particular case of the limit of energies .Gε [aε ] with varying .aε , characterizing their limits. In particular, when .aε = a we obtain the .convergence to energy (1.9). This limit is used to provide lower and upper bounds for the general case. The main result of the book is contained in Chap. 5, where the general compactness and integral-representation Theorem 5.1 is proved using a variation of the localization method of .-convergence, which is possible since, even though convolution-type functionals are non-local, their non-locality ‘vanishes’ as .ε → 0. An important technical result formalizing this observation is Lemma 5.1, which states, in terms of functionals (1.6), that it is not restrictive to deal with kernels a with bounded support, up to a truncation argument. Section 5.5 deals with the convergence of minimum problems with Dirichlet boundary conditions. Note that for convolution-type functionals such conditions must be imposed on a neighbourhood of size of order .ε of the boundary. Chapter 6 specializes the description of the .-limit in the case of periodically oscillating energies, using

References

7

the integral-representation result, the truncation argument and the convergence of minimum problems to obtain homogenization formulas for the energy function of the .-limit, both of asymptotic type (Theorem 6.1 and formula (6.7)) taking into account interactions within cubes of diverging size and on periodic functions in the convex case (Theorem 6.2 and formula (6.18)). Note that in the latter we take into account interactions .u(x) − u(y) with x in the periodicity set and y in the whole space. In Chap. 7 we consider functionals of the form (1.19) and prove their equivalence to the corresponding functionals (1.15) when .Tε approaches the identity up to an error of order .o(ε), under some technical conditions on .fε . This result is then applied to the analysis of energies defined on point clouds in Chap. 7.2. Chapter 8 contains a homogenization theorem in the stochastic setting (Theorem 8.2), whose proof generalizes the arguments utilized for the deterministic homogenization result, using a subadditive ergodic theorem by Krengel and Pyke (Theorem 8.1) to characterize an asymptotic homogenization formula. Finally, in Chap. 9 we study the convergence of the gradient flows associated to our energies in the convex case. A general approach by minimizing movements allows to deduce in particular the convergence of gradient flows in the case of the homogenization to the gradient flow of the homogenized limit, which is a standard parabolic equation (Theorem 9.3).

References 1. Alberti, G., Bellettini, G.: A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies. Eur. J. Appl. Math. 9, 261–284 (1998) 2. Alicandro, R., Cicalese, M.: A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36, 1–37 (2004) 3. Alicandro, R., Cicalese, M., Gloria, A.: Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. 200, 881–943 (2011) 4. Alicandro, R., Braides, A., Cicalese, M., Solci, M.: Discrete Variational Problems with Interfaces. Cambridge University Press, Cambridge (2023) 5. Bellido, J.C., Mora-Corral, C., Pedregal, P.: Hyperelasticity as a -limit of peridynamics when the horizon goes to zero. Calc. Var. Partial Differ. Equ 54, 1643–1670 (2015) 6. Bellido, J.C., Cueto, J., Mora-Corral, C.: -convergence of polyconvex functionals involving s-fractional gradients to their local counterparts. Calc. Var. Partial Differ. Equ. 60(7) (2021) 7. Blanc, X., Le Bris, C., Lions, P.-L.: The energy of some microscopic stochastic lattices. Arch. Ration. Mech. Anal. 184, 303–339 (2007) 8. Blanc, X., Le Bris, C., Lions, P.-L.: Stochastic homogenization and random lattices. J. Math. Pures Appl. (9) 88, 34–63 (2007) 9. Bobaru, F., Foster, J.T., Geubelle, P.H., Silling, S.A.: Handbook of Peridynamic Modeling. Advances in Applied Mathematics. CRC press, Boca Raton (2016) 10. Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Optimal Control and Partial Differential Equations, pp. 439–455. IOS Press, Amsterdam (2001) 11. Braides, A.: Discrete-to-continuum variational methods for lattice systems. In: Proceedings of the International Congress of Mathematicians–Seoul, vol. IV, pp. 997–1015. Kyung Moon Sa, Seoul (2014) 12. Braides, A., Kreutz, L.: An integral-representation result for continuum limits of discrete energies with multibody interactions. SIAM J. Math. Anal. 50, 1485–1520 (2018)

8

1 Introduction

13. Braides, A., Piatnitski, A.: Homogenization of surface and length energies for spin systems. J. Funct. Anal. 264, 1296–1328 (2013) 14. Braides, A., Piatnitski, A.: Homogenization of random convolution energies. J. Lond. Math Soc. (2) 104, 295–319 (2021) 15. Braides, A., Piatnitski, A.: Homogenization of convolution energies in periodically perforated domains. Adv. Calc. Var. 15, 351–368 (2022) 16. Braides, A., Solci, M.: Geometric Flows on Planar Lattices. Pathways in Mathematics. Birkhäuser/Springer, Cham (2021) 17. Braides, A., Chiadò Piat, V., D’Elia, L.: An extension theorem from connected sets and homogenization of non-local functionals. Nonlinear Anal. 208, 112316 (2021) 18. Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Cham (2016) 19. Caroccia, M., Chambolle, A., Slepˇcev, D.: Mumford-shah functionals on graphs and their asymptotics. Nonlinearity 33, 3846–3888 (2020) 20. Crook, O.M., Hurst, T., Schönlieb, C.-B., Thorpe, M., Zygalakis, K.C.: PDE-inspired algorithms for semi-supervised learning on point clouds. arXiv preprint, arXiv:1909.10221v1 21. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012) 22. Diehl, P., Lipton, R., Wick, T., Tyagi, M.: A comparative review of peridynamics and phasefield models for engineering fracture mechanics. Comput. Mech. 69, 1259–1293 (2022) 23. Du, Q., Zhou, K.: Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM: Math. Model. Numer. Anal. 45, 217–234 (2011) 24. Finkelshtein, D., Kondratiev, Y., Kutoviy, O.: Semigroup approach to birth-and-death stochastic dynamics in continuum. J. Funct. Anal. 262, 1274–1308 (2012) 25. García Trillos, N., Slepˇcev, D.: Continuum limit of total variation on point clouds. Arch. Ration. Mech. Anal. 220, 193–241 (2016) 26. Gobbino, M.: Finite difference approximation of the Mumford-Shah functional. Commun. Pure Appl. Math. 51, 197–228 (1998) 27. Gobbino, M., Mora, M.G.: Finite-difference approximation of free-discontinuity problems. Proc. R. Soc. Edinb. Sect. A 131, 567–595 (2001) 28. Kondratiev, Y., Kutoviy, O., Pirogov, S.: Correlation functions and invariant measures in continuous contact model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11, 231–258 (2008) 29. Kreisbeck, C., Zappale, E.: Loss of double-integral character during relaxation. SIAM J. Math. Anal. 53, 351–385 (2021) 30. Lipton, R.: Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117, 21–50 (2014) 31. Mengesha, T., Du, Q.: On the variational limit of a class of nonlocal functionals related to peridynamics. Nonlinearity 28, 3999–4035 (2015) 32. Mora-Corral, C., Tellini, A.: Relaxation of a scalar nonlocal variational problem with a doublewell potential. Calc. Var. Partial Differ. Equ. 59(67) (2020) 33. Piatnitski, A., Remy, E.: Homogenization of elliptic difference operators. SIAM J. Math. Anal. 33, 53–83 (2001) 34. Piatnitski, A., Zhizhina, E.A.: Homogenization of biased convolution type operators. Asymptot. Anal. 115, 241–262 (2019) 35. Ponce, A.C.: A new approach to Sobolev spaces and connections to -convergence. Calc. Var. Partial Differ. Equ. 19, 229–255 (2004) 36. Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000) 37. Silling, S.A., Lehoucq, R.B.: Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008) 38. Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010) 39. Silling, S.A., Epton, M.A., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)

Chapter 2

Convolution-Type Energies

Abstract In this chapter we formalize the assumptions on our families of convolution-type functionals. Such assumptions are stated in terms of some growth and integrability conditions. We explain and comment these hypotheses comparing them with the corresponding assumptions for families of local integral functionals commonly used in the literature. Keywords Nonlocal functionals · Convolution kernels · Growth conditions · Non-degeneracy · Kernels of polynomial decay · Sign-changing kernels

2.1 Notation In these notes .d, m ∈ N will be fixed natural numbers denoting the dimension of the reference and target spaces of the functions we consider, respectively, and d .p > 1 will be a growth exponent. We let . ⊂ R be a bounded open set with Lipschitz boundary. Note that for many results this regularity assumption on . may be removed up to considering local arguments. For a generic point .x ∈ Rd we write .x = (x1 , . . . , xd ), we let .{ej }dj =1 be the canonical basis of .Rd . We write .Sd−1 = {x ∈ Rd : |x| = 1}. We let .t and .t denote the lower and upper integer part of .t ∈ R, respectively. Given .x ∈ Rd , we let .x denote the vector in .Zd whose components are the integer parts of the components of x; that is, .x = (x1 , . . . , xd ). Analogously d .x = (x1 , . . . , xd ). If .x, y ∈ R then .|x| denotes the norm of x and .x, y the scalar product between x and y. .Rm×d denotes the space of .m × d matrices with real entries; if .M ∈ Rm×d and .x ∈ Rd then .Mx ∈ Rm is defined by the usual row-by-column product. We use .SO(d) ⊂ Rd×d to denote the group of rotations of .Rd . We let .Br (x) (if .x = 0, simply .Br ) be the open ball of centre x and radius r. If A and B are subsets of .Rd then dist.(x, A) = inf{|z − x| : z ∈ A} and dist.(A, B) = inf{|z − x| : z ∈ A, x ∈ B} denote the distance of x from A and from A to B, respectively. .A() will be the family of all open subsets of . and .Areg () the subfamily of open subsets with Lipschitz boundary. By .A  B we mean that the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_2

9

10

2 Convolution-Type Energies

closure of A is a compact subset of B. The Lebesgue d-dimensional measure and the Hausdorff .(d − 1)-dimensional measure in .Rd are denoted by .Ld and .Hd−1 respectively. We also denote the Lebesgue measure of a set .E ⊂ Rd by .|E|. Given .g : Rm×d → R we use the notation  ∂g(M)  .∇g(M) = ∈ Rm×d ∂Mi,j i∈{1,...,m} j ∈{1,...,d} for the gradient of g. Given a matrix-valued map . :  → Rm×d we let the divergence of . be the map .Div() :  → Rm with  .

Div()

 i

= div(i ) for every i = 1, . . . , m,

where .i denotes the i-th row of .. We use standard notation for Lebesgue and Sobolev spaces, and their local versions. If u is an integrable function on a measurable set .E ⊂ , uE :=

.

1 |E|

 u(x)dx E

denotes the average of u on E. If .A  B, a cut-off function .ϕ between A and B is a C ∞ -function with .0 ≤ ϕ ≤ 1, .ϕ = 0 on .∂B and .ϕ = 1 on A. We use standard notation for .-convergence [1], indicating the topology with respect to which it is performed. Unless otherwise stated, the letter C denotes a generic strictly positive constant independent of the parameters of the problem taken into account.

.

2.2 Setting of the Problem and Comments Given .ε > 0 and .fε :  × Rd × Rm → [0, +∞) a positive Borel function, we introduce the non-local functional .Fε : Lp (; Rm ) → [0, +∞] defined as 

 Fε (u) :=

.

Rd

 u(x + εξ ) − u(x)  fε x, ξ, dx dξ ε ε (ξ )

where .ε (ξ ) := {x ∈  : x + εξ ∈ } (see Fig. 2.1).

Fig. 2.1 The set .ε (ξ )

(2.1)

2.2 Setting of the Problem and Comments

11

Note that definition (2.1) is equivalent to (1.15) up to the change of variable y = x + εξ . The reason to rewrite our energies in this apparently more complicated way is in order to state more clearly the hypotheses in the following, highlighting the ‘microscopic interaction length’ .ξ . A particular class of functionals of the form (2.1), which will be used in the following as comparison energies, is introduced below.

.

Definition 2.1 (The Convolution Functionals .Gε [a] and .Grε ) Given a measurable function .a : Rd → [0, +∞), we set  Gε [a](u) :=

 a(ξ )

.

Rd

 u(x + εξ ) − u(x) p    dx dξ.  ε ε (ξ )

(2.2)

Moreover, we define the local versions .Gε [a](u, A) as for .Fε in (2.3). In the case in which .a = χBr we will simply write .Grε instead of .Gε [χBr ]. Remark 2.1 (Origin-Symmetric Kernels) The functionals .Gε [a] do not change if we replace the kernel a with its symmetric part; that is, Gε [a](u) = Gε [a sym ](u),

.

where

a sym (ξ ) :=

a(ξ ) + a(−ξ ) . 2

This can be easily seen by rewriting .Gε [a] as in (1.6) and switching the roles of x and y. We will study the behaviour of energies .Fε as .ε → 0, under proper assumptions on .fε , by proving a compactness result with respect to .-convergence in the .Lp topology. In this context the role of convolution functionals .Gε [a] will be the analog of that of the integral of the p-th power of the gradient as a comparison energy for local integral functional with p-growth. For any .u ∈ Lp (; Rm ) and .A ∈ A() we also introduce a local version of the functionals in (2.1) by setting  Fε (u, A) :=



.

Rd

 u(x + εξ ) − u(x)  dx dξ, fε x, ξ, ε Aε (ξ )

(2.3)

where .Aε (ξ ) := {x ∈ A : x + εξ ∈ A}. In what follows we use the standard notation F  (u, A) := - lim inf Fε (u, A),

.

ε→0

F  (u, A) := - lim sup Fε (u, A) ε→0

for the upper and lower .-limits (cf. [1]) performed with respect to the strong topology of .Lp (; Rm ). Remark 2.2 Note that in this setting the upper and lower .-limits of .Fε (·, A) performed with respect to the strong .Lp (A; Rm ) topology or with respect to the strong .Lp (; Rm ) topology are the same. Indeed, for any sequence .uε converging

12

2 Convolution-Type Energies

to .u ∈ Lp (; Rm ) strongly in .Lp (A; Rm ), we can take u˜ ε (x) =

.

 uε (x)

if x ∈ A if x ∈ \A,

u(x)

which converges to u strongly in .Lp (; Rm ) and leaves the energy unchanged; that is, .Fε (uε , A) = Fε (u˜ ε , A).

2.3 Assumptions The main assumption on our energy densities .fε introduced above is that they are controlled by convolution energies of p growth. In order to maintain the greatest generality and applicability of our results, we consider the following set of hypotheses: there exist two strictly positive constants .r0 , c0 and four families of non-negative functions .ψε,1 , ρε,1 : Br0 → [0, +∞) and .ψε,2 , ρε,2 : Rd → [0, +∞) satisfying  .

 ρε,1 (ξ ) dξ +

lim sup Br0

ε→0

Rd

 ρε,2 (ξ ) dξ < +∞,

(2.4)

 .

lim sup ε→0

Rd

ψε,2 (ξ )|ξ |p dξ < +∞,

ψε,1 (ξ ) ≥ c0 ,

.

for a.e. ξ ∈ Br0 ,

(2.5) (2.6)

 ψε,2 (ξ ) dξ = +∞ then there exists c1 > 0 such that

if lim sup .

Br0

ε→0

(2.7) ψε,2 (ξ ) ≤ c1 ψε,1 (ξ ), for a.e. ξ ∈ Br0

such that ψε,1 (ξ )|z|p − ρε,1 (ξ ) ≤ fε (x, ξ, z)

.

for a.e. ξ ∈ Br0 ;

fε (x, ξ, z) ≤ ψε,2 (ξ )|z|p + ρε,2 (ξ )

.

for a.e. ξ ∈ Rd

(H0) (H1)

for a.e. .x ∈  and every .z ∈ Rm . Moreover for any .δ > 0 there exists .rδ > 0 such that  . lim sup (ψε,2 (ξ )|ξ |p + ρε,2 (ξ )) dξ < δ. (H2) ε→0

Rd \Brδ

2.3 Assumptions

13

Note that, when .fε (x, ξ, z) = aε (ξ )|z|p the hypotheses above hold if .aε satisfies (2.5), (2.6) and (H2). In the sequel, with a slight abuse of notation, we still denote with .ψε,1 and .ρε,1 their zero extension to the whole .Rd . In what follows, when we refer to any of the assumptions (H0)–(H2) we will always implicitly assume that the functions .ρε,1 , ρε,2 , ψε,1 , ψε,2 satisfy (2.4)–(2.7). For any .A ∈ A() and for all sufficiently small .ε, we have that by hypotheses (H0) and (H1), the localized functionals satisfy Gε [ψε,1 ](u, A) − C|A| ≤ Fε (u, A) ≤ Gε [ψε,2 ](u, A) + C|A| .

.

(2.8)

By (2.6), the expression above implies c0 Grε0 (u, A) − C|A| ≤ Fε (u, A) ≤ Gε [ψε,2 ](u, A) + C|A| .

.

(2.9)

Remark 2.3 The hypotheses above can be compared with the standard p-growth assumptions for a family of integral functionals .  fε (x, ∇u) dx, which read a1 (|z|p − a0 ) ≤ fε (x, z) ≤ a2 (1 + |z|p )

(2.10)

.

for some positive constants .a0 , a1 , and .a2 . The polynomial lower-bound in (2.10) implies the boundedness of the .Lp norm of the gradients of functions with equibounded energies and hence provides weak compactness in .W 1,p spaces. Analogously, we will show that condition (H0) and (2.6) provides strong compactness in .Lp of functions with equibounded energies and yields that any limit function is in .W 1,p (; Rm ). Condition (H1) and (2.5) are the analog of the polynomial upperbound in (2.10) and ensure that the .-limits are finite on .W 1,p -functions. Condition (H2) is crucial to deduce the locality of the .-limits, in that it forbids relevant longrange interactions, and has no direct analog in terms of condition (2.10). We spend now a few words to motivate also assumption (2.7) which, at a first glance, may appear mysterious. Loosely speaking, such a condition yields that the behaviour of the kernels .ψε,1 and .ψε,2 is the same for short-range interaction; i.e., as .|ξ | → 0. This ensures that any sequence with bounded energy .uε , by (2.8) also satisfies .Gε [ψε,2 ](uε ) ≤ c1 Gε [ψε,1 ](uε ) ≤ C. Eventually, we notice that in the case in which the integral on a ball centered in zero of .ψε,2 is equibounded, the pointwise control .ψε,2 ≤ c1 ψε,1 is not needed to bound .Gε [ψε,2 ] thanks to Lemma 4.1. Remark 2.4 (Convolution Functionals with Kernels of Polynomial Decay) A simple class of kernels .ψε,1 , .ψε,2 complying with (2.5)–(2.7) and (H2) are those satisfying  ψε,1 (ξ ) = |ξ |

.

−α1

χB1 (ξ ),

ψε,2 (ξ ) = C

|ξ |−α2

|ξ | < 1

|ξ |−β2

|ξ | ≥ 1

14

2 Convolution-Type Energies

for some .C > 1, .0 ≤ α1 ≤ α2 < p + d < β2 with α1 =

 0

.

α2

if α2 < d otherwise.

For the validity of the integral-representation result in Theorem 5.2 condition (H0) can be weakened requiring only that: (H0. ) the computation of .F  (u, A) and .F  (u, A) can be restricted to families m p  .{uε } ⊂ L (; R ) satisfying, for every open .A  A, c0 Grε0 (uε , A ) ≤ Gε [ψε,1 ](uε , A ) ≤ C(Fε (uε , A) + |A|)

.

(2.11)

when .ε is small enough, where C is a constant depending on .A ∈ A(). Such condition is clearly implied by (H0) and (2.6). Assuming (H0. ) in place of (H0) allows to include in our analysis a wide varieties of cases which are not covered by (H0), such as convolution energies on perforated domains, where (2.11) is obtained by an extension theorem (see Sect. 6.4). In the two following examples we exhibit two further classes of convolution functionals satisfying (2.11). Remark 2.5 (Control from Below Provided by a Translated Kernel) We now show that for the validity of (2.11) it suffices that .fε (x, ·, z) be controlled from below by an interaction kernel not necessarily centered in the origin. More precisely, assume that there exist .r0 , c0 > 0 and .ξ0 ∈ Rd such that c0 (|z|p − ρε (ξ − ξ0 )) ≤ fε (x, ξ, z)

.

if |ξ − ξ0 | ≤ r0

(2.12)

(translated kernel). Take .r < r0 /2 and an open .A  A. For every .η ∈ Br (ξ0 ), by Jensen’s inequality we have Grε (u, A ) ≤ 2p−1



.

    u(x + ε(ξ + η)) − u(x) p  dx dξ  ε Br A       u(x + ε(ξ + η)) − u(x + εξ ) p +  dx dξ .  ε Br A

Note that, for .ε small enough, the previous expression is well defined since .A + ε(ξ + η) ⊂ A, for every .ξ and .η as above. Averaging with respect to .η we get Grε (u, A ) ≤

.



    u(x + ε(ξ + η)) − u(x) p   dx dξ dη ε Br (ξ0 ) Br A       u(x + ε(ξ + η)) − u(x + εξ ) p  dx dξ dη .  ε Br (ξ0 ) Br A

2p−1  |Br |  +



2.3 Assumptions

15

By the change of variable .ξ  = ξ + η and from the fact that .Br (η) ⊂ Br0 (ξ0 ) we have      1  u(x + ε(ξ + η)) − u(x) p .  dx dξ dη  |Br | Br (ξ0 ) Br A ε       u(x + εξ ) − u(x) p ≤  dx dξ  .  ε Br0 (ξ0 ) A Using the change of variable .x  = x + εξ and the fact that .A + εξ ⊂ Aε (η) for every .η ∈ Br (ξ0 ), .ξ ∈ Br we also get .

1 |Br |

 Br (ξ0 )



    u(x + ε(ξ + η)) − u(x + εξ ) p   dx dξ dη ε Br A    u(x  + εη) − u(x  ) p   ≤   dx  dη . ε Br (ξ0 ) Aε (η)

Gathering all the inequalities above, for any open .A  A we obtain that Grε (u, A ) ≤ 2p Gε [χBr0 (ξ0 ) ](u, A)

.

for every .ε small enough and (2.12) yields (2.11). Remark 2.6 (Sign-Changing Kernels) In case of convolution(-type) energies the assumption that the kernels be non-negative can be relaxed to some extent. Note that this has no direct counterpart in condition (2.10) for integral functionals, since the negativeness of .fε on a set of x of positive measure would give a .-limit identically equal to .−∞. A simple example is obtained by taking r and .ξ0 ∈  .r0 as in the previous remark,  Rd such that .|ξ0 | > 2r0 and .fε (x, ξ, z) = χBr0 (ξ0 ) (ξ ) − γ χBr (0) (ξ ) |z|p , with −p . This function is negative for .|ξ | < r. Nevertheless, by the previous .γ < 2 remark, we obtain      u(x + εξ ) − u(x) p −p .(2 − γ)  dξ dx  ε Rd Br   u(x + εξ ) − u(x) ) dξ dx, ≤ fε (x, ξ, d d ε R R which gives a bound for the convolution energies on the left-hand side for families with equibounded energies on the whole .Rd . From this bound it is possible to derive compactness properties. Note however that it does not immediately imply condition (2.11) for the localized energies, so that arguments requiring local estimates, such as integral-representation theorems, must be reworked.

16

2 Convolution-Type Energies

We may also treat the case when the domain is not the whole .Rd if we make some assumptions on the integration domain; e.g. convexity. Let A be an open convex set. Applying Jensen’s inequality and switching the roles of x and y we get  .

 u(y) − u(x) p    dx dy  ε {(x,y)∈A×A : ε 0 be fixed. We drop the dependence on M for simplicity of notation. Set   .ϕ ˜j (ξ ) = min M, inf ϕk (ξ ) ,

.

k≥j

which converges almost everywhere to   .ϕ(ξ ) = min M, lim inf ϕj (ξ ) . j →+∞

By Egorov’s theorem for every .δ > 0 there exists .Eδ ⊂ Rd with .μ(Rd \ Eδ ) < δ such that .ϕ˜j χEδ converge uniformly to .ϕχEδ on .Rd . By Lusin’s theorem (see for instance [1, Theorem 1.45 and Remark 1.46]) there exist .vjδ ∈ Cc (Rd ; [0, M]) and compacts .Kjδ with .μ(Rd \ Kjδ ) < 2−j δ such that .vjδ = ϕ˜j χEδ on .Kjδ . Hence,  δ δ d δ .v ≡ ϕ ˜j χEδ for every j on the compact .K δ = ∞ j =1 Kj and .μ(R \ K ) < δ. This j implies that .vjδ converge uniformly on .K δ to .ϕχEδ . By the weak.∗ convergence of .μaε and (3.12) we get  .

lim

j →+∞ Rd

aεj (ξ )|ξ |p ϕ˜j (ξ )χEδ (ξ ) dξ



≥

K δ ∩Eδ

 ≥

K δ ∩Eδ

ϕ(ξ ) dμ(ξ ) 

  min M, ξ

Ay,ξ

 |u y,ξ (s)|p ds dy dμ(ξ ).

3.1 The -Limit of Gε [aε ]

25

Multiplying both members of (3.11), with .uj in place of u, by .aεj (ξ ), integrating in ξ and then taking the limit as .j → +∞, we obtain

.

 .

lim inf Gεj [aεj ](uj , A) ≥

j →+∞

 Rd



ξ

Ay,ξ

|u y,ξ (s)|p ds dy dμ(ξ )

(3.13)

by the arbitrariness of .δ and M. If .u ∈ W 1,p (A; Rm ), for almost every .ξ ∈ Rd \ {0} and .y ∈ ξ u y,ξ (s) = ∇u(x)ξˆ ,

x = y + s ξˆ

.

then Fubini’s Theorem leads to (3.7). Thus it remains to prove that if the left-hand side of (3.13) is finite then .u ∈ W 1,p (A; Rm ). From (3.13)   . |u y,ξ (s)|p ds dy < +∞ (3.14) ξ

Ay,ξ

for .μ-a.e. .ξ ∈ Rd . Since .span(supp(μ)) = Rd , (3.14) holds for linearly independent points .ξ1 , . . . , ξd ∈ Rd , from which the conclusion follows.  Proposition 3.2 Let .aε : Rd → [0, +∞) be a family of non-negative functions satisfying (3.2). Then, for every .A ∈ Areg () and .u ∈ W 1,p (A; Rm )  - lim sup Gε [aε ](u, A) ≤ C

|∇u(x)|p dx,

.

(3.15)

A

ε→0

for some constant .C > 0. Suppose in addition that, for every .δ > 0 there exists rδ such that (3.3) holds and that the measures .μaε as in (3.1) weakly.∗ converge to a measure .μ such that .span(supp(μ)) = Rd . Then, for every .A ∈ Areg () and 1,p (A; Rm ) .u ∈ W   .- lim sup Gε [aε ](u, A) ≤ |∇u(x)ξˆ |p dμ(ξ ) dx. (3.16) .

A Rd

ε→0

Proof By a density argument, we may restrict to the case .u ∈ Cc∞ (Rd ; Rm ). By Remark 4.1 (see Chapter 4), in order to prove (3.15) it is sufficient to estimate the upper limit of .Gε [aε χBT ](u, A), for some .T > 0. For every .x ∈ Rd we have u(x + εξ ) − u(x) = . ε



1

∇u(x + sεξ )ξ ds,

0

and by Jensen’s inequality and Fubini’s Theorem we get  Gε [aε χBT ](u, A) ≤

 aε (ξ )|ξ |

.

BT



p Aε (ξ ) 0

1

|∇u(x + sεξ )|p dsdx dξ

26

3 The .-Limit of a Class of Reference Energies



 ≤

aε (ξ )|ξ |p

1

0

BT





aε (ξ )|ξ |

≤

|∇u(x)|p dx dξ + o(1).

p

BT

|∇u(x)|p dx ds dξ

A+BεT

A

Taking the .lim sup as .ε → 0, by (3.2) we obtain (3.15). We now prove (3.16) under the additional assumption (3.3) and the weak.∗ convergence of .μaε to .μ. We split .Gε [aε ](u, A) as follows  Gε [aε ](u, A) =

 aε (ξ )

.

Brδ

 u(x + εξ ) − u(x) p     dx dξ ε Aε (ξ )    u(x + εξ ) − u(x) p   aε (ξ ) +   dx dξ c ε Br Aε (ξ ) δ

for any .δ > 0, where .rδ is defined as in assumption (3.3). Expanding .u(x) at the first order when .|ξ | < rδ we get 

 aε (ξ )

.

Brδ

 u(x + εξ ) − u(x) p     dx dξ ε Aε (ξ )   aε (ξ ) = Brδ

|∇u(x)ξ |p dx + o(1) dξ Aε (ξ )

and for .|ξ | > rδ by assumption (3.3) 



.

Brc

aε (ξ )

δ

 u(x + εξ ) − u(x) p     dx dξ ε Aε (ξ )   p aε (ξ ) (∇uL∞ (Rd ) |ξ |)p dx dξ ≤ |A|∇uL∞ (Rd ) δ. ≤ Brcδ

A

Hence, gathering the inequalities above we obtain  Gε [aε ](u, A) ≤

 |∇u(x)ξ |p dx dξ + Cδ

aε (ξ )

.

Brδ

(3.17)

A

for some .C > 0. Letting .ε → 0 in (3.17), we get   .

lim sup Gε [aε ](u, A) ≤ ε→0

|∇u(x)ξˆ |p dμ(ξ ) dx + Cδ

A Brδ

and the arbitrariness of .δ implies (3.16).



References

27

From the right-hand side inequality in (2.9), (2.11), Propositions 3.1 and 3.2 the following estimates hold. Proposition 3.3 Given .A ∈ Areg (), let .{Fε (·, A)} be the family of functionals defined by (2.3) and assume that (H0. ), (H1) and (H2) hold. If .F (u, A) is finite, then .u ∈ W 1,p (A; Rm ). Moreover for every .u ∈ W 1,p (A; Rm ) we have F (u, A) ≥ c (∇uLp (A) − |A|), .

(3.18)

F (u, A) ≤ C(∇uLp (A) + |A|),

(3.19)

.

p

p

for some positive constants .c, C.

References 1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000) 2. Braides, A.: Approximation of Free-Discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer-Verlag, Berlin (1998) 3. Ponce, A.C.: A new approach to Sobolev spaces and connections to -convergence. Calc. Var. Partial Differ. Equ. 19, 229–255 (2004)

Chapter 4

Asymptotic Embedding and Compactness Results

Abstract In this section we include some results that extend corresponding results in Sobolev spaces to the case of convolution energies. In Theorem 4.2 we prove the compactness of sequences of functions for which both the .Lp norms and the energies are uniformly bounded, whose proof is a non-local counterpart of that of the classical Riesz-Fréchet-Kolmogorov Theorem. We will see that the role played by the .Lp norm of gradients in the standard compact immersion results of Sobolev spaces is played in our context by the energies .Grε . In Propositions 4.1 and 4.2 we show the validity of Poincaré-type inequalities. Keywords Extension theorem · Short-range interactions · Long-range interactions · .Lp -compactness · Poincaré-type inequalities

Before proving compactness results in the next sections, we provide some preliminary results of independent interest concerning extension operators and the possibility of controlling long-range interactions with short-range interactions. Subsequently, we prove some asymptotic analogs of embedding and compactness results for Sobolev spaces (see e.g. [2]).

4.1 An Extension Result By mimicking a standard procedure of extension of Sobolev functions, we provide the following corresponding result.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_4

29

30

4 Asymptotic Embedding and Compactness Results

Theorem 4.1 (Extension) Let A be any open set with Lipschitz boundary with .∂A bounded and let .r > 0. Then there exist an open set .A˜  A, a linear continuous map ˜ Rm ) E : Lp (A; Rm ) → Lp (A;

.

and three positive constants .C = C(A), r1 = r1 (r, A), .ε0 = ε0 (r, A) such that for all .u ∈ Lp (A; Rm ), Eu = u

.

p ˜ m) Lp (A;R

Eu

.

a.e. in A ,

  r ˜ ≤ C up p + Grε1 (Eu, A) L (A;Rm ) + Gε (u, A)

for all .ε < ε0 . Proof We follow the construction of the extension of functions in fractional Sobolev spaces (see [1]). From the boundedness and the Lipschitz regularity of the boundary we can find a finite open covering .{Ui }ni=1 of .∂A and Lipschitz invertible maps −1 ∞ .Hi : Q → Ui with .DHi L∞ (Q) ≤ L and .DH i L (Ui ) ≤ L, such that Hi (Q+ ) = A ∩ Ui ,

.

Hi (Q− ) = Ac ∩ Ui ,

Hi (Q0 ) = ∂A ∩ Ui ,

where .Q = (−1, 1)d , .Q± = {x ∈ Q : ±x1 > 0} and .Q0 = {x ∈ Q : x1 = 0}. Let n ∞ d .{ϕi } i=0 ⊂ C0 (R ) be a partition of the unity, that is .Vi := supp(ϕi )  Ui and n  .

ϕi (x) = 1, for every x ∈ A,

i=0

 where .A\ ni=1 Ui ⊂ U0 ⊂ A is an open set. Define the maps .Ri : Ac ∩Ui → A∩Ui as follows Ri (x) = Hi (−y1 , y2 , . . . , yd ),

.

where .Hi−1 (x) = y = (y1 , . . . , yd ) ∈ Q− . Note that .Ri are invertible Lipschitz maps of Lipschitz constant less than .L2 . Given .u ∈ Lp (A; Rm ), define  ui (x) :=

u(x)

x ∈ A ∩ Ui

u(Ri (x))

x ∈ Ac ∩ Ui ,

.

Then .u˜ :=

n

˜i i=0 u

extends u on .A˜ =

n

i=0 Ui .

u˜ i (x) := ϕi (x)ui (x). For every .0 ≤ i ≤ n we have



 |ui (x)|p dx ≤ C

.

Ui

Ui ∩A

|u(x)|p dx,

(4.1)

4.1 An Extension Result

31 p

p

where .C > 0 depends only on A, which yields .u ˜ p ˜ m ≤ CuLp (A;Rm ) . Now, L (A;R ) we show that for every .1 ≤ i ≤ n Grε1 (ui , Ui ) ≤ CGrε (u, Ui ∩ A)

.

(4.2)

with .r1 = r/(1 + 2L2 ), which is trivial when .i = 0. Using the change of variable .y = x + εξ , it is convenient to rewrite the energy functionals as Grε1 (ui , Ui ) =

.

1 εd



 Ui

u (y) − u (x) p i i dy dx ε Ui ∩Bεr1 (x)

≤ Grε (u, Ui ∩ A).   u (y) − u (x) p 1 i i + d dy dx. ε Ui ∩A (Ui \A)∩Bεr1 (x) ε   u (y) − u (x) p 1 i i + d dy dx. ε Ui \A Ui ∩A∩Bεr1 (x) ε   u (y) − u (x) p 1 i i + d dydx. ε Ui \A (Ui \A)∩Bεr1 (x) ε

(4.3) (4.4) (4.5) (4.6)

For every .1 ≤ i ≤ n and .x ∈ Ui ∩ A, we claim that .|Ri (y) − x| ≤ εr for any y ∈ (Ui \A) ∩ Bεr1 (x). Indeed, if .z ∈ Ui ∩ Bεr1 (x) ∩ ∂A then .Ri (z) = z and therefore

.

|Ri (y) − x| ≤ |Ri (y) − Ri (z)| + |z − x| ≤ L2 |y − z| + |z − x| ≤ (2L2 + 1)εr1 .

.

Thus, .Ri ((Ui \A) ∩ Bεr1 (x)) ⊂ Ui ∩ A ∩ Bεr (x), and, after the change of variable y = Ri (y), we obtain

.





.

Ui ∩A

u (y) − u (x) p i i dy dx ε (Ui \A)∩Bεr1 (x)   ≤ L2d Ui ∩A

u(y ) − u(x) p dy dx, ε Ui ∩A∩Bεr (x)

that controls the term in (4.4). Applying the same argument, we obtain an analogous estimate for the integral in (4.5). By the changes of variable .y = Ri (y) and .x = Ri (x) we also get  .

Ui \A



u (y) − u (x) p i i dy dx ε (Ui \A)∩Bεr1 (x)   ≤ L4d Uj ∩A

u(y) − u(x) p dy dx ε Ui ∩A∩Bεr (x)

32

4 Asymptotic Embedding and Compactness Results

and the integral in (4.6) is controlled as well. Hence (4.2) holds. Set ε0 = ε0 (r, A) =

.

1 min dist(Vi , Uic ) > 0. r1 0≤i≤n

For every .ε < ε0 and .0 ≤ i ≤ n, by (4.2) and (4.1), summing and subtracting ϕi (x + εξ )ui (x) we get

.





n 

  ˜ ≤ C Grε (u, A) + up p Grε1 (u˜ i , A) L (A;Rm ) .

u (x + εξ ) − u (x) p i i dx dξ ε Br1 (Ui )ε (ξ )   ϕ (x + εξ ) − ϕ (x) p i i . p−1 |ui (x)|p +2 dx dξ ε Br1 (Ui )ε (ξ )   p ≤ C Grε (u, Ui ∩ A) + uLp (Ui ∩A;Rm ) . (4.7) Hence, for every .ε < ε0 from (4.7) we obtain ˜ ≤ 2p−1 Grε1 (u˜ i , A)

˜ ≤ Grε1 (u, ˜ A)

.

i=0

Eventually, we notice that the map .E : u → u˜ is linear by definition of .u˜ which concludes the proof.  Note that, the theorem provides an extension to the whole space .Rd . Indeed, by construction we have that for every .u ∈ Lp (A; Rm ), .Eu ∈ Lp (Rd ; Rm ) with ˜ and .Grε1 (Eu, A) ˜ = Grε1 (Eu, Rd ) for every .ε < ε0 . .supp(Eu)  A

4.2 Control of Long-Range Interactions with Short-Range Interactions With the following key result we show that interactions of any admissible range can be suitably controlled by the short-range energy .Grε . Lemma 4.1 For every .r > 0 there exists a positive constant C such that, for any open set .E ⊂  and for every .ξ ∈ Rd , .ε > 0 such that ε r < dist(E + (0, ε)ξ, Rd \ )

.

and .u ∈ Lp (; Rm ), there holds  u(x + εξ ) − u(x) p . dx ≤ C(|ξ |p + 1)Grε (u, Eε,ξ ) ε E with .Eε,ξ = E + (0, ε)ξ + Bεr and .(0, ε)ξ = {sξ : s ∈ (0, ε)}.

(4.8)

4.2 Control of Long-Range Interactions with Short-Range Interactions

33

Proof First, note that condition (4.8) implies that .Eε,ξ ⊂  and thus the terms √ in the inequality above are well defined. For notational reasons we set .ε = εr/ d + 3. Let .Rξ ∈ SO(d) be a rotation matrix such that .Rξ e1 = ξ/|ξ |. We introduce j the lattice .Lε := {Rξ i : i ∈ ε Zd } and, for any .j ∈ Lε define .Qε := d j + Rξ (−ε /2, ε /2) and set



E˜ ε,ξ :=

.

Qjε : Qjε ∩ (E + (0, ε)ξ ) = ∅ .

j ∈Lε

Let .k = ε|ξ |/ε  + 1, and, for any .j0 ∈ Lε and .0 ≤ l ≤ k − 1, define .jl = j j0 + lε |ξξ | . Denote by .xl any point in .Qεl and, for the sake of simplicity of notation, j

j

xk = x0 + εξ and .Qεk = Qε0 + εξ . Using the inequality

.

k u(x ) − u(x ) p  k 0 u(xl ) − u(xl−1 ) p ≤ k p−1 ε ε

.

l=1

and integrating in every variable we get  .

u(x + εξ ) − u(x ) p 0 0 dx0 j0 ε Qε ≤

 k  k p−1  u(xl ) − u(xl−1 ) p dxl dxl−1 j j (ε )d ε Qεl−1 Qεl l=1

≤

 k  u(y) − u(x ) p k p−1  l−1 dy dxl−1 . j (ε )d ε Qεl−1 Bεr (xl−1 ) l=1

j

j

Note that in the second line of the inequality above we have used that .Qεk ⊂ Qεk−2 ∪ j Qεk−1 (see Fig. 4.1). Then, by the change of variable .y = xl−1 + εξ  .

j

Qε0

u(x + εξ ) − u(x ) p 0 0 dx ε ≤

ε d ε

k p−1 

≤ Ck p−1 Br

k  



u(x + εξ ) − u(x ) p l−1 l−1 dxl−1 dξ jl−1 ε B Q r ε l=1  u(x + εξ ) − u(x) p dx dξ , ε Qε (j0 )

 jl with .Qε (j0 ) = k−1 l=1 Qε and C a positive constant depending on r and d. Since the sets .{Qε (j0 ) : j0 ∈ Lε } overlap at most .k − 1 times, by summing over .j0 ∈ Lε

34

4 Asymptotic Embedding and Compactness Results

j

Fig. 4.1 The picture shows the cubes .Qεl and the corresponding centers .jl ∈ Lε , .0 ≤ l ≤ k. The arrow represents the vector .εξ

j

such that .Qε0 ∩ E = ∅ we get that  u(x + εξ ) − u(x) p dx ≤ C(|ξ |p + 1)Gr (u, E˜ ε,ξ ). . ε ε E Since .dist(E + (0, ε)ξ, Rd \ E˜ ε,ξ ) < εr then .E˜ ε,ξ ⊂ Eε,ξ and the result follows.



As a consequence of Lemma 4.1 and Theorem 4.1 we infer the following result. Corollary 4.1 For any open set .A ∈ Areg () and .r > 0 there exist two positive constants .C = C(r, A) and .ε0 = ε0 (r, A) such that  .

u(x + εξ ) − u(x) p   dx ≤ C(|ξ |p + 1) Gr (u, A) + up p m) , ε L (A;R ε Aε (ξ )

for every .ξ ∈ Rd , .u ∈ Lp (A; Rm ), and .ε < ε0 . Remark 4.1 (short-range control) Let .ψε,2 be as in (2.5) and let .r, r > 0 be fixed. Then, Lemma 4.1 implies that there exist a positive constant .C = C(r, r ) such that for every .u ∈ Lp (; Rm ) and .A ∈ A() there holds Gε [ψε,2 χRd \B ](u, A) ≤ CGrε (u, A ),

.

r

for any .A ∈ A() with .A  A, for every .ε < 2r1 dist(A, (A )c ). If moreover .ψε,2 L1 (Rd ) ≤ C then Lemma 4.1 provides a control also of shortrange interactions; namely, for any given .r > 0 there exists .C = C(r) such that r .Gε [ψε,2 ](u, A) ≤ CGε (u, A ) for every u, .A, A and .ε as above.

4.3 Compactness in Lp Spaces We now discuss the compactness in the strong .Lp topology of sequences of functions with uniformly bounded energy and show that their limits are in the corresponding Sobolev space. In the scalar case and for bounded domains A, an

4.3 Compactness in Lp Spaces

35

analogous result has been proved in [3, Theorems 1.2 and 1.3] for energies with radially-symmetric kernels. Theorem 4.2 Let A be any open set of .Rd with bounded Lipschitz boundary. Let m p .{uε }ε ⊂ L (A; R ) be such that for some .r > 0 .

 sup uε Lp (A;Rm ) + Grε (uε , A) < +∞. ε>0

If A is unbounded, assume in addition that for any .η > 0 there exists .rη > 0 such that .

sup uε Lp (A\Brη ) < η.

(4.9)

ε>0

Then, given .εj → 0, .{uεj }j is relatively compact in .Lp (A; Rm ) and every limit of a converging subsequence is in .W 1,p (A; Rm ). ˜ Rm ) such that Proof By Theorem 4.1, there exists .A˜  A, .r˜ > 0 and .u˜ ε ∈ Lp (A; .u ˜ ε = uε on A and p ˜ m) Lp (A;R

u˜ ε 

.

r ˜ ≤ Cuε  p + Grε˜ (u˜ ε , A) L (A;Rm ) + CGε (uε , A). p

˜ and let .{φη }η0

Given .εj < r˜ −1 η and setting .ηj = η/(˜r εj )˜r εj ≥ η, from (4.10) we get in particular u˜ εj ∗ φηj L∞ (Aη ;Rm ) ≤ Cη

.

∇(u˜ εj ∗ φηj )L∞ (Aη ;Rm ) ≤ Cη

d(1−p ) p

,

d(1−p ) −1 p

.

Hence by Ascoli-Arzelá’s Theorem, for every fixed .η the sequence .{u˜ εj ∗ φηj }j is precompact in .C(Aη ; Rm ) and therefore it is totally bounded, that is there exists a m finite set of functions .{gk }L k=1 ⊂ C(Aη ; R ) such that for every .j ∈ N 1

u˜ εj ∗ φηj − gk Lp (Aη ;Rm ) ≤ |Aη | p u˜ εj ∗ φηj − gk L∞ (Aη ;Rm ) < η

.

(4.14)

4.4 Poincaré Inequalities

37

for some .k ∈ {1, . . . , L}. So, by (4.12), (4.13) and (4.14), denoting by .g˜ k the extension of .gk which equals zero outside .Aη , we have uεj − g˜ k Lp (A;Rm ) ≤ uεj − u˜ εj ∗ φηj Lp (A;Rm ) + u˜ εj ∗ φηj − g˜ k Lp (A;Rm )

.

= uεj − u˜ εj ∗ φηj Lp (A;Rm ) + u˜ εj ∗ φηj − gk Lp (Aη ;Rm ) + u˜ εj ∗ φηj Lp (A\Aη ) ≤ ηj

1 1 Gr˜ (u˜ ε , A) p + 2η ≤ Cη; r˜ d/p+1 εj j

i.e., .{uεj }j is totally bounded and hence relatively compact in .Lp (A; Rm ). Finally Proposition 3.1 yields that every limit function is in .W 1,p (A; Rm ).  Remark 4.2 From the previous theorem we deduce a compactness result on any open set A (without regularity assumptions on the boundary) with respect to the local .Lp -topology. Namely, that every bounded sequence .{uε } with bounded .Grε energy on A is precompact in .Lp (A ; Rm ) for every .A  A. Indeed, it suffices to apply the previous theorem with a set .A with Lipschitz boundary in the place of A, with .A  A  A.

4.4 Poincaré Inequalities We complete this section with the asymptotic analogs of Poincaré inequalities. Similar results can be found e.g. in [3]. Proposition 4.1 (Poincaré Inequality) Let .r > 0 and let A be an open set in .Rd such that .A ⊆ (a, b)ν + ν for some .a, b ∈ R and .ν ∈ Sd−1 , where .(a, b)ν = {sν : s ∈ (a, b)} and . ν = {x ∈ Rd : x, ν = 0}. Then there exists a positive constant .C = C(r, A) such that  . |u(x)|p dx ≤ CGrε (u, A) A

for every .ε > 0 and for any .u ∈ Lp (A; Rd ) with .u(x) = 0 for almost every .x ∈ A with .dist(x, Rd \ A) ≤ εr. Proof We identify u with its extension to the whole .Rd that equals zero outside A. It is not restrictive to consider .ν = e1 and up to a change of variables we may r and for every .j ∈ {0, . . . , N := 1/r ε} assume .a = 0 and .b = 1. Set .r := √d+3 and .l ∈ Zd−1 denote by .xjl an independent variable lying in

l l d−1 Qj,l . ε := (j r ε, (j + 1)r ε) × Qε , with Qε := l + (0, r ε)

.

38

4 Asymptotic Embedding and Compactness Results

By the boundary assumption, for any .0 ≤ k ≤ N we have u(xkl ) =

.

k  (u(xjl ) − u(xjl −1 )). j =1

Thus, by Jensen’s inequality we get |u(xkl )|p ≤ k p−1

k 

.

j =1

|u(xjl ) − u(xjl −1 )|p ,

l , and, integrating in .x0l , x1l , . . . , xN



(r ε)d(N −1)

.

Qk,l ε

|u(xkl )|p dxkl

≤ (r ε)d(N −2) k p−1

k  

 j −1,l

j,l

j =1 Qε

Qε

|u(xjl ) − u(xjl −1 )|p dxjl dxjl −1 .

j,l

Now, since .Qε ⊂ Brε (xjl −1 ), we have  .

Qk,l ε

|u(xkl )|p dxkl

 k  k p−1  ≤ d |u(y) − u(xjl −1 )|p dydxjl −1 j −1,l (r ε) Qε Brε (xjl −1 ) j =1

=

k p−1 (r ε)d

N p−1 ≤ d (r )







(0,kr ε)×Qlε



(0,1)×Qlε

|u(y) − u(x)|p dy dx Brε (x)

|u(x + εξ ) − u(x)|p dξ dx. Br

By summing over .0 ≤ k ≤ N and .l ∈ Zd−1 both the left- and the right-hand sides we get  |u(x)|p dx ≤

.

A

1 (r )d+p

  A Br

u(x + εξ ) − u(x) p dξ dx ε

1 = d+p Grε (u, A), (r ) which proves the claim.



Proposition 4.2 (Poincaré-Wirtinger Inequality) Let .r > 0 and let A be a bounded connected open set of .Rd with Lipschitz boundary. Then for every

4.4 Poincaré Inequalities

39

measurable set .E ⊂ A with .|E| > 0 there exists a positive constant .C = C(r, A, E) such that  . |u(x) − uE |p dx ≤ CGrε (u, A) A

for any .u ∈ Lp (A; Rm ) and .ε ∈ (0, 1). Proof We argue by contradiction. Suppose that for any positive integer j there exists .εj > 0 and .uj ∈ Lp (A; Rm ) such that  .

A

|uj (x) − (uj )E |p dx > j Grεj (uj , A).

Thus, letting uj − (uj )E , uj − (uj )E Lp (A;Rm )

u˜ j :=

.

we have .u˜ j Lp (A;Rm ) ≡ 1, .(u˜ j )E ≡ 0 and Grεj (u˜ j , A)
0, up to passing to a further subsequence, .u˜ j u weakly in .Lp (A; Rm ) and so

.

.

u˜ j (x + εj ξ ) − u˜ j (x) u(x + ε0 ξ ) − u(x)

weakly in Lp (A; Rm ) εj ε0

for any .ξ ∈ Br . For every open set .A  Aε0 (ξ ), Fatou’s Lemma yields 0 = lim inf Grεj (u˜ j , A)

.

j →∞

 ≥

lim inf

Br j →∞

 u˜ j (x + εj ξ ) − u˜ j (x) p dx dξ. ε A

j

40

4 Asymptotic Embedding and Compactness Results

Hence, by the arbitrariness of .A , we have that .Grε0 (u, A) = 0 and therefore u is  constant almost everywhere on A, which again leads to a contradiction. By means of the Poincaré-Wirtinger inequality we can improve Theorem 4.2. Indeed, we recover the compactness result by assuming the uniform boundedness of the mean values of .uε . Corollary 4.2 Let A be any open bounded set of .Rd with Lipschitz boundary. Let m p .E ⊂ A be a measurable set with .|E| > 0. Let .{uε }ε ⊂ L (A; R ) be such that for some .r > 0 .

 sup |(uε )E | + Grε (u, A) < +∞. ε>0

Then the same conclusion of Theorem 4.2 holds. Proof It is sufficient to prove that .uε Lp (A;Rm ) is uniformly bounded. By Proposition 4.2 we have p

p

uε Lp (A;Rm ) ≤ 2p−1 uε −(uε )E Lp (A;Rm ) +2p−1 |A||(uε )E |p ≤ C(Grε (uε , A)+1)

.

for some constant C depending on A and r and the claim follows.



References 1. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012) 2. Leoni, G.: A First Course in Sobolev Spaces. Graduate Students in Mathematics, vol. 181, 2nd edn. American Mathematical Society, Providence (2017) 3. Ponce, A.C.: An estimate in the spirit of Poincaré’s inequality. J. Eur. Math. Soc. 6, 1–15 (2004)

Chapter 5

A Compactness and Integral-Representation Result

Abstract The main result of this chapter is a compactness and integralrepresentation result for the .-limits of the families .{Fε (·, A)}ε , which we can obtain through a convolution version of the localization method of .-convergence. A key point is that it is possible to limit the analysis to finite-range convolutions through a truncation argument. Keywords Direct method of .-convergence · Compactness · Integral representation · Truncations · Dirichlet boundary conditions · Boundary-value problems · Euler-Lagrange equations

5.1 The Integral-Representation Theorem We now prove that from any family of functionals .{Fε }ε we can extract a .converging sequence, whose limit can be represented as a local integral functional. Its proof will follow a strategy devised by De Giorgi and described, e.g., in [2, Chapter 9]. It consists in considering the localized functionals  Fε (u, A) :=

.

Rd



 u(x + εξ ) − u(x)  dx dξ, fε x, ξ, ε Aε (ξ )

already introduced in (2.3), and in proving properties of their .-limit both in terms of the variable u and the set variable A. The result is stated as follows. Theorem 5.1 (Compactness and Integral Representation) Given . a bounded open set with Lipschitz boundary, let .Fε , .Fε (·, ·) be defined by (2.1) and (2.3), respectively, and let assumptions (H0. ), (H1) and (H2) be satisfied. Then, for every .εj → 0 there exists a subsequence .{εjk } ⊂ {εj } and a Carathédory function m×d → [0, +∞) which is quasiconvex in the second variable and .f0 :  × R satisfies the growth condition C0 (|M|p − 1) ≤ f0 (x, M) ≤ C1 (|M|p + 1)

.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_5

(5.1) 41

42

5 A Compactness and Integral-Representation Result

for almost every .x ∈  and every .M ∈ Rm×d , such that ⎧ ⎨ f0 (x, ∇u(x))dx p .(L )- lim Fεj (u, A) = A k ⎩ k→+∞ +∞

if u ∈ W 1,p (A; Rm )

(5.2)

otherwise,

for every .A ∈ Areg (). In order to prove Theorem 5.1 we will show that, up to subsequences, the .-limit F (u, A) exists for all .A ∈ Areg (), and that we can apply the following result.

.

Theorem 5.2 (Theorem 9.1 [2]) Let . be a bounded open set, and let .F : W 1,p (; Rm ) × A() → [0, +∞) satisfy: (i) for any .A ∈ A() .F (u, A) = F (v, A) if .u = v almost everywhere on A; (ii) for any .u ∈ W 1,p (; Rm ) the set function .F (u, ·) is a restriction of a Borel measure on .A(); (iii) there exists a constant .C > 0 and .a ∈ L1 () such that  .F (u, A) ≤ C (a(x) + |∇u(x)|p ) dx; A

(iv) .F (u + z, A) = F (u, A) for any .u ∈ W 1,p (; Rm ), .z ∈ Rm and .A ∈ A(); (v) .F (·, A) is weakly lower semicontinuous for any .A ∈ A(). Then there exists a Carathéodory function .f :  × Rm×d → [0, +∞), quasiconvex in the second variable, with .0 ≤ f (x, M) ≤ c(a(x) + |M|p ) for almost every .x ∈ A and every .M ∈ Rm×d , such that  F (u, A) =

f (x, ∇u(x))dx

.

A

for every .A ∈ A() and .u ∈ W 1,p (; Rd ). We postpone the proof of Theorem 5.1 to the end of the section as it will be a direct consequence of some propositions that show that the limit functionals satisfy the hypotheses of Theorem 5.2.

5.2 Truncated-Range Functionals In this subsection we introduce the ‘ truncated functionals’ obtained by limiting the range of interaction in (2.1) to a fixed threshold .T > 0. We will show that, to some extent, the limit as .T → +∞ and the .-limit as .ε → 0 commute. This result will allow to limit our analysis to truncated functionals in the proofs of the results of

5.2 Truncated-Range Functionals

43

the following sections, in particular in that of Theorem 5.1, leading to significant simplifications. Definition 5.1 (The Truncated Functionals .FεT ) For any .A ∈ A() and .T > 0 the functional .FεT (·, A) : Lp (A; Rm ) → [0, +∞] is defined by  T .Fε (u, A)



:= BT

 u(x + εξ ) − u(x)  dx dξ . fε x, ξ, ε Aε (ξ )

(5.3)

Note that Proposition 3.3 clearly applies also to the truncated functionals .F T (·, A), since they comply with all the hypotheses of Sect. 2.3. In what follows we use the notation F ,T (u, A) := - lim inf FεT (u, A),

.

ε→0

F ,T (u, A) := - lim sup FεT (u, A). ε→0

Lemma 5.1 Let .Fε (·, A) and .FεT (·, A) be defined by (2.3) and (5.3), respectively, and let assumptions (H0)–(H2) be satisfied. Then for every .A ∈ Areg () and .u ∈ Lp (; Rm ) F  (u, A) = lim F ,T (u, A)

.

T →+∞

and

F  (u, A) = lim F ,T (u, A). T →+∞

T

In particular, given .Tj → +∞ such that .Fε j (·, A) .-converge to .F Tj (·, A) as .ε → 0 for every .j ∈ N, - lim Fε (u, A) = lim F Tj (u, A)

.

ε→0

j →+∞

for every .u ∈ Lp (; Rm ). Proof Note first that, since .FεT (u, A) ≤ Fε (u, A) for every .u ∈ Lp (; Rm ) and .A ∈ A(), one inequality in the statement is trivial. Thanks to Proposition 3.3, it is sufficient to prove the opposite inequality for every .u ∈ W 1,p (A; Rm ). Hence, let m p .uε → u in .L (; R ). Without loss of generality we may assume that .Fε (uε , A) is uniformly bounded. We have that    uε (x + εξ ) − uε (x)  T dx dξ, .Fε (uε , A) = Fε (uε , A) + fε x, ξ, ε BTc Aε (ξ ) and from assumption (H1) we get Fε (uε , A) ≤ FεT (uε , A)    u (x + εξ ) − u (x) p   ε   ε + ψε,2 (ξ )  + ρε,2 (ξ ) dx dξ. ε BTc Aε (ξ )

.

(5.4)

44

5 A Compactness and Integral-Representation Result

Thanks to Corollary 4.1, for .ε small enough 



.

BTc

≤



 u (x + εξ ) − u (x) p   ε  ε  ψε,2 (ξ )  + ρε,2 (ξ ) dx dξ ε Aε (ξ )

BTc





p ψε,2 (ξ )C(|ξ |p + 1) Grε0 (uε , A) + uε Lp (A;Rm ) + ρε,2 (ξ )|A| dξ . (5.5)

Since by (2.9) .Grε0 (uε , A) is bounded, by (5.4), (5.5) and (H2) we have Fε (uε , A) ≤ FεT (uε , A) + Cδ + o(1)

.

for every .T > rδ , where .rδ is chosen as in (H2), and the thesis follows letting first .ε and then .δ tend to 0. 

5.3 Fundamental Estimates In what follows, with a slight abuse of notation, .F  (·, ·) will denote the .-.lim sup of both the family of functionals .{Fε }ε and the sequence .{Fεj }j for any .εj → 0. A crucial step in order to apply Theorem 5.2 is provided by the following two propositions. Proposition 5.1 (Subadditivity) Let .Fε (·, ·) be defined by (2.3) and assume that (H0)–(H2) hold. Let .A, B ∈ A() and let .A , B  ∈ A() with .A  A and .B   B. Then F  (u, A ∪ B  ) ≤ F  (u, A) + F  (u, B)

.

(5.6)

for every .u ∈ Lp (; Rm ). Proof Without loss of generality we may suppose that .F  (u, A) and .F  (u, B) are finite. Moreover, since .F  (u, ·) is an increasing set function, we may assume that reg (). Let .u , v both converge to u in .Lp (; Rd ) and be such that   .A , B ∈ A ε ε .

lim Fε (uε , A) = F  (u, A),

ε→0

lim Fε (vε , B) = F  (u, B).

ε→0

Note that, by (2.8) and (2.9), .Gε [ψε,1 ](uε , A), .Gε [ψε,1 ](vε , B), .Grε0 (uε , A) and r0 d  .Gε (vε , B) are uniformly bounded. Let .R := dist(A , R \A) and, fixed .N ∈ N, set Ai = {x ∈ A : dist(x, A ) < iR/N}, 1 ≤ i ≤ N.

.

5.3 Fundamental Estimates

45

Let .ϕ i be a cut-off function between .Ai and .Ai+1 , with .|∇ϕ i | ≤ 2N/R, and set m i i i i p .wε := uε ϕ + vε (1 − ϕ ). Note that .wε → u in .L (; R ) for any .i ∈ {1, . . . , N}. i i By adding and subtracting .ϕ (x)uε (x + εξ ) + (1 − ϕ (x))vε (x + εξ ) we have wεi (x + εξ ) − wεi (x) = ϕ i (x)(uε (x + εξ ) − uε (x)) + (1 − ϕ i (x))(vε (x + εξ ) − vε (x))

.

+ (ϕ i (x + εξ ) − ϕ i (x))(uε (x + εξ ) − vε (x + εξ )). (5.7) Note that  i .wε (x

+ εξ ) − wεi (x)

=

uε (x + εξ ) − uε (x),

if x ∈ (Ai )ε (ξ )

vε (x + εξ ) − vε (x),

if x ∈ (\Ai+1 )ε (ξ ),

(5.8)

while, having set

i Sε,ξ := (A ∪ B  )ε (ξ ) \ (Ai )ε (ξ ) ∪ (\Ai+1 )ε (ξ ) ,

.

by (5.7) and Jensen’s inequality, using the notation .Dξε g(x) = (g(x + εξ ) − g(x))/ε for the sake of brevity, we get |Dξε wεi (x)|p ≤ 2p−1 ϕ i (x)|Dξε uε (x)|p + 2p−1 (1 − ϕ i (x))|Dξε vε (x)|p + 2p−1 |Dξε ϕ i (x)|p |uε (x + εξ ) − vε (x + εξ )|p   ≤ 2p−1 |Dξε uε (x)|p + |Dξε vε (x)|p

.

+ 2p−1

 2N p R

(5.9)

|ξ |p |uε (x + εξ ) − vε (x + εξ )|p

i (see Fig. 5.1). for every .x ∈ Sε,ξ Now, we consider the truncated functionals .FεT introduced in Definition 5.1. By (5.8), (5.9) and assumption (H1) we have

FεT (wεi , A ∪ B  )

.

= FεT (uε , Ai ∩ (A ∪ B  )) + FεT (vε , (\Ai+1 ) ∩ B  )   fε (x, ξ, Dξε wεi (x)) dx dξ + BT

i Sε,ξ

≤ Fε (uε , A) + Fε (vε , B)

46

5 A Compactness and Integral-Representation Result

i . The dashed lines are the boundaries of .A and Fig. 5.1 The shaded region represents the set .Sε,ξ i In this picture .ξ = e1

.Ai+1 .



 +C



i Sε,ξ

BT

ψε,2 (ξ )(|Dξε uε (x)|p + |Dξε vε (x)|p ) + ρε,2 (ξ ) dx dξ .



(5.10)



+ CN p

ψε,2 (ξ ) BT

i Sε,ξ

|ξ |p |uε (x + εξ ) − vε (x + εξ )|p dx dξ.

(5.11)

The integral in (5.11) can be estimated from above uniformly in T as follows 

 ψε,2 (ξ )

.

BT

i Sε,ξ

 ≤

Rd

|ξ |p |uε (x + εξ ) − vε (x + εξ )|p dx dξ p

ψε,2 (ξ )|ξ |p uε − vε Lp (;Rm ) dξ p

≤ C uε − vε Lp (;Rm ) ; hence it tends to zero as .ε → 0, since .uε and .vε both converge to u in .Lp (; Rm ). i ⊂ (A  Note that, if .|ξ | < T then .Sε,ξ i+1 \Ai + (−ε, ε)ξ ) ∩ B ; hence, N−4 .

i=1

i Sε,ξ ⊂ (AN−2 \A ) ∩ B  .

(5.12)

5.3 Fundamental Estimates

47

i } intersect at most pairwise for .ε small enough. Thus, from Moreover, the sets .{Sε,ξ i (5.12) for .ε small enough we get



N −4 

ψε,2 (ξ )

.

i Sε,ξ

BT

i=1

ε

|Dξ uε (x)|p + |Dξε vε (x)|p dx dξ





≤2

ψε,2 (ξ ) BT

AN−2 ∩B 

ε

|Dξ uε (x)|p + |Dξε vε (x)|p dx dξ.

Here, we proceed in two different ways according to the dichotomy of (2.7). If  lim supε→0 Rd ψε,2 (ξ )dξ < +∞ then we apply Lemma 4.1 on .uε and .vε with   .E = AN −2 ∩ B for any .|ξ | < T , so that .Eε,ξ ⊂ AN −1 ∩ (B + Bε(r0 +T ) ). Thus for .ε small enough .Eε,ξ ⊂ A ∩ B and we get .



N −4 

ψε,2 (ξ )

.

BT

i=1

i Sε,ξ

ε

|Dξ uε (x)|p + |Dξε vε (x)|p dx dξ 

≤ (Grε0 (uε , A) + Grε0 (vε , B)) for some .C  > 0. If instead .lim supε→0 Lemma 4.1 we obtain 

N −4 

ψε,2 (ξ )

.

i=1

BT

i Sε,ξ

 Rd

Cψε,2 (ξ )(|ξ |p + 1)dξ ≤ C  , BT

ψε,2 (ξ )dξ = +∞ by (2.7) and applying

ε

|Dξ uε (x)|p + |Dξε vε (x)|p dx dξ



≤ 2c1 Gε [ψε,1 ](uε , A) + Gε [ψε,1 ](vε , B)  Cψε,2 (ξ )(|ξ |p + 1)dξ ≤ C  , + (Grε0 (uε , A) + Grε0 (vε , B)) BT \Br0

where the last inequality is a consequence of (2.5) and the fact that the integration domain is far from the origin. We can choose an index .1 ≤ kε ≤ N − 4 such that  .

BT





ψε,2 (ξ )(|Dξε uε (x)|p + |Dξε vε (x)|p ) + ρε,2 dx dξ ≤

kε Sε,ξ

C , N −4

uniformly in T , and - lim sup FεT (u, A ∪ B  ) ≤ - lim sup FεT (wεkε , A ∪ B  )

.

ε→0

ε→0

≤ F  (u, A) + F  (u, B) +

C . N −4

(5.13)

48

5 A Compactness and Integral-Representation Result

Letting first .N → +∞ and then .T → +∞, by Lemma 5.1 we get the conclusion. 

Remark 5.1 Reasoning as in Proposition 5.1 we can also derived the following inequality for the .- lim inf F  (u, A ∪ B  ) ≤ F  (u, A) + F  (u, B) .

.

(5.14)

Proposition 5.2 (Inner Regularity) Let .Fε (·, ·) be defined by (2.3) and assume that (H0)–(H2) hold. Then .

sup F  (u, A ) = F  (u, A)

A A

for every .A ∈ Areg () and .u ∈ Lp (; Rm ) ∩ W 1,p (A; Rm ). Proof Since .F  (u, ·) is an increasing set-function, it suffices to prove that .

sup F  (u, A ) ≥ F  (u, A).

A A

To this end we argue as in the proof of Proposition 5.1. For any .δ > 0 let .Aδ  A be an open set such that p

|A \ Aδ | + Du Lp (A\A ) < δ

.

δ

and let .A ∈ A() be such that .Aδ  A  A. Let .uε , vε ∈ Lp (; Rm ) both converge to u in .Lp (; Rm ) and be such that .

lim Fε (uε , A ) = F  (u, A ),

ε→0

lim Fε (vε , A \ Aδ ) = F  (u, A \ Aδ ).

ε→0

Thus, by Proposition 3.3 we get Fε (vε , A \ Aδ ) ≤ F  (u, A \ Aδ ) + oε (1) ≤ Cδ + oε (1).

.

(5.15)

Set .R := dist(Aδ ,  \ A ), .Ai = {x ∈ A : dist(x, Aδ ) < iR/N}, and let .ϕi and .wεi be defined as in the proof of Proposition 5.1. Set i Sε,ξ := Aε (ξ ) \ ((Ai )ε (ξ ) ∪ (A \ Ai+1 )ε (ξ )) ⊂ (Ai+1 \ Ai + (−ε, ε)ξ ) ∩ Aε (ξ ).

.

5.4 Proof of the Integral-Representation Theorem

49

Consider first the finite-range interaction energies .FεT (·, ·). Reasoning as in the proof of Proposition 5.1, we can find .2 ≤ kε ≤ N − 4 such that FεT (wεkε , A)

.



= FεT (uε , Akε ) + FεT (vε , A \ Akε +1 ) + ≤ Fε (uε , A ) + oε (1) +



BT

kε Sε,ξ

fε (x, ξ, Dξε wεk (x))dx dξ

C + Cδ. N −4

Letting first .ε → 0 and then .N → +∞ we get F ,T (u, A) ≤ F  (u, A ) + Cδ ≤ sup F  (u, A ) + Cδ.

.

A A

The result follows from the arbitrariness of .δ and T and Lemma 5.1.



Remark 5.2 We can also prove the inner regularity of .F  . Indeed, we may consider m m p p .uε , vε ∈ L (; R ) both converge to u in .L (; R ) and be such that .

lim inf Fε (uε , A ) ≥ F  (u, A ), ε→0

lim Fε (vε , A \ Aδ ) = F  (u, A \ Aδ ).

ε→0

By (5.14) and reasoning as in Proposition 5.2 we get .

sup F  (u, A ) = F  (u, A).

A A

Note that, when dealing with local functionals, the fundamental estimates immediately implies the inner regularity of .F  (u, ·), F  (u, ·) as done in [2]. In the non-local framework, it is important to be ‘far’ from .A ∩ ∂B because otherwise we can have interactions with points outside B that cannot be controlled by .Grε0 (vε , B).

5.4 Proof of the Integral-Representation Theorem This section is devoted to the proof of Theorem 5.1. We divide the proof in two steps, dealing first with the case in which (H0) holds and with the general case. Step 1 Assume that (H0) holds. The compactness properties of .-convergence, Proposition 5.2, Remark 5.2 and [2, Theorem 10.3] yield the existence of a subsequence .(εjk ) such that the .-limit (Lp )- lim Fεjk (u, A) =: F (u, A)

.

k→∞

50

5 A Compactness and Integral-Representation Result

exists for any .(u, A) ∈ Lp (; Rm ) × Areg (). By Proposition 3.3, .F (u, A) = +∞ if and only if .u ∈ W 1,p (A; Rm ). We now introduce the inner-regular extension of .F (u, ·) on the whole family .A() defined by F˜ (u, A) := sup{F (u, A ) : A ∈ Areg (), A  A}.

.

Since, by Proposition 5.2, .F˜ (u, A) = F (u, A) for any .u ∈ W 1,p (A; Rm ) and .A ∈ Areg (), it remains to check that .F˜ satisfies all the hypotheses of Theorem 5.2. .F˜ (u, ·) is clearly increasing. By Remark 2.2, hypothesis (i) trivially holds. Proposition 3.3 yields (iii). Since .Fε (·, A) depends only on incremental ratios, it is translation invariant and so does .F˜ (·, A); thus, (iv) is satisfied. By the properties of .-convergence .F (·, A) is lower semicontinuous with respect to the .Lp (; Rm ) topology (see for instance [1, Proposition 1.28]). Thus, Proposition 3.3 yields the weak lower semicontinuity of .F (·, A) with respect to the .W 1,p (; Rm ) topology. By its definition, .F˜ (·, A) inherits the same property, thus (v) holds. As a consequence of Propositions 5.1 and 5.2, .F˜ (u, ·) is subadditive, superadditive on disjoint sets and inner regular. Hence, by the De Giorgi-Letta measure criterion (see [2]), .F˜ (u, ·) is the restriction on .A() of a Borel measure, thus (ii) is satisfied and the thesis follows. Step 2 Assume that only (H0. ) holds. For any .n ∈ N let .Fε,n : Lp (; Rm ) × A() → [0, +∞) be defined by Fε,n (u, A) := Fε (u, A) +

.

1 Gε [ψε,1 ](u, A). n

Since the family of functionals .Fε,n satisfies (H0)–(H2) for every .n ∈ N, by Step 1 and a diagonalization argument there exist a subsequence .(εjk ) and a non increasing sequence of functions .fn :  × Rm×d → [0, +∞), .n ∈ N, quasiconvex in the second variable and satisfying (5.1), such that ⎧ ⎨ fn (x, ∇u) dx p .(L )- lim Fεj ,n (u, A) = A k ⎩ k→∞ +∞

if u ∈ W 1,p (A, Rm ), otherwise.

for any .A ∈ Areg (). We claim that (5.2) holds with .f0 (x, M) := infn∈N fn (x, M). Indeed, since .Fε ≤ Fε,n for every .ε > 0 and .n ∈ N, we have ⎧ ⎨ f0 (x, ∇u) dx  .F (u, A) ≤ F (u, A) := A ⎩ +∞

if u ∈ W 1,p (A, Rm ), otherwise.

It remains to prove that F (u, A) ≤ F  (u, A).

.

(5.16)

5.5 Convergence of Minimum Problems

51

By Proposition 3.3, it suffices to prove (5.16) for .u ∈ W 1,p (A; Rm ). Let then .uk → u in .Lp (; Rm ) and be such that .

lim inf Fεjk (uk , A) = F  (u, A). k→+∞

Given .A  A, by (H0. ) we may assume that .Gεjk [ψε,1 ](uk , A ) is uniformly bounded. Hence   C . f0 (x, ∇u) dx ≤ fn (x, ∇u) dx ≤ lim inf Fεjk ,n (uk , A ) ≤ F  (u, A) + . k→+∞ n A A Thus (5.16) follows from the arbitrariness of .n ∈ N and .A  A. The proof is then complete.



Remark 5.3 Theorem 5.1 clearly applies also to the truncated energies .FεT . Suppose that for any .(u, A) ∈ W 1,p (; Rm ) × Areg () and .T > 0 there exists  - lim FεT (u, A) =

.

ε→0

A

f0T (x, ∇u(x))dx.

(5.17)

Then, by Lemma 5.1 and monotone convergence, we infer that  - lim Fε (u, A) =

f0 (x, ∇u(x))dx

.

ε→0

A

where for almost every .x0 ∈  and every .M ∈ Rm×d f0 (x0 , M) := lim f0T (x0 , M).

.

(5.18)

T →+∞

5.5 Convergence of Minimum Problems In this section we prove the convergence of minimum problems under Dirichlet boundary conditions. p

Definition 5.2 (Boundary-Value Problems) For any .g ∈ Lloc (Rd ; Rm ), .A ∈ Areg () and .r > 0 we set   Dr,g (A) := u ∈ Lp (; Rm ) : u(x) = g(x) for a.e. x ∈  , dist(x, Rd \A) < r (5.19)

.

52

5 A Compactness and Integral-Representation Result r,g

and define the functionals .Fε

: Lp (; Rd ) × Areg () → [0, +∞) 

r,g .Fε (u, A)

:=

Fε (u, A)

if u ∈ Dεr,g (A)

+∞

otherwise.

(5.20)

When dealing with the affine function .g(x) = Mx for some .M ∈ Rm×d we will use the notation .Dr,M and .Fεr,M . Proposition 5.3 Let .A ∈ Areg (), let .Fε (·, A) be defined by (2.3) and assume that (H0)–(H2) hold. Let .εj → 0 and let .f0 :  × Rm×d → [0, +∞) be such that ⎧ ⎨ f0 (x, ∇u(x) dx p .(L )- lim Fεj (u, A)= A ⎩ j →+∞ +∞

if u ∈ W 1,p (A; Rm )

=: F (u, A).

otherwise

1,p

Then, given .g ∈ Wloc (Rd ; Rm ) and .r > 0, the corresponding sequence of r,g functionals .Fεj defined in (5.20) .-converges with respect to the .Lp (; Rm ) topology to the functional  F (u, A) :=

.

g

1,p

F (u, A)

if u − g ∈ W0 (A; Rm )

+∞

otherwise.

(5.21)

r,g

Proof Since .Fε (u, A) ≥ Fε (u, A), in order to prove the .-.lim inf inequality it r,g suffices to show that if .uj → u in .Lp (A; Rm ) and .supj Fεj (uj , A) is finite, then 1,p

u − g ∈ W0 (A; Rm ). Denote by .u˜ j and .u˜ the extension of .uj and u on the whole d .R obtained by setting .u ˜ j = g, .j ∈ N, and .u˜ = g on .Rd \A. Let .A˜ be an open set such that .A˜  A and note that, by (2.9), for every .r  ≤ min{r0 , r/2} we have .

 r r ˜ ≤ Grε (uj , Ar/2 ˜ Grεj (u˜ j , A) ε ) + Gεj (g, A \ Aεj ) j

.



≤ C(Fεj (uj , A) + |A|) + Grεj (g, A˜ \ Arεj ) ≤ C, ρ ˜ Rm ), by where .Aε := {x ∈ A : dist(x, \A) > ερ}. Since .u˜ j → u˜ in .Lp (A; 1,p ˜ Rm ) and thus .u − g ∈ W (A; Rm ). Proposition 3.1 we get that .u˜ ∈ W 1,p (A; 0 By a density argument it suffices to prove the .-.lim sup inequality for .u ∈ W 1,p (; Rm ) such that .spt(u − g)  A. Given such a u, let .uj converge to u in m p .L (; R ) such that .

lim Fεj (uj , A) = F (u, A).

j →∞

With an argument analogous to the one used in the proof of Propositions 5.1 and 5.2, given .δ > 0, we can find a suitable cut-off function .ϕj with .spt(ϕj )  A such that,

5.5 Convergence of Minimum Problems

53

having set .vj := ϕj uj +(1−ϕj )u, we have that .vj still converge to u in .Lp (; Rm ) and Fεj (vj , A) ≤ Fεj (uj , A) + δ.

.

Since .vj ∈ Dεj r,g (A) for j large enough, we get .

lim sup Fεr,g (vj , A) ≤ F (u, A) + δ j j →∞



and the arbitrariness of .δ leads to the desired inequality.

As a consequence of Propositions 5.3, 4.1 and Theorem 4.2, we derive the following result of convergence of minimum problems with Dirichlet boundary data. Proposition 5.4 (Convergence of Boundary-Value assumptions of Proposition 5.3 there holds .

Problems) Under

the

1,p

lim inf{Fεj (u, A) : u ∈ Dεj r,g (A)} = min{F (u, A) : u − g ∈ W0 (A; Rm )}.

j →∞

(5.22) Moreover, if .uj ∈ Dεj r,g (A) is a converging sequence such that .

lim Fεj (uj , A) = lim inf{Fεj (u, A) : u ∈ Dεj r,g (A)},

j →∞

j →∞

1,p

then its limit is a minimizer for .min{F (u, A) : u − g ∈ W0 (A; Rm )}. Proof Note that .

inf{Fεj (u, A) : u ∈ Dεj r,g (A)} ≤ Fεj (g, A) ≤ C.

Hence, by the properties of .-convergence (see for instance [1, Theorem 1.21]), we r,g only need to prove the equi-coerciveness of the family .{Fεj (·, A)}εj in the strong r,g m m p p .L (A; R ) topology. Let then .{uj }j ⊂ L (; R ) be such that .Fεj (uj , A) ≤ C. Reasoning as in the proof of Proposition 5.3, from assumption (H0) and (2.6) we  deduce that .Grεj (uj , A) ≤ C for every .r  ≤ min{r0 , r/2}. Proposition 4.1 yields that  .



    p |uj (x) − g(x)|p dx ≤ CGrεj (uj − g, A) ≤ C Grεj (uj , A) + ∇g Lp () ≤ C.

Hence, we may apply Theorem 4.2 and deduce that .{uj }j is precompact in the strong Lp (; Rm ) topology. 

.

Remark 5.4 It is worth noting that infima in (5.22) are indeed minima if the functions .fε are convex in the last variable, and hence .Fε are weakly lower

54

5 A Compactness and Integral-Representation Result

semicontinuous in .Lp (; Rm ) (see also the next section; we refer e.g. to [5] for lower-semicontiuity issues in the context of peridynamics). Note that if .fε are not convex in the last variable, then the lower-semicontinuous envelope of .Fε is not a double integral [3, 4].

5.6 Euler-Lagrange Equations We complete the analysis of convergence of minimum problems of the previous section by studying Euler-Lagrange equations associated to functionals .Fε and F , under suitable regularity assumptions on the densities .fε .

5.6.1 Regularity of Functionals Fε We consider functionals .Fε : Lp (; Rm ) → [0, +∞) as defined in (2.1). We preliminary note that .Fε are strongly continuous in the .Lp (; Rm )-topology provided that .fε is continuous in the last variable and (H1) holds with .ψε,2 , 1 d .ρε,2 ∈ L (R ) for every .ε > 0. Indeed, let .uj → u in .Lp (; Rm ) as .j → +∞; then, for almost every .ξ ∈ Rd there holds   uj (x + εξ ) − uj (x)  dx . lim fε x, ξ, j →+∞ ε (ξ ) ε   u(x + εξ ) − u(x)  = dx fε x, ξ, ε ε (ξ ) and (H1) yields  .

 uj (x + εξ ) − uj (x)  p dx ≤ C uj Lp (;Rm ) ψε,2 (ξ ) + ||ρε,2 (ξ ). fε x, ξ, ε ε (ξ )

Hence, the Dominated Convergence Theorem implies the continuity of .Fε . The next result shows that, assuming .C 1 -regularity in the last variable of the density and a control of the growth of .∇z fε , the functionals .Fε are indeed differentiable in the Gateaux sense. Lemma 5.2 Let .fε be .C 1 regular in the last variable, and let (H1) hold with .ψε,2 , 1 d .ρε,2 ∈ L (R ). If |∇z fε (x, ξ, z)| ≤ ψ(ξ )|z|p−1 + ρ(ξ ),

.

(5.23)

5.6 Euler-Lagrange Equations

55

for some .ρ ∈ L1 (Rd ) and .ψ ∈ L∞ (Rd ), then .Fε is Gateaux differentiable and its Gateaux differential is given by DFε (u)   x − y u(x) − u(y)   y − x u(y) − u(x)  1 −∇z fε x, dy. ∇z fε y, = d+1 , , ε ε ε ε ε  (5.24) .

Proof Let .hε,u (x) denote the right-hand side of (5.24). For every .v ∈ Lp (; Rm ), exploiting the symmetric roles of x and y we get Fε (u + tv) − Fε (u) t→0 t   y − x u(y) − u(x)  v(y) − v(x) 1 , ,  dx dy ∇z fε x, = d ε × ε ε ε  = hε,u (x)v(x) dx. lim

.



Condition (5.23) yields that |hε,u (x)| ≤

.

2 εd+1

 ψ 

 y − x  u(y) − u(x) p−1 y − x    dy. +ρ   ε ε ε

If .p ≥ 2 we get |hε,u (x)| ≤

.

ε

 C  p−1 d p (;Rm ) + ε |u(x)| ; + u L d+p

if instead .1 < p < 2 then Jensen’s inequality implies |hε,u (x)| ≤

.

C  εd+p

 |u − u(x)|p−1 + εd .

In both cases, .hε,u ∈ Lp−1 (; Rm ); thus, .Fε is Gateaux differentiable and (5.24) holds true. 

We remark that, in the case of .fε convex in the last variable, (5.23) is a consequence of (H1). We will use this fact in the next section.

5.6.2 Relations with Minimum Problems Provided suitable regularity (and convexity) conditions on .fε , minimizers for functionals .Fε are weak solutions to associated Euler-Lagrange equations. In particular,

56

5 A Compactness and Integral-Representation Result

the convergence of minimum problems proved in Sect. 5.5 implies convergence of solutions to the Euler-Lagrange equations of the .-limit, in the strictly convex case. Proposition 5.5 Let .fε be convex and of class .C 1 in the last variable, and let (H0) and (H1) hold with .ψε,2 ∈ L1 (Rd ) ∩ L∞ (Rd ) and .ρε,2 ∈ L1 (Rd ). Let .r > 0 and εr,g () be as in Definition 5.2. Then there exist m p .g ∈ L (; R ) be given and let .D solutions to the minimum problem .

min

u∈Dεr,g ()

Fε (u)

(5.25)

and every minimizer .uε ∈ Dεr,g () solves  .

 x − y u(x) − u(y)   y − x u(y) − u(x)  − ∇z fε x, dy = 0 ∇z fε y, , , ε ε ε ε  (5.26)

for almost every .x ∈ (εr), where .(εr) = {x ∈  : dist(x, Rd \ ) > εr}. Proof By the strong continuity and convexity, .Fε is lower-semicontinuous with respect to the weak .Lp (; Rm ) topology. By assumption (H0) and applying Poincaré inequality (Proposition 4.1) as done in the proof of Proposition 5.3, the set {u ∈ Dεr,g () : Fε (u) ≤ C}

.

is bounded and therefore .Fε is weakly coercive. This ensures that the minimum problem (5.25) has solutions thanks to the Direct Method of the Calculus of Variations. The convexity of .fε and assumption (H1) implies (5.23) with .ψ = ψε,2 ; thus, .Fε r,g is Gateaux differentiable and (5.24) holds. The restriction of .Fε to .Dε () is also εr,0 differentiable and its differential is defined on .D (); thus, by minimality  DFε (uε )(x), v(x)dx = 0

.



for every .v ∈ Dεr,0 () and the result follows.



We recall well-known properties of local functionals as those that are cluster points (in the sense of .-convergence) of .Fε . Remark 5.5 Consider .F : W 1,p (; Rm ) → [0, +∞) as defined in the statement of Theorem 5.1; that is,  .F (u) = f0 (x, ∇u(x)) dx 

5.6 Euler-Lagrange Equations

57

where .f0 :  × Rm×d → [0, +∞) is quasiconvex in the second variable and complies with the growth condition (5.1). Then, an application of the Direct Method of the Calculus of Variations in the weak .W 1,p ()-topology yields the existence of minimizers for the problem .

1,p

min{F (u) : u − g ∈ W0 (; Rm )},

where .g ∈ W 1,p (; Rm ) is given. If we assume in addition that f is .C 1 in the last variable and that |∇M f0 (x, M)| ≤ C(|M|p−1 + 1)

.

then F is Gateaux differentiable on .W 1,p (; Rm ). Indeed, for every .v 1,p W0 (; Rm ) F (u + tv) − F (u) = . lim t→0 t

∈

 ∇M f0 (x, ∇u(x)), ∇v(x)dx 

which is linear in v and bounded thanks to the growth condition. Let .u0 be a minimizer of F , then applying the divergence theorem to the right-hand side above we get by minimality that

.

 Div(∇M f (·, ∇u0 )) = 0 in  u0 = g

(5.27)

on ∂

in the sense of distribution. Proposition 5.6 Let .fε be strictly convex and of class .C 1 in the last variable and let (H0)–(H2) hold with .ψε,2 ∈ L1 (Rd ) ∩ L∞ (Rd ) and .ρε,2 ∈ L1 (Rd ). Let .r > 0 and .g ∈ W 1,p (; Rm ) be given and let .Dεr,g () be as in Definition 5.2. Then Eqs. (5.26) and (5.27) are uniquely solved by .uε and u respectively and .uε converges to u in the weak .W 1,p (; Rm )-topology as .ε → 0. Proof By Theorem 5.1 and the convexity of .fε we have  - lim Fε (u) = F (u) =

f0 (x, ∇u(x))dx

.

ε→0



for some .f0 strictly convex in the second variable and complying with (5.1). From Proposition 5.5 and Remark 5.5 and the strict convexity of .fε and .f0 the equations have unique (weak) solutions .uε and .u0 respectively such that Fε (uε ) = εr,g min Fε

.

D

()

and

F (u0 ) =

min

1,p

g+W0 (;Rm )

F.

58

5 A Compactness and Integral-Representation Result r,g

In the end, we remark that the family .Fε is equi-coercive, as shown in the proof of Proposition 5.4. This, thanks to the convergence of minimum problems provided by Proposition 5.4, implies that for any .εj → 0 there exists a subsequence .{εj } such that .uεj converges to .u0 , and the result is proved. 

References 1. Braides, A.: -Convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications, vol. 22. Oxford University Press, Oxford (2002) 2. Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Oxford Lecture Series in Mathematics and Its Applications, vol. 12. The Clarendon Press/Oxford University Press, New York (1998) 3. Kreisbeck, C., Zappale, E.: Loss of double-integral character during relaxation. SIAM J. Math. Anal. 53, 351–385 (2021) 4. Mora-Corral, C., Tellini, A.: Relaxation of a scalar nonlocal variational problem with a doublewell potential. Calc. Var. Partial Differ. Equ. 59(67) (2020) 5. Pedregal, P.: Weak lower semicontinuity and relaxation for a class of non-local functionals. Rev. Mat. Complut. 29, 485–495 (2016)

Chapter 6

Periodic Homogenization

Abstract In this chapter we study the homogenization of convolution functionals under an assumption of periodicity in the space variable of the energy densities. The limit energy density is characterized by an asymptotic nonlocal homogenization formula, which reduces to a non-local cell-problem formula when the energy density is convex in the last variable. In the case of homogeneous integrands the homogenization formula simplify only in the convex case. In the last sections of the chapter we prove the homogenization result in perforated domains using an extension theorem. Keywords Periodic homogenization · Asymptotic formula · Cell-problem formula · Relaxation · Extension theorems · Lipschitz domains · Perforated domains

6.1 A Homogenization Theorem Let .f : Rd × Rd × Rm → [0, +∞) be a Borel function such that .f (·, ξ, z) is d m d .[0, 1] -periodic in the first variable for every .ξ ∈ R and .z ∈ R . Throughout this chapter, we will assume that fε (x, ξ, z) = f

.

x ε

 , ξ, z

(6.1)

in (2.1). In this setting, assumptions (H0)–(H2) are straightforward consequence of the following growth conditions on f : for a.e. .x ∈ Rd and every .z ∈ Rm ψ1 (ξ )|z|p − ρ1 (ξ ) ≤ f (x, ξ, z),

for a.e. ξ ∈ Br0 ,

(6.2)

f (x, ξ, z) ≤ ψ2 (ξ )|z|p + ρ2 (ξ ),

for a.e. ξ ∈ Rd ,

(6.3)

.

.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_6

59

60

6 Periodic Homogenization

where .ρ1 , ψ1 : Br0 → [0, +∞) and .ψ2 , ρ2 : Rd → [0, +∞) are such that 

 ρ1 (ξ ) dξ +

.

Br0

 Rd

ρ2 (ξ ) dξ < +∞, ψ1 (ξ ) ≥ c0 ,

Rd

ψ2 (ξ )|ξ |p dξ < +∞,

for a.e. ξ ∈ Br0

.

(6.4) (6.5)

requesting additionally that  ψ2 (ξ ) dξ = +∞ then there exists c1 > 0 such that

if Br0

.

(6.6) ψ2 (ξ ) ≤ c1 ψ1 (ξ ) for a.e. ξ ∈ Br0 .

In the sequel we will use the notation .QR (x0 ) = x0 + (0, R)d and the shorthand .QR if .x0 = 0 The main result of this chapter is stated in the following theorem. Theorem 6.1 (Homogenization Theorem) Let .Fε be defined by (2.1), with .fε given by (6.1), and let (6.2)–(6.6) be satisfied. Then for every .M ∈ Rm×d the limit (asymptotic homogenization formula)  1 .fhom (M) := lim inf R→∞ R d



 f (x, y − x, v(y) − v(x))dx dy

QR

 : v ∈ D1,M (QR ) ,

QR

(6.7)

where .D1,M (QR ) is defined by (5.19), exists and defines a quasiconvex function m×d → [0, +∞) satisfying .fhom : R c(|M|p − 1) ≤ fhom (M) ≤ C(|M|p + 1).

.

(6.8)

Moreover, ⎧ ⎨ fhom (∇u(x))dx p .(L )- lim Fε (u) =  ⎩ ε→0 +∞

if u ∈ W 1,p (; Rm ) otherwise.

The proof of Theorem 6.1 relies on the results stated in the following two propositions. The first one provides the independence of the limit energy densities on the space variable, the second one the existence of the limit in (6.7) in the case of truncated energies.

6.1 A Homogenization Theorem

61

Proposition 6.1 Under the assumptions of Theorem 6.1, let .εj → 0 and let .f0 : ×Rm×d → [0, +∞) be a Carathéodory function such that for every .A ∈ Areg () and .u ∈ W 1,p (A; Rm ) there holds  p .(L )- lim Fεj (u, A) = f0 (x, ∇u(x))dx. j →+∞

A

Then .f0 is independent of the first variable. Proof It is sufficient to prove that F (Mx, Br (y)) = F (Mx, Br (y  ))

.

(6.9)

for every .M ∈ Rm×d , .y, y  ∈  and .r > 0 such that .Br (y), Br (y  ) ⊂ . We will prove that .F (Mx, Br  (y)) ≤ F (Mx, Br (y  )) for all .r  < r. By the inner regularity of .F (Mx, ·) provided by Proposition 5.2 we get (6.9) by switching the roles of y and .y  . Let .uj → Mx in .Lp (Br (y  ); Rm ) be such that .

lim Fεj (uj , Br (y  )) = F (Mx, Br (y  )),

j →+∞



y − y  

y − y  + εj M and set .vj (x) := uj x − εj . Since .Br  (y) − εj εj

y − y  εj ⊂ Br (y  ) when .εj is small enough, .vj → Mx in .Lp (Br  (y); Rm ). εj Moreover, by the periodicity assumption on f , we have that Fεj (vj , Br  (y))   =

.

Rd

 =

Rd



f

(Br  (y))εj (ξ )

f (Br  (y))εj (ξ )

x εj x εj

, ξ, −

vj (x + εj ξ ) − vj (x)  dx dξ εj

y − y  vj (x + εj ξ ) − vj (x)  , ξ, dx dξ. εj εj

and, through the change of variable .x = x  + εj (y − y  )/εj , we get  Fεj (vj , Br  (y)) ≤

.

Rd

 (Br (y  ))εj (ξ )

= Fεj (uj , Br (y  )).

f

 x εj

, ξ,

uj (x  + εj ξ ) − uj (x  )   dx dξ εj

62

6 Periodic Homogenization

Finally, letting .j → +∞, we obtain F (Mx, Br  (y)) ≤ lim inf Fεj (vj , Br  (y))

.

j →+∞

≤ lim Fεj (uj , Br (y  )) = F (Mx, Br (y  )) j →+∞



and the claim.

Proposition 6.2 Let .f : Rd × Rd × Rm → [0, +∞) be a Borel function .[0, 1]d periodic in the first variable such that assumptions (6.3) and (6.4) hold. Let .T > 0 and set f (x, ξ, z) if |ξ | < T , T .f (x, ξ, z) = (6.10) 0 otherwise. Then, for every .r ≥ T the limit  1 inf R→∞ R d

T fhom (M) := lim

.



 f T (x, y − x, v(y) − v(x))dx dy

QR

 : v ∈ Dr,M (QR ) ,

QR

(6.11)

exists and it is finite for every .M ∈ Rm×d , with .Dr,M (QR ) defined as in Definition 5.2. Proof For every .R > 0 we set  T .F1 (v, QR )



:=

f T (x, y − x, v(y) − v(x))dx dy, QR

HR (M) :=

.

QR

  1 T r,M inf F (v, Q ) : v ∈ D (Q ) R R 1 Rd

and let .uR ∈ Dr,M (QR ) be such that .

1 T 1 F (uR , QR ) ≤ HR (M) + . Rd 1 R

For any .S > R, define .SR := S/RR and d 

S  −1 , LR,S := R 0, 1, . . . , R

.

SR,S :=

h∈LR,S

∂(h + QR ) .

6.1 A Homogenization Theorem

63

Hence, we define vS (x) :=

.

uR (x − h) + Mh if x ∈ h + QR , h ∈ LR,S otherwise .

Mx

Note that .f T (x, y − x, vS (y) − vS (x)) = 0 only if .x, y ∈ h + QR , for some T .h ∈ LR,S , or .x, y ∈ SR,S , where STR,S := {x ∈ QS : dist(x, SR,S ) < T } ∪ (QS \ QSR ).

.

In the latter case, since .r ≥ T , .vS (x) = Mx and .vS (y) = My. Hence, from the definition of .vS we get  F1T (vS , QS ) ≤ .

+



STR,S

STR,S

f T (x, y − x, My − Mx)dx dy

  h∈LR,S

 f T (x, y − x, uR (y − h) − uR (x − h))dx dy.

h+QR

h+QR

(6.12) By the periodicity of .f (·, ξ, z) we have that   .

k∈LR,S

=

 f T (x, y − x, uR (y − h) − uR (x − h))dx dy h+QR

h+QR

S d   R

QR



(6.13)



f T (x, y − x, uR (y) − uR (x))dx dy . QR

By (6.3) and (6.4), we get 



.

STR,S

STR,S

f T (x, y − x, My − Mx)dx dy ≤ C(|M|p + 1)(T + 1)S d−1

S R

 +R . (6.14)

Gathering (6.12)–(6.14), from the definition of .uR we obtain .

F1T (vS , QS ) ≤

 S

S d   +R . R d HR (M) + R d−1 + C(|M|p + 1)(T + 1)S d−1 R R (6.15)

Finally, by using .vS as a test function in the definition of .HS (M), (6.15) gives .

HS (M) ≤

1 R R d S d R d−1 S d p + . H (M) + + C(|M| + 1)(T + 1) R Sd R Sd R R S

64

6 Periodic Homogenization

By taking the limit first as .S → +∞ and then as .R → +∞ we get .

lim sup HS (M) ≤ lim inf HR (M) R→+∞

S→+∞

that yields the existence of the limit. Finally, .HR (M) ≤ F1T (Mx, QR )/R d ≤ C(|M|p + 1), hence the limit is finite.

Proof (of Theorem 6.1) We split the proof in two steps, dealing first with the case of truncated energies and then with the general case. Step 1 For .T > 0 let .FεT (·, ·) be as in Definition 5.1. Given .εj → 0, by Theorem 5.1 and Proposition 6.1 there exist a subsequence (not relabelled) and m×d → [0, +∞) such that .f0 : R  (Lp )- lim FεTj (u, A) =

.

j →+∞

f0 (∇u(x)) dx =: F T (u, A) A

T , for every .A ∈ Areg () and .u ∈ W 1,p (A; Rm ). We now prove that .f0 = fhom T where .fhom is defined in (6.11). Since .f0 is quasiconvex and satisfies the growth conditions (5.1), given .x0 ∈  and .r > 0 such that .Qr (x0 ) ⊂ , we have

f0 (M) =

.

 1 min d r



1,p

Qr (x0 )

f0 (∇u(x))dx : u − Mx ∈ W0 (Qr (x0 ); Rm )



for every .M ∈ Rm×d and then, by Proposition 5.4, we get  T  1 sεj ,M inf F (u, Q (x )) : u ∈ D (Q (x )) r 0 r 0 ε j j →∞ r d

f0 (M) = lim

.

for any .s > 0. Without loss of generality we may assume .x0 = 0 and use the notation .Qr = Qr (0). Setting .v(x) = u(εj x)/εj and using the changes of variable   T .x = x/εj and .y = x + ξ , we rescale .Fε as j  T .Fε (u, Qr ) j



=

f 

BT

(Qr )εj (ξ )



=

f BT



= εjd

(Qr )εj (ξ )



QRj

QRj

x εj x εj

, ξ,

u(x + εj ξ ) − u(x)  dx dξ εj

, ξ, v

  x  dx dξ +ξ −v εj εj

x

f T (x  , y − x  , v(y) − v(x  ))dx  dy,

6.1 A Homogenization Theorem

65

where .Rj := r/εj and .f T is defined in (6.10). Thus, f0 (M) = lim

.

j →∞



1 Rjd

 f T (x, y − x, v(y)

inf QRj

QRj

 − v(x))dx dy : v ∈ Ds,M (QRj ) . By the arbitrariness of .s > 0 and Proposition 6.2 we eventually get T f0 (M) = fhom (M)

.

 1 inf R→∞ R d





= lim

f T (x, y − x, v(y) − v(x))dx dy QR

QR

 : v ∈ D1,M (QR ) ,

which in particular proves the claim of Theorem 6.1 when .f ≡ f T . Step 2 By the previous step, Lemma 5.1 and Remark 5.3 we infer that  (Lp )- lim Fε (u, A) = lim (Lp )- lim FεT (u, A) =

.

T →+∞

ε→0

ε→0

A

∞ fhom (∇u(x))dx

for every .A ∈ Areg () and .u ∈ W 1,p (A; Rm ), where for every .M ∈ Rm×d ∞ T fhom (M) := lim fhom (M).

.

T →+∞

∞ (M) = f Hence, the result will follow if we prove that .fhom hom (M). Set

 1 inf d R



 1 = lim sup d inf R→∞ R



 fhom (M) = lim inf

.

R→∞

 f (x, y − x, v(y) − v(x))dx dy QR

 : v ∈ D1,M (QR ) ,

 .fhom (M)

QR

 f (x, y − x, v(y) − v(x))dx dy QR

 : v ∈ D1,M (QR ) .

QR

66

6 Periodic Homogenization

T (M) ≤ f  (M) for every .T > 0, it suffices to prove that .f  (M) ≤ Since .fhom hom hom ∞ fhom (M). Now, define T .HR (M)

 1 = d inf R

HR (M) =

.



 1 inf Rd



QR



 f T (x, y − x, v(y) − v(x))dx dy : v ∈ D1,M (QR ) , QR

 QR

 f (x, y − x, v(y) − v(x))dx dy : v ∈ D1,M (QR ) ,

QR

and let .uR ∈ D1,M (QR ) be such that   1 1 . f T (x, y − x, uR (y) − uR (x))dx dy ≤ HRT (M) + . d R QR QR T Note that, by (6.2)–(6.5), we get that .

  1   1 r0 1 1 T T + 1 ≤ C + 1 G (u , Q ) ≤ C H (M) + F (Mx, Q ) + R R R R 1 Rd 1 T Rd T ≤ C(|M|p + 1),

where C is a constant independent of T and R, so that, by Lemma 4.1, we get that 1 . Rd

 |uR (x + ξ ) − uR (x)|p dx ≤ C(|ξ |p + 1)(|M|p + 1). (QR )1 (ξ )

By taking .uR as a test function for the minimum problem defining .HR (M), we then have   1 1 T + d .HR (M) ≤ HR (M) + f (x, ξ, uR (x + ξ ) − uR (x))dx dξ T R BTc (QR )1 (ξ ) 1 ≤ HRT (M) + T     1 ψ2 (ξ )(|uR (x + ξ ) − uR (x)|p ) + ρ2 (ξ ) dx dξ + d c R BT (QR )1 (ξ )   1 ≤ HRT (M) + + C(|M|p + 1) ψ2 (ξ )(|ξ |p + 1) dξ + C ρ2 (ξ ) dξ. T BTc BTc By assumption (6.3) taking the limit first as .R → +∞ and then as .T → +∞ we get the conclusion.



6.2 The Convex Case

67

As a straightforward consequence of Theorem 6.1, Propositions 5.3 and 5.4, we deduce the following results about .-convergence and convergence of minimum problems for periodic functionals subject to Dirichlet boundary conditions. Proposition 6.3 Under the assumptions of Theorem 6.1, given any .g ∈ 1,p r,g Wloc (Rd ; Rm ) and .r > 0, let .{Fε (·, )} be the family of functionals defined in (5.20). Then ⎧ ⎨ fhom (∇u(x))dx if u − g ∈ W 1,p (; Rm ) 0 r,g .- lim Fε (u, ) = (6.16)  ⎩ ε→0 +∞ otherwise. Proposition 6.4 Under the assumptions of Theorem 6.1, for any .g 1,p Wloc (Rd ; Rm ) and .r > 0 there holds  rε,g . lim inf{Fε (u) : u ∈ D ()} = min fhom (∇u(x))dx ε→0

∈



 1,p : u − g ∈ W0 (; Rm ) .

(6.17)

Moreover, if .εj → 0 and .uj ∈ Drεj ,g () is a converging sequence such that .

lim Fεj (uj ) = lim inf{Fεj (u) : u ∈ Drεj ,g ()},

j →∞

j →∞

 then its limit is a minimizer for .min 

 1,p fhom (∇u(x))dx : u − g ∈ W0 (; Rm ) .

6.2 The Convex Case In this section we show that, analogously to the homogenization of integral functionals, in the convex case the asymptotic formula (6.7) reduces to a cellproblem formula. Theorem 6.2 (Convex-Homogenization Theorem) Under the hypotheses of Theorem 6.1, assume in addition that .f (y, ξ, ·) is convex for every .y, ξ ∈ Rd . Then the function .fhom defined by (6.7) satisfies for every .M ∈ Rm×d    .fhom (M) = inf f (x, y − x, v(y) − v(x))dx dy : v ∈ D#,M (Q1 ) , Rd

Q1

(6.18) where p

D#,M (Q1 ) = {u ∈ Lloc (Rd ; Rm ) : u − Mx is Q1 -periodic}.

.

68

6 Periodic Homogenization

Proof For brevity of notation, we denote by .f # (M) the right-hand side of (6.18). We first prove that .fhom (M) ≤ f # (M). Given .δ > 0, let .v ∈ D#,M (Q1 ) be such that   . f (x, y − x, v(y) − v(x))dx dy ≤ f # (M) + δ, Rd

Q1

  and set .uε (x) := εv xε . Then .uε → Mx in .Lp (; Rm ) and, by Theorem 6.1 and the periodicity of f , we have ||fhom (M) ≤ lim sup Fε (uε ) ≤ ||(f # (M) + δ).

.

ε→0

The conclusion follows by the arbitrariness of .δ > 0. It remains to prove that fhom (M) ≥ f # (M).

.

(6.19)

Note that, reasoning as in Step 2 of the proof of Theorem 6.1, we get that .

lim f #,T (M) = f # (M),

T →+∞

where .f #,T (M) is defined by the right-hand side of (6.18) with .f T in place of f . Hence it suffices to prove (6.19) with .f T . Let .R ∈ N and, for any function T ,M (Q ), let .u ∈ D#,M (Q ) be the function defined by .v ∈ D R 1 u(x) :=

.

1 Rd



v(x ˜ + i),

i∈[0,R)d ∩Zd

where .v˜ = w(x) ˜ + Mx and .w˜ denotes the periodic extension of .v(x) − Mx outside QR . From the convexity of .f (x, ξ, ·) we get

.





f #,T (M) ≤

f (x, ξ, u(x + ξ ) − u(x))dx dξ BT

.

≤ =

1 Rd 1 Rd

Q1

 i∈QR



∩Zd



 f (x, ξ, v(x ˜ + ξ ) − v(x))dx ˜ dξ

BT

i+Q1



f (x, ξ, v(x ˜ + ξ ) − v(x))dx ˜ dξ. BT

QR

(6.20)

6.3 Relaxation of Convolution-Type Energies

69

Since for every .v ∈ DT ,M (QR ) there holds 



.

BT

QR \(QR )1 (ξ )

f (x, ξ, v(x + ξ ) − v(x))dx dξ 



≤

QR \(QR )1 (ξ )

BT

f (x, ξ, Mξ )dx dξ ≤ C(|M|p + 1)T R d−1 ,

by taking the infimum in (6.20) we get  1   f (x, ξ, v(x + ξ ) − v(x))dx dξ R d BT (QR )1 (ξ )  CT : v ∈ DT ,M (QR ) + . R

f #,T (M) ≤ inf

.

Then, passing to the limit as .R → +∞ we obtain the desired inequality.



Example 6.1 (Quadratic Forms) A well-known property of .-convergence is the fact that the .-limit of non-negative quadratic forms is still a non-negative quadratic form (see [7, Theorem 11.10]). Hence, under the hypotheses of Theorem 6.1, if .f (x, ξ, z) is a non-negative quadratic form of the type f (y, ξ, z) = A(y, ξ )z, z

.

where .A : Rd × Rd → Rm×m is .[0, 1]d -periodic then fhom (M) = Ahom M, M   = inf

.

v∈D#,M (Q1 ) Rd

A(x, ξ )(v(x + ξ ) − v(x)), v(x + ξ ) − v(x)dx dξ,

Q1

with .Ahom ∈ T2 (Rm×d ), where .T2 (Rm×d ) denotes the space of .(0, 2)-tensors on m×d . .R

6.3 Relaxation of Convolution-Type Energies In the case of energies independent of .ε; that is when fε (x, ξ, z) = f (ξ, z),

.

(6.21)

energies of the form  Fε (u) :=

.

Rd



 u(x + εξ ) − u(x)  dx dξ f ξ, ε ε (ξ )

(6.22)

70

6 Periodic Homogenization

can be regarded as a convolution version of homogeneous functionals and the related homogenization formula  1 inf R→∞ R d





fhom (M) := lim

.

f (y − x, v(y) − v(x))dx dy QR

: v ∈ D1,M (QR )



QR

(6.23)

can be considered as a kind of relaxation formula. Remark 6.1 If the function f is also convex, then the computation of .fhom is trivial, and  .fhom (M) = f (ξ, Mξ ) dξ. (6.24) Rd

Indeed, we can apply Theorem 6.2. After the change of variable .y = x + ξ , (6.18) reads    .fhom (M) = inf f (ξ, v(x + ξ ) − v(x)) dx dξ : v ∈ D#,M (Q1 ) . Rd

Q1

Hence, for every .v ∈ D#,M (Q1 ), Jensen’s inequality yields that 

f (ξ, v(x + ξ ) − v(x)) dx dξ ≥

.

Rd



 Q1

Rd

f (ξ, Mξ ) dξ.

Thus, by taking the infimum over v we obtain  fhom (M) ≥

.

Rd

f (ξ, Mξ ) dξ.

The converse inequality comes by taking .v(x) = Mx as a test function and the claim follows. A particular case is when .f (ξ, z) is a non-negative quadratic form of the type f (ξ, z) = A(ξ )z, z,

.

in which case  fhom (M) =

.

Rd

A(ξ )Mξ, Mξ  dξ.

In particular, if .A(ξ ) = a(ξ )I we recover the result in Theorem 3.1 with .aε (ξ ) = a(ξ ).

6.3 Relaxation of Convolution-Type Energies

71

More generally, if we do not assume the convexity of .f (ξ, ·), we have the following bounds f ∗∗ (M) ≤ fhom (M) ≤ Qf (M),

(6.25)

.

where, given .g : Rd × Rm → [0, +∞), we set  g(M) :=

.

Rd

M ∈ Rm×d ,

g(ξ, Mξ ) dξ

f ∗∗ (ξ, ·) denotes the convex envelope of .f (ξ, ·) and .Qf (·) denotes the quasiconvex envelope of .f (·). Indeed the first inequality follows from (6.24), since

.

f ∗∗ (ξ, z) ≤ f (ξ, z),

z ∈ Rd .

.

To prove the second inequality, notice that, by taking .v(x) = Mx as a test function in (6.7), we get fhom (M) ≤ f (M),

.

from which the inequality follows, since .fhom (·) is quasi-convex. In the next two one-dimensional examples we show that both inequalities in (6.25) may be strict. Example 6.2 (.f ∗∗ (M) < fhom (M)) Let .Fε be defined by (2.1), with .d = m = 1, . = (0, 1) and .fε (x, ξ, z) = f (ξ, z), where ⎧ z ⎪ ⎪ ⎪f0 ξ ⎪ ⎪ z ⎪ ⎪ ⎨ f1 .f (ξ, z) :=  ξz  ⎪ ⎪ f ⎪ 2 ⎪ ⎪ ξ ⎪ ⎪ ⎩0

if 0 < |ξ | < if

1 2

1 2

≤ |ξ | < 1

if 1 ≤ |ξ | < 2 otherwise,

with f0 (z) = z2 ,

.

f1 (z) = (|z| − 1)2 ,

f2 (z) =

1 2 z . 2

We get that .f ∗∗ (z)

=z + 2

if |z| < 1

z2 (|z| − 1)2

+ z2

if |z| ≥ 1.

72

6 Periodic Homogenization

We now show that fhom (z) > f ∗∗ (z) for all z ∈ (−1, 0) ∪ (0, 1).

.

To this end, we suitably bound .Fε from below with discrete energies whose limit energy density is explicit. More precisely, let .Fεdisc (u, (a, b)) be the functional defined on discrete functions .u : εZ ∩ (0, 1) → R and localized on each interval .(a, b) ⊆ (0, 1) as Fεdisc (u, (a, b)) =

2 



.

εjfj

 u(k + j ε) − u(k)  jε

j =1 k,k+j ε∈εZ∩(a,b)

The discrete functions u are identified with their piecewise interpolation on the elementary cells of the lattice .εZ; that is, with a little abuse of notation, u(x) = u(k) if x ∈ k + (0, ε), k ∈ εZ ∩ (0, 1).

.

Hence .Fεdisc can be regarded as defined on a subset of .L2 (0, 1). In [5] it was proved that  - lim Fεdisc (u, (a, b)) =

b

.

ε→0

(g0 )∗∗ (u ) dx,

u ∈ W 1,2 ((a, b),

a

where the .-limit is performed with respect to the .L2 -convergence and g0 (z) :=

.

1 inf{f1 (z1 ) + f1 (z2 ) : z1 + z2 = z} + 2f2 (z). 2

With our choice of .f1 and .f2 , .(g0 )∗∗ can be explicitly computed, giving

(g0 )∗∗ (z) =

.

⎧ 2 ⎪ ⎪ ⎨2z

|z| − 18 ⎪ ⎪ ⎩(|z| − 1)2 + z2

if |z| ≤ if

1 4

< |z|
z2 .

.

Given .u ∈ L1loc (R), set uε,t (x) := u(εt + ε x/ε).

.

1 4 3 4

(6.26)

6.3 Relaxation of Convolution-Type Energies

73

As a consequence of a more general result we will exploit in the following (see (6.29)), we have that if .uε → u in .L1loc (R), then (uε )ε,t → u in L1loc (R) for a.e. t ∈ (0, 1).

.

(6.27)

We can now prove the claim. Let .z ∈ R and let .uε → zx in .L2 (0, 1) such that .

lim Fε (uε ) = fhom (z).

ε→0

Since .f (−ξ, −z) = f (ξ, z), we have 



 u (x + εξ ) − u (x) 2 ε ε dx dξ εξ 0 ε (ξ )  1   u (x + εξ ) − u (x)  ε ε f1 +2 1 εξ ε (2ξ ) 2  u (x + 2εξ ) − u (x)  ε ε dx dξ + 2f2 2εξ  1  u (x + εξ ) − u (x) 2 2 ε ε ≥2 dx dξ εξ 0 ε (ξ )

Fε (uε ) ≥ 2

.

1 2

  uε (x + εξ ) − uε (x)  f1 1 εξ lεξ 2 l=0  u (x + 2εξ ) − u (x)  ε ε dx dξ + 2f2 2εξ 

+2

1 1/εξ −3  (l+1)εξ

Through the change of variable .x = εξ(l + t), .t ∈ (0, 1), we then get for .r < 1 and ε small enough

.



1 2

Fε (uε ) ≥ 2



.

0

 u (x + εξ ) − u (x) 2 ε ε dx dξ εξ ε (ξ )

 +2

1  1 1/εξ −3 1 2

0

l=0

  u (εξ(l + t + 1)) − u (εξ(l + t))  ε ε εξ f1 εξ

 u (εξ(l + t + 2)) − u (εξ(l + t))  ε ε + 2f2 dt dξ 2εξ  1  u (x + εξ ) − u (x) 2 2 ε ε ≥2 dx dξ εξ 0 ε (ξ )

74

6 Periodic Homogenization

 +2

1 1 1 2

0

disc Fεξ ((uε )εξ,t , (0, r)) dt dξ

=: I1ε + I2ε . We can apply now Theorem 3.1. Since .uε → zx in .L2 (0, 1) we deduce that ε ε 2 .lim inf I 1 ≥ z . By (6.26), (6.27) and Fatou’s Lemma, we get that .lim inf I2 ≥ ε→0

ε→0

r(g0 )∗∗ (z) for every .r < 1. Hence, we conclude that fhom (z) ≥ z2 + (g0 )∗∗ (z) > f ∗∗ (z)

.

whenever .z ∈ (−1, 0) ∪ (0, 1). Example 6.3 (.fhom (M) < Qf (M)) Let .Fε be defined by (2.1), with .d = m = 1,  = (0, 1) and .fε (x, ξ, z) = f (ξ, z), where

.

⎧ z ⎪ f1 ⎪ ⎪ ⎨ ξ  z .f (ξ, z) := f ⎪ 2 ξ ⎪ ⎪ ⎩ 0

if 0 < |ξ | ≤ 1 if k ≤ |ξ | ≤ k + 1 otherwise,

with k to be chosen and f1 (z) = (|z| − 1)2 ,

.

f2 (z) = z2 .

Note that f (z) = 2(|z| − 1)2 + 2z2 ;

.

hence, ∗∗

Qf (z) = (f ) (z) =

.

We now show that for .k >

if |z| ≤ 1/2

1 2(|z| − 1)2

+ 2z2

√ 6 fhom (0) < 1 = (f )∗∗ (0).

.

To this end, let .u : R → R be 2-periodic and such that u(x) =

.

x

if 0 ≤ x ≤ 1

2−x

if 1 ≤ x ≤ 2

if |z| > 1/2.

6.3 Relaxation of Convolution-Type Energies

75

and set .uε (x) := εu(ε−1 x). Note that .uε 0 weakly in .H 1 (0, 1), hence fhom (0) ≤ lim sup Fε (uε ).

.

ε→0

We get that 

1/ε 

Fε (uε ) =2

1  (j +2)ε

f1

.

0

j =0, j even



k+1 

+2

jε

(j +2)ε

f2 k

jε

 u (x + εξ ) − u (x)  ε ε dx dξ εξ

  u (x + εξ ) − u (x)  ε ε dx dξ + o(1), εξ

thus  .

lim Fε (uε ) = 2

ε→0

1 2

 u(x + ξ ) − u x)  ( dx dξ ξ 0 0  k+1  2  u(x + ξ ) − u( x)  +2 dx dξ =: S1 + S2 . f2 ξ k 0 f1

A direct computation shows that S1 =

.

while .S2 ≤

2 . k2

1 2



1

−1

1 , 3

Hence fhom (0) ≤

.

whenever .k >

f1 (z) dz =

4 1 + 2 1 and for every .ξ ∈ B1 \ {0} ⎧  p ⎨1 + |z| |ξ | .fε (x, ξ, z) = f (ξ, z) := ⎩ 0

if z = ±ξ if z = ±ξ.

76

6 Periodic Homogenization

By testing the minimum problem defining .fhom with the identity function and its opposite, we immediately get that fhom (I ) = fhom (−I ) = 0,

.

where I is the identity matrix in .R2×2 . We now show that .fhom (0) > 0, from which we get that .fhom is not convex. The proof relies on a suitable lower bound of .Fε with discrete energies, for which an analogous result was proven in [2]. Let us introduce some notation. Given .ξ ∈ R2 \ {0}, let .Lξ be the lattice in .R2 defined by Lξ = Zξ ⊕ Zξ ⊥ ,

.

where .ξ ⊥ := (−ξ2 , ξ1 ). We now introduce the functionals .Fε (u, A) defined on discrete functions .u : εLξ ∩ (0, 1)2 → R2 and localized on each open set A in .R2 , as ξ,disc



Fεξ,disc (u, A) =



.

ξ  ∈{ξ,ξ ⊥ ,ξ +ξ ⊥ } k∈Aε,ξ

 u(k + εξ  ) − u(k)  , (ε|ξ |)2 f ξ  , ε 

where, for .ξ  ∈ Lξ \ 0, we have set 

Aε,ξ := εLξ ∩ A ∩ (A − εξ  ).

.

The discrete functions u can be regarded as .Lp functions by identifying them with their piecewise interpolation on the elementary cells of the lattice .εLξ , that is, with a little abuse of notation, u(x) = u(k) if x ∈ k + εQξ , k ∈ εLξ ∩ (0, 1)2 ,

.

where Qξ := [0, 1)ξ ⊕ [0, 1)ξ ⊥ .

.

In [2], it was proved that for any open set .A ⊂ R2 with Lipschitz boundary  e ,disc disc .- lim Fε 1 (u, A) = fhom (∇u) dx, u ∈ W 1,p (A; R2 ), ε→0

A

where the .-limit is performed with respect to the .Lp -convergence, and that disc (0) > 0 (see [2] Theorem 7.1). Then, for any .ξ ∈ R2 \ {0} fhom

.

 - lim Fεξ,disc (u, A) =

.

ε→0

A

disc fhom (RξT ∇uRξ ) dx,

where .Rξ is the rotation in .R2×2 such that .Rξ e1 =

u ∈ W 1,p (A; R2 ), ξ |ξ | .

(6.28)

6.3 Relaxation of Convolution-Type Energies

77

Regarding the convergence of proper discrete interpolation of .L1 functions, in [3] the following result was proved. Given .u ∈ L1loc (R2 ; R2 ), set for .y ∈ Qξ Tyε,ξ u(x) := u(εy + ε x/εξ ),

.

where

zξ := z · ξ ξ + z · ξ ⊥ ξ ⊥ .

.

If .uε → u in .L1loc (R2 ; R2 ), then Tyε,ξ uε → u in L1loc (R2 ; R2 ) for a.e. y ∈ Qξ

.

(6.29)

(see [3, Lemma 2.11]). We are now in a position to prove the claim. Let .uε → 0 in .Lp ((0, 1)2 ; R2 ) such that .

lim Fε (uε ) = fhom (0).

ε→0

We easily get the following lower bound for a square .Q   and .ε small enough 1 .Fε (uε ) ≥ 3 ≥

1 3



 B √2



Q

2

B √2 2

 ξ  ∈{ξ,ξ ⊥ ,ξ +ξ ⊥ }

  ξ

k∈Kε

 u (x + εξ  ) − u (x)  ε ε dx dξ f ξ , ε 

k+εQξ

ξ  ∈{ξ,ξ ⊥ ,ξ +ξ ⊥ }

 u (x + εξ  ) − u (x)  ε ε dx dξ, f ξ , ε

where Kεξ := {k ∈ εLξ : k + εQξ ⊂ Q }.

.

Through the change of variable .x = k + εy, .y ∈ Qξ , we then get for a square   .Q  Q and .ε small enough Fε (uε ) ≥    1 3 B √2 Qξ ξ

.

 ⊥ ⊥ k∈Kε ξ ∈{ξ,ξ ,ξ +ξ }

2

≥

1 3



 B√

2 2

1 |ξ |2



Qξ

 u (k + εy + εξ  ) − u (k + εy)  ε ε dy dξ ε2 f ξ  , ε

Fεξ,disc (Tyε,ξ uε , Q ) dy dξ .

78

6 Periodic Homogenization

By (6.28), (6.29), Fatou’s Lemma and the arbitrariness of .Q , we conclude that fhom (0) ≥

.

π disc f (0) > 0. 6 hom

6.4 An Extension Lemma from Periodic Lipschitz Domains We prove the existence of an extension operator for non-local functionals defined on general connected domains. For the reader convenience we recall the definition of Lipschitz boundary that will be used in the proof of Lemma 6.1. Definition 6.1 An open set .E ⊂ Rn has Lipschitz boundary at .x ∈ ∂E if .∂E is locally the graph of a Lipschitz function, in the sense that there exist a coordinate system .(y1 , . . . , yd ) obtained by a translation and a rotation, a Lipschitz function . of .d − 1 variables, and an open cube .Ux in the y-coordinates, centred at x, such that .E ∩ Ux = {y : yd < (y1 , . . . , yd−1 )} and that .∂E splits .Ux into two connected sets, .E ∩ Ux and .Ux \ E. If this property holds for every .x ∈ ∂E with the same Lipschitz constant L and side length .ρ of .Ux , we say that E has Lipschitz boundary. Throughout the following two sections, for the sake of brevity, we will adopt the following notation to indicate a neighbourhood of thickness .r > 0 of the diagonal in .Rd × Rd , Dr = {(x, y) ∈ Rd × Rd : |x − y| < r}.

.

Theorem 6.3 (Extension Theorem) Let E be a periodic open subset of .Rd with Lipschitz boundary and let . be a bounded open subset of .Rd . Then, there exist .R = R(E) > 0 and .k0 > 0 such that for all .ε > 0 there exists a linear and continuous extension operator .Tε : Lp ( ∩ εE) → Lp () such that for all .r > 0 and for all .u ∈ Lp ( ∩ εE), Tε u = u almost everywhere in  ∩ εE,

.



(6.30)

 |Tε u|p dx ≤ c1

.

(εk0 )

|u|p dx,

(6.31)

|u(x) − u(y)|p dx dy,

(6.32)

∩εE

 |Tε u(x) − Tε u(y)|p dx dy

.

((εk0 )×(εk0 ))∩DεR



≤ c2 (r) ((εk0 )×(εk0 ))∩Dεr

6.4 An Extension Lemma from Periodic Lipschitz Domains

79

where we use notation (λ) := {x ∈  : dist(x, ∂) > λ}.

.

(6.33)

The positive constants .c1 and .c2 depend on E and d and, in addition, .c2 depends also on r, but both are independent of .ε. We will prove the theorem in a special case, when E = Rd \ (Zd + K),

.

where .K is a compact set with .C 2 -boundary such that .(i + K) ∩ (j + K) = ∅ if .i, j ∈ Zd and .i = j . The case with separate compact perforations contains the relevant arguments due to the non-local form of the functionals separated from the issues on perforated domains that are common with local functionals. The case of a general E is more technical, involving a partition-of-unity argument as in the case of local integral functionals on perforated domains treated by Acerbi et al. [1] (for details see [6]). Lemma 6.1 Let A be a connected bounded set with Lipschitz boundary. For any r > 0 there exists a constant .cr > 0 such that the following inequality holds

.



 |u(η) − u(ξ )|p dξ dη ≤ cr

.

A×A

|u(η) − u(ξ )|p dξ dη.

(6.34)

(A×A)∩Dr

If L, .ρ are constants as in Definition 6.1 then the constant .cr depends on A only through its diameter and such constants. Proof Since for any function u the integral on the right-hand side of (6.34) is an increasing function of r, it is sufficient to prove (6.34) for r positive and small enough. Since A has a Lipschitz boundary and is connected, with fixed .r > 0 there exists 1 .r1 ∈ (0, r) and .ν ∈ (0, 1/2] that only depends on the Lipschitz constant of A such 2 that for any two points .η , η ∈ A there is a discrete path from .η to .η ; i.e., a set of points .η = η0 , η1 , . . . , ηN , η = ηN +1 , that possesses the following properties: (a) .|ηj +1 − ηj | ≤ r1 for .j = 0, 1, . . . , N; (b) for all .j = 1, . . . , N the ball .B νr1 (ηj ) is contained in A; ¯ 1 , diam(A)) such that .N ≤ N¯ for all .η , .η ∈ A. (c) there exists .N¯ = N(r Indeed, since A is a bounded set with Lipschitz boundary, it has a Lipschitz continuous boundary. Then there exists a constant .r2 = r2 (L, ρ, r) > 0 such that 1 1 .r2 < 2 r, .r2 < 2d ρ min(1, L), and the set .A(r2 ), defined as in (6.33), is connected. r2 We choose .r1 = 8(L+1) and denote .ZA = {z ∈ √r1 Zd : z ∈ Ar2 }. By construction d .Br1 (x) ⊂ A for any .x ∈ A(r2 ), and for any .z1 and .z2 in .ZA there exists a path  diam(A) d .z1 = η1 , . . . , ηN = z2 in .ZA such that .|ηj +1 − ηj | ≤ r1 and .N ≤ . Also, r1 by construction, for any .x ∈ A \ A(r2 ) there exists a path .x = η˜ 0 , . . . , η˜ N˜ such

80

6 Periodic Homogenization

that .η˜ N˜ ∈ ZA , .|η˜ j +1 − η˜ j | ≤ r1 , .N˜ ≤ 16(L + 1)d, and .B

r1 2(L+1)

(η˜ j ) ⊂ A for all

j = 1, . . . , N˜ . This implies the existence of a path that has properties .(a)–.(c). Writing

.

u(ξ0 ) − u(ξN +1 ) = u(ξ0 ) − u(ξ1 ) + u(ξ1 ) − . . . − u(ξN ) + u(ξN ) − u(ξN +1 ),

.

where .ξj denotes a point in .Bνr1 (ηj ) for .1 ≤ j ≤ N, we get  .

(A∩Bνr1 (η ))×(A∩Bνr1 (η ))

C = (νr1 )dN

|u(ξ0 ) − u(ξN +1 )|p dξ0 dξN +1 





... Bνr1 (η1 )

Bνr1 (ηN ) (A∩Bνr1

(η ))×(A∩B

νr1

(η ))

 u(ξ ) − u(ξ1 )

p +u(ξ1 ) − . . . − u(ξN ) + u(ξN ) − u(η) dξ0 dξN +1 dξN . . . dξ1 ≤C

(N + 1)p − 1 (νr1 )dN

= C (N + 1)p−1 ≤ C(N¯ + 1)p





 ... A∩Bνr1 (η0 )

N +1 A∩Bνr1 (ηN + 1 ) j =1

N +1   j =1

(A∩Bνr1 (ηj ))×(A∩Bνr1 (ηj −1 ))

|u(ξj ) − u(ξj − 1 )|p dξN + 1 . . . dξ0

|u(ξj ) − u(ξj −1 )|p dξj dξj −1

|u(η) − u(ξ )|p dξ dη. (A×A)∩Dr

Covering A with a finite number of balls of radius .νr1 and summing up the last inequality over all pairs of these balls gives the desired estimate (6.34).

Proof (of Theorem 6.3 for Compact Perforations) We apply our arguments separately to each connected component of .Rd \ E. With fixed .τ > 0 chosen below we consider a connected component .K of .Rd \ E, and set A := {ξ ∈ Rd \ K : dist(ξ, ∂K) < τ } and A := {ξ ∈ K : dist(ξ, ∂K) < τ }.

.

Since .K is bounded and .C 2 , we may fix .τ > 0 small enough and an invertible mapping .R from A to .A such that .

1 |R(ξ  ) − R(ξ  )| ≤ |ξ  − ξ  | ≤ 2|R(ξ  ) − R(ξ  )| 2

for all .ξ  , ξ  ∈ A. Slightly abusing the notation we call this mapping a reflection. In what follows for the sake of brevity we use the notation .ξR = R−1 (ξ ) for .ξ ∈ A .

6.4 An Extension Lemma from Periodic Lipschitz Domains

81

We set 

1 .u ¯A =  |A |

A

u(ξR ) dξ.

Let .ϕ be a .C ∞ function such that .0 ≤ ϕ ≤ 1, .ϕ = 1 in A and in a neighbourhood of .∂K, .ϕ = 0 in a neighbourhood of .∂A \ ∂K. We define .v(ξ ) as follows ⎧ if ξ ∈ A ⎨ u(ξ ) .v(ξ ) = ϕ(ξ )u(ξR ) + (1 − ϕ(ξ ))u¯ A if ξ ∈ A ⎩ u¯ A if ξ ∈ K \ A . Letting .k0 = diam(Q1 ) =

√ d and .R = min(r, k0 , τ ), we have 

 |v(η) − v(ξ )|p dξ dη =

.

(A×A)∩DR

|u(η) − u(ξ )|p dξ dη

(6.35)

(A×A)∩DR

and    ∂R(ζ )   dζ dη |u(η) − u(ζ )|p  ∂ζ  A×A  ≤ CR |u(η) − u(ζ )|p dζ dη. (6.36)





|v(η) − v(ξ )|p dξ dη ≤

.

(A×A )∩DR

A×A

      ∂R(ζ )  ) Here we have used the fact that the Jacobian . ∂R(ζ ∂ζ  is a bounded function: . ∂ζ  ≤ CR . Next, taking into account the relation v(ξ )−v(η) = (ϕ(ξ )−ϕ(η))(u(ξR )− u¯ A )+ϕ(η)(u(ξR )−u(ηR )) if η ∈ A , ξ ∈ A

.

we obtain  |v(η) − v(ξ )|p dξ dη

.

(A ×A )∩DR



≤



A ×A

|u¯ A − u(ξR )|p dξ dη +

A ×A

|u(ηR ) − u(ξR )|p dξ dη.

Since .u¯ A is the average of the function .u(ξR ) over .A , then 

 |u¯ A − u(ξR )| dξ dη ≤ p

.

A ×A

A ×A

|u(ηR ) − u(ξR )|p dξ dη.

82

6 Periodic Homogenization

This yields 



.

(A ×A )∩DR

|v(η) − v(ξ )|p dξ dη ≤ 2 CR2

|u(η) − u(ξ )|p dξ dη.

(6.37)

A×A

Finally, 



.

(K\A )∩DR

A

|v(η) − v(ξ )|P dξ dη  ≤

 K\A

A

  ≤ K \ A  ≤ CR

|ϕ(ξ )|p |u¯ A − u(ξR )|p dξ dη

 A

|u¯ A − u(ξR )|p dξ

   K \ A   |A |

|u(η) − u(ξ )|p dξ dη.

A×A

Combining the last inequality with (6.35), (6.36) and (6.37) we conclude that 

 |v(η)−v(ξ )| dξ dη ≤ C

|u(η)−u(ξ )|p dξ dη

p

.

((K∪A)×(K∪A))∩DR

(6.38)

A×A

We may now apply Lemma 6.1. By (6.34) we obtain  |v(η) − v(ξ )|p dξ dη

.

((K∪A)×(K∪A))∩DR



|u(η) − u(ξ )|p dξ dη.

≤C (A×A)∩DR

After rescaling, this inequality reads  |v(η) − v(ξ )|p dξ dη

.

(ε(K∪A)×ε(K∪A))∩Dεr



|u(η) − u(ξ )|p dξ dη

≤ C1

(6.39)

(εA×εA)∩Dεr

Summing up the last inequality over all the inclusions in .(k0 ε), choosing .Tε u = v we obtain (6.32). Condition (6.30) and inequality (6.31) are straightforward consequences of the definition of v.



6.5 Homogenization on Perforated Domains

83

Corollary 6.1 Let .uε be a family of functions in .Lp ( ∩ εE) such that there exists .c > 0 and .r > 0 such that .uε Lp (∩εE) ≤ c and 



.

Br

 u (x + εξ ) − u (x) p ε   ε  dx dξ ≤ c,  ε (∩εE)ε (ξ )

(6.40)

for all .ε > 0, with .( ∩ εE)ε (ξ ) = {x ∈  ∩ εE : x + εξ ∈  ∩ εE}. Then, for any sequence .εj → 0 as .j → +∞, and for any open subset .   the set .{Tεj uεj }j is relatively compact in .Lp ( ) and every its limit point is in .W 1,p (). Proof Let .uε be such that .||uε ||Lp (∩εE) ≤ c and (6.40) hold for every .ε > 0. From Theorem 6.3, the extended functions .Tε uε satisfy the estimates  |Tε uε |p dx ≤ c

.

(6.41)

(εk0 )

and 1 .

 |Tε uε (y) − Tε uε (x)|p dy dx

εd+p

((εk0 )×((εk0 ))∩DεR





≤ c2 (r) Br

(∩E)ε (ξ )

   uε (x + εξ ) − uε (x) p  dx dξ ≤ c ,    ε

for some .R > 0 independent of .ε. The latter, after the change of variables .y = x + εξ , is equivalent to 



.

(εk0 ) BR

   Tε uε (x + εξ ) − Tε uε (x) p  dξ dx ≤ c,    ε

which corresponds to 



.

(εk0 ) BR

   wε (x + εξ ) − wε (x) p  dξ dx ≤ c    ε

(6.42)

for .wε = Tε uε . Using Theorem 4.2 for .wε and (6.41), (6.42), we can conclude that for any sequence .εj → 0 as .j → +∞, and for any open subset . ⊂⊂ , .Tεj uεj is relatively compact in .Lp ( ) and every its limit point is in .W 1,p ().



6.5 Homogenization on Perforated Domains In this section we present an application of the Extension Theorem 6.3 to the homogenization of non-local functional. Specifically, we consider a periodic integrand .h : Rd × Rd × Rm → [0, +∞); i.e., a Borel function such that .h(·, ξ, z)

84

6 Periodic Homogenization

is .[0, 1]d -periodic for all .ξ ∈ Rd and .z ∈ Rm and satisfies the following growth conditions: there exist positive constants .c0 , c1 , r0 and non-negative function . ψ : Rd → [0, +∞) such that h(x, ξ, z) ≤ ψ(ξ )(|z|p + 1).

(6.43)

.

h(x, ξ, z) ≥ c0 (|z|p − 1)

for all |ξ | ≤ r0

(6.44)

with  .

Rd

ψ(ξ )|ξ |p dξ ≤ c1 .

(6.45)

Let . ⊂ Rd be an open set with Lipschitz boundary. For any .ε > 0, we introduce the non-local functional .Hε : Lp (; Rm ) → [0, +∞] defined as  Hε (u) =



 h

.

Rd

(∩εE)ε (ξ )

 u(x + εξ ) − u(x) x , ξ, dx dξ. ε ε

(6.46)

Note that the integration in (6.46) is performed for .x, ξ such that both x and .x + εξ belong to the perforated domain . ∩ εE. Conditions (6.43)–(6.45) guarantee that functionals .Hε are estimated from above and below as in Sect. 2.3. Thanks to Corollary 6.1, our functionals .Hε are equi-coercive with respect to p the .Lloc ()-convergence upon identifying functions with their extensions from the perforated domain. More precisely, from each sequence .{uε } with equi-bounded energy .Hε (uε ) we can extract a subsequence such that the corresponding extensions p converge in .Lloc to some limit .u ∈ W 1,p (). This is implied by Corollary 6.1 applied with .r = r0 to each component of the vector-valued functions .uε , upon noting that (6.44) implies (6.40). We now may state the homogenization result for the functional .Hε with respect p to the .Lloc (; Rm ) convergence. Theorem 6.4 (Homogenization on Perforated Domains) The functionals .Hε p defined by (6.46) .-converge with respect to .Lloc (; Rm )-convergence to the functional ⎧ ⎨ h (∇u(x)) dx if u ∈ W 1,p (; Rm ) hom .Hhom (u) = ⎩  +∞ otherwise,

(6.47)

with .hhom satisfying the asymptotic formula hhom () =

.

 1 inf T →+∞ T d lim



 (0,T )d ∩E

(0,T )d ∩E

h(x, y − x, v(y) − v(x)) dx dy :

v(x) = x if dist(x, ∂(0, T )d ) < k0

 (6.48)

6.5 Homogenization on Perforated Domains

85

for all . ∈ Rm×d and .k0 as in Theorem 6.3. Furthermore, if h is convex in the third variable, the following cell-problem formula holds .

hhom () = inf





h(x, y − x, v(y) − v(x)) dx dy

(0,1)d ∩E

E

 : v(x) − x is 1-periodic .

(6.49)

Proof We will prove Theorem 6.4 reducing to Theorem 6.1 (i.e., to .E = Rd ) by a perturbation argument. For every .δ ≥ 0 we set hδ (x, ξ, z) = χE (x)χE (x + ξ ) h(x, ξ, z) + δχBR0 (ξ )|z|p ,

.

where .R0 > 0 is fixed but arbitrary, and  Hεδ (u) =



 hδ

.

Rd

ε (ξ )

 x u(x + εξ ) − u(x) dx dξ , ξ, ε ε

is defined for .u ∈ Lp (; Rm ). Note that .Hεδ ≥ Hε , and for .δ = 0 we have .Hε0 = Hε . In the following, for any open set A and .δ ≥ 0, we also consider the ‘localized’ functionals     x u(x + εξ ) − u(x) δ , ξ, dx dξ. .Hε (v, A) = hδ ε ε Rd Aε (ξ ) If .δ = 0 we write .Hε (v, A) in the place of .Hε0 (v, A). Theorem 6.1 ensures that for all .δ > 0 there exists the .-limit δ Hhom (u) = - lim Hεδ (u)

.

ε→0

with domain .W 1,p (; Rm ), on which it is represented as  δ Hhom (u) =

.



hδhom (∇u) dx.

The energy density .hδhom satisfies hδhom () =

.

 1 inf T →+∞ T d





lim

(0,T )d

(0,T )d

hδ (x, y − x, v(y) − v(x)) dx dy :

 v(x) = x if dist(x, ∂(0, T )d ) < r ,

86

6 Periodic Homogenization

for any fixed .r > 0, and .

c1 (||p − 1) ≤ hδhom () ≤ c2 (1 + ||p )

with .c1 , c2 independent of .δ, for .δ ∈ [0, 1]. Note that the independence of .c1 from .δ is an immediate consequence of the Extension Theorem 6.3. Indeed, let .uδε → x be such that hδhom () = lim Hεδ (uδε , (0, 1)d ).

.

ε→0

Applying Corollary 6.1 with . = (0, 1)d , we deduce that .Tε uδε converge to .x locally in .(0, 1)d (in particular the convergence is strong e.g. in .( 14 , 34 )d ). Hence, using (6.44), the Extension Theorem (with .r0 in place of r), and the liminf inequality of the .-limit we have lim. Hεδ (uδε , (0, 1)d )

ε→0

≥ lim inf Hε (uδε , (0, 1)d ) ε→0

 1   δ δ p |u (x) − u (y)| dxdy − 1 ε ε ε→0 εp+d ((0,1)d ∩εE)2 ∩Dr0  1   c0 lim inf p+d |Tε uδε (x) − Tε uδε (y)|p dxdy − 1 ≥ c2 (r0 ) ε→0 ε (( 14 , 34 )d )2 ∩DR 1  c0 min d cR , 1 (||p − 1), ≥ c2 (r0 ) 2

≥ c0 lim inf

where in the last inequality we have used that - lim

.

ε→0

1 εp+d





(( 14 , 34 )d )2 ∩DR

|v(x) − v(y)|p dxdy = cR

( 14 , 34 )d

|∇v|p dx,

where .cR is as in (3.5). Since .hδhom is increasing with .δ, we may define h0 () = inf hδhom () = lim hδhom (),

.

δ>0

δ→0+

and deduce (here we use the usual notation for the upper .-limit) that  h0 (∇u) dx ≥ - lim sup Hε (u) .

.



ε→0

(6.50)

6.5 Homogenization on Perforated Domains

87

If .u ∈ W 1,p (; Rm ) and .uε → u with .supε Hε (uε ) < +∞ then for all fixed . compactly contained in ., if .R0 < R, upon identifying .uε with its extension given by the Extension Theorem, we obtain that, 



.

BR0

   uε (x + εξ ) − uε (x) p  dx dξ ≤ c,    ε ( )ε (ξ )

so that .

lim inf Hε (uε ) ≥ lim inf Hε (uε ,  ) ≥ lim inf Hεδ (uε ,  ) − δc. ε→0

ε→0

ε→0

From this inequality we obtain (in terms of the lower .-limit)  - lim inf Hε (u) ≥

h0 (∇u)dx

.

ε→0



by the arbitrariness of .δ and .  . Hence, recalling (6.50), we have proved that  .- lim Hε (u) = h0 (∇u)dx, ε→0



and in particular that the .-limit exists as .ε → 0 (no subsequence is involved) and it can be represented as an integral functional with a homogeneous integrand. Note moreover that the lower-semicontinuity of the .-limit implies that .h0 is quasiconvex (see [4]). We now prove that .h0 coincides with .hhom given by the asymptotic formula. First, note that   1 .h0 () ≥ lim sup inf h(x, y − x, v(y) − v(x)) dx dy : d (0,T )d ∩E (0,T )d ∩E T →+∞ T  v(x) = x if dist(x, ∂(0, T )d ) < r . (6.51) If we take .r = k0 , we obtain a lower bound for .h0 . To prove the opposite inequality, for any diverging sequence .{Tj } we can consider (almost-)minimizers .vj of the problems in (6.51) with .r = k0 and .T = Tj . By Theorem 6.3 (applied component-wise and with .ε = 1) with . = (0, T )d and . = p k0 , T − k0 )d ; Rm ) ( k20 , Tj − k20 )d , we can consider the extended functions  .vj ∈ L (( j 2 2 d with  .vj = vj on . = (0, T ) ∩ E and  .

(

k0 k0 d 2 ,Tj − 2 ) ∩DR

| vj (ξ ) −  vj (η)|p dξ dη

≤ c2 (r0 )

 (0,Tj )d ∩E)2 ∩Dr0

|vj (ξ ) − vj (η)|p dξ dη ≤ c Tjd (1 + ||p )

88

6 Periodic Homogenization

for some .c > 0 independent of j . Upon choosing a larger .k0 > 2 we may suppose that . k20  + 1 < k0 so that we may consider .wj ∈ Lp ((0, Tj − n)d ; Rm ), where k0 .n = 2

2  + 2, defined by    k     k  0 0 + 1 (1, . . . , 1) − + 1 (1, . . . , 1). wj (x) =  vj x + 2 2

.

Having set .εj = (Tj − n)−1 we can consider the scaled functions uj (x) = εj wj

.

x . εj

By the boundedness of the energies above and noting that there exists .c > 0 such that .wj (x) = x if .x ∈ E and dist.(x, ∂(0, Tj − n)d ) < c, upon extracting a 1,p subsequence, we may suppose that .uj → u and .u ∈ x + W0 ((0, 1)d ; Rm ). We may then use the quasiconvexity inequality for .h0 to obtain  h0 () ≤

.

(0,1)d

h0 (∇u)dx

≤ lim inf Hεδj (uj , (0, 1)d ) j

≤ lim inf Hεj (uj , (0, 1)d ) + cδ j

≤ lim inf j

1 H1 (wj , (0, Tj − n)d ) + cδ (Tj − n)d

1 H1 (vj , (0, Tj )d ) + cδ (Tj − n)d   1 = lim inf inf h(x, y − x, v(y) − v(x)) dx dy j (Tj − n)d (0,Tj )d ∩E (0,Tj )d ∩E  : v(x) = x if dist(x, ∂(0, Tj )d ) < k0 + cδ   1 = lim inf d inf h(x, y − x, v(y) − v(x)) dx dy j Tj (0,Tj )d ∩E (0,Tj )d ∩E  : v(x) = x if dist(x, ∂(0, Tj )d ) < k0 + cδ. ≤ lim inf j

By the arbitrariness of .δ and of the sequence .Tj we obtain the desired upper bound for .h0 , which, together with (6.51), proves the asymptotic formula.

References

89

In the convex case, again by Theorem 6.1, we may repeat the arguments used to get (6.51) to obtain the lower bound for .h0 .

h0 () ≥ inf





(0,1)d ∩E

h(x, y − x, v(y) − v(x)) dx dy : E

 v(x) − x is 1-periodic .

(6.52)

Note that this implies that the right-hand side is bounded from above by .c2 (1+||p ). Now, let v be an (almost) minimizing function for (6.52), and set .vε (x) = εv( xε ). After applying Theorem 6.3 to any set . compactly containing .(0, 1)d to possibly redefine .vε outside .εE, we can suppose that .vε converge in .Lp ((0, 1)d ; Rm ) to .x and that  1 |vε (x) − vε (y)|p dx dy ≤ c(1 + ||p ). . εp+d ((0,1)d ×(0,1)d )∩DεR0 We then estimate hδhom () ≤ lim inf Hεδ (vε )

.

ε→0

 ≤

(0,1)d ∩E

 h(x, y − x, v(y) − v(x)) dx dy + cδ(1 + ||p ). E

Taking the limit as .δ → 0, we obtain the converse inequality of (6.52), and conclude the proof.



References 1. Acerbi, E., Chiadò Piat, V., Dal Maso, G., Percivale, D.: An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18, 481–496 (1992) 2. Alicandro, R., Cicalese, M.: A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36, 1–37 (2004) 3. Alicandro, R., Focardi, M., Gelli, M.S.: Finite-difference approximation of energies in fracture mechanics. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 29, 671–709 (2000) 4. Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals, volume 12 of Oxford Lecture Ser. Math. Appl. The Clarendon Press/Oxford University Press, New York (1998) 5. Braides, A., Gelli, M.S., Sigalotti, M.: The passage from nonconvex discrete systems to variational problem in Sobolev spaces: the one-dimensional case. Proc. Steklov Inst. Math. 236, 395–414 (2002) 6. Braides, A., Chiadò Piat, V., D’Elia, L.: An extension theorem from connected sets and homogenization of non-local functionals. Nonlinear Anal. 208, 112316 (2021) 7. Dal Maso, G.: An Introduction to -convergence. Progr. Nonlinear Differential Equations Appl. Birkhäuser, Boston (1993)

Chapter 7

A Generalization and Applications to Point Clouds

Abstract In this chapter we prove a .-convergence result for functionals defined on point clouds. To this end, we first generalize the homogenization theorem for a family of convolution-type functionals where the Lebesgue measure is replaced by a measure with continuous density and we prove that it is .-asymptotically equivalent to a family of perturbed continuous functionals obtained by composing also the densities with suitable transportation maps. We then define discrete energies on point clouds whose number of points is going to infinity and prove that such a family .-converges to the same .-limit of the perturbed functionals. Keywords Point clouds · Discrete functionals · Transportation maps · Voronoi cells · .-asymptotic equivalence

7.1 Perturbed Convolution-Type Functionals We consider here a generalization of the class of functionals defined in (2.1), obtained by replacing the Lebesgue measure with a measure .μ = ρ(x)Ld , with 0 .ρ ∈ C () and satisfying 0 < c ≤ ρ(x) ≤ C

.

for every x ∈ .

(7.1)

More precisely, given such a .ρ, we set Fε [ρ](u) :=

.

1 εd

  

 y − x u(y) − u(x)  , ρ(y)ρ(x)dx dy. fε x, ε ε 

(7.2)

In the periodic case, that is when .fε satisfies (6.1), a generalization of Theorem 6.1 is provided by the following result.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_7

91

92

7 A Generalization and Applications to Point Clouds

Theorem 7.1 Let .Fε [ρ] be defined by (7.2) with .fε satisfying (6.1), .ρ ∈ C 0 () and such that (7.1) holds. Then, under the assumptions of Theorem 6.1, ⎧ ⎨ fhom (∇u(x))ρ 2 (x)dx p .(L )- lim Fε [ρ](u) =  ⎩ ε→0 +∞

if u ∈ W 1,p (; Rm ), otherwise,

where .fhom is defined by (6.7). Proof We highlight only the main differences with respect to the proof of Theorem 6.1. Note that, by (7.1), .Fε [ρ] satisfies all the assumptions of Theorem 5.1. Hence, given .εj → 0, there exists a subsequence (not relabelled) such that  (Lp )- lim Fεj [ρ](u, A) =

.

j →+∞

f0 (x, ∇u(x)) dx. A

The characterization of non-homogeneous quasiconvex functions by their minima (see [2, Theorem II]) yields that for every .M ∈ Rm×d and for almost every .x0 ∈   1 1,p m .f0 (x0 , M) = lim min f (x, ∇u(x))dx : u − Mx ∈ W (Q (x ); R ) . 0 r 0 0 r→0 r d Qr (x0 ) Then, proceeding as in the proof of Theorem 6.1 and using the continuity of .ρ, we obtain that f0 (x0 , M) = ρ 2 (x0 )fhom (M).

.

Since .f0 does not depend on .(εj )j , we get the conclusion.

 

We study now a perturbed version of the class of functionals defined in (7.2), obtained by composing the energy densities with a family of transportation maps. Specifically, we consider the family of perturbed functionals .Eε : Lp (; Rm ) → [0, +∞] defined by Eε (u) :=

.

1 εd

  

 T (y) − T (x) u(y) − u(x)  ε ε , ρ(y)ρ(x)dx dy. fε x, ε ε  (7.3)

Note that in Sect. 6.4 we denote by .Tε the extension operator .Tε : Lp ( ∩ εE) → Lp (). Here .Tε :  →  is a measurable map (representing a transportation map in Sect. 7.2). We also assume that .fε fulfills assumptions (H0)–(H2), with .ψε,2 satisfying the additional hypothesis ψε,2 (ξ ) := ψ ε (|ξ |) , where ψ ε : [0, +∞) → [0, +∞) is non increasing

.

(7.4)

7.1 Perturbed Convolution-Type Functionals

93

and .ψ ε (0) < c, for some constant .c > 0. Notice that, under these additional assumptions we have that .lim supε→0 Rd ψε,2 (ξ ) dξ < +∞. Remark 7.1 Set  T (x + εξ ) − T (x)  ε ε , z ρ(x)ρ(x + εξ ) gε (x, ξ, z) = fε x, ε

.

(7.5)

and note that, under the assumptions above on .fε and .ρ, .gε satisfies (H0)–(H2) provided the following regularity conditions are fulfilled by .Tε : |Tε (y) − Tε (x)| ≤ C (|y − x| + ε),

.

|Tε (y) − Tε (x)| ≥ C

|y − x|,

if |y − x| ≤ εr .

if |y − x| ≥ εr

,

(7.6) (7.7)

for some .C , C

, r , r

positive constants. Indeed, if .r0 is a positive constant such that .r0 < r and .C (r0 + 1) < r0 , then by (7.6)  T (x + εξ ) − T (x)  ε  ε    ≤ r0 for every ξ ∈ Br0

ε

.

and assumption (H0) on .fε yields that the same assumption is satisfied by .gε with another choice of the constants. Moreover, denoting  T (x + εξ ) − T (x)  ε   ε ψ˜ ε (ξ ) := sup ψ ε   , ε x∈

.

(7.8)

from conditions (7.7) and the monotonicity of .ψ ε we get  .

Rd

ψ˜ ε (ξ )|ξ |p dξ ≤



 Br

ψ ε (0)|ξ |p dξ +

Brc

ψ ε (C

|ξ |)|ξ |p dξ

which yields that condition (2.5) is satisfied by .ψ˜ ε , since it is satisfied by .ψε,2 . Analogously it can be shown that .ψ˜ ε satisfies (H2). Hence, under the assumptions (7.6) and (7.7), energies as in (7.3) belong to the class of functionals satisfying the hypotheses of Theorem 5.1. Notice that conditions (7.6) and (7.7) hold in particular if . Tε − id ∞ ≤ Cε. In the next one-dimensional example we show that the asymptotic behaviour of Eε could be degenerate if (7.4) is not satisfied.

.

Example 7.1 (A Pathological Problem) Assume that in (7.3) . = (0, 1), .ρ ≡ 1 and fε (x, ξ, z) = a(ξ )|z|p with .a : R → [0, +∞) defined as follows

.

a(ξ ) =

.

1 ξ ∈ (−1, 1) ∪ Q 0 otherwise.

94

7 A Generalization and Applications to Point Clouds

Given .λε → 0, let .Tε : (0, 1) → (0, 1) be such that .Tε ((0, 1)) ⊂ εQ and . Tε − id ∞ ≤ λε . In particular (7.6) and (7.7) are satisfied if .λε = o(ε). We may construct −1 such maps as follows: for any .k ∈ {0, . . . , λ−1 ε  − 1}, let .qk,ε ∈ Q ∩ ε λε [k, k +  −1 1) ∩ (0, ε ) and set Tε (x) := εqk,ε if x ∈ λε [k, k + 1) for some k ∈ {0, . . . , λ−1 ε  − 1}.

.

Since .a = χ(−1,1) almost everywhere, .Gε [a] = G1ε ; thus, by Theorem 3.1, .1,p (0, 1). Whereas, since .(T y − T x)/ε ∈ Q, . lim Gε [a](u) < +∞ for any .u ∈ W ε ε ε→0 Eε reads

.

 Eε (u) =

+∞  1∧(1−εξ )  u(x

.

−∞

0∨(−εξ )

 

+ εξ ) − u(x) p  dx dξ , ε

From which we deduce that - lim Eε (u) =

.

ε→0

0

if u = 0 in (0, 1),

+∞ otherwise.

In the next proposition we show that if . Tε − id ∞ = o(ε) and .fε (x, ·, z) satisfies a suitable continuity assumption uniformly with respect to .ε and x, then the functionals .Eε defined by (7.3) are asymptotically equivalent in the sense of the .-convergence to the functionals .Fε [ρ] defined by (7.2). Proposition 7.1 Let .Fε [ρ] and .Eε be defined by (7.2) and (7.3), respectively, with fε satisfying (H0)–(H2) and (7.4). Assume in addition that:

.

(i) there exists a family of positive functions .{ωh }h>0 ⊂ L1loc (Rd ) such that .ωh → 0 in .L1loc (Rd ) and .

sup sup |fε (x, ξ + v, z) − fε (x, ξ, z)| ≤ ωh (ξ )|z|p

x∈ |v|≤h

for every .ε > 0 and .z ∈ Rm ;  T − id    ε (ii) . → 0.  ∞ L (;Rd ) ε Then (Lp )- lim inf Eε (u) = (Lp )- lim inf Fε [ρ](u),

.

ε→0

ε→0

(Lp )- lim sup Eε (u) = (Lp )- lim sup Fε [ρ](u).

.

ε→0

ε→0

(7.9)

7.1 Perturbed Convolution-Type Functionals

95

Proof Notice that, since both .Fε [ρ] and .Eε satisfy the hypotheses of Proposition 3.3, we have that (Lp )- lim Eε (u) = (Lp )- lim Fε [ρ](u) = +∞, u ∈ Lp (; Rm )\W 1,p (; Rm ).

.

ε→0

ε→0

Hence, by (H0)–(H1), it suffices to prove that Eε (uε ) = Fε [ρ](uε ) + o(1)

(7.10)

.

for any sequence .(uε )ε such that .Grε (uε ) is uniformly bounded for some .r ≤ r0 ∧ r0 , where .r0 and .r0 refer to assumption (H0) for .Fε [ρ] and .Eε , respectively. By Lemma 5.1, we may reduce to prove (7.10) in the case .fε (x, ξ, z) = 0 for every .x ∈ , .|ξ | > T and .z ∈ Rm , for some .T > 0. Let then .uε be such that r .supε>0 Gε (uε ) < +∞. We have Eε (uε ) = Fε (uε ) + Rε (uε ),

.

where 



  v(x + εξ ) − v(x)  gε x, ξ, ε ε (ξ )  v(x + εξ ) − v(x)  − fε x, ξ, dx dξ , ε

Rε (v) :=

.

BT

with .gε defined by (7.5). Set  T − id    ε hε :=  .  ∞ L (;Rd ) ε

.

By (i) and Lemma 4.1 we get 

 |Rε (uε )| ≤ C

.

BT

 u (x + εξ ) − u (x) p ε   ε ω2hε (ξ )  dx dξ ε ε (ξ ) ≤ C(R p + 1)Grε (uε )

which goes to zero as .ε → 0 by (ii).

 BT

ω2hε (ξ )dξ,  

As a straightforward consequence of the previous proposition and Theorem 7.1, we obtain the following result.

96

7 A Generalization and Applications to Point Clouds

Corollary 7.1 Under the assumptions of Proposition 7.1, assume in addition that fε satisfies (6.1). Then

.

⎧ ⎨ fhom (∇u(x))ρ 2 (x)dx p .(L )- lim Eε (u) =  ⎩ ε→0 +∞

if u ∈ W 1,p (; Rm ), otherwise,

where .fhom is defined by (6.7).

7.2 Application to Functionals Defined on Point Clouds ε The case in which .Tε () = Xnε := {xi }ni=1 ⊂ , has already been studied in the context of variational methods for Machine Learning, when dealing with discrete convolution-type energies of the form

.

nε 1 1  ε ai,j |u(xi ) − u(xj )|p , εp n2ε

(7.11)

i,j =1

that are the discrete version of energies (7.3) when .Tε are transportation maps from  to .Xnε , see [4] when .p = 1 and [1] when .p > 1. Therein, .Xnε denotes a point cloud obtained by refining random samples of a given probability measure .μ  Ld , having continuous density bounded from above and below by two positive constants. In this subsection, we will prove a .-convergence result for a generalized version of discrete energies as in (7.11) defined on point clouds. In particular, we will recover the convergence result provided in [1]. Before setting the problem, we recall, for the reader’s convenience, some useful notions about point-cloud models. Let .μ = ρ(x)Ld be a probability measure supported on ., such that .ρ ∈ C 0 () and satisfies (7.1). Given .(X, σ, P) a probability space, we consider a sequence of random variables

.

xi : X  ω → xi (ω) ∈ ,

.

i ∈ N,

that are i.i.d. according to the distribution .μ. Then, given .n ∈ N, we say that the set Xn (ω) = {xi (ω)}ni=1 is a point cloud obtained as samples from a given distribution .μ. In the following, we will drop the dependence on .ω for the sake of simplicity of notation, unless otherwise specified. To any point cloud .Xn we associate its empirical measure .

1 δx i . n n

μn =

.

i=1

It is well known that .μn weak.∗ converge to .μ as .n → +∞ .P-almost surely.

(7.12)

7.2 Application to Functionals Defined on Point Clouds

97

Let .nε ∈ N be such that .

lim nε = +∞

ε→0

and let .f : Rd × Rm → [0, +∞) be a Borel function that is convex in the second variable. We then consider the family of functionals (“dis” stands for discrete) Eεdis (u) =

.

nε  x − x u(x ) − u(x )  1  i j i j , , f d 2 ε ε ε nε i,j =1

defined on functions .u : Xnε → Rm . Note that .Eεdis can be written in terms of .μnε as    y − x u(y) − u(x)  1 dis , dμnε (x) dμnε (y). .Eε (u) = f (7.13) d ε   ε ε Let .Eε be defined by (7.3) with fε (x, ξ, z) = f (ξ, z).

(7.14)

.

If .Tε :  → Xnε is a transportation map between .μnε and .μ; that is, .(Tε )# μ = μnε , where .T# μ denotes the push-forward of .μ by T , then we may identify any m −1 .u : Xnε → R with its piecewise constant interpolation on .Tε (xi ), .i = 1, . . . , nε , and, by (7.13), Eεdis (u) = Eε (u)

.

for every u ∈ P C(Xnε ),

(7.15)

where   P C(Xnε ) := u :  → Rm | u is constant on Tε−1 (xi ) for every 1 ≤ i ≤ nε .

.

With a slight abuse of notation we assume that .Eεdis is defined on the whole space m p .L (; R ) by setting ⎧ nε  x − x u(x ) − u(x )   ⎪ i j i j ⎪ ⎨ 1 , , f d 2 dis ε ε ε nε .Eε (u) = i,j =1 ⎪ ⎪ ⎩+∞

u ∈ P C(Xnε )

(7.16)

otherwise.

In the next theorem we show that .Eεdis and .Eε are asymptotically equivalent in the sense of .-convergence under the assumptions of Proposition 7.1. Theorem 7.2 Let .Xnε be a family of point clouds obtained as samples from .μ and let .Tε :  → Xnε be a transportation map between .μnε and .μ, where .μnε is

98

7 A Generalization and Applications to Point Clouds

defined in (7.12). Let .Eεdis be defined by (7.16) and let .Eε be defined by (7.3) with d .fε satisfying (7.14) and .f (ξ, ·) convex for any .ξ ∈ R . If .fε and .Tε satisfy the assumptions of Proposition 7.1, then (Lp )- lim Eεdis (u) = (Lp )- lim Eε (u) ε→0

.

=

⎧ ⎨

 Rd

ε→0

f (ξ, ∇u(x)ξ )ρ 2 (x) dx dξ

⎩ +∞

if u ∈ W 1,p (; Rm ),



otherwise. (7.17)

Proof The second equality in (7.17) is a straightforward consequence of Corollary 7.1, Theorem 6.2 and Remark 6.1. Since .Eεdis ≥ Eε , in order to prove the first equality in (7.17) it suffices to prove that given .u ∈ W 1,p () we can find m p .(uε )ε ⊂ P C(Xnε ) such that .uε → u in .L (; R ) and   dis . lim sup Eε (uε ) ≤ f (ξ, ∇u(x)ξ )ρ 2 (x) dx dξ. (7.18) Rd

ε→0



By a density argument it suffices to prove (7.18) for .u ∈ C ∞ (Rd ; Rm ). Fix such a function u and note that, by the regularity assumptions on f and assumption (ii) of Proposition 7.1, we have that   . lim Eε (u) = f (ξ, ∇u(x)ξ )ρ 2 (x) dx dξ. (7.19) ε→0

Rd

Let .uε ∈ P C(Xnε ) be defined by uε (xi ) :=

.

1 |Vεi |



 u(y) dy Vεi

i ∈ {1, . . . , nε },

where, for .i = 1, . . . , nε , we have set .Vεi = Tε−1 (xi ). Note that .Vεi ⊆ Brε (xi ), where .rε := Tε − I d ∞ . Hence, . uε − u L∞ (;Rm ) ≤ Crε → 0 by assumption (ii) of Proposition 7.1 By the convexity of .f (ξ, ·), we get f

.

 x − x u (x ) − u (x )  i j ε i ε j , ε ε   x − x 1 1 u(y) u(x)  i j =f , i dy − j dx ε |Vε | Vεi ε |Vε | Vεj ε    x − x u(y) 1 u(x)  1 i j , − j dx dy f ≤ i |Vε | Vεi ε ε |Vε | Vεj ε    x − x u(y) − u(x)  1 i j ≤ , dy dx f j ε ε |V i ||Vε | Vεi Vεj ε

for any .1 ≤ i, j ≤ nε .

(7.20)

7.2 Application to Functionals Defined on Point Clouds

99

Note that, since .Tε is a transportation map between .μ and .με , by the continuity of .ρ we get .

1 = (ρ(xi ) + o(1)) nε , |Vεi |

uniformly in .1 ≤ i ≤ nε . Hence, by (7.20) we get dis .Eε (uε )

nε  x − x u (x ) − u (x )  1  i j ε i ε j , = d 2 f ε ε ε nε i,j =1

≤

  nε  x − x u(y) − u(x)  1  1 i j , dy dx f j d 2 j i i ε ε ε nε i,j =1 |Vε ||Vε | Vε Vε

=

nε    x − x u(y) − u(x)  1  i j , ρ(xi )ρ(xj ) dy dx + o(1) f j εd ε ε Vεi Vε i,j =1

= Eε (uε ) + o(1),  

and the conclusion follows from (7.19).

Remark 7.2 In [3], extending previous results, it has been proved that for .d ≥ 2 almost surely there exists a family of transportation maps .Tn (ω) :  → Xn (ω) between .μ and .μn such that 0 < c ≤ lim inf

.

n→+∞

Tn (ω) − id L∞ Tn (ω) − id L∞ ≤ lim sup ≤ C, ln ln n→+∞

where ⎧ 3/4 ⎪ ⎨ (log n) n1/2 .ln =  log n 1/d ⎪ ⎩ n

if d = 2, if d ≥ 3.

Hence, if .nε → +∞ and .

ln ε = 0, ε→0 ε lim

(7.21)

the corresponding transportation maps .Tε = Tnε are such that . Tε − id L∞ = o(ε) and, in particular, assumptions (ii) of Proposition 7.1 is satisfied. Hence if (7.21) is satisfied the .-convergence result stated in Theorem 7.2 holds almost surely, thus extending the convergence result provided in [1], where the analysis is limited to energy densities of the form .f (ξ, z) = a(|ξ |)|z|p , with .a(·) non increasing.

100

7 A Generalization and Applications to Point Clouds

References 1. Crook, O.M., Hurst, T., Schönlieb, C.-B., Thorpe, M., Zygalakis, K.C.: Pde-inspired algorithms for semi-supervised learning on point clouds (2019). Preprint, arXiv:1909.10221v1 2. Dal Maso, G., Modica, L.: Integral functionals determined by their minima. Rend. Semin. Mat. Univ. Padova 76, 255–267 (1986) 3. García Trillos, N., Slepˇcev, D.: On the rate of convergence of empirical measures in ∞transportation distance. Canad. J. Math. 67, 1358–1383 (2015) 4. García Trillos, N., Slepˇcev, D.: Continuum limit of total variation on point clouds. Arch. Ration. Mech. Anal. 220, 193–241 (2016)

Chapter 8

Stochastic Homogenization

Abstract In this section we consider random energies of convolution type and prove that, under stationarity and ergodicity assumptions, the .-limit of such energies is almost surely a deterministic integral functional whose integrand can be characterized through an asymptotic formula. Keywords Stochastic homogenization · Statistically homogeneous random functions · Ergodicity Let .(X, σ, P) be a standard probability space with a measure-preserving ergodic dynamical system .τy , .y ∈ Rd . We recall that .{τy } is a collection of measurable invertible maps .τy : X → X such that – .τy+y  = τy ◦ τy  for all .y, y  ∈ Rd , .τ0 = Id, – .P(τy (A)) = P(A) for all .A ∈ σ and .y ∈ Rd , – .τ : X × Rd → X is measurable, where .Rd is equipped with the Borel .σ -algebra. The ergodicity of .τ means that for every .A ∈ σ such that .τy (A) = A for all .y ∈ Rd there holds either .P(A) = 0 or .P(A) = 1. Consider now a random function .f defined as a measurable map f : X × Rd × Rm → [0, +∞),

.

where .Rd and .Rm are equipped with the Borel .σ -algebra. We assume that there exist two positive constants .r0 and .c0 and four random functions .ρ1 (ω), ψ1 (ω) : Br0 → [0, +∞) and .ψ2 (ω), ρ2 (ω) : Rd → [0, +∞) such that the following conditions hold .P-almost surely: ψ1 (ω)(ξ )|z|p − ρ1 (ω)(ξ ) ≤ f(ω, ξ, z),

for a.e. ξ ∈ Br0 ,

(8.1)

f(ω, ξ, z) ≤ ψ2 (ω)(ξ )|z|p + ρ2 (ω)(ξ ),

for a.e. ξ ∈ Rd ,

(8.2)

.

.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_8

101

102

8 Stochastic Homogenization

ρ1 ∈ L1 (X × Br0 )

and

.

ψ1 (ω)(ξ ) ≥ c0 ,

for a.e. ξ ∈ Br0 ,

(8.3)

 .

C1 (·) :=

Rd

ψ2 (·)(ξ )|ξ |p + ρ2 (·)(ξ ) dξ ∈ L1 (X, P),

(8.4)

requesting additionally that  ψ2 (ω)(ξ ) dξ = +∞ then there exists c1 > 0 such that

if .

Br0

(8.5) ψ2 (ω)(ξ ) ≤ c1 ψ1 (ω)(ξ ) for a.e. ξ ∈ Br0 .

Letting .f (ω)(y, ξ, z) = f(τy ω, ξ, z), we introduce a family of stochastic nonp local energy functionals .Fε (ω) : Lloc (Rd ; Rm )×A(Rd ) → [0, +∞), .ε > 0, defined by  Fε (ω)(u, A) :=

 f (ω)

.

Rd

x

Aε (ξ )

ε

, ξ,

u(x + εξ ) − u(x)  dx dξ. ε

(8.6)

By construction the densities f are statistically homogeneous random functions of x that is f (ω)(x + y, ξ, z) = f (τy ω)(x, ξ, z),

.

for every ω ∈ X, x, y, ξ ∈ Rd , z ∈ Rm . (8.7)

In Theorem 8.2 below we prove a homogenization theorem for the functionals Fε . Our proof of a homogenization formula relies on a subadditive ergodic theorem, a result that we recall below preceded by the definition of multiparameter stationary stochastic processes.

.

Definition 8.1 (Subadditive Process) Let .V be the family of all finite subset of the lattice .Zd,+ := {0, 1, . . . }d . A real valued process . : V → L1 (X, P) is called a subadditive process if it satisfies the following conditions: (i) it is stationary, that is for any .j ∈ Zd,+ and any finite collection .{V1 , . . . , VN } ⊂ V the joint law of .{ (V1 + j ), . . . , (VN + j )} is the same as the joint law of .{ (V1 ), . . . , (VN )}; (ii) it is subadditive, that is . (V1 ∪ V2 ) ≤ (V1 ) + (V2 ) for any disjoint .V1 and .V2 in .V; (iii) there holds  1 . inf

({0, 1, . . . n}d )(ω) dP(ω) > −∞. n∈N X nd A proof of the following theorem can be found in [3, Theorem 1].

8 Stochastic Homogenization

103

Theorem 8.1 (Subadditive Ergodic Theorem) Let .B0 be a family of Borel subsets of .[0, 1]d such that .

sup{|∂B + Bδ | : B ∈ B0 } → 0,

as δ → 0.

Then, for every subadditive process . : V → L1 (X, P) there exists a real random variable .φ ∈ L1 (X, P) such that .

   1   sup  d ((N B) ∩ Zd,+ ) − |B|φ  : B ∈ B0 → 0 almost surely as N → + ∞. N (8.8)

The stochastic homogenization theorem is then stated as follows. This result extends [1, Theorem 6.1], where quadratic convolution energies with random coefficients are studied. Theorem 8.2 (Stochastic Homogenization Theorem) Let .f be a random function satisfying (8.1)–(8.5). Let .Fε be defined as in (8.6) with .f (ω)(y, ·, ·) = f(τy ω, ·, ·) a statistically homogeneous random function according to (8.7). Then for .P-almost every .ω ∈ X and for every .M ∈ Rm×d the limit  1 inf F1 (ω)(u, QR ) : u ∈ D1,M (QR ) , d R→+∞ R

fhom (ω)(M) = lim

.

(8.9)

where .D1,M (QR ) is defined by (5.19), exists and defines a quasiconvex function m×d → [0, +∞) satisfying .fhom (ω) : R c(ω)(|M|p − 1) ≤ fhom (M) ≤ C(ω)(|M|p + 1),

.

(8.10)

where .c(·), C(·) ∈ L1 (X, P). Moreover, for .P-almost every .ω ∈ X and for every reg () there holds .A ∈ A ⎧ ⎨ fhom (ω)(∇u(x))dx u ∈ W 1,p (A; Rm ) .- lim Fε (ω)(u, A) = F (ω)(u, A) := A ⎩ ε→0 +∞ otherwise. If, in addition, the dynamical system .τ is ergodic, then .fhom (ω)(·) is constant almost surely and satisfies fhom (ω)(M) ≡ fhom (M) .

1 R→+∞ R d



 inf F1 (ω)(u, QR ) : u ∈ D1,M (QR ) dP(ω).

:= lim

X T .Fε (ω)

(8.11)

Proof For any given .T > 0 let be the truncation functional of .Fε (ω) as defined in (5.3). By Theorem 5.1, for .P-almost every .ω ∈ X and for every sequence .εj → 0 there exists a subsequence (not relabelled) and a Carathéodory function

104

8 Stochastic Homogenization

f0T (ω) :  × Rm×d → [0, +∞), which is quasiconvex in the second variable, such that for every .A ∈ Areg () and .u ∈ W 1,p (A; Rm )

.

 p

(L )- lim

.

j →+∞

FεTj (ω)(u, A)

= F (ω)(u, A) := T

A

f0T (ω)(x, ∇u(x))dx.

Arguing as in the proof of Theorem 6.1 and using the characterization of nonhomogeneous quasiconvex functions by their minima (see [2, Theorem II]), for every .M ∈ Rm×d and for almost every .x0 ∈  we have f0T (ω)(x0 , M)

.

 1 1,p min F T (ω)(u, Qr (x0 )) : u − Mx ∈ W0 (Qr (x0 ); Rm ) d r→0 r  T 1 −1 1,M −1 inf F (ω)(v, R Q (r x )) : v ∈ D (R Q (r x )) = lim lim j 1 0 j 1 0 1 r→0 j →+∞ R d j

= lim

= lim lim

1

r→0 j →+∞ R d j

 inf F1T (ω)(v, Rj Q1 (r −1 x0 )) : v ∈ DT ,M (Rj Q1 (r −1 x0 )) ,

with .Rj = r/εj . Set for any .A ∈ A(Rd )  T T T ,M .H (ω)(M, A) := inf F1 (ω)(v, A) : v ∈ D (A) . T : X → [0, +∞) such that for any .x¯ ∈ Rd We now show that there exists .φM

.

lim

1 T T H (ω)(M, RQ1 (x)) ¯ = φM (ω). Rd





R→+∞

Set H˜ T (ω)(M, A) := inf

.

BT

 f (ω)(x, ξ, v(x + ξ ) − v(x)) dx dξ : v ∈ DT ,M (A) .

A

(8.12) Since |H˜ T (ω)(M, RQ1 (x)) ¯ − H T (ω)(M, RQ1 (x))| ¯ ≤ 2dC1 (ω)(|M|p + 1)T R d−1 ,

.

it suffices to show that .

1 ˜T T ¯ = φM (ω). H (ω)(M, RQ1 (x)) R→+∞ R d lim

(8.13)

For any .A ∈ V denote .QA = j ∈A Q1 (j ) and . T (M, A)(ω) = H˜ T (ω)(M, QA ). Note that . T (M, A) ∈ L1 (X, P) for any .A ∈ V by conditions (8.2) and (8.4), and that by (8.12), . T is subadditive, according to Definition 8.1 (ii). Moreover,

References

105

since f is statistically homogeneous, . T is stationary, according to Definition 8.1 (i). Condition (iii) of Definition 8.1 is trivially satisfied, since f takes values in T we deduce that there .[0, +∞). Hence, by applying Theorem 8.1 to the process .

T 1 exists .φM ∈ L (X, P) such that for every .N > 0 .

   H˜ T (ω)(M, RQ (x)) 1   T : |x| ≤ N = 0, lim sup  − φ (ω)  M R→+∞ Rd

which in particular yields (8.13). This on the one hand implies that .fωT (x0 , M) does not depend on the first variable and, on the other hand, that it does not depend on the subsequence .{εj }. Hence, the whole family .FεT (ω) .-converges to .F T (ω). Arguing as in the proof of Theorem 6.1 we get the same result for .Fε (ω). If .τ is ergodic, then T T are deterministic, so is .f (M) and (8.11) holds. .f (M) and .F   We complete this section by providing a typical example of a random density f = f(ω, ξ, z).

.

Example 8.1 (A Random Density) Let .Y be a Poisson point process in .Rd of intensity 1; that is, .Y is a point process such that for any bounded Borel set A in .Rd the number of points of .Y in A has a Poisson distribution with parameter .|A|, and for any N and any disjoint bounded Borel sets .A1 , . . . , AN the random variables .#(A1 ∩ Y), . . . , #(AN ∩ Y) are independent. Let .ϕ be a .C0∞ (Rd ) function such that .ϕ ≥ 0. We set  −1     f (ω)(x, ξ, z) = χB1 (ξ )(1 + |z|)p + 1 + ϕ(x − Yj ) ϕ(x − Yj )

.

j

j

× h(ξ )(1 + |z|)p ,

where .Y(ω) = ∞ j =1 Yj (ω), .p > 1, and .h(·) is a non-negative function that satisfies the following upper bound: h(ξ ) ≤

.

C (1 + |ξ |)d+p+δ

for some .δ > 0. In this example .f(ω, ξ, z) = f (ω)(0, ξ, z) and, by Remark 2.4, the growth assumptions are satisfied.

References 1. Braides, A., Piatnitski, A.: Homogenization of random convolution energies. J. Lond. Math Soc. (2) 104, 295–319 (2021) 2. Dal Maso, G., Modica, L.: Integral functionals determined by their minima. Rend. Semin. Mat. Univ. Padova 76, 255–267 (1986) 3. Krengel, U., Pyke, R.: Uniform pointwise ergodic theorems for classes of averaging sets and multiparameter subadditive processes. Stochastic Process. Appl. 26, 289–296 (1987)

Chapter 9

Application to Convex Gradient Flows

Abstract So far, we dealt with static problems by studying the .-convergence of families of functionals .{Fε }. Here, by taking advantage of the .-convergence results proved, we analyze some dynamical aspects by considering gradient flows for a convex family .{Fε }, and prove the stability of the gradient flows with respect to .convergence by applying the minimizing movements scheme along this family of functionals. Keywords Gradient flows of convex energies · Stability by .-convergence · Minimizing movements along sequences of functionals · Homogenized gradient flows · p-Laplacian evolution equation

9.1 The Minimizing-Movement Approach to Gradient Flows We first recall some definitions and relevant results to our end (see [1] and [5, Chapter 8]). Let .F : L2 (; Rm ) → [0, +∞] be a weakly lower semicontinuous proper (that is, not identically equal to .+∞) functional. We introduce a time-scale parameter .τ > 0 and we solve the sequence of minimum problems starting from the initial value .u0 ∈ L2 (; Rm ); that is, ⎧    1 ⎪ u − uτ 2 2 ⎨uτn ∈ arg min F (u) + n−1 L (;Rm ) , n ≥ 1 2τ . (9.1) u∈L2 (;Rm ) ⎪ ⎩uτ ≡ u . 0 0 By applying the direct method of Calculus of Variations we get that the solutions uτn exist for any .n ∈ N. The sequence .{uτn }n≥0 is called a discrete solution of the scheme (9.1) (implicit Euler scheme). A discrete solution extends to an interpolation curve .uτ : [0, +∞) → L2 (; Rm ) defined by

.

uτ (t) := uτn ,

.

t ∈ [(n − 1)τ, nτ ).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7_9

(9.2)

107

108

9 Application to Convex Gradient Flows

Definition 9.1 (Minimizing Movements) A curve .u : [0, +∞) → L2 (; Rm ) is called a minimizing movement for F from .u0 if .uτ , defined as in (9.1) and (9.2), up to subsequences, converge to u as .τ → 0 uniformly on compact sets of .[0, +∞). In this chapter we will deal with convex functionals F for which weak lower semicontinuity is equivalent to strong lower semicontinuity. Moreover, by the convexity of F , the functional .u → F (u) + cu − v2L2 (;Rm ) is strictly convex for every fixed .c > 0 and .v ∈ L2 (; Rm ); hence, the solutions .uτn are also unique for any .n ∈ N. Then there exists a unique minimizing movement for F from .u0 , belonging to .C 1/2 ([0, +∞); L2 (; Rm )) (see [5, Theorem 11.1]). If .F (u) < +∞ then a subgradient of F at u is a function .ϕ ∈ L2 (; Rm ) satisfying the following inequality F (v) ≥ F (u) +

ϕ(x), v(x) − u(x) dx ,

.

(9.3)



for every .v ∈ L2 (; Rm ). For each .u ∈ L2 (; Rm ) we denote by .∂F (u) the set of all subgradients of F at u. The subdifferential of F is the multivalued mapping .∂F which assigns the set .∂F (u) to each u. The domain of .∂F is given by the set m 2 .dom ∂F = {u ∈ L (; R ) | ∂F (u) = ∅}. If .u ∈ dom ∂F then there exists a unique element of .∂F (u) having minimal norm that is denoted by ∂ 0 F (u) := arg min ϕL2 (;Rm ) ,

(9.4)

.

ϕ∈∂F (u)

see for instance [1, Section 1.4]. Definition 9.2 ([1, Definition 1.3.2, Remark 1.3.3]) A locally absolutely continuous map .u : [0, +∞) → L2 (; Rm ) is called a curve of maximal slope for F if it satisfies the energy identity F (u(t1 )) − F (u(t2 )) =

.

1 2



t2

t1

u (s)2L2 (;Rm ) ds +

1 2



t2 t1

∂ 0 F (u(s))2L2 (;Rm ) ds, (9.5)

for any interval .[t1 , t2 ] ⊂ [0, +∞). 1,2 Note that u, as in Definition 9.2, satisfies (9.5) if and only if .u ∈ Wloc ([0, +∞); and it is solution to the gradient flow equation

m 2 .L (; R ))

u (t) = −∂ 0 F (u(t)),

.

for almost every t > 0,

(9.6)

see for instance [1, Corollary 1.4.2]. Lemma 9.1 Let .F : L2 (; Rm ) → [0, +∞] be a proper, convex and lowersemicontinuous functional. Then, for every .u0 ∈ L2 (; Rm ), with .F (u0 ) < +∞,

9.1 The Minimizing-Movement Approach to Gradient Flows

109

1,2 there exists a unique minimizing movement .u ∈ Wloc ([0, +∞); L2 (; Rm )) for F from .u0 which is the unique solution to the gradient flow (9.6) with initial condition .u(0) = u0 .

Proof We already observed that for such functional F there exists a unique minimizing movement .u ∈ C 1/2 ([0, +∞); L2 (; Rm )). By Ambrosio et al. [1, Theorem 2.3.3], we have that the minimizing movement is a curve of maximal slope,   which concludes the proof. Instead of a single functional, we now consider a family of functionals .Fε : L2 (; Rm ) → [0, +∞] for .ε > 0, that are proper, lower semicontinuous and convex, and .uε0 ∈ L2 (; Rm ) a given family of initial data. We apply the minimizing-movement scheme (9.1) with .Fε in place of F and, similarly, we get that there exists a unique discrete solution .{uτ,ε n }n≥0 and an interpolation curve τ,ε : [0, +∞) → L2 (; Rm ) as in (9.1) and (9.2), respectively, depending on .u the parameter .ε. Definition 9.3 (Minimizing Movements Along Families of Functionals) Consider .uε0 → u0 in .L2 (; Rm ) and let .{τε }ε>0 be a family of positive parameters such that .τε → 0 as .ε → 0. A curve .u : [0, +∞) → L2 (; Rm ) is called a minimizing movement along .{Fε }ε>0 from .uε0 at rate .τε if, up to subsequences, .uτε ,ε converges to u, as .ε → 0, on compact subsets of .[0, +∞). Note that, in general, the minimizing movements u may depend on the rate .τε (see e.g. [5, Example 8.2] and [4]). This does not occur for convex families of functionals, for which gradient flows are stable with respect to .-convergence, as stated in the following result (see [5, Theorem 11.2]). Theorem 9.1 (Stability of Minimizing Movements Along Convex Functionals) Let H be a Hilbert space and let .φε : H → [0, +∞] be a equi-coercive family of convex functionals .-converging to .φ and let .x0ε → x0 be such that .supε φε (x0ε ) < +∞. Suppose that for every .ε > 0 there exists a minimizing movement of .φε from ε .x . Then 0 (i) the family of minimizing movements of .φε from .x0ε converge to the minimizing movement of .φ from .x0 ; (ii) for every rate .τε the minimizing movement along the sequence .Fε from .x0ε coincides with the same minimizing movement of .φ from .x0 . The following theorem states the stability of the .-limit of convolution-type energies with respect to gradient flows; for technical reasons, we assume .p ≥ 2. Theorem 9.2 (Convergence of Gradient Flows) Let .p ≥ 2 and let .Fε be defined as in (2.1) with .fε convex in the last variable. Assume that (H0)–(H2) hold and that .Fε .(Lp )-converge to .F : Lp (; Rm ) → [0, +∞], as .ε → 0. Let .{uε0 } ⊂ Lp (; Rm ) be a given family of initial data such that .

sup Fε (uε0 ) < +∞, ε>0

uε0 → u0 in L2 (; Rm )

(9.7)

110

9 Application to Convex Gradient Flows

as .ε → 0. Let .u¯ and .uε be the minimizing movements for F from .u0 and for .Fε from ε .u , respectively. Then, for every .τε → 0 as .ε → 0, we have that .u ¯ : [0, +∞) → 0 W 1,p (; Rm ) is the unique minimizing movement along .{Fε }ε>0 from .uε0 at rate .τε and satisfies 1,2 u¯ ∈ C 0 ([0, +∞); Lp (; Rm )) ∩ Wloc ([0, +∞); L2 (; Rm )) .

.

(9.8)

Moreover, we have that .uε are solutions to the gradient flows for .Fε ; i.e.,

.

(uε ) (t) = −∂ 0 Fε (uε (t)) for almost every t > 0 uε (0) = uε0 ,

u¯ is solution to the gradient flow for F ; i.e.,

.

.

u (t) = −∂ 0 F (u(t)) for almost every t > 0 u(0) = u0

and .{uε } converges to .u¯ as follows .

lim uε (t) = u(t), ¯

ε→0 .

in Lp (; Rm ) for every t > 0,

1,2 weakly in Wloc ([0, +∞); L2 (; Rm )).

lim uε = u, ¯

ε→0

(9.9) (9.10)

Proof We extend the functionals .Fε and F to .L2 (; Rm ) by setting .Fε (u) = +∞ and .F (u) = +∞ for every .u ∈ L2 (; Rm ) \ Lp (; Rm ). Note that, by the assumptions on .fε we have that the functionals .Fε are convex and lower semicontinuous in .L2 (; Rm ) and the same holds for F since it is a .-limit. By Corollary 4.2 every .{uε }ε>0 ⊂ L2 (; Rm ) satisfying .

sup(uε L2 (;Rm ) + Fε (uε )) < +∞

(9.11)

ε>0

is precompact in .Lp (; Rm ) and therefore in .L2 (; Rm ). Moreover, every limit is in 1,p (; Rm ). Hence, we get that .{F } is .L2 -equicoercive and .(L2 )-converges to .W ε F , as .ε → 0. Note that at fixed .ε > 0 the functionals .Fε are lower semicontinuous and coercive with respect to the weak .L2 -topology, which is sufficient to deduce the existence of the corresponding minimizing movements. We can then apply Theorem 9.1. Thus, there exists a unique minimizing movement along .{Fε } from ε .u at rate .τε which coincides with .u, ¯ and .{uε } converge uniformly to .u¯ on compact 0 1,2 subset of .[0, +∞). By Lemma 9.1 we have that .u¯ ∈ Wloc ([0, +∞); L2 (, Rm )) m 1,p and .u(t) ¯ ∈ W (; R ) for every .t ∈ [0, +∞). Moreover, the decreasing behavior of .Fε along .uε , (9.5) and (9.7) give (9.10). It remains to prove (9.9) and the

9.2 Homogenized Flows for Convex Energies

111

continuity of .u¯ with respect to the .Lp -topology. By the monotonicity of .F (u(t)) ¯ and (3.18), we infer that ∇ u(t) ¯ Lp (;Rm ) ≤ C(F (u0 ) + 1)

.

for every .t ≥ 0. Therefore, by the strong .L2 -continuity of .u, ¯ we get that .u(s) ¯ → u(t) ¯ weakly in .W 1,p (; Rm ) as .s → t, which, in particular, gives (9.8). Finally, since ε .{u (t)} satisfies (9.11) for every .t > 0, then (9.9) is implied by (9.10).  

9.2 Homogenized Flows for Convex Energies In this section we apply the results obtained in Theorem 9.2 to the periodichomogenization case. We describe the homogenized gradient flow in (9.14) and (9.15) under Neumann and Dirichlet boundary conditions, respectively. Theorem 9.3 (Homogenized Gradient Flows) Let .p ≥ 2, let .Fε be defined as in (2.1) with .fε convex and .C 2 in the last variable and let (6.1)–(6.6) hold. Let ε m p .{u } ⊂ L (; R ) be a given family of initial data satisfying (9.7). Then, 0 ∂ Fε (u)(x) =

.

0



1 εd+1

 y x − y u(x) − u(y) , , dy ε ε ε   x y − x u(y) − u(x) 1 , , dy , − d+1 ∇z f ε ε ε ε  ∇z f

(9.12)

Theorem 9.2 holds and the family of solutions .{uε } to the gradient flows

.

∂t uε = −∂ 0 Fε (u), in (0, +∞) × , uε (0) = uε0 ,

(9.13)

1,2 converges weakly in .Wloc ([0, +∞); L2 (; Rm )) to the solution to the gradient flow

⎧ ⎪ ⎪ ⎨∂t u = Div(∇fhom (∇u)), in (0, +∞) × , . ∇fhom (∇u) ν = 0, on ∂, ⎪ ⎪ ⎩u(0) = u , 0

with .fhom defined by (6.18).

(9.14)

112

9 Application to Convex Gradient Flows

Proof By Theorem 6.2 we have that ⎧ ⎨ fhom (∇u(x))dx p .(L )- lim Fε (u) = F (u) :=  ⎩ ε→0 +∞

u ∈ W 1,p (; Rm ) otherwise .

For every .u, v ∈ Lp (; Rm ), arguing as in the proof of Proposition 5.2, the first variation of .Fε is given by .

 δFε (x), v(x) dx  δu  x y − x u(y) − u(x) v(y) − v(x)  1 , , , dy dx. ∇z f = d ε   ε ε ε ε

Thus (9.12) follows by the symmetric roles of x and y. For the limit functional F we have that  1,p .dom ∂F = u ∈ W (; Rm ) : Div(∇fhom (∇u)) ∈ L2 (; Rm ),  ∇fhom (∇u)ν = 0 on ∂ and, for every .u ∈ dom ∂F , .∂F (u) is single-valued and there holds ∂ 0 F (u) = −Div(∇fhom (∇u)) .

.

Hence the thesis follows by applying Theorem 9.2.

 

Reasoning as in the proof of Theorem 9.3, by Proposition 5.3 we obtain an analogue result for the homogenized gradient flow with Dirichlet boundary r,g conditions. Note that, the .Lp -equicoerciveness of the family .{Fε }ε>0 , defined in (5.20) and satisfying (H0), has been already shown in the proof of Proposition 5.4. 1,p

Theorem 9.4 For fixed .p ≥ 2, .g ∈ Wloc (Rd ; Rm ), and .r > 0, let .Drε,g () be defined by (5.19). Let .Fε and .{uε0 } ⊂ Drε,g () satisfy the hypotheses of Theorem 9.3 r,g and let .Fε be defined by (5.20). Then, Theorem 9.2 holds and the family of solutions .{uε } to the gradient flows ⎧ ε 0 ε ⎪ ⎪ ⎨∂t u = −∂ Fε (u ), in (0, +∞) × , . uε = g, in {x ∈  | dist(x, c ) < rε}, ⎪ ⎪ ⎩uε (0) = uε , 0

9.2 Homogenized Flows for Convex Energies

113

1,2 with .∂ 0 Fε defined by (9.12), converges weakly in .Wloc ([0, +∞); L2 (; Rm )) to the solution to the gradient flow

⎧ ⎪ ⎪ ⎨∂t u = Div(∇fhom (∇u)), in (0, +∞) × , . u = g, on ∂, ⎪ ⎪ ⎩u(0) = u ,

(9.15)

0

where .fhom is defined by (6.18). An interesting application of Theorem 9.3 is provided by the following example, which is then made more specific in Remark 9.1. For similar results related to pLaplacian evolutions see also [2, 3]. Example 9.1 (Evolution of Purely-Convolution Operators) Let .a ∈ Cc∞ (B1 ) be a non-negative function. We consider functionals .Fε as in the statement of Theorem 9.3 with .f (y, ξ, z) = a(ξ )|z|p /p. For every .u ∈ dom ∂Fε , formula (9.12) reads as    y − x x −y 1 0 + a |u(y) − u(x)|p−2 (u(y) − u(x)) dy a .∂ Fε (u) = − εd+p  ε ε and, by Theorem 9.3, the family of solutions .{uε } to the gradient flows for .Fε converges, as in (9.10), to the solution .u¯ to the gradient flow for the functional ⎧1 ⎨ a(ξ ) |∇u(x)ξ |p dx dξ p B1 .F (u) :=  ⎩ +∞

u ∈ W 1,p (; Rm ) otherwise,

that is given by the following system of equations ⎧  ⎪ ∂t u = Div a(ξ ) |∇u ξ |p−2 (Du ξ ⊗ ξ ) dξ in (0, +∞) ×  ⎪ ⎪ ⎪ B1 ⎨ . a(ξ ) |∇u ξ |p−2 (∇u ξ ⊗ ξ ) ν dξ = 0 on ∂ ⎪ ⎪ B ⎪ 1 ⎪ ⎩ u(0) = u0 .

(9.16)

We now consider the case .p = 2 and .m = 1. By Example 6.1, the .-limit is given by the quadratic form ⎧ ⎨ 1 A ∇u(x), ∇u(x) dx hom 2  .F (u) := ⎩ +∞

u ∈ W 1,2 () otherwise

114

9 Application to Convex Gradient Flows

where .Ahom satisfies (Ahom )i,j =

a(ξ )ξi ξj dξ,

.

for every 1 ≤ i, j ≤ d .

B1

Hence, the homogenized gradient flow takes the form ⎧   ⎪ ⎪ ⎨∂t u = div Ahom ∇u . Ahom ∇u, ν = 0 ⎪ ⎪ ⎩u(0) = u .

in (0, +∞) ×  on ∂

0

Remark 9.1 (Approximation of p-Laplacian Evolution Equation) In the scalar case m = 1, if we assume the kernel a to be also radially symmetric then .F (u) = cp  |∇u(x)|p dx where

.

cp :=

a(ξ )|ξ1 |p dξ ,

.

B1

as shown in Example 3.1. Thus, formula (9.16) reduces to ⎧ ⎪ ⎪ ⎨∂t u = cp p u in (0, +∞) ×  . ∇u, ν = 0 on ∂ ⎪ ⎪ ⎩u(0) = u . 0

References 1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures Math. ETH Zürich. Birkhäuser Verlag, Basel (2008) 2. Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: A nonlocal p-Laplacian evolution equation with Neumann boundary conditions. J. Math. Pures Appl. (9) 90, 201–227 (2008) 3. Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: A nonlocal p-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal. 40, 1815–1851 (2009) 4. Ansini, N., Braides, A., Zimmer, J.: Minimizing movements for oscillating energies: the critical regime. Proc. R. Soc. Edinb. Sect. A 149, 719–737 (2019) 5. Braides, A.: Local Minimization, Variational Evolution and -convergence, volume 2094 of Lecture Notes in Math. Springer, Cham (2014)

Index

A Asymptotic homogenization formula, 60, 84

B Boundary-value problem, 51

C Cell-problem formula, 67, 85 Compactness theorem, 34 Convex envelope, 71 Convolution functional, 2, 11 Curve of maximal slope, 108

D Discrete energies, 3 Discrete solution, 107

E Empirical measure, 96 Energy identity, 108 Ergodic dynamical system, 101 Euler-Lagrange equations, 54 Evolution of convolution operators, 113 Extension from a Lipschitz domain, 78 Extension theorem, 30, 78

F Fundamental estimate, 44

G Gamma-limit, 11 Gateaux differentiability, 55 Gradient flow, 108 Growth condition, polynomial, 12

H Homogenization theorem, 60 Homogenization theorem, convex –, 67 Homogenized flow, 111 Homogenized gradient flow, 111

I Implicit Euler scheme, 107 Inner regularity, 48

K Kernels of polynomial decay, 13 Kernel with changing sign, 15

L Lipschitz boundary, 78 Locality assumption, 21 Localization, 11, 17 Long-range interactions, 32

M Minimizing movement, 108, 109 Mumford-Shah energy, 3

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Alicandro et al., A Variational Theory of Convolution-Type Functionals, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-99-0685-7

115

116 N Non-local functional, 10 Non-local problems, 3

O Origin-symmetric kernel, 11

P Perforated domain, 80 Peridynamics, 1, 54 Periodic Lipschitz domain, 78 p-Laplacian evolution equation, 114 Poincaré inequality, 37 Poincaré-Wirtinger inequality, 38 Point cloud, 5, 96 Polynomial growth condition, 12, 13

Q Quadratic forms, 69 Quasi-convex envelope, 71 Quasiconvex function, 60

Index R Radially-symmetric kernel, 20 Random density, 105 Random function, 101 Relaxation, 69 Relaxation formula, 70

S Short-range interactions, 32 Statistically homogeneous function, 102 Subadditive ergodic theorem, 103 Subadditive process, 102 Subdifferential, 108 Subgradient, 108

T Translated kernel, 14 Transportation map, 92 Truncated functional, 43