Energetic Relaxation to Structured Deformations: A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics 9811987998, 9789811987991

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SpringerBriefs on PDEs and Data Science José Matias · Marco Morandotti · David R. Owen

Energetic Relaxation to Structured Deformations A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics

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

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José Matias • Marco Morandotti • David R. Owen

Energetic Relaxation to Structured Deformations A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics

José Matias Instituto Superior Técnico University of Lisbon Lisbon, Portugal

Marco Morandotti Department of Mathematical Sciences Politecnico di Torino Turin, Italy

David R. Owen Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA, USA

ISSN 2731-7595 ISSN 2731-7609 (electronic) SpringerBriefs on PDEs and Data Science ISBN 978-981-19-8799-1 ISBN 978-981-19-8800-4 (eBook) https://doi.org/10.1007/978-981-19-8800-4 Mathematics Subject Classification: 49J45, 74A60, 74G65, 74Q05, 74KXX © 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

Behind this book is a tale of many cities, principally Lisbon, Pittsburgh, Trieste, Turin, Salerno, Prague, and Udine, whose academies and broader cultures inspired the writing process and much of the content herein and to which, in order that these precious resources not be forgotten or taken for granted, we dedicate this volume. Behind this book is a tale of precious people, Diane, Gisela, and Mikel, whose closeness and unconditional support sustained our spirits. To them, too, we dedicate this volume.

Preface

The original idea for this book arose during the final stages of the first author’s habilitation defense, at which time the members of the jury suggested that it would be useful to collect the most recent mathematical treatment of mechanical problems in the context of structured deformations in a single volume, including both the theory and applications. Broadly speaking, structured deformations provide a mathematical framework to capture the effects at the macroscopic level of geometrical changes at submacroscopic levels. The availability of this framework leads naturally to the enrichment of the energies and force systems that underlie variational and field-theoretic descriptions of important physical phenomena without having to commit at the outset to any of the existing prototypical mechanical theories, such as elasticity or plasticity. More specifically, in classical descriptions of geometrical changes at the macroscale, a single field g and its gradient .∇g suffice to characterize the deformations of a continuous body (possibly together with higher order gradients of g). Structured deformations provide an additional geometrical field G of the same tensorial character as .∇g in such a way that G captures the contributions at the macrolevel of smooth submacroscopic changes, while the difference .∇g − G captures the effects of submacroscopic disarrangements, that is slips and separations that occur at the submacroscopic level. In our setting, the mathematical description of the energy of a body and the force systems acting on the body contains not only .∇g but also G or the difference .∇g − G. Thus it is the task of variational arguments to provide minimization problems sufficient to determine not only g but also G. Through this additional field, some of the standard fields that are typical in elasticity or plasticity theories can be recovered, but are not essential to our broader development. Moreover, by introducing a third object .Γ having the tensorial character of .∇ 2 g and .∇G, problems where gradient disarrangements account for the energetics of deformations including bending can be addressed. A pair .(g, G) will be called a first-order structured deformation, whereas a triple .(g, G, Γ ) will be called a second-order structured deformation. Central to the task of providing minimization problems in the context of firstorder structured deformations is the task of assigning an energy .I (g, G) to each vii

viii

Preface

first-order structured deformation .(g, G). In the past 25 years, the theory of energetic relaxation to structured deformations has emerged as a mathematically rich and broadly applicable setting for providing such an assignment, one that can be extended to cover second-order structured deformations as well. It is our intention to provide the reader with a concise presentation of the mathematical theory of energetic relaxation to structured deformations that allows one to arrive at variational problems in the context of structured deformations and to present instances of such variational problems available in the recent literature. We believe that gathering together these elements will afford both an introduction to the subject and a sampling of current topics of research. We have also written, at the end of the book, a chapter devoted to an outlook for future work that is relevant to both mathematical and mechanical points of view. The structure of the book reflects which of the two types of structured deformations is to be considered. Thus, after we introduce the mathematical preliminaries in Chap. 2, Chap. 3 contains the theory of energetic relaxation to first-order structured deformations, that is with objects of the type .(g, G) that only require consideration on the first derivatives of elements of the associated spaces of deformations. By contrast, Chap. 4 contains the theory of energetic relaxation to second-order structured deformations: here we enrich the theory by introducing the additional field .Γ in such a way that the resulting triples .(g, G, Γ ) require consideration of derivatives up to order two of elements of the associated spaces of deformations. Finally, Chap. 5 provides topics for further research relevant to both Chaps. 3 and 4. We believe this book will be accessible to both researchers and graduate students with a background in applied mathematics or engineering, who are knowledgeable about variational methods and continuum mechanics. Lisbon, Portugal Turin, Italy Pittsburgh, PA, USA October 27, 2022

José Matias Marco Morandotti David R. Owen

Acknowledgements

The authors warmly thank the hospitality of the Department of Mathematical Sciences and of the Center for Nonlinear Analysis at Carnegie Mellon University in Pittsburgh, the Departamento de Matemática of Instituto Superior Técnico in Lisbon, and the Dipartimento di Scienze Matematiche “G. L. Lagrange” of Politecnico di Torino, together with the hospitality of DeustoTech, where this book was written. The authors acknowledge financial support from: – CAMGSD, IST-ID projects UIDB/04459/2020 and UIDP/04459/2020, from FCT (Portuguese foundation for science and technology) – Dipartimenti di Eccellenza 2018–2022 (E11G18000350001), from MIUR (Italian Ministry for University and Research) – GNAMPA project 2019 Analysis and optimisation of thin structures – Starting grant per giovani ricercatori (53_RIL18MORMAR), from Politecnico di Torino – PRIN 2020 Mathematics for Industry 4.0 (2020F3NCPX), from the Italian Ministry of University and Research MM is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM) and of the Integrated Additive Manufacturing (IAM) group at Politecnico di Torino.

ix

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 First-Order Structured Deformations: L∞ -Theory .. . . . . . . . . . . . . . . . . . . 1.1.1 Examples of First-Order Structured Deformations.. . . . . . . . . . . 1.1.2 Energetics: The L∞ -Theory and Beyond ... . . . . . . . . . . . . . . . . . . . 1.2 Second-Order Structured Deformations: L∞ -Theory .. . . . . . . . . . . . . . . . 1.2.1 Examples of Second-Order Structured Deformations.. . . . . . . . 1.2.2 Energetics: The L∞ -Theory and Beyond ... . . . . . . . . . . . . . . . . . . . 1.3 Scope of the Present Contribution .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 3 5 6 7 11 12 13

2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Measure Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Function Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Γ -Convergence and Relaxation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 The Global Method for Relaxation .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

15 15 18 21 23 25

3 Energetic Relaxation to First-Order Structured Deformations.. . . . . . . . 3.1 Spaces of First-Order Structured Deformations, Approximation Theorems, and Representation of Relaxed Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Relaxation in SBV . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Relaxation in BV . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Other Approximation Results . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Relaxation of Purely Interfacial Energies... . . . . . . . . . . . . . . . . . . . 3.2.2 Other Special Settings for Relaxation . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 Dimension Reduction in the Context of Structured Deformations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.4 Optimal Design of Fractured Media . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.5 Relaxation of Non-local Energies .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

27

27 28 42 51 53 54 59 64 77 81

xi

xii

Contents

3.2.6 Periodic Homogenization of Structured Deformations . . . . . . . 86 3.2.7 Hierarchical First-Order Structured Deformations .. . . . . . . . . . . 97 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 4 Energetic Relaxation to Second-Order Structured Deformations .. . . . . 4.1 Spaces of Second-Order Structured Deformations, Approximation Theorems, and Representation of Relaxed Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Relaxation in SBV 2 . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Relaxation in SBH . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Outlook for Applications . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

107

5 Outlook for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Microdegree and Micromixing . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Multiscale Differential Geometry . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 Toward a Multiscale Enrichment of Classical Hypersurfaces in RN+1 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Structured Hypersurfaces in RN+1 in an L∞ -Setting .. . . . . . . . 5.2.3 L∞ -Approximation of Structured Insertions .. . . . . . . . . . . . . . . . . 5.2.4 L∞ -Approximation of Other Structured Hypersurfaces .. . . . . 5.2.5 SBV 2 - and SBH -Approximation of Structured Hypersurfaces in RN+1 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

129 129 136

107 109 120 127 128

136 140 144 148 150 152

Chapter 1

Introduction

1.1 First-Order Structured Deformations: L∞ -Theory In classical theories of the mechanics of continua, the macroscopically observable changes in geometry of a continuous body in N-dimensional physical space are identified with smooth, injective mappings g from a reference region Ω ⊂ RN into RN , typically with N ∈ {1, 2, 3}. The mechanical response at a point x in Ω to such a “classical” deformation g of the body depends upon the nature of the material composing the body. Elastic bodies provide the simplest example of mechanical response: the energy stored per unit volume at x depends only on the N × N matrix ∇g(x), the classical gradient at x of the smooth deformation g of Ω. Among the variety of models of material behavior that describe departures from elastic response, many incorporate the effects of geometrical changes at submacroscopic levels, i.e., changes that are not observable with the naked eye. In this spirit, Del Piero and Owen [4] envisioned a way to account for the effects at the macroscopic level of not only smooth submacroscopic geometrical changes, but also the effects of non-smooth geometrical changes (disarrangements) at both submacroscopic and at macroscopic levels. Their multiscale geometrical theory of first-order structured deformations provides deformations of the form (κ, g, G), in which κ is a surface-like subset of Ω such that Ω \ κ is an open, “piecewise fit” region [4, Definition 3.1], g : Ω \ κ → RN is a smooth, injective mapping, and in which G : Ω \ κ → RN×N is a continuous mapping with values in the set of N × N real matrices RN×N . In cases where g does not extend to a continuous mapping on all of Ω, the mapping g captures discontinuities at the macroscopic level such as the opening of cracks in a body, and the set κ plays the role of the crack site. Each structured deformation (κ, g, G) is required to satisfy the following accommodation inequality (see [4, Definition 5.1, (Std 3)]): there exists C > 0 such that C < det G(x)  det ∇g(x)

.

for every x ∈ Ω \ κ.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. Matias et al., Energetic Relaxation to Structured Deformations, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-19-8800-4_1

(1.1)

1

2

1 Introduction

The accommodation inequality is intended to rule out the interpenetration of matter by requiring that volume changes associated with smooth submacroscopic geometrical changes cannot exceed volume changes associated with smooth macroscopic geometrical changes, and the following considerations of an important subclass of structured deformations validate this intention. Del Piero and Owen introduced the class of simple deformations (κ, g, ∇g), i.e., the structured deformations for which G = ∇g. Specifying a simple deformation thus requires only the specification of the pair (κ, g), so that simple deformations differ from the classical deformations of continuum mechanics in that simple deformations admit discontinuous changes in geometry such as relative sliding or separation of two parts of the body. Consequently, simple deformations capture the geometrical changes associated with fracture at the macroscopic level. The importance of simple deformations within the class of structured deformations emerges through the following approximation theorem [4, Theorem 5.8]: for every first-order structured deformation (κ, g, G) there exists a sequence {(κn , un , ∇un )} of simple deformations such that .

lim un = g,

n→∞

lim ∇un = G,

n→∞

(1.2)

where both limits are taken in the sense of L∞ , i.e., essentially uniform convergence. (The sequence {(κn , un , ∇un )} can be chosen so that, in addition to (1.2), the sets κn converge to the set κ in a precise sense, an issue that need not concern us further here.) Because of the second limit in (1.2), G is a limit of the matrix fields ∇un describing smooth geometrical changes of pieces of the body. These pieces of the body generally become smaller and more numerous as the index n increases. Accordingly, G captures the contributions at the macrolevel of smooth submacroscopic geometrical changes and is called the (first-order) deformation without disarrangements for the structured deformation (κ, g, G). In this light, the inequality det G(x)  det ∇g(x) in (1.1) indeed means that volume changes associated with smooth submacroscopic geometrical changes cannot exceed the smooth macroscopic volume changes associated with g. The role of the deformation without disarrangements G for a structured deformation (κ, g, G) is complemented by the matrix-valued field M := ∇g − G. On the one hand, the approximation theorem allows us to write M = ∇ lim un − lim ∇un

.

n→∞

n→∞

for each sequence {(κn , un , ∇un )} of simple deformations as in (1.2), so that M captures the degree to which the operations ∇ and limn→∞ fail to commute. On the other hand, as was established in [4, Section 6], [5, 16], the matrix M measures the contributions to the macroscopic deformation g of the jump discontinuities [un ] of the approximating simple deformations (κn , un , ∇un ) in the limit as n

1.1 First-Order Structured Deformations: L∞ -Theory

3

tends to infinity. Consequently, M is called the (first-order) deformation due to disarrangements for (κ, g, G), and the formula ∇g = G + M

.

(1.3)

emerges as an additive decomposition of the macroscopic deformation gradient ∇g into the deformation without disarrangements G and the deformation due to disarrangements M. The first-order Taylor approximation, the additive decomposition formula (1.3), and the second relation in (1.2) yield a refined approximation for the translations in the deformed body g(Ω), namely g(y) − g(x) = G(x)(y − x) + M(x)(y − x) + o(y − x),

.

(1.4)

in which the translation without disarrangements G(x)(y − x), arising from limits of gradients ∇un , and the translation due to disarrangements M(x)(y − x), arising from jumps in un , can be identified. More details about the identification relations determining G and M can be found in [5, Section 4].

1.1.1 Examples of First-Order Structured Deformations We wish here to indicate the ability of first-order structured deformations (κ, g, G) to capture a range of multiscale geometrical changes and to illustrate how the approximating deformations arising in the approximation theorem provide a submacroscopic view of geometrical changes. To do so we restrict attention to the case of “submacroscopic rearrangements”, i.e., G = I , with I the identity matrix, and we refer the reader to [4, Sections 4 and 6] for a wider range of examples. (Note that for N = 1, the 1 × 1 matrix I is identified with the number 1.) It is useful to note using (1.3) that the macroscopic displacement field, d : x → g(x) − x, in the case of submacroscopic rearrangements has gradient ∇d = ∇g − I = ∇g − G = M, i.e., the displacement gradient ∇d, measuring changes in geometry relative to initial geometry, arises entirely from submacroscopic disarrangements. Example 1.1 (Stretching via Submacroscopic Void Formation) We put N = 1, Ω = (0, 1), κ = Ø, g(x) = 2x, and G(x) = 1 for each x ∈ Ω, so that g stretches the interval (0, 1) uniformly into the interval (0, 2). Therefore, ∇g(x) = 2 and ∇d = M(x) = G(x) = 1 for every x ∈ Ω. Consequently, in this case, the increase in length of the region Ω caused by (g, G) is due to disarrangements, alone. To further substantiate this assertion, we note that the following “broken ramp sequence” [4, (3.8)] of simple deformations (κn , un , ∇un ) satisfies the convergence conditions in (1.2) for (κ, g, G), and (κn , un , ∇un ) breaks the interval (0, 1) into n − 1 congruent subintervals, each of which is rigidly displaced from its adjacent

4

1 Introduction

(d)

(c)

(b)

(a)

Fig. 1.1 The first four steps of the approximating sequence un . The graph of the function g is in black, whereas the graphs of the functions u1 , . . . , u4 are in blue. The dots on the x-axis form the discontinuity sets of the un ’s, that is, the sets κn

neighbor(s) by an amount 1/n, thus creating n − 1 “voids” whose individual lengths 1/n total (n − 1)/n (Fig. 1.1): k k+1 k 1. If μ : E → [0, +∞], it is called a positive measure. Given a measure μ on (X, E ) and a set E ∈ E , we define the restriction of μ to E as μ E(F ) := μ(E ∩ F ) for every F ∈ E . Given λ, μ two measures on (X, E ), we say that λ is absolutely continuous with respect to μ, and write λ  μ, if μ(E) = 0 ⇒ |λ|(E) = 0 for all E ∈ E , where |λ| denotes the total variation of the measure λ, defined as |λ|(E) := sup

 ∞

.

j =1

|λ(Ej )| : Ej ∈ E are pairwise disjoint and E =

∞ 

 Ej . .

j =1

If λ, μ are two positive measures on (X, E ), they are called mutually singular, and we write λ ⊥ μ, if there exists E ∈ E such that μ(E) = λ(E \ E) = 0. General measures λ, μ : E → R are called mutually singular if their total variations |λ| and |μ| are. We will be mostly concerned with Borel measures, i.e., the case in which X is a bounded, connected, open subset Ω of RN and E is the Borel σ -algebra generated by the open subsets of Ω, which we denote by B(Ω). If a Borel measure is finite on compact sets, it is called a Radon measure. We denote the set of finite R -valued Radon measures on Ω by M (Ω; R ). As a consequence of the Riesz Representation Theorem, the space M (Ω; R ) can  viewed as the dual of C0 (Ω; R ) under the pairing μ, ϕ :=  be i=1 Ω ϕi dμi . The dual norm is given by the total variation of μ, which, in © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. Matias et al., Energetic Relaxation to Structured Deformations, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-19-8800-4_2

15

16

2 Mathematical Preliminaries

this case, can be characterized as follows: for every U ∈ A (Ω), |μ|(U ) := sup



 ϕi dμi : ϕ ∈ Cc (U ; R ), ϕL∞  1 .

.

Ω

i=1

We will be concerned with sequences of measures, so that it is useful to introduce appropriate notions of convergence. A sequence {μn } ⊂ M (Ω; R ) weakly* ∗ converges to μ ∈ M (Ω; R ), μn  μ if



.

lim

n→∞ Ω

ϕ dμn =

for all ϕ ∈ C0 (Ω; R ).

ϕ dμ, Ω

A weak* compactness criterion for finite Radon measures holds: a sequence {μn } ⊂ M (Ω; R ) such that supn {|μn |(Ω)} < +∞ has a weakly* converging subsequence. The map μ → |μ|(Ω) is lower semicontinuous with respect to the weak* convergence. Given a measure μ ∈ M (Ω; R ) the Radon-Nikodým Theorem grants that there exists a unique pair of measures μa and μs , such that μa  L N , μs ⊥ L N and μ = μa + μs . Moreover, there exists a unique function a ∈ L1 (Ω; R ), called the density of μ with respect to L N and denoted by dμ/dL N , such that μa = aL N . For functionals defined on measures, (semi)continuity results go under the name of Reshetnyak Theorems. For every U ∈ B(Ω) we define the measure

μ(U ) :=



.

1 + |a(x)|2 dx + |μs |(U ).

U

Given a sequence of measures μn = an L N + μsn in M (Ω; R ) and μ = aL N + μs ∈ M (Ω; R ), we say that μn ·-strictly converges to μ if ∗

μn  μ in M (Ω; R )

.

and μn (Ω) → μ(Ω).

We say that a function Φ ∈ C(R ) has at most linear growth at infinity if there exists a constant C > 0 such that |Φ(ξ )|  C(1 + |ξ |) for all ξ ∈ R . We say that a function Φ : Ω × R → [0, +∞) belongs to the class E(Ω × R ) if .

lim

x →x ξ →ξ t →+∞

Φ(x , tξ ) t

exists for every x ∈ Ω and every ξ ∈ S −1 .

(2.1)

Given μ = aL N + μs ∈ M (Ω; R ) and Φ : Ω × R → [0, +∞) continuous, let

 dμs (x) d|μs |(x), Φ ∞ x, d|μs | Ω



I (μ) :=

Φ(x, a(x)) dx +

.

Ω

2.1 Measure Theory

17

where Φ ∞ is the recession function of Φ at infinity, defined by Φ ∞ (x, ξ ) := lim sup

.

x →x ξ →ξ t →+∞

Φ(x , tξ ) t

(2.2)

for every x ∈ Ω and ξ ∈ S −1 , and extended to R by positive 1-homogeneity. Notice that if Φ ∈ E(Ω × R ), then by (2.1) Φ ∞ is a limit, so that Φ ∞ (x, ξ ) = lim

.

x →x ξ →ξ t →+∞

Φ(x , tξ ) t

(2.3)

for every x ∈ Ω and ξ ∈ S −1 and extended to R by positive 1-homogeneity. We refer the reader to [23, Section 2.4] for a detailed description of the class E(Ω ×R ). We collect in the following statement the Reshetnyak [27] upper semicontinuity and continuity Theorems. Theorem 2.1 (Reshetnyak) Let μn = an L N + μsn be a sequence in M (Ω; R ) and let μ = aL N + μs ∈ M (Ω; R ) be such that μn ·-strictly converges to μ. (i) (upper semicontinuity, [5, Corollary 2.11]) If Φ ∈ C(Ω ×R ) has linear growth at infinity, then Φ ∞ is given by (2.2) and I (μ)  lim supn→∞ I (μn ). (ii) (continuity, [24, Theorem 4]) If Φ ∈ E(Ω × R ), then Φ ∞ is given by (2.3) and I (μn ) → I (μ), as n → ∞. We recall a by now classical result of Fonseca and Malý [22], which refines the De Giorgi–Letta criterion [17] to establish sufficient conditions under which a functional is the restriction to open sets of a bounded Radon measure. Theorem 2.2 (Fonseca–Maly [22]) Let X be a locally compact Hausdorff space, let Π : A (X) → [0, +∞] be a set function, and let μ be a finite Radon measure on X satisfying (i) for every U, V , Z ∈ A (X) such that U ⊂⊂ V ⊂⊂ Z, the following nested subadditivity property holds: Π(Z)  Π(V ) + Π(Z \ U ); (ii) for every U ∈ A (X) and for every ε > 0 there exists Uε ∈ A (X) such that Uε ⊂⊂ U and Π(U \ Uε )  ε; (iii) Π(X)  μ(X); (iv) for every U ∈ A (X), there holds Π(U )  μ(U ). Then Π is the restriction of the finite Radon measure μ to the open subsets of X.

18

2 Mathematical Preliminaries

2.2 Function Spaces We assume that the reader is familiar with Lebesgue and Sobolev spaces. We say that u ∈ L1 (Ω; Rd ) is a function of bounded variation, and we write u ∈ BV (Ω; Rd ), if its distributional derivative Du ∈ M (Ω; Rd×N ). The space BV (Ω; Rd ) is a Banach space when endowed with the norm uBV (Ω;Rd ) := uL1 (Ω;Rd ) + |Du|(Ω).

.

(2.4)

It turns out that convergence with respect to this norm is too strong for practical applications, so that it is customary to consider weak* convergence in BV , which has the advantage of being the appropriate notion for compactness properties (see [4, page 124]). We say that a sequence {un } ⊂ BV (Ω; Rd ) weakly* converges to a ∗ function u ∈ BV (Ω; Rd ), in symbols un  u, if un → u

in L1 (Ω; Rd )

.

and

∗

Dun  Du

in M (Ω; Rd×N ).

The Radon-Nikodým Theorem applied to Du ∈ M (Ω; Rd×N ) grants the existence and uniqueness of two mutually singular measures D a u and D s u such that Du = D a u + D s u. We denote dDu/dL N by ∇u, so that Du = ∇u L N + D s u.

(2.5)

.

The measure D s u can be further split into the sum of two contributions, D j u measuring the discontinuities of u, and D c u measuring the Cantor-like behavior of Du. In particular, denoting by Su the set of points x ∈ Ω for which there exist two vectors a, b ∈ Rd and a unit vector ν ∈ SN−1 , normal to Su at x, such that a = b and

lim

ε→0+

1 εN

.

limε→0+

1 εN



{y∈x+εQν :(y−x)·ν>0} {y∈x+εQν :(y−x)·ν 0 depending only on N such that

N N−1 Du . = f L + βH Su , |β(x)| dH N−1 (x)  CN f L1 (Ω;Rd×N ) . Su ∩Ω

(2.16)

2.3 Γ -Convergence and Relaxation

21

Lemma 2.1 ([12, Lemma 2.9]) Let u ∈ BV (Ω; Rd ). Then there exist piecewise constant functions u¯ n ∈ SBV (Ω; Rd ) such that u¯ n → u in L1 (Ω; Rd ) and

|Du|(Ω) = lim |D u¯ n |(Ω) = lim

.

n→∞

n→∞ S ∩Ω u¯ n

|[u¯ n ](x)| dH N−1 (x).

(2.17)

Lemma 2.2 (Matias [26, Lemma 4.3]) Let Ω ⊂ RN be a bounded open set with Lipschitz boundary and let M ∈ RN×N . There exist a constant C(N) > 0, independent of M and Ω, and u ∈ SBV (Ω; RN ) such that: (1) u|∂Ω = 0; (2) ∇u(x) = M for L N -a.e. x ∈ Ω; (3) |D s u|(Ω)  C(N) M L N (Ω). In the context of BH the analogues of Theorem 2.4 and Lemma 2.1 are the following statements. Theorem 2.5 ([21, Theorem 1.4]) Let Ω ⊂ RN be an open set and f ∈ L1 (Ω; RN×N sym ). Then there exist u ∈ BH (Ω) and a constant CN > 0 depending only on N such that D 2 u = f L N + [∇u] ⊗ ν∇u H N−1 S∇u , and



(|u(x)| + |∇u(x)|) dx +

.

Ω

S∇u ∩Ω

|[∇u](x)| dH N−1  CN

|f (x)| dx. Ω

(2.18) Lemma 2.3 ([20, Lemma 3.3]) Let Π ∈ L1 (Ω; Rd×N×N ) and for every δ > 0 let {Uiδ } ⊂ A (Ω) be a countable family such that Uiδ ∩ Ujδ = Ø for every δ i, j ∈ N with i = j , L N (Ω \ ∪i Uiδ

) = 0, and sup

i diam Ui  δ. For i ∈ N, let Πiδ : Uiδ → Rd×N×N be such that Πiδ dx = Π dx, and set Π δ (x) := Uiδ Uiδ

∗ Πiδ (x)χU δ (x). If supδ ||Π δ ||L1 (Ω;Rd×N×N ) < +∞, then Π δ L N  ΠL N as i

i

δ→

0+ .

2.3 Γ -Convergence and Relaxation We recall now the basics of Γ -convergence: this is a notion of convergence, introduced by De Giorgi and Franzoni [16] (see also [15]), which is very important in the context of the calculus of variations to study the convergence of (sequences of) variational functionals by identifying their variational limit. We refer the reader to [8, 13] for treatises on the topic, and we collect here the most important definitions and results.

22

2 Mathematical Preliminaries

Let X be a metric space and let {Fn } be a sequence of functions Fn : X → R. Definition 2.1 ([8, Definition 1.5]) We say that the sequence {Fn } Γ -converges in X to F : X → R if for all x ∈ X we have (i) (lim inf inequality) for every sequence {xn } converging to x F (x)  lim inf Fn (xn );

.

(2.19)

n→∞

(ii) (lim sup inequality) there exists a sequence {xn } converging to x such that F (x)  lim sup Fn (xn ).

(2.20)

.

n→∞

The function F is called the Γ -limit of {Fn }, and we write F Γ - limn→∞ Fn .

=

When X is an arbitrary topological space (in particular, it is not a metric space), a more general, topological, definition of Γ -convergence can be given in terms of the topology of X. We refer the reader to [8, Section 1.4] and [13, Definition 4.1] for the details. It is not difficult to see that inequalities (2.19) and (2.20) imply that     F (x) = inf lim inf Fn (xn ) : xn → x = inf lim sup Fn (xn ) : xn → x ,

.

n→∞

n→∞

(2.21) stating that the Γ -limit exists if and only if the two infima in (2.21) are equal. Remark 2.2 All the definitions and results presented above can be generalized to the case of families of functionals, indexed by a continuous parameter ε. A family of functions {Fε } Γ -converges in X to F : X → R as ε → 0+ if, for every sequence εn → 0+ , the functions {Fεn } Γ -converge to F in the sense of Definition 2.1 (see, e.g., [8, Section 1.9]). One of the most important results of the theory of Γ -convergence is that, under appropriate compactness hypotheses, Γ -convergence implies convergence of minima. In fact, let F : X → R be the Γ -limit of a family of functionals {Fε } defined on X. If uε is a minimizer for Fε in X and uε → u in X as ε → 0+ , then u is a minimizer for F in X and F (u) = limε→0+ Fε (uε ). The first infimization problem in (2.21) defines the Γ -lower limit at x ∈ X, in symbols Γ -lim infn→∞ Fn (x), of a sequence of functionals {Fn }. In view of this, the relaxation of a functional F can be viewed as the Γ -lower limit of a family of constant family of functionals   sc− F (x) := inf lim inf F (xn ) : xn → x = Γ - lim inf F (x).

.

n→∞

n→∞

2.4 The Global Method for Relaxation

23

2.4 The Global Method for Relaxation One of the central problems in the calculus of variations is to determine whether an abstract functional F : X × A (Ω) → [0, +∞], defined on pairs (u; U ), where X is a function space and A (Ω) is the collection of open sets of a domain Ω, admits an integral representation. We will assume that X = BV (Ω; Rd ), since many abstract and practical applications such as free-discontinuity problems fall within this setting. In this context, the problem is formulated as the search for densities H , h, and η such that, once integrated over U , measure the contribution of each of the three mutually singular terms (see (2.7)) of the distributional derivative Du. Dependencies of H , h, and η on the spatial variable x or on the function u itself are possible according to the problem at hand. The fundamental paper that addressed this problem in the space BV is [7], whereas, for the sake of completeness, we point out that analogous results for functionals with superlinear growth have been proved in [6]. The minimal introduction to this theory that we present here is preparatory for the study of relaxation of energies to second-order structured deformations in the SBH context contained in Sect. 4.1.2. The functionals considered in [7] are of the form F : BV (Ω; Rd ) × A (Ω) → [0, +∞] satisfying the following assumptions. Assumptions 2.1 We require that the functional F satisfies: (1) for each u ∈ BV (Ω; Rd ), F (u; ·) is the restriction to A (Ω) of a Radon measure; (2) for each U ∈ A (Ω), F (·; U ) is L1 (U ; Rd )-lower semicontinuous; (3) there exists C > 0 such that for all (u, U ) ∈ BV (Ω; Rd ) × A (Ω),   0  F (u; U )  C L N (U ) + |Du|(U ) .

.

Remark 2.3 We observe that condition (2) implies that F is local, see [7, Remark 2.1], that is for all u, v ∈ BV (Ω; Rd ) and U ∈ A (Ω), if u = v for L N -a.e. x ∈ U , then F (u; U ) = F (v; U ) for all U ∈ A (Ω). The main idea of the global method for relaxation is to show that it is possible to reconstruct F (u; U ) in terms of a set function m(u; ·) defined on Lipschitz subdomains of Ω; in more detail, for such a set U , the function m(u; ·) is defined by  m(u; U ) := inf F (v; U ) : v ∈ BV (Ω; Rd ), v|∂U = u|∂U .

.

Condition (3) can be strengthened to include coercivity of F , namely (3 ) there exists C > 0 such that for all (u, U ) ∈ BV (Ω; Rd ) × A (Ω),   C −1 |Du|(U )  F (u; U )  C L N (U ) + |Du|(U ) .

.

(2.22)

24

2 Mathematical Preliminaries

which turns out both to be a crucial assumption for some results and to have impactful implications in the applications. Moreover, to characterize the density of F with respect to the Cantor part D c u of the distributional derivative of u, it will be necessary to assume the following condition: (4) there exists a modulus of continuity ωF : [0, +∞) → [0, +∞), with ωF (s) → 0 as s → 0+ such that    F (u(· − z) + b; z + U ) − F (u; U )  ωF (|b| + |z|)(L N (U ) + |Du|(U )

.

for every (u, U, b, z) ∈ BV (Ω; Rd ) × A (Ω) × Rd × RN such that z + U ⊂ Ω. The main results in [7] are the following two theorems. Theorem 2.6 (Integral Representation in SBV [7, Theorem 3.7]) Let (1), (2), and (3 ) hold. Then for every u ∈ SBV (Ω; Rd ) and U ∈ A (Ω)

F (u; U ) =

f (x, u(x), ∇u(x)) dx

.

U



+

Su ∩U

g(x, u+ (x), u− (x), νu (x)) dH N−1 (x),

where, for every x0 ∈ Ω, b, λ1 , λ2 ∈ Rd , A ∈ Rd×N , and ν ∈ SN−1 ,   m b + aA (· − x0 ); Q(x0 , ε) .f (x0 , b, A) := lim sup ,. εN ε→0   m sλ1 ,λ2 ,ν (· − x0 ); Qν (x0 , ε) . g(x0 , λ1 , λ2 , ν) := lim sup εN−1 ε→0

(2.23a) (2.23b)

Theorem 2.7 (Integral Representation in BV [7, Theorem 3.12]) Let (1), (2), (3 ), and (4) hold. Then for every u ∈ BV (Ω; Rd ) and U ∈ A (Ω)

F (u; U ) =

f (x, u(x), ∇u(x)) dx

.

U



+

+

Su ∩U

g(x, u+ (x), u− (x), νu (x)) dH N−1 (x)

 dD c u (x) d|D c u|(x) f ∞ x, u(x), d|D c u| U

where f and g are given by (2.23) and f ∞ is the recession function of f with respect to its last variable (compare with (2.2)) f ∞ (x0 , b, A) := lim sup

.

t →+∞

f (x0 , b, tA) . t

(2.24)

References

25

The growth conditions (3) or (3 ) suggest that the functional F depends on, or at least is affected by, variations of the function u ∈ BV (Ω; Rd ). By the structure formula (2.7) for the distributional gradient of a BV function, it is then natural to think that there should be three different terms, in the integral representation, measuring the contributions of each of the three mutually singular terms D a u, D j u, and D c u. Since D a u = ∇uL N , the bulk energy density f is found by measuring the contribution of the gradient which is absolutely continuous with respect to the Lebesgue measure. Such gradients computed at a point x0 are represented by matrices, and the simplest function that is determined by a constant matrix A = ∇u(x0 ) is the linear function aA . Formula (2.23a) says that f (x0 , b, A) is obtained by analysing the behavior of the measure m defined in (2.22) over smaller and smaller cubes centered at x0 and tested on functions having the same boundary values of the affine function given by x → b + aA (x − x0 ). The rescaling by εN = L N (Q(x0 , ε)) is the natural rescaling for volume terms. Similarly, since D j u = [u] ⊗ νu H N−1 Su , the surface energy density is found by measuring the contribution of jump-type discontinuity across the interface Su . Looking closely, the simplest jump is that determined by a piecewise constant function across jumping in the direction perpendicular to the discontinuity set Su , and it is the one defined, at a point x0 , by the values of the traces λ = u+ (x0 ) and ξ = u− (x0 ), and the normal ν = νu (x0 ). Formula (2.23b) says that g(x0 , λ1 , λ2 , ν) is obtained by analysing the behavior of the measure m defined in (2.22) over smaller and smaller cubes centered at x0 and tested on functions having the same boundary values of the elementary jump function given by x → sλ1 ,λ2 ,ν (x). The rescaling by εN−1 = H N−1 (∂Q(x0 , ε)) is the natural rescaling for surface terms. The Cantor part D c u of the measure Du is the most obscure to grasp and to treat. Despite the rank-one structure highlighted in (2.8), this measure is not supported on hyperplanes (since it is singular with respect to H N−1 ) and its contribution to the functional F emerges only through the recession function f ∞ . For A given by the rank-one matrix (2.8) evaluated at x0 , the recession function f ∞ (x0 , b, A) in (2.24) singles out the linear behavior of f at infinity along the direction bu (x0 ) ⊗ νu (x0 ) in the space of d × N matrices.

References 1. G. Alberti: A Lusin type Theorem for gradients. J. Funct. Anal., 100 (1991), 110–118. 2. G. Alberti: Rank-one property for derivatives of functions with bounded variation. Proc. Royal Soc. Edinburgh A, 123 (1993), 237–274. 3. L. Ambrosio: A compactness theorem for a special class of functions of bounded variation. Boll. Un. Mat. Ital., 3-B (1989), 857–881. 4. L. Ambrosio, N. Fusco, and D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press 2000.

26

2 Mathematical Preliminaries

5. M. Baía, M. Chermisi, J. Matias, and P. M. Santos: Lower semicontinuity and relaxation of signed functionals with linear growth in the context of A -quasiconvexity. Calc. Var. 47 (2013), 465–498. 6. G. Bouchitté, I. Fonseca, G. Leoni, and L. Mascarenhas: A global method for relaxation in W 1,p and in SBVp . Arch. Rational Mech. Anal., 165(3) (2002), 187–242. 7. G. Bouchitté, I. Fonseca, and L. Mascarenhas: A global method for relaxation. Arch. Rational Mech. Anal., 145 (1998), 51–98. 8. A. Braides: Γ -convergence for Beginners. Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, 2002. 9. M. Carriero, A. Leaci, and F. Tomarelli: Special bounded hessian and elastic-plastic plate. Rend. Accad. Naz. Sci. XL Mem. Mat., 16(5) (1992), 223–258. 10. M. Carriero, A. Leaci, and F. Tomarelli: A second-order model in image segmentation: Blake and Zisserman Functional. Progress in Nonlinear Diff. Equations, 25 (1996), 57–72. 11. M. Carriero, A. Leaci, and F. Tomarelli: Second-order variational problems with free discontinuity and free gradient discontinuity. Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat. Dept. Math. Seconda Univ. Napoli, Caserta, 14 (2004), 135–186. 12. R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103. 13. G. Dal Maso: An Introduction to Γ -convergence. Progress in Nonlinear Differential Equations and Their Applications Vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. 14. E. De Giorgi and L. Ambrosio: Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei, 82 (1988), 199–210. 15. E. De Giorgi and G. Dal Maso: Γ -convergence and calculus of variations. Mathematical theories of optimization, Lecture Notes in Math. 979 (1983), 121–143. 16. E. De Giorgi and T. Franzoni: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6) (1975), 842–850. 17. E. De Giorgi and G. Letta: Une notion générale de convergence faible des fonctions croissantes d’ensemble. Ann. Scuola Sup. Pisa, 33 (1977), 61–99. 18. F. Demengel: Fonctions à hessien borné . Ann. Inst. Fourier (Grenoble), 34 (1984), 155–190. 19. H. Federer: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969. 20. I. Fonseca, A. Hagerty, and R. Paroni: Second-order structured deformations in the space of functions of bounded hessian. J. Nonlinear Sci., 29(6) (2019), 2699–2734. 21. I. Fonseca, G. Leoni and R. Paroni: On hessian matrices on the space BH . Commun. Contemp. Math., 7 (2005), 401–420. 22. I. Fonseca and J. Malý: Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré. Anal. Non Linéaire, 14(3) (1997), 309–338. 23. J. Kristensen and F. Rindler: Characterization of generalized gradient Young measures generated by sequences in W 1,1 and BV , Arch. Rational Mech. Anal. 197(2) (2010), 539– 598. 24. J. Kristensen and F. Rindler: Relaxation of signed integral functionals in BV . Calc. Var. 37 (2010), 29–62. 25. A. Massaccesi and D. Vittone: An elementary proof of the rank-one theorem for BV functions. J. Eur. Math. Soc. (JEMS), 21(10) (2019), 3255–3258. 26. J. Matias: Differential inclusions in SBV0 (Ω) and applications to the calculus of variations. J. Convex Anal. 14(3) (2007), 465–477. 27. Y. G. Reshetnyak: Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968), 1039–1045. Translation of Sibirsk Mat. Z. 9 (1968), 1386–1394. 28. R. Temam: Problémes Mathématiques en Plasticité. Gauthier-Villars, 1983.

Chapter 3

Energetic Relaxation to First-Order Structured Deformations

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems, and Representation of Relaxed Energies The variational approach to continuum mechanics involves the introduction of an energy functional which associates with any deformation u of a body an energy, whose minimizers are the equilibrium configurations of the body (possibly subject to external loading). The typical energy functional that is considered features a bulk contribution, measuring the deformation (gradient) throughout the whole body, and an interfacial contribution, accounting for the energy needed for fracturing the body. The general form of such an energy, for simple deformations u : Ω → Rd , is



E(u) =

W (∇u(x)) dx +

.

Ω

Su ∩Ω

ψ([u](x), νu (x)) dH N−1 (x),

(3.1)

where W : Rd×N → [0, +∞) is the (possibly non-linear and non-convex) bulk energy density and ψ : Rd × SN−1 → [0, +∞) is the interfacial energy density. The singular set Su is the discontinuity set of the deformation u and νu (x) is the unit normal vector to Su at x (see Sect. 2.2). The explicit dependence of ψ on the normal νu models the presence of anisotropies in the material. Energy (3.1) contains two very well known and studied models in mechanics: by taking W (ξ ) = 12 |ξ |2 and ψ(λ, ν) ≡ 0, one has the classical Dirichlet energy for linear elasticity; by taking W (ξ ) = 12 |ξ |2 and ψ(λ, ν) = 1, one has the Griffiths model for fracture mechanics [24, 35, 37]: indeed, the second term reduces to H N−1 (Su ), measuring the length or the area (for N = 2 and 3, respectively) of the crack. Assigning an energy to a structured deformation (g, G) is not a straightforward task. The proposal in [20] is to give the structured deformation (g, G) the energy of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. Matias et al., Energetic Relaxation to Structured Deformations, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-19-8800-4_3

27

28

3 Energetic Relaxation to First-Order Structured Deformations

the most economical approximation, namely, for every (g, G), we define   I (g, G) := inf lim inf E(un ) : un  (g, G) ,

.

n→∞

(3.2)

where  indicates a suitable sense of convergence yet to be specified: un → g, ∇un → G. In particular, in view of Sect. 2.3, it is evident that the process described in the right-hand side of (3.2) is a relaxation process; therefore a suitable function space for structured deformations must be introduced and a suitable notion of convergence must be specified. In this section, we present two different variational approaches, due to Choksi and Fonseca [20] and Baía et al. [8], respectively, to the program on energetics described in Sect. 1.1.2 for first-order structured deformations. Of the two, the one by Choksi and Fonseca sets the basis for the variational approach to the energetics for structured deformations and is cast in the SBV setting for the initial energy (3.1) and the convergences in (3.2) are strong in L1 (Ω; Rd ) for the deformation un and weak* in M (Ω; Rd×N ) for ∇un (see (3.4)); the one by Baía, Matias, and Santos is cast in BV setting for an initial energy including also a dependence on the second-order gradient (see (3.32)) and the convergence in (3.2) involves strong convergences in L1 (Ω; Rd ) and L1 (Ω; Rd×N ) for the deformation and the firstorder gradient, respectively (see (3.30)). In each of the following subsections, we present the approximation theorems, the definition of I (g, G) (3.2) in the specific contexts of those subsections, and the integral representations for the relaxed energy I (g, G) of the form:

I (g, G) =

h([g](x), νg (x)) dH N−1 (x).

H (∇g(x), G(x)) dx+

.

Ω

(3.3)

Ω∩Sg

3.1.1 Relaxation in SBV The results contained in this subsection have been established by Choksi and Fonseca in [20]. We begin with the following definition. Definition 3.1 The space of (Rd -valued) first-order structured deformations on a domain Ω ⊂ RN is SD(Ω; Rd × Rd×N ) := SBV (Ω; Rd ) × L1 (Ω; Rd×N ).

.

It is endowed with the natural norm induced by the product structure (g, G)SD(Ω;Rd ×Rd×N ) := gBV (Ω;Rd ) + GL1 (Ω;Rd×N ) ,

.

which is going to be denoted by (g, G)SD when no domain specification is needed.

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

29

Moreover, we let SDp (Ω; Rd × Rd×N ) := SBV (Ω; Rd ) × Lp (Ω; Rd×N ). The statement we give of the following approximation theorem is in the spirit of [20, Theorem 2.12] and [56, Theorem 1.2] (which, in turn, are the (S)BV formulation of [28, Theorem 5.8]). Theorem 3.1 (Approximation Theorem [46, Proposition 2.1]) Let Ω ⊂ RN and (g, G) ∈ SD(Ω; Rd × Rd×N ). Then there exists a sequence {un } ⊂ SBV (Ω; Rd ) such that un → g

.

in L1 (Ω; Rd )

and

∗

∇un  G

in M (Ω; Rd×N )

(3.4)

and such that, for all sufficiently large n ∈ N, and for a constant C > 0 depending only on N, |Dun |(Ω)  C(g, G)SD(Ω;Rd ×Rd×N ) .

.

(3.5)

In particular, this implies that ∗

D s un  (∇g − G)L N + D s g

.

in M (Ω; Rd×N ).

(3.6)

∗

We write un − (g, G) whenever un satisfies (3.4). SD

Proof By Theorem 2.4, there exists h ∈ SBV (Ω; Rd ) such that ∇h = ∇g − G and such that |D s h|(Ω)  CN ∇g−GL1 (Ω;Rd×N ) , (see (2.16)). By Lemma 2.1, we can find a sequence of piecewise constant functions hn approaching h in the L1 (Ω; Rd ) norm and such that |Dhn |(Ω)| → |Dh|(Ω). Then, we see that the sequence of functions un := g − h + hn ,

.

satisfies (3.4). Indeed, since hn → h in L1 (Ω; Rd ), we have that un → g in L1 (Ω; Rd ). Moreover, ∇hn = 0 since hn is piecewise constant, so that ∇un = ∇g − (∇g − G) = G, and also the second convergence in (3.4) is satisfied. In particular, for p > 1, if G ∈ Lp (Ω; Rd×N ), then ∇un ∈ Lp (Ω; Rd×N ) forms a constant sequence {∇un } in that space, namely {G}. Moreover, by the triangle inequality, the inequality in (2.16), and (2.17), we have that, for C = 3(1 + CN ), |Dun |(Ω)  C(g, G)SD(Ω;Rd ×Rd×N ) ,

.

so that (3.5) is proved for a suitable tail of {un }. The convergence of un → g in L1 (Ω; Rd ) implies that Dun converges to Dg in the sense of distributions, whereas (3.5) ensures the existence of a weakly* converging subsequence {unk }

30

3 Energetic Relaxation to First-Order Structured Deformations ∗

∗

such that Dunk  Dg in M (Ω; Rd×N ). Since ∇un  G in M (Ω; Rd×N ), we have ∗

D s unk  (∇g − G)L N + D s g

.

in M (Ω; Rd×N ),

which is (3.6) for the subsequence {unk }. Moreover, by the metrisability on compact sets of the weak* convergence, and by Urysohn’s principle, we conclude that the whole sequence {un } satisfies (3.6). The proof is concluded. ∗

Remark 3.1 The convergence − of (3.4) implies the following peculiar fact: the SD

requirement that the gradients of approximating sequences {un } are constrained to converge to the given field G which is not necessarily equal to ∇g is achieved at the cost of the difference Dg − GL N being obtained as the limit of the singular measures D s un , see (3.6). In particular, regardless of the regularity of g (namely, if it belongs or not to W 1,1 (Ω; Rd )), this forces the approximating functions un to be sought after in SBV (Ω; Rd ) \ W 1,1 (Ω; Rd ) if ∇g = G. See Remark 3.3 below for more comments in this direction. The following corollary of Theorem 3.1 is immediate. Corollary 3.1 (Approximation Theorem in SDp [2, Theorem 2.5]) For every (g, G) ∈ SDp (Ω; Rd × Rd×N ) there exists a sequence un ∈ SBV (Ω; Rd ) such that un − (g, G), namely SDp

un → g

.

in L1 (Ω; Rd )

and

∇un  G

in Lp (Ω; Rd×N ).

(3.7)

Moreover, there exists C > 0 such that, for all n ∈ N,   |Dun |(Ω)  C gBV (Ω;Rd ) + GLp (Ω;Rd×N ) .

.

(3.8)

In particular, this implies that, up to a subsequence, (3.6) holds true. Example 3.1 (Example 1.1 Revisited) Let Ω = (0, 1), d = N = 1, and let (g, G) be given by g(x) = 2x, G(x) ≡ 1 for every x ∈ Ω. The structured deformation describes a stretching of the domain Ω to double its size, through a deformation that behaves like the identity at sub-macroscopic levels. This can be achieved by introducing discontinuities, in the following way. Let un : Ω → R be defined by un (x) = x +

.

k , n

for

k k+1 x< , n n

k = 0, . . . , n − 1.

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

31

It is not difficult to see that un → g in L1 (0, 1) and ∇un = u n ≡ 1, so that (3.4) holds. The peculiarity of this example emerges when looking at the distributional derivatives Dun = 1L 1

Ω+

.

n−1

1 δk/n n k=1

which features a non-trivial jump part (see Fig. 1.1). Notice that, in the limit n → ∞, the height 1/n of the jumps vanishes and of the singular  the 1total variation n−1 part of Dun (see (2.5) and (2.9)), given by n−1 k=1 n = n , tends to 1. On the other hand, the singular sets Sun are the discrete sets given by points of the form {k/n : k = 1, . . . , n − 1}, so that the (counting) measure H 0 (Sun ) = n − 1 diverges as n → ∞; in particular, it is not uniformly bounded with respect to n. Finally, notice that M(x) = g (x) − G(x) = 1 emerges as the (weak*) limit (in the sense of measures) of the singular parts D s un . Example 3.2 (Example 1.2 Revisited) Let d = N = 3, Ω = (0, 1)3 , and let g(x) = (x1 + x3 , x2 , x3 ) be a simple shear, and G(x) ≡ I , the identity matrix, for every x ∈ Ω. An approximating sequence can be obtained by implementing the broken ramp to approximate the shear in the following way   k un (x) = x1 + , x2 , x3 n

.

for

k k+1  x3 < , n n

k = 0, . . . , n − 1.

Also in this case we obtain un → g in L1 ((0, 1)3 ; R3 ) and ∇un ≡ I so that (3.4) holds; the distributional derivative Dun is given by Dun = I L 3

.

Ω+

n−1

1 δk/n (x3 )e1 ⊗ e3 . n k=1

The idea is that the approximating sequence breaks Ω into n layers of thickness 1/n each, and slides them on top of one another by an amount of 1/n, as if it were a deck made of n cards. The same comments for the previous example hold in this case as well; we invite the reader to complete the details. Once the notion of convergence (3.2) has been specified via (3.4) for structured deformations in SD(Ω; Rd × Rd×N ), we are ready to tackle the relaxation (3.2). The study of the relaxation process is contained in Theorem 3.2, which provides the announced integral representation (3.3). To deduce the explicit form of the relaxed energy densities H and h, suitable assumptions on the initial bulk and interfacial energy densities W and ψ are needed. Following the approach of [20], we first present Theorem 3.2 under a rather strong set of hypotheses with respect to possible applications, and later comment in Remark 3.5 how to weaken them. In contrast to the statements in [20, Theorems 2.16 and 2.17], we allow here for the explicit dependence of the densities W and ψ on the

32

3 Energetic Relaxation to First-Order Structured Deformations

space variable x. This generalization is important from the mechanical point of view, in order to account for the physical requirement of frame indifference, see, e.g., [46]. Specifically, without adding an explicit dependence on the spatial variable x in the energy densities W and ψ in (3.1), the requirement of frame indifference of the initial energy E turns out to limit unnecessarily the relaxed energy I in (3.3), thereby limiting the class of energies available for subsequent analysis and applications. Assumptions 3.1 Let p  1 and let W : Ω ×Rd×N → [0, +∞) and ψ : Ω ×Rd × SN−1 → [0, +∞) be continuous functions satisfying the following conditions 1. there exists C > 0 such that, for all x ∈ Ω and A, B ∈ Rd×N ,   |W (x, A) − W (x, B)|  C|A − B| 1 + |A|p−1 + |B|p−1 ;

.

2. there exists a continuous function ωW : [0, +∞) → [0, +∞) with ωW (s) → 0 as s → 0+ such that, for every x, x0 ∈ Ω and A ∈ Rd×N , |W (x, A) − W (x0 , A)|  ωW (|x − x0 |)(1 + |A|p );

.

3. if p = 1, there exist C, T > 0 and 0 < α < 1 such that, for all x ∈ Ω and A ∈ Rd×N with |A| = 1,     ∞ . W (x, A) − W (x, tA)   C , for all t > T ,  tα  t with W ∞ denoting the recession function at infinity of W with respect to A, see (2.2); 4. there exist c, C > 0 such that, for all x ∈ Ω, λ ∈ Rd , and ν ∈ SN−1 , c|λ|  ψ(x, λ, ν)  C|λ|;

.

5. (positive 1-homogeneity) for all x ∈ Ω, λ ∈ Rd , ν ∈ SN−1 , and t > 0 ψ(x, tλ, ν) = tψ(x, λ, ν);

.

6. (sub-additivity) for all x ∈ Ω, λ1 , λ2 ∈ Rd , and ν ∈ SN−1 , ψ(x, λ1 + λ2 , ν)  ψ(x, λ1 , ν) + ψ(x, λ2 , ν);

.

7. there exists a continuous function ωψ : [0, +∞) → [0, +∞) with ωψ (s) → 0 as s → 0+ such that, for every x0 ∈ Ω, λ ∈ Rd , and ν ∈ SN−1 , |ψ(x, λ, ν) − ψ(x0 , λ, ν)|  ωψ (|x − x0 |)|λ|.

.

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

33

Remark 3.2 Assumptions 3.1-4 and 6 imply Lipschitz continuity of the function λ → ψ(x, λ, ν), i.e., for every (x, ν) ∈ Ω × SN−1 and for every λ1 , λ2 ∈ Rd , |ψ(x, λ1 , ν) − ψ(x, λ2 , ν)|  Cψ |λ1 − λ2 |.

.

Given (g, G) ∈ SD(Ω; Rd × Rd×N ) and U ∈ A (Ω), let us define the class   ∗ RpCF (g, G; U ) := {un } ⊂ SBV (U ; Rd ) : un − (g, G), sup ∇un Lp < + ∞

.

SD

n∈N

(3.9) of admissible sequences for (g, G) ∈ SD(Ω; Rd ×Rd×N ), with uniformly bounded gradients in the Lp (U ; Rd×N ) norm. To each function u ∈ SBV (Ω; Rd ) we associate the initial energy



E(u) :=

W (x, ∇u(x)) dx +

.

Ω

Su ∩Ω

ψ(x, [u](x), νu (x)) dH N−1 (x).

(3.10)

The energy assigned to a structured deformation (g, G) ∈ SD(Ω; Rd × Rd×N ) is the relaxation of the energy (3.10) in the class RpCF (g, G; Ω), that is   Ip (g, G) := inf lim inf E(un ) : {un } ∈ RpCF (g, G; Ω) .

.

n→∞

(3.11)

We also define   Ip∞ (g, G) := inf lim inf E(un ) : {un } ∈ RpCF (g, G; Ω), sup un L∞ < +∞

.

n→∞

n∈N

which, owing to the following lemma, allows us to truncate the function un of admissible sequences in RpCF , for p > 1. Lemma 3.1 ([20, Lemma 2.20]) Let p > 1, let g ∈ SBV (Ω; Rd ) ∩ L∞ (Ω; Rd ), and let Assumptions 3.1-1 and 4 hold. Then Ip (g, G) = Ip∞ (g, G) for every (g, G) ∈ SD(Ω; Rd × Rd×N ). Remark 3.3 Assume that the sequence {∇un Lp } is uniformly bounded for p > 1 (by (3.9), this means that the sequence {un } is admissible for the energy Ip in (3.11)). If ∇g = G, this implies that H N−1 (Sun ) → +∞ as n → ∞, as it can be seen by applying Theorem 2.3 with Φ(t) = t p , in light of Lemma 3.1. This has the mechanical interpretation that the jump discontinuities diffuse in the bulk and contribute to creating volume energy; in [28], the part of the bulk in which ∇g = G is referred to as the micro-fractured zone. Moreover, in the case ∇g = G, Theorem 2.3 and Lemma 3.1 imply that if the growth Assumption 3.1-1 holds for p > 1, then the surface energy density ψ cannot be taken with sublinear growth. This justifies the choice of the coercivity condition on ψ in Assumption 3.1-4.

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3 Energetic Relaxation to First-Order Structured Deformations

Remark 3.4 Note that, for p = 1, by the Approximation Theorem 3.1, the set RpCF (g, G; Ω) is non-empty. For the case p > 1, this assertion requires the additional hypothesis that G ∈ Lp (Ω; Rd×N ), as the proof of Theorem 3.1 indicates. Moreover, by (3.5), Assumptions 3.1-1 and 4, we have that, for every (g, G) ∈ SD(Ω; Rd × Rd×N ), the energy Ip (g, G) < +∞. For the sake of being concise, we introduce the symbol δ1 (p) as the Kronecker delta computed at p, namely δ1 (p) = 1 if p = 1 and zero otherwise, and use it as a selector between the cases p = 1 and p > 1. To prepare for the integral representation Theorem 3.2, which will establish that the energy (3.11) is local and has an integral representation in terms of a bulk energy and a surface energy, we introduce the classes of competitors for the cell formulae for the relaxed bulk and surface energy densities. For A, B ∈ Rd×N , let aA (x) := Ax

.

and Cpbulk(A, B)

 := u ∈ SBV (Q; Rd ) : u|∂Q (x) = aA (x),  ∇u dx = B, |∇u| ∈ Lp (Q) ;



.

(3.12)

Q

for λ ∈ Rd and ν ∈ SN−1 let  Cpsurf (λ, ν) := u ∈ SBV (Qν ; Rd ) : u|∂Qν (x) = sλ,0,ν (x),  δ1 (p)C1 (u) + (1 − δ1 (p))C(u) ,

.

(3.13)

where the function sλ1 ,λ2 ,ν is defined by sλ1 ,λ2 ,ν (x) :=

.

 λ1

if x · ν  0,

λ2

if x · ν < 0,

(3.14)

and the conditions C1 (u) and C(u) are

.C1 (u) ⇐⇒ ∇u dx = 0 and C(u) ⇐⇒ ∇u(x) = 0 for L N -a.e. x ∈ Qν . Qν

We state now the integral representation theorem for the relaxed energies Ip defined in (3.11). It generalizes the results contained in [20, Theorems 2.16 and 2.17] to the inhomogeneous case considered here. To make some formulae lighter, it is useful to

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

35

introduce the following notation: for x0 ∈ Ω, U ∈ A (Ω), and W ∞ defined in (2.2), we let

surf := .Ex (u; U ) ψ(x0 , [u](x), νu (x)) dH N−1 (x); . (3.15a) 0

Ex0 (u; U ) :=

Su ∩U

W (x0 , ∇u(x)) dx +

U

=

Exrec (u; U ) := 0

U

Su ∩U

W (x0 , ∇u(x)) dx + Exsurf (u; U ); . 0 W ∞ (x0 , ∇u(x)) dx +

U

= U

ψ(x0 , [u](x), νu (x)) dH N−1 (x) (3.15b)

ψ(x0 , [u](x), νu (x)) dH N−1 (x)

Su ∩U

W ∞ (x0 , ∇u(x)) dx + Exsurf (u; U ). 0

(3.15c)

We will omit the subscript x0 when the energy E does not depend explicitly on x, that is, when it has the expression (3.1). Theorem 3.2 Let p  1; let W : Ω × Rd×N → [0, +∞) and ψ : Ω × Rd × SN−1 → [0, +∞) satisfy Assumptions 3.1; let (g, G) ∈ SD(Ω; Rd ×Rd×N ) and let Ip (g, G) be given by (3.11). Then there exist Hp : Ω × Rd×N × Rd×N → [0, +∞) and hp : Ω × Rd × SN−1 → [0, +∞) such that



I. p (g, G) =

Hp (x, ∇g(x), G(x)) dx + Ω

hp (x, [g](x), νg (x)) dH N−1 (x).

Sg ∩Ω

(3.16) For all x0 ∈ Ω and A, B ∈ Rd×N , .

 Hp (x0 , A, B) := inf Ex0 (u; Q) : u ∈ Cpbulk (A, B) ;

(3.17)

for all x0 ∈ Ω, λ ∈ Rd , and ν ∈ SN−1 ,  hp (x0 , λ, ν) := inf δ1 (p)Exrec (u; Qν ) + (1 − δ1 (p))Exsurf (u; Qν ) : 0 0 . u ∈ Cpsurf (λ, ν) .

(3.18)

Proof (Sketch of the Proof of Theorem 3.2) We provide an outline of the proof of the theorem and refer the reader to [20] for the full details. We discuss first the case where there is no dependence on x (that is, we consider the energy in (3.1) instead of the one in (3.10)) and point out later how to easily adapt the proof in [20] to the general case, in light of the extra hypotheses listed in Assumptions 3.1-2 and 7.

36

3 Energetic Relaxation to First-Order Structured Deformations

Step 1—Upper Bound for Ip (g, G) By Assumption 3.1-4, the continuity of W , ∗

and since ∇un  G, it is immediate to show that there exists a constant C > 0 depending only on N such that 

 W (G(x)) dx + (g, G)SD ,

Ip (g, G)  C

.

Ω

which, in turn, implies that Ip (g, G) is finite whenever Lemma 2.18 and Remark 2.19]).

 Ω

W (G) is (see [20,

Step 2—Localization Let E(u; U ) be the energy in (3.1) restricted to U ∈ A (Ω) and let   CF .Ip (g, G; U ) := inf lim inf E(un ; U ) : {un } ∈ Rp (g, G; U ) . n→∞

It is proved that the set function Ip (g, G; ·) is the restriction to A (Ω) of a bounded positive Radon measure, and that it is absolutely continuous with respect to L N + |D s g| [20, Proposition 2.22]. The main tools to prove this are Theorem 2.2 and a truncation argument to apply Lemma 3.1. Step 3—Sequential Characterization of Hp (A, B) and hp (λ, ν) For A, B ∈ Rd×N , λ ∈ Rd , and ν ∈ SN−1 , let   ˜ p (A, B) := inf lim inf E(un ; Q) : {un } ∈ RpCF (aA , B; Q) , . .H (3.19a) n→∞   h˜ 1 (λ, ν) := inf lim inf E rec (un ; Qν ) : {un } ∈ R1CF (uλ,ν , 0; Qν ) , (3.19b) . n→∞   h˜ p (λ, ν) := inf lim inf E surf (un ; Qν ) : {un } ∈ R˜ pCF (uλ,ν , 0; Qν ) . (3.19c) n→∞

Definition (3.19c) is for the case p ∈ (1, +∞], where   L1 Lp R˜ pCF . (uλ,ν , 0, Qν ) := {un } ⊂ SBV (Qν ; Rd ) : un → uλ,ν , ∇un → 0 .

(3.20)

Then Hp (A, B) = H˜ p (A, B) [20, Proposition 3.1] and hp (λ, ν) = h˜ p (λ, ν) [20, Propositions 4.1 and 4.2], for all p ∈ [1, +∞].

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

37

Step 4—Radon-Nikodým Derivatives The blow-up method of Fonseca-Müller [33, 34] is used to conclude the proof. It amounts to proving that dIp (g, G; ·) (x) = Hp (∇g(x), G(x)), for L N -a.e. x ∈ Ω, . (3.21a) dL N dIp (g,G; ·) 1 hp ([g](x),νg (x)), for H N−1 -a.e. x ∈ Sg . (x) = N−1 |[g](x)| d(|[g]|H Sg ) .

(3.21b) Formula (3.21a) is proved in [20, Theorem 3.2] and formula (3.21b) in [20, Theorems 4.4 and 4.5]. Both are obtained by showing lower and upper bounds for the Radon-Nikodým derivatives in the left-hand sides of (3.21). Step 4.1—Lower Bounds Consider U ∈ A (Ω); for each admissible sequence {un } ∈ RpCF (g, G; U ), the sequence of Radon measures W (∇un )L N

.

U + ψ([un ], νun ) dH N−1 (Sun ∩ U )

is uniformly bounded in L1 (U ), and therefore it has a weak* limit μ ∈ M + (U ). The “” inequality in (3.21a) follows by proving that .

dμ (x 0 )  Hp (∇g(x 0 ), G(x 0 )), dL N

for L N -a.e. x 0 ∈ Ω.

This is obtained by showing that the admissible sequence {un } can be massaged to obtain an admissible sequence for H˜ p (∇g(x 0 ), G(x 0 )) in (3.19a), and by using the sequential characterization Hp = H˜ p in the line below (3.20). The reasoning for proving the “” inequality in (3.21b) is similar. Step 4.2—Upper Bounds To prove the “” inequality in (3.21a), consider an admissible sequence for the sequential characterization of H˜ p (∇g(x 0 ), G(x 0 )). This can be turned into an admissible sequence belonging to RpCF (g, G; Ω) by adding a suitable fast-oscillating function, and an application of the Riemann– Lebesgue Lemma concludes the argument. The reasoning for proving the “” inequality in (3.21b) is analogous, by using in addition a classical argument by Ambrosio et al. [6, Proposition 4.8] that allows one to first prove the inequality for functions g of the form g(x) = λχF (x), where χF is the characteristic function of a set of finite perimeter F and then extend it to general function g ∈ SBV (Ω; Rd ). This strategy requires continuity and semicontinuity properties of hp [20, Proposition 4.3]. So far, we have proved that the relaxed energy (3.2) has the integral representation (3.3), where the relaxed energy densities H and h are provided by (3.17) and (3.18), respectively, with no explicit dependence on x.

38

3 Energetic Relaxation to First-Order Structured Deformations

Step 5—Extension of the Results for Explicit Dependence on x Extending the results to the initial energy (3.10) and thus providing the integral representation (3.16) for the relaxed energy (3.11) is easily done following the arguments in [9] and relying on Assumptions 3.1-2 and 7. Indeed, these hypotheses allow us to “freeze” x0 before deriving formulae (3.21). Remark 3.5 We make some comments on the results of Theorem 3.2 and on some possible ways to weaken the hypotheses. 1. The form of the integral representation of the relaxed energy (3.11) provided by formula (3.16) is structurally the same both for p = 1 and for p > 1: it features a bulk energy and an interfacial energy. The reader will notice that the differences in the hypotheses and conclusions, depending on whether p = 1 or p > 1, arises wherever δ1 (p) appears in the statement of Theorem 3.2. 2. If p = 1, we notice that in formula (3.18) the recession function at infinity W ∞ defined in (2.2) appears, to account for concentration phenomena arising when taking the limit of functions in L1 . 3. Note that in the set of hypotheses, there is no request of coercivity of W . In fact, the bulk energy density of a crystal may have potential wells (at matrices where W vanishes). The coercivity assumption is obviated through the choice of the classes of competitors in (3.9). In the proof provided in [20] coercivity of W is artificially imposed and then removed in the proof of the lower bound for H (Step 4.1): the technique for doing so is first to prove the lower bound for a coercive W (i.e., there exists C > 0 such that C|A|  W (A)) and then to relax the condition by approximating a non-coercive W by Wε (·) := W (·) + ε| · |p and by taking the limit as ε → 0 (see Cases 1 and 2 in the proof of [20, Theorem 3.2]). 4. Coercivity (the bound from below in Assumptions 3.1-4) and positive 1-homogeneity of ψ (see Assumptions 3.1-5) may rule out some relevant physical settings, in particular some arising from fracture mechanics (see [20, Remark 3.3]). If we impose the extra condition that admissible sequences remain uniformly bounded in BV norm, we do not need to impose coercivity. Therefore, we can relax Assumptions 3.1-4 by requiring that 0  ψ(x, λ, ν)  C|λ|.

.

(3.22)

Notice that, if ψ does not depend explicitly on x, condition (3.22) needs not be imposed as an assumption, since it follows immediately from the continuity and the positive 1-homogeneity of ψ: indeed, continuity and Assumptions 3.1-5 imply that ψ(0, ν) = 0 for every ν ∈ SN−1 and so, for every (λ, ν) ∈ (Rd \ {0}) × SN−1 , we have 0  ψ(λ, ν) = |λ|ψ

.

 λ  , ν  C|λ|, |λ|

where C = maxSd−1 ×SN−1 ψ, which is finite by continuity.

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

39

Assumptions 3.1-5 can be relaxed, for p > 1, replacing ψ by ψ0 defined as ψ0 (x, λ, ν) := lim sup

.

t →0+

ψ(x, tλ, ν) , t

(3.23)

in the relaxed bulk density Hp defined in (3.17) (one has to define Exsurf with ψ0 0 in place of ψ). In fact, it is the linear behavior in fixed directions at (amplitude equal to) zero of the initial interfacial energy density ψ which contributes to the new relaxed bulk density. Under hypotheses (3.22) and (3.23), Theorem 3.2 still holds, provided the necessary adjustments are done, namely, one defines Exsurf 0 with ψ0 in place of ψ, replaces the class RpCF (g, G; U ) defined in (3.9) by  ∗ CF (g, G; U ) := {un } ⊂ SBV (U ; Rd ) :un − R (g, G), p SD

.

   sup ∇un Lp + un BV < +∞ , n∈N

and defines the functional Ip (g, G) in (3.11) by taking the relaxation in the class CF (g, G; Ω). In particular, the cell formulae (3.17) and (3.18) hold, so that the R p integral representation (3.16) still holds unchanged. 5. The cell formula (3.18) corrects formula (2.17) in [20], where the dependence on the normal ν was mistakenly omitted, as already noted in [52, Theorem 3] and [57, formula (4)]. The source of the omission is an error in the last equation of the proof of Proposition 4.2 in [20] that otherwise does not affect the validity of the conclusions of that proposition. Points 3. and 4. above stress the importance of having a uniform control on ∇un Lp and on un BV to ensure compactness of the approximating sequences: they can be either obtained as a consequence of coercivity of the energy densities (at the expenses of the range of the physical applications that the model can comprise) or imposed as a request on the classes of functions on which the relaxation is performed.

3.1.1.1 Some Properties of the Relaxed Energy In this section, we collect some properties of both the relaxed energy density Ip defined in (3.11) and of the relaxed energy densities Hp and hp defined in (3.17) and (3.18), respectively. We direct the reader to [20, Section 5] for full details. We start with a lower semicontinuity result for Ip with respect to the convergence ∗

−, under the assumption that the bulk energy density W is coercive. SD

Proposition 3.1 Let W : Ω × Rd×N → [0, +∞) and ψ : Ω × Rd × SN−1 → [0, +∞) satisfy Assumptions 3.1, let W be coercive in the second variable, and let

40

3 Energetic Relaxation to First-Order Structured Deformations

(gn , Gn ), (g, G) ∈ SD(Ω; Rd × Rd×N ) be such that gn → g in L1 (Ω; Rd ) and ∗ Gn  G in M (Ω; Rd×N ). Then, for every p ∈ [1, +∞], Ip (g, G; Ω)  lim inf Ip (gn , Gn ; Ω).

.

n→∞

Proof The proof can be obtained adding the dependence on x, following the arguments in [9], to the proof of [20, Proposition 5.1] for the homogeneous case. Proposition 3.2 Let W : Ω × Rd×N → [0, +∞) and ψ : Ω × Rd × SN−1 → [0, +∞) satisfy Assumptions 3.1. Then the relaxed bulk energy density (x, A, B) → Hp (x, A, B) defined in (3.17) is uniformly continuous in A and B. The notion of inf-convolution (introduced by Moreau [50, 51]; see also [9, Section 2], and [20, Section 5] for the special case with no explicit x-dependence) is useful when dealing with interacting energy densities, as is the case with the definition of Hp in (3.17). We also give the definition of quasiconvex envelope. Definition 3.2 (Inf-Convolution and Quasiconvex Envelope) Rd×N → [0, +∞) and ψ : Ω × Rd × SN−1 → [0, +∞).

Let W : Ω ×

1. The inf-convolution W ψ of W and ψ is defined by  (W ψ)(x, A) := inf W (x, A − a ⊗ b) + ψ0 (x, a, b) : a ∈ Rd , b ∈ SN−1 ,

.

where ψ0 is defined in (3.23). 2. The quasiconvex envelope QW of W is defined by 

QW (x, A) := inf

 W (x, A + ∇φ(y)) dy : φ ∈

.

Ω

W01,∞ (Ω; Rd )

.

Proposition 3.3 Let W : Ω × Rd×N → [0, +∞) and ψ : Ω × Rd × SN−1 → [0, +∞) satisfy Assumptions 3.1 and let W be coercive in the second variable. Let (g, G) ∈ SD(Ω; Rd × Rd×N ) and let p  1. Then



.

inf

G∈Lp (Ω;Rd×N ) Ω

Hp (x, ∇g(x), G(x)) dx =

inf

Ω B∈Rd×N

Hp (x, ∇g(x), B) dx. (3.24)

Moreover, if p = 1, then for every x ∈ Ω and A ∈ Rd×N .

inf

B∈Rd×N

H1 (x, A, B) = Q(W ψ)(x, A).

(3.25)

Equality (3.24) is a statement of commutativity between the evaluation of the bulk contribution to the relaxed energy density Ip and the minimization with respect to the second variable. Mechanically, the left-hand side of (3.24) entails finding

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

41

globally optimal deformations without disarrangements G for a given macroscopic deformation g; the right-hand side entails integrating the field obtained pointwise by minimizing the relaxed bulk energy density with respect to the matrix variable B corresponding to the values of G. Equality (3.25) is a characterization of the optimal relaxed bulk energy density when the initial bulk energy density has linear growth: one must compute the quasiconvex envelope of the inf-convolution of W and ψ. The following result concerns the search for the optimal microstructure for a given macroscopic deformation g ∈ SBV (Ω; Rd ). In particular, if g ∈ W 1,1 (Ω; Rd ), since there is no surface contribution and Sg = Ø, the relaxed energy only has a bulk contribution. Proposition 3.4 Let W : Ω × Rd×N → [0, +∞) and ψ : Ω × Rd × SN−1 → [0, +∞) satisfy Assumptions 3.1 and let W be coercive in the second variable. Then 1. if p > 1 and g ∈ SBV (Ω; Rd ),

I (g) =

inf

Ω B∈Rd×N

Hp (x, ∇g(x), B) dx



.

(3.26)

hp (x, [g](x), νg (x)) dH N−1 (x),

+ Ω∩Sg

2. if p = 1 and g ∈ BV (Ω; Rd ),

I (g) =

Q(W ψ)(x, ∇g(x)) dx Ω

.



Q(W ψ)∞ (x, [g](x) ⊗ νg (x)) dH N−1 (x)

+

+

Ω∩Sg

(3.27)

 dD c g (x) d|D c |(x), Q(W ψ)∞ x, c g| d|D Ω

where   I (g) := inf lim inf E(un ) : un ∈ SBV (Ω; Rd ), un → g in L1 (Ω; Rd )

.

n→∞

is the relaxation of the initial energy E of (3.10) with respect to the L1 topology. In particular, if p = 1 and g ∈ W 1,1 (Ω; Rd ), then (3.27) reduces to

I (g) =

Q(W ψ)(x, ∇g(x)) dx.

.

(3.28)

Ω

Proposition 3.4 goes in the direction of establishing a contact between the theory of structured deformations and classical deformations, uniquely described by the

42

3 Energetic Relaxation to First-Order Structured Deformations

macroscopic quantity g. In this respect, we havethe following result, which ties the relaxed energy functional Ip (g, ∇g) with g → Ω QW (∇g(x)) dx. Proposition 3.5 Let Hp be defined by (3.17). 1. The function Rd×N  A → Hp (A, A) is quasiconvex and Hp (A, A)  QW (A). In particular, if g ∈ W 1,1 (Ω; Rd ), then



Ip (g, ∇g) =

Hp (x, ∇g(x), ∇g(x)) dx 

.

Ω

QW (x, ∇g(x)) dx.

(3.29)

Ω

2. If g ∈ W 1,1 (Ω; Rd ) and if either W is convex, or W is quasiconvex with linear growth (namely, there exists c, C > 0 such that c|A|  W (x, A)  C|A| for every x ∈ Ω) and if ψ(x, λ, ν)  W ∞ (x, λ ⊗ ν), then

Ip (g, ∇g) =

W (x, ∇g(x)) dx.

.

Ω

3. A matrix A ∈ Rd×N is such that W ∗∗ (x, A) < QW (x, A) if and only if there exist a constant α ∈ R such that

.Ip (g, ∇g) < QW (x, ∇g(x)) dx, Ω

where g = aA , ψ(·) = α|·|, and W ∗∗ is the convex analysis bipolar function of W . Remark 3.6 We notice that the very definition of Hp in (3.17) implies that Hp (x, A, A)  QW (x, A), from which (3.29) follows immediately for functions g ∈ W 1,1 (Ω; Rd ), namely for deformations not featuring cracks. In this case, the relaxation process described in (3.11) assigns a deformation g ∈ W 1,1 (Ω; Rd ) a lower energy than the standard quasiconvexification process of the right-hand side in (3.29). Conditions 2. and 3. in Proposition 3.5 characterize cases in which either equality or the strict inequality in (3.29) hold.

3.1.2 Relaxation in BV Counterparts of the results of Theorem 3.2 were obtained by Baía et al. in [8], where, motivated by the study of equilibrium configurations of thin defective crystalline structures, a class of energies depending on the second-order gradient was introduced and an integral representation result for the relaxation to structured deformations was proved in the full BV setting. The process, which is in spirit analogous to that of (3.2), starts from a different initial energy and is carried out for generalized structured deformations. Therefore, we need to define the space of

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

43

generalized structured deformations, the new initial energy, and the meaning we give to the convergence in (3.2). Definition 3.3 The space of (Rd -valued) strong first-order structured deformations on a domain Ω ⊂ RN is SSD(Ω; Rd × Rd×N ) := SBV 2 (Ω; Rd ) × SBV (Ω; Rd×N ).

.

It is endowed with the natural norm induced by the product structure (g, G)SSD(Ω;Rd ×Rd×N ) := gBV 2 (Ω;Rd ) + GBV (Ω;Rd×N ) ,

.

which is going to be denoted by (g, G)SSD when no domain specification is needed. (See below (2.9) and (2.12) for definitions of ·BV and ·BV 2 .) We are going to state and prove the approximation theorem for strong structured deformations. Theorem 3.3 (Approximation Theorem [8]) Let Ω ⊂ RN and (g, G) ∈ SSD(Ω; Rd × Rd×N ). Then there exists a sequence {un } ⊂ SBV 2 (Ω; Rd ) such that un → g

.

in L1 (Ω; Rd )

and

∇un → G

in L1 (Ω; Rd×N )

(3.30)

and such that, for all sufficiently large n ∈ N, and for a constant C > 0 depending only on N, |Dun |(Ω)  C(g, G)SSD(Ω;Rd ×Rd×N ) .

.

(3.31)

In particular, this implies that (3.6) holds. We write un −→ (g, G) whenever un satisfies (3.30). SSD

Proof Because SSD(Ω; Rd × Rd×N ) ⊂ SD(Ω; Rd × Rd×N ), for each structured deformation (g, G) ∈ SSD(Ω; Rd × Rd×N ) we may choose the sequence {un } ⊂ SBV (Ω; Rd ) in the proof of Theorem 3.1 that satisfies not only (3.4), (3.5), and (3.6), but also satisfies ∇un = G for all n ∈ N. Since G ∈ SBV (Ω; Rd×N ), then {un } ⊂ SBV 2 (Ω; Rd ) and (3.30) follows immediately. The bound (3.31) follows from (3.5) since ·SD  ·SSD on their common domain.

44

3 Energetic Relaxation to First-Order Structured Deformations

Motivated by the above-mentioned application to thin defective crystalline materials, we introduce the following initial energy, defined for functions u ∈ SBV 2 (Ω; Rd )



2 E(u) := W (∇u(x), ∇ u(x)) dx + ψ (1) ([u](x), νu (x)) dH N−1 (x) .

Su ∩Ω

Ω



+

S∇u ∩Ω

ψ (2) ([∇u](x), ν∇u (x)) dH N−1 (x), (3.32)

where the densities W : Rd×N × Rd×N×N → [0, +∞), ψ (1) : Rd × SN−1 → [0, +∞), and ψ (2) : Rd×N × SN−1 → [0, +∞) are continuous functions. The presence of the new interfacial term with density ψ (2) causes the initial energy to depend upon both the jumps of u and the jumps of ∇u. We assume that, apart from the dependence on x, both ψ (1) and ψ (2) satisfy Assumptions 3.1-4, 5, and 6 (with Rd×N in place of Rd for ψ (2) ) and we prescribe suitable conditions on W , so that the new set of hypotheses on W , ψ (1) , and ψ (2) is the following. Assumptions 3.2 Let W : Rd×N × Rd×N×N → [0, +∞), ψ (1) : Rd × SN−1 → [0, +∞), and ψ (2) : Rd×N × SN−1 → [0, +∞) be continuous functions satisfying the following conditions 1. there exists C > 0 such that, for all x ∈ Ω and A ∈ Rd×N and L ∈ Rd×N×N , .

1 |L| − C  W (A, L)  C(1 + |L|); C

2. there exists C > 0 such that, for every A, B ∈ Rd×N and L, M ∈ Rd×N×N , |W (A, L) − W (B, M)|  C(|A − B| + |L − M|);

.

3. there exist C, T > 0 and 0 < α < 1 such that, for all x ∈ Ω and A ∈ Rd×N and L ∈ Rd×N×N with |L| = 1,    ∞ W (A, tL)  C .W (A, L) − for all t > T ,   tα ,  t W ∞ denoting the recession function at infinity of W with respect to L, namely W ∞ (A, L) := lim supt →+∞ W (A, tL)/t (compare with (2.2)); 4. there exist c, C > 0 such that, for all λ ∈ Rd and ν ∈ SN−1 , c|λ|  ψ (1) (λ, ν)  C|λ|;

.

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

45

5. (positive 1-homogeneity) for all λ ∈ Rd , ν ∈ SN−1 , and t > 0 ψ (1) (tλ, ν) = tψ (1) (λ, ν);

.

6. (sub-additivity) for all λ1 , λ2 ∈ Rd and ν ∈ SN−1 , ψ (1) (λ1 + λ2 , ν)  ψ (1) (λ1 , ν) + ψ (1) (λ2 , ν);

.

7. there exist c, C > 0 such that, for all Γ ∈ Rd×N and ν ∈ SN−1 , c|Γ |  ψ (2) (Γ, ν)  C|Γ |;

.

8. (positive 1-homogeneity) for all Γ ∈ Rd×N , ν ∈ SN−1 , and t > 0 ψ (2) (tΓ, ν) = tψ (2) (Γ, ν);

.

9. (sub-additivity) for all Γ1 , Γ2 ∈ Rd and ν ∈ SN−1 , ψ (2) (Γ1 + Γ2 , ν)  ψ (2) (Γ1 , ν) + ψ (2) (Γ2 , ν).

.

Given (g, G) ∈ SSD(Ω; Rd × Rd×N ) and U ∈ A (Ω), we define the class   R BMS (g, G; U ) := {un } ⊂ SBV 2 (U ; Rd ) : un −→ (g, G)

.

SSD

of admissible sequences for (g, G). Notice that, unlike in the definition of the class RpCF (g, G; U ) in (3.9), we do not impose any uniform bound on the gradients because we are assuming that ∇un converges to G strongly in L1 . The energy assigned to a strong structured deformation (g, G) ∈ SSD(Ω; Rd × d×N R ) is the relaxation of the energy (3.32) in the class R BMS (g, G; Ω), that is   I (g, G) := inf lim inf E(un ) : {un } ∈ R BMS (g, G; Ω) .

.

n→∞

(3.33)

We now introduce collections of admissible deformations for the BV context that are counterparts of those defined in (3.12) and (3.13) for the SBV setting. For all A, B, Γ1 , Γ2 ∈ Rd×N , L ∈ Rd×N×N , λ ∈ Rd , and ν ∈ SN−1 , we let  C bulk,(1)(A; Q) := u ∈ SBV 2 (Q; Rd ) : u|∂Q = 0,

.

∇u = A for a.e. x ∈ Q ; . (3.34a)  C bulk,(2) (L; Q) := w ∈ SBV (Q; Rd×N ) : w|∂Q (x) = aL (x) ; . (3.34b)  C surf,(1)(λ, ν; Q) := u ∈ SBV 2 (Qν ; Rd ) : u|∂Qν (x) = sλ,ν (x),

46

3 Energetic Relaxation to First-Order Structured Deformations

∇u = 0 for a.e. x ∈ Qν ; . (3.34c)  C surf,(2)(Γ1 , Γ2 , ν; Q) := w ∈ SBV (Qν ; Rd×N ) : w|∂Qν (x) = sΓ1 ,Γ2 ,ν (x) . (3.34d) Similarly, the following relations define the BV counterparts of the initial energies in (3.15) for the SBV setting. For U ∈ A (Ω), we let

E surf,(1) (u; U ) :=

.

(2) EB (w; U )



Su ∩U

:=

ψ (1) ([u](x), νu (x)) dH N−1 (x), .

(3.35a)

W (B, ∇w(x)) dx U



+

Sw ∩U

E

(2)

(w; U ) :=

ψ (2) ([w](x), νw (x)) dH N−1 (x), .

(3.35b)

W (w(x), ∇w(x)) dx U



+

Sw ∩U

E

rec,(2)

ψ (2) ([w](x), νw (x)) dH N−1 (x), .

(3.35c)

W ∞ (w(x), ∇w(x)) dx

(w; U ) := U



+

Sw ∩U

ψ (2) ([w](x), νw (x)) dH N−1 (x).

(3.35d)

Theorem 3.4 ([8, Theorem 3.2]) Let W : Rd×N × Rd×N×N → [0, +∞), ψ (1) : Rd × SN−1 → [0, +∞), and ψ (2) : Rd×N × SN−1 → [0, +∞) satisfy Assumptions 3.2; let (g, G) ∈ SSD(Ω; Rd × Rd×N ) and let I (g, G) be given by (3.33). Then there exist H (1) : Rd×N → [0, +∞), H (2) : Rd×N × Rd×N×N → [0, +∞), h(1) : Rd × SN−1 → [0, +∞), and h(2) : Rd×N × Rd×N × SN−1 → [0, +∞) such that

 (1)  H (∇g(x) − G(x)) + H (2)(G(x), ∇G(x)) dx I (g, G) = Ω



.

+

+

Sg ∩Ω

SG ∩Ω

h(1) ([g](x), νg (x)) dH N−1 (x) h(2) (G+ (x), G− (x), νG (x)) dH N−1 (x),

(3.36)

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

47

where for all A ∈ Rd×N , L ∈ Rd×N×N ,   H (1)(A) = inf E surf,(1)(u; Q) : u ∈ C bulk,(1)(A; Q) , .   H (2)(B, L) = inf EB(2)(w; Q) : w ∈ C bulk,(2) (L; Q) , .

(3.37a) (3.37b)

(2)

with C bulk,(1), C bulk,(2) as in (3.34) and with E surf,(1) , EB as in (3.35); for all λ ∈ Rd , Γ1 , Γ2 ∈ Rd×N , and ν ∈ SN−1 ,   (1) .h (λ, ν) = inf E surf,(1) (u; Qν ) : u ∈ C surf,(1)(λ, ν; Q) ; . (3.38a)   h(2) (Γ1 , Γ2 , ν) = inf E rec,(2) (w; Qν ) : w ∈ C surf,(2)(Γ1 , Γ2 , ν; Q) , (3.38b) with C surf,(1) , C surf,(2) as in (3.34) and with E surf,(1) , E rec,(2) as in (3.35). Proof (Sketch of the Proof of Theorem 3.4) We divide the proof into three steps. Step 1—Decomposition We claim that the following additive decomposition holds I (g, G) = I (1) (g, G) + I (2) (G),

(3.39)

.

where   I (1) (g, G) := inf lim inf E surf,(1)(un ; Ω) : {un } ⊂ SBV 2 (Ω; Rd ), un −→ (g, G)

.

n→∞

SSD

and   L1 I2 (G) := inf lim inf E (2) (wn ; Ω) : {wn } ⊂ SBV (Ω; Rd×N ), wn → G .

.

n→∞

The inequality I (g, G)  I (1) (g, G) + I (2) (G) follows immediately from the properties of the infimum. To prove the reverse inequality, let us consider sequences {un } ⊂ SBV 2 (Ω; Rd ) and {wn } ⊂ SBV (Ω; Rd×N ) such that un −→ (g, G), SSD

L1

wn → G and I (1) (g, G) = lim E surf,(1) (un ; Ω)

.

n→∞

and

I (2) (G) = lim E (2) (wn ; Ω). n→∞

By Theorem 2.4, for every n ∈ N there exists a function hn ∈ SBV 2 (Ω; Rd ) such that ∇hn = wn − ∇un ; by Lemma 2.1, for each n ∈ N, there exists a sequence {h¯ n,m } ⊂ SBV (Ω; Rd ) of piecewise constant functions converging to hn in L1 (Ω; Rd ). Upon the extraction of a diagonal sequence, the function u˜ n := un + hn + h¯ n,m(n)

.

48

3 Energetic Relaxation to First-Order Structured Deformations

is an admissible function for the definition of I (g, G). Indeed,

.

  lim u˜ n = lim un + hn + h¯ n,m(n) = lim un = g, n→∞ n→∞   lim ∇ u˜ n = lim ∇un + ∇hn = lim wn = G, n→∞

n→∞

n→∞

n→∞

both limits being taken in the L1 sense. Therefore, u˜ n −→ (g, G). Using u˜ n SSD

in (3.33), we have I (g, G)  lim inf E(un )  lim E surf,(1) (un ; Ω) + lim sup E surf,(1)(hn ; Ω) n→∞

n→∞

+ lim sup E

n→∞

surf,(1)

n→∞

n→∞

.

(h¯ n,m(n) ; Ω) + lim E (2) (wn ; Ω)

I (1) (g, G) + I (2) (G) + C lim sup n→∞

|wn (x) Ω

− ∇un (x)| dx, where we have used the subadditivity and linear growth of ψ (1) , the estimate in (2.16), and (2.17). Recalling the convergences wn → G and ∇un → G in L1 , we obtain the desired inequality I (g, G)  I (1) (g, G) + I (2) (G). Step 2—Integral Representations Following an analogous reasoning to that of the proof of Theorem 3.2, we can prove the following integral representation for I (1) :

I

.

(1)

(g, G) =

H

(1)

(∇g(x) − G(x)) dx +

Ω

Sg ∩Ω

h(1) ([g](x), νg (x)) dH N−1 (x),

with H (1) and h(1) defined in (3.37a) and (3.38a), respectively. We notice that Lemma 2.2 is used to construct the admissible functions for H (1). A direct application of the global method for relaxation, see [12, Theorem 4.2.2], allows us to derive the following integral representation for I (2) :

I (2) (G) =

H (2)(G(x), ∇G(x)) dx

.

Ω



+

SG ∩Ω

h(2) (G+ (x), G− (x), νG (x)) dH N−1 (x),

with H (2) and h(2) defined in (3.37b) and (3.38b), respectively.

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

49

Remark 3.7 The additive decomposition (3.39) is reflected in the representation (3.36) in both the bulk term and in the interfacial term. The contribution of H (1)(∇g(x) − G(x)) to the bulk relaxed density captures energy stored due to submacroscopic disarrangements through [un ], while the contribution of H (2)(G(x), ∇G(x)) captures both energy stored without submacroscopic disarrangements through ∇un and energy stored due to submacroscopic gradient disarrangements through [∇un ]. The contributions of h(1) and h(2) to the interfacial relaxed energy capture energy stored due to disarrangements [g] at the macrolevel and disarrangements [G] at the macrolevel, respectively. We now introduce a larger class of structured deformations allowing for the presence of a Cantor-like part in the distributional derivatives, that is, we consider functions in BV -like spaces without the requirement that the Cantor part of the distributional derivative vanish. Definition 3.4 The space of (Rd -valued) generalized first-order structured deformations on a domain Ω ⊂ RN is GSD(Ω; Rd × Rd×N ) := BV 2 (Ω; Rd ) × BV (Ω; Rd×N ).

.

It is endowed with the natural norm induced by the product structure (g, G)GSD(Ω;Rd ×Rd×N ) := gBV 2 (Ω;Rd ) + GBV (Ω;Rd×N ) ,

.

which is going to be denoted by (g, G)GSD when no domain specification is needed. (See (2.4) and (2.12) for the definitions of ·BV and ·BV 2 .) For the same initial energy E introduced in (3.32) and under the same convergence of sequences un −→ (g, G) introduced in (3.30), we can now establish an SSD

integral representation theorem for the relaxed energy I (g, G) defined in (3.33) for generalized structured deformations (g, G) ∈ GSD(Ω; Rd × Rd×N ). Let EBrec,(2) (v; U ) :=



W ∞ (B, ∇v(x)) dx +

.

U

Sv ∩U

ψ (2) ([v](x), νv (x)) dH N−1 (x).

Theorem 3.5 ([8, Theorem 4.1]) Let W : Rd×N × Rd×N×N → [0, +∞), ψ (1) : Rd × SN−1 → [0, +∞), and ψ (2) : Rd×N × SN−1 → [0, +∞) satisfy Assumptions 3.2; let (g, G) ∈ GSD(Ω; Rd × Rd×N ) and let I (g, G) be given

50

3 Energetic Relaxation to First-Order Structured Deformations

by (3.33). Then there exist H (1) : Rd×N → [0, +∞), H (2) : Rd×N × Rd×N×N → [0, +∞), h(1) : Rd × SN−1 → [0, +∞), and h(2) : Rd×N × Rd×N × SN−1 → [0, +∞) such that

 (1)  H (∇g(x) − G(x)) + H (2)(G(x), ∇G(x)) dx I (g, G) = Ω



+

+

.

Sg ∩Ω

h(1) ([g](x), νg (x)) dH N−1 (x)

SG ∩Ω

h(2) (G+ (x), G− (x), νG (x)) dH N−1 (x)

(3.40)

dD c g (x) d|D c g|(x) H + − d|D c g| Ω 

dD c G (x) d|D c G|(x), + (H (2))∞ G(x), c G| d|D Ω



(1)

where H (1), H (2), h(1) , and h(2) are defined in (3.37) and (3.38), and where, for B ∈ Rd×N and L ∈ Rd×N×N ,   rec,(2) (H. (2))∞ (B, L) := inf EB (u; Q) : v ∈ C bulk,(2) (L; Q) . Proof (Sketch of the Proof of Theorem 3.5) As in the proof of Theorem 3.4, the decomposition (3.39) of I (g, G) still holds. Due to the structured deformation, (g, G) being in BV -like spaces, the integral representations have the form

I (2) (G) =

H (2)(G(x), ∇G(x)) dx Ω



+

.



SG ∩Ω

+

(H Ω

h(2) (G+ (x), G− (x), νG (x)) dH N−1 (x)  dD c G (x) d|D c G|(x), G(x), ) d|D c G|

(2) ∞

which can be derived in the same way as in the proof of Theorem 3.4, and



I (1) (g, G) =

H (1)(∇g(x) − G(x)) dx + Ω



.

+

Sg ∩Ω

h(1) ([g](x), νg (x)) dH N−1 (x)

 dD c g (x) d|D c g|(x). H (1) − d|D c g| Ω

3.1 Spaces of First-Order Structured Deformations, Approximation Theorems,. . .

51

The formula above can be obtained via a sequential characterization of the localization of the energy I (1) (g, G): for (g, G) ∈ GSD(Ω; Rd × Rd×N ), U ∈ A (Ω), let  I˜(1) (g, G; U ) = inf lim inf I (1) (gn , Gn ; U ) :{gn } ⊂ SBV 2 (U ; Rd ), n→∞

{Gn } ⊂ SBV (U ; Rd×N ),  L1 L1 gn → g, Gn → G .

.

As usual, the proof of I (1) (g, G; U ) = I˜(1) (g, G; U ) relies on proving suitable upper and lower bounds for the Radon-Nikodým derivatives with respect to the measures L N , H N−1 Sg , and |D c g|. For the last one, it is necessary to invoke Alberti’s rank-one Theorem [1] and Reshetnyak’s continuity Theorem 2.1(ii) (see the proof of [8, Theorem 4.1] for the details). Because SSD(Ω; Rd × Rd×N ) is a subspace of GSD(Ω; Rd × Rd×N ), one expects that the representation (3.36) for I (g, G) with (g, G) in the smaller space should be recovered from the representation (3.40) for I (g, G), with (g, G) in the larger space, when one sets the Cantor parts of the measures Dg and DG equal to zero. Doing so sets both the measures |D c g| and |D c G| equal to zero, and the last two integrals in (3.40) vanish, so that (3.36) is indeed recovered.

3.1.3 Other Approximation Results Two additional approximation results were obtained by Šilhavý in [56] for structured deformations (g, G) ∈ SD(Ω; Rd × Rd×N ) := BV (Ω; Rd ) × L1 (Ω; Rd×N ),

.

thus allowing for the deformation map to have non-zero Cantor part in the distributional derivative. The first result is a generalization of the Approximation Theorem 3.1 to structured deformations in SD(Ω; Rd × Rd×N ); the second result is a different proof that does not rely on Alberti’s Theorem 2.4. The statement of the first result stems from the observation that the construction by means of Alberti’s Theorem 2.4 works in a broader setting than that of Theorem 3.1, in particular, if Ω ⊂ RN is an open and bounded set, for every (g, G) ∈ L1 (Ω; Rd ) × L1 (Ω; Rd×N ) there exists a sequence {un } ⊂ SBV (Ω; Rd ) such that un → g

.

in L1 (Ω; Rd )

and

∇un = G

in Ω,

as can be easily seen by inspecting the proof of Theorem 3.1.

(3.41)

52

3 Energetic Relaxation to First-Order Structured Deformations

Theorem 3.6 ([56, Theorem 1.2]) Given (g, G) ∈ SD(Ω; Rd × Rd×N ), there exists a sequence {un } ⊂ SBV (Ω; Rd ) satisfying (3.41) and such that .

sup |Dun |(Ω) < +∞.

(3.42)

n∈N

This theorem has the immediate corollary that ∗

Dun  Dg

.

in M (Ω; Rd×N ),

which is clearly a consequence of (3.42). As pointed out in [56], Theorem 3.6 also shows that the advantage gained by using (g, G) ∈ SD(Ω; Rd × Rd×N ) versus (g, G) ∈ L1 (Ω; Rd )×L1 (Ω; Rd×N ) is precisely the availability of approximations un ∈ SBV (Ω; Rd ) that satisfy, in addition to (3.41), the bound (3.42). Theorem 3.7 ([56, Theorem A.1]) Given (g, G) ∈ SD(Ω; Rd × Rd×N ), there exist two sequences {vn }, {wn } ⊂ SBV (Ω; Rd ) such that vn → g

.

wn → 0

in L1 (Ω; Rd ) 1

and

in L (Ω; R )

and

sup |Dvn |(Ω) < +∞

and

d

n∈N

∇vn = 0 ∇wn → G

in Ω, .

(3.43a) 1

in L (Ω; R

d×N

), . (3.43b)

sup |Dwn |(Ω) < +∞,

(3.43c)

n∈N

so that the sequence un := vn + wn ∈ SBV (Ω; Rd ) is such that un → g

.

in L1 (Ω; Rd )

sup |Dun |(Ω) < +∞

and and

∇un → G ∗

Dgn  Dg

in L1 (Ω; Rd×N ), .

(3.44a)

in M (Ω; Rd×N ). (3.44b)

n∈N

Proof (Sketch of the Proof of Theorem 3.7) The proof is achieved by suitably extending the structured deformation (g, G) to zero outside of Ω. Denoting this extension by (g0 , G0 ), the first component is approximated by convolution with the standard mollifier, and this is in turn approximated by a sequence of functions {vn } ∈ SBV (Ω; Rd ); the functions wn are constructed as a sequence of piecewise affine functions on a suitable tiling of the space RN , with constant gradient given by the average of G0 on each tile. This construction guarantees that (3.43) hold, so that (3.44) is obtained by the definition of un . In both Theorems 3.1 and 3.7, the additional requirement that Ω is an admissible domain [56, Definition 3.10] is imposed which provides alternate paths to the bounds (3.42) and (3.43c) than the use of (2.17).

3.2 Applications

53

3.2 Applications We present several applications of the relaxation process of an initial energy defined on deformation functions un to an energy defined on structured deformations (g, G), by means of formulae like (3.11) or (3.33), and we give an integral representation for the relaxed energy in each of the applications that reflects the special features in that application. As mentioned in Sect. 1.1.2, the passage from an energy defined on un to one defined on (g, G) may be viewed as an upscaling from micro- to macrolevels, and the variety of applications treated reflects the richness of this upscaling process. In Sect. 3.2.1, we consider the case in which W ≡ 0, so that the initial energy has the expression

E(u) :=

.

Su ∩Ω

ψ(x, [u](x), νu (x)) dH N−1 (x).

Explicit formulae for the relaxed energy densities have first been obtained in [52] and in [10] for specific energy densities ψ. Šilhavý generalized these results to a wider class of interfacial energies, and both the specific cases in [10, 52] and general result [57] are discussed in Sect. 3.2.1. We consider two other special cases in Sect. 3.2.2, namely, how the convexity of the initial bulk energy density W affects the relaxed energy densities and a motivation for the addition of a contribution of non-local nature to the energy. In Sect. 3.2.3, we present a problem of dimension reduction in the context of structured deformations, namely we find an expression for the relaxed energy I (g, G), where (g, G) is a structured deformation defined on a two-dimensional region ω that represents, in the limit ε → 0, the cross section of the thin body Ωε := ω × (−ε/2, ε/2). In this context, the relaxation to structured deformations and the zero-thickness limit operations can be performed in either order. We provide integral representation theorems for both cases, and we show that the relaxed energies are the same in the case of a specific initial energy accounting only for the interfacial term (as in Sect. 3.2.1). Moreover, for that specific purely interfacial initial energy, an interesting comparison can be done with the simultaneous procedure of relaxation to structured deformations and dimension reduction performed in [49]. In Sect. 3.2.4, we study an optimal design problem for fractured media, namely bodies where two different materials (or two phases of the same one) coexist and the energy includes a perimeter term in order to minimize fragmentation. This problem is interesting since the initial energy functional depends not only on the deformation u, but also on the phase χ. The relaxation procedure will keep track of both the convergence of un  (g, G) and the convergence χn → χ in a suitable sense. In Sect. 3.2.5, taking inspiration from the discussion in Sect. 3.2.2.2, we couple the initial energy (3.11) with a non-local energy of the form

  Ψ x, (D s u ∗ αr )(x) dx,

E αr (u) =

.

Ωr

54

3 Energetic Relaxation to First-Order Structured Deformations

where αr is a convolution kernel averaging out the contributions of the jumps D s u on a ball of radius r > 0, and Ψ is an energy density for this non-local effect. We first prove an integral representation result for the relaxation of E αr (u) to I αr (g, G), and then study the limit as r → 0+ . In Sect. 3.2.6, we address periodic homogenization in the context of structured deformations. This turns out to be a delicate situation when combining the techniques from Sect. 3.1 with those from homogenization theory (see, e.g., [14]). The integral representation result obtained is similar to that one might expect by looking at Theorem 3.11, but the relaxed bulk and surface energy densities are obtained via an asymptotic cell formula (see (3.105) and (3.106) below). Finally, in Sect. 3.2.7, we study hierarchical structured deformations, a situation in which multiple submacroscopic levels are singled out, and at each of them a relaxation procedure is performed. These multiple upscalings track the effects at the macro-level of all the layered sub-macroscopic contributions.

3.2.1 Relaxation of Purely Interfacial Energies The explicit computation of Hp and hp from the cell formulae (3.17) and (3.18) for the integral representation (3.16) or from the cell formulae (3.37) and (3.38) for the integral representations (3.36) and (3.40) is not always easy to accomplish, although doing so would be useful in order to minimize the energy (3.16) in the spaces of structured deformations. With this in mind, Owen and Paroni [52] evaluated explicitly the right-hand side of the cell formulae (3.17), (3.18), after making the choices (for the cases d = N and p > 1) W ≡0

.

 ψ(x, λ, ν) = (λ · L(x)ν)±

and

or

ψ(x, λ, ν) = |λ · L(x)ν|,

(3.45)

where L : Ω → RN×N is continuous and invertible for every x ∈ Ω. (The cell formula for h in [52, Theorem 3] has a misprint that was corrected in equation (1–17) of [10].) With the choices (3.45), the energy densities Hp and hp in (3.17) and (3.18), respectively, have the explicit expressions [52, Theorem 4] hp (x, λ, ν) = ψ(x, λ, ν),

.

(3.46a)

and Hp (x, A, B) = (L(x) · (A − B))±

.

or Hp (x, A, B) = |L(x) · (A − B)|,

which become Hp (A, B) = (tr(A − B))±

.

or Hp (A, B) = | tr(A − B)|

(3.46b)

3.2 Applications

55

when L(x) = I is the constant map with value the identity (see [52, Section 5]). These formulae show that the surface energy density remains the same under relaxation and that the relaxed bulk energy density depends explicitly on the disarrangement tensor M = ∇g − G. In light of (3.45), the geometrical interpretation of (3.46b) is the following: • (tr M)+ , the positive part of the trace of the disarrangement tensor M, is a volume density of disarrangements due to submacroscopic separations; • (tr M)− , the negative part of the trace of the disarrangement tensor M, is a volume density of disarrangements due to submacroscopic switches and interpenetrations; • the absolute value | tr M| is a volume density of all three of these non-tangential disarrangements: separations, switches, and interpenetrations. In [10] formulas (3.46b) were re-proved in a manner that captures two results at the same time: it recovers explicit expressions for the relaxed bulk and interfacial energy densities obtained in [52] and shows the equivalence of two minimum problems (see (3.50) below). The proof was carried out without explicit dependence on x, but it can be easily adapted to the x-dependent case by including Assumptions 3.1-7 and by means of the techniques borrowed from [9]. The result is contained in the following theorem. Theorem 3.8 ([10]) Let W ≡ 0 and for λ ∈ RN , ν ∈ SN−1 let h(λ, ν) = f (λ · ν), where f : R → R is one of the functions f (t) = (t)± , f (t) = |t|. Then, for any structured deformation (g, G) ∈ SD(Ω; Rd × Rd×N ), the relaxed energy (3.16) has the following integral representation

  f tr(∇g(x) − G(x)) dx +

I (g, G) =

.

Ω

Sg ∩Ω

  f [g(x)] · νg (x) dH N−1 (x). (3.47)

Proof (Sketch of the Proof of Theorem 3.8 for the Bulk Term in (3.47)) Let us recall (3.17) and notice that, under the assumption that W ≡ 0, the energy E(u; Q) defined in (3.15b) becomes the energy E surf (u; Q) defined in (3.15a), where we have omitted the dependence on x0 , since we are proving the result for spatially homogeneous energies. Recalling (3.12), we have, for every A, B ∈ RN×N ,  Hp (A, B) = inf E surf (u; Q) : u ∈ Cpbulk(A, B) .   inf E surf (u; Q) : u ∈ S C bulk(B − A) ,

(3.48)

where, for evert M ∈ RN×N ,  S. C bulk(M) := u ∈ SBV (Ω; RN ) : u|∂Q (x) = 0, ∇u = M a.e. in Q .

(3.49)

56

3 Energetic Relaxation to First-Order Structured Deformations

It is clear from the definition of the set of competitors S C bulk (compare with the set C bulk,(1) in Sect. 3.1.2, from which it differs only for the regularity imposed on the functions) that if u ∈ S C bulk (B − A), then the function x → u(x) + aA (x) belongs to Cpbulk (A, B) and this justifies the inequality in (3.48). Let us consider the function f (t) = |t| in the hypothesis (the proof for f (t) = (t)± will follow). Use of the Gauss–Green formula and (3.48) yield 

| tr(A − B)|  inf

|[u](x) · νu (x)| dH 

.

 inf

 N−1

(x) : u ∈

N−1

(x) : u ∈ S C

Q∩Su

|[u](x) · νu (x)| dH

Cpbulk(A, B) bulk

 (B − A) .

Q∩Su

(3.50) If we can bound the last line of (3.50) from above by | tr(A − B)| then, by (3.48), Hp (A, B) and the two minima in (3.50) will have the common value | tr(B − A)|. To prove that 

.

 |[u](x) · νu (x)| dH N−1 (x) : u ∈ S C bulk(B − A)  | tr(A − B)|,

inf Q∩Su

(3.51) it is necessary to construct competitors u(n) m for the infimum in (3.51). Since, in general, B − A = 0, the function x → aB−A (x) cannot be used. To comply with both the zero the boundary datum on ∂Q and with the requirement that ∇u = B − A a.e. in Q, we will construct a sequence of competitors in SBV (Q; RN ) which is admissible for the infimum problem. The idea is to consider piece-wise affine functions whose gradients are M = B − A and that vanish in the proximity of the boundary ∂Q. To do so, we proceed by considering, for every n  1, the frame  .Fn := Q\ 1 −

2 Q n+2

and by applying Lemma 2.2: this gives us a function u(n) ∈ SBV (Fn ; RN ) (n) satisfying u∂ Fn = 0, ∇u(n) = M a.e. in Fn , and

.

Fn ∩Su(n)

  |[u(n) ]|(x) dH N−1 (x)  C(N)M 1 − 1 −

2 n+2

N ,

(3.52)

where C(N) > 0 is a constant depending only on the dimension. The righthand side of (3.52) vanishes as n → ∞ because of the thinness of the frame Fn . This construction cannot be replicated in the main part of the cube, so a different strategy is needed. The idea is to tessellate the region Q \ F n with a

3.2 Applications

57

family Cn,m of cubes with side-length 1/m so that it will be possible to define  (n)  2 Q → RN whose gradient is M almost everywhere functions um : 1 − n+2 and such that they are competitors for the infimum problems in the left-hand side of (3.51). For technical reasons, the cubes cannot have the faces parallel to Q, but the optimal construction involves a rotation matrix R, to be determined, by which an orthonormal basis made of eigenvectors of (M + M  )/2 is rotated in an optimal way depending on M. These rotated cubes are those which form the family Cn,m . Necessarily, some of those will overlap with ∂Fn , and we set a function un,m (x) = 0 if x belongs to those cubes. For all of the remaining cubes, we construct k the piecewise affine function whose gradient is M. More precisely, denoting by cn,m k , for k = 1, . . . , K the center of the cube Cn,m n,m , belonging to the family Cn,m , we set    k ) if x ∈ 1 − 2 Q ∩ C k for some k ∈ {1, . . . , Kn,m }, M(x − cn,m n+2 .un,m (x) :=  n,m 2 0 otherwise in 1 − n+2 Q. Finally, we let (n) .um (x)

:=

 u(n) (x) un,m (x)

if x ∈ Fn  if x ∈ 1 −

2 n+2

 Q

and we seek an estimate for

.

Q∩Su

N−1 |[u(n) (x). m ](x) · νu (x)| dH

By resorting to the notion of isotropic vectors (see [21]), it is possible to determine a rotation R such that, up to an error vanishing as n, m → ∞, the integral above is bounded by | tr M| = | tr(A − B)|, thus completing the verification of (3.51) and the proof of (3.47) for the bulk term with f (·) = | · |. Contemporaneously to [10], Šilhavý [57] proved a characterization of the relaxed bulk and interfacial energy densities arising from purely interfacial initial energies. The bulk relaxed energy is shown to coincide with the subadditive envelope of the initial interfacial energy, while the relaxed interfacial energy is the restriction of that envelope to rank-one tensors. Moreover, it is shown that the minimizing sequence required to define the relaxed bulk energy in the relaxation scheme of [20] can be taken in the more restrictive class required in the relaxation scheme of [8], thus establishing the equivalence of the two relaxation schemes departing from pure interfacial energies. From this characterization, the results in [10] are simple consequences. The main result in [57] reads as follows:

58

3 Energetic Relaxation to First-Order Structured Deformations

Theorem 3.9 Let W ≡ 0 and p ∈ [1, +∞). Let ψ : RN × SN−1 → [0, +∞) satisfy Assumptions 3.1-5 and 6, (3.22) (all three of them with no x-dependence), and the symmetry condition ψ(−λ, −ν) = ψ(λ, ν)

.

for all (λ, ν) ∈ RN × SN−1 .

(3.53)

Then, for every A, B ∈ RN×N , λ ∈ RN , and ν ∈ SN−1 , the relaxed bulk and interfacial energy densities H (A, B) := Hp (A, B) and h(λ, ν) := hp (λ, ν) defined in (3.17) and (3.18), respectively, are given by H (A, B) = Φ(A − B),

.

h(λ, ν) = Φ(λ ⊗ ν),

where Φ is a subadditive and positively 1-homogeneous function given by one of the following equivalent expressions: 1. Φ is the largest subadditive function on RN×N which is below ψ on dyads of the form λ ⊗ ν with (λ, ν) ∈ RN × SN−1 , namely, for every M ∈ RN×N  Φ(M) = sup Θ(M) : Θ is subadditive on RN×N and Θ(λ ⊗ ν)  ψ(λ, ν) for every (λ, ν) ∈ RN × SN−1 ;

.

(3.54a)

2. for every M ∈ RN×N , Φ(M) = inf

 m

ψ(λi , νi ) : (λi , νi ) ∈ RN × SN−1 for every i = 1, . . . , m

i=1

.

and

m

 λi ⊗ νi = M ;

i=1

(3.54b) 3. for every M ∈ RN×N ,  Φ(M) = inf E surf (u; Q) : u ∈ S C∗bulk(M) ,

.

(3.54c)

where  S C∗bulk(M) := u ∈ SBV (Q; RN ) : u|∂Q (x) = aM (x), ∇u = 0 a.e. in Q ;

.

4. for every M ∈ RN×N ,  Φ(M) = inf E surf (u; Q) : u ∈ C∗bulk(M) ,

.

(3.54d)

3.2 Applications

59

where  bulk .C∗ (M)

 ∇u = 0 ;



:= u ∈ SBV (Q; R ) : u|∂Q (x) = aM (x), N

Q

5. finally, if M = λ ⊗ ν is a rank-one matrix, then   Φ(λ. ⊗ ν) = inf E surf (u; Qν ) : u|∂Qν (x) = sλ,ν (x), ∇u = 0 a.e. in Q . (3.54e) Remark 3.8 We would like to remark that the last three formulae (3.54c), (3.54d), and (3.54e) for Φ are similar, both in spirit and in form, to those of Theorems 3.2 and 3.5. The only difference is that in (3.54c) and (3.54d) the infimization is taken on the classes of competitors S C∗bulk and C∗bulk, where the conditions on the boundary of the cube and on the gradient are swapped with respect to the conditions defining the class S C bulk in (3.49). The novelty in Šilhavý’s Theorem 3.9 is twofold: the discovery of formulae (3.54a) and (3.54b) and the equivalence of all of the expressions in (3.54). In particular, the equivalence of (3.54c) and (3.54d) generalizes the conclusion that all of the terms in the inequalities (3.50) and (3.51) equal | tr(A − B)|. Moreover, the formula (3.54c) permits an explicit formula to be derived for the bulk density H (1) for the integral representation in the context of strong structured deformations (Theorem 3.4, relations (3.36) and (3.37a)). Finally, Theorems 3.2 and 3.9 yield new formulae for the relaxed interfacial densities even when the initial energy is not purely interfacial. Corollary 3.2 Let p > 1; let W : RN×N → [0, +∞) satisfy Assumption 3.1-1 (without x-dependence); let ψ : RN × SN−1 → [0, +∞) satisfy Assumptions 3.1-5 and 6 (without x-dependence), (3.22), and the symmetry condition (3.53). Then hp (λ, ν) = Φ(λ ⊗ ν)

.

for all (λ, ν) ∈ RN × SN−1 ,

where Φ is as in Theorem 3.9. Proof The desired formula is an immediate consequence of (3.18) and (3.54e).

3.2.2 Other Special Settings for Relaxation In this section we present a special case of the theory discussed so far: the case of a convex bulk energy density W (see Sect. 3.2.2.1). Moreover, in Sect. 3.2.2.2, we present a proposal in a one-dimensional setting for including a contribution to the initial energy that takes into account an averaged contribution of the discontinuities of the deformation. This sets the basis and provides motivation for the study of the

60

3 Energetic Relaxation to First-Order Structured Deformations

relaxation of initial energies of non-local nature to be developed in Sect. 3.2.5 in general dimensions. 3.2.2.1 The Case of a Convex W The analysis underlying Theorem 3.9 along with the hypothesis of convexity of the initial bulk energy density leads to the following result, first provided in [46, Remark 5.6] for the case p = 1, and presented and proved here for the case p ∈ [1, +∞). Proposition 3.6 Assume that W : Ω × RN×N → R satisfies Assumptions 3.1-1 and 2 and is convex in the second variable; assume that ψ : Ω × RN × SN−1 → [0, +∞) satisfies Assumptions 3.1-4, 5, 6, and 7, and the symmetry condition ψ(x, . −λ, −ν) = ψ(x, λ, ν)

for all x ∈ Ω and (λ, ν) ∈ RN × SN−1 .

(3.55)

Then, the cell formula (3.17) becomes Hp (x0 , A, B) = W (x0 , B)  

. ψ(x0 , [u](x), νu (x)) dH N−1 (x) : u ∈ Cpbulk(A, B) + inf

(3.56)

Q∩Su

for every x0 ∈ Ω and A, B ∈ RN×N . Moreover, the infimum on the right-hand side is given by the expressions  Hp (x0 , A, B) − W (x0 , B) = sup Θ(x0 , A − B) : Θ(x0 , ·) : RN×N → [0, +∞) .

is subadditive and Θ(x0 , λ ⊗ ν)  ψ(x0 , λ, ν) for all λ ∈ RN and ν ∈ SN−1 . (3.57)

Proof We omit the explicit appearance of the point x0 ∈ Ω that appears on both sides of the desired decomposition and that remains fixed throughout the proof. Let A, B ∈ RN×N and u ∈ Cpbulk(A, B) be given. The cell formula (3.17) along with

3.2 Applications

61

Jensen’s inequality and continuity of W yield the inequalities



W (∇u(x)) dx + Q .

ψ([u](x), νu (x)) dH N−1 (x) Q∩Su







∇u(x) dx +

W Q

ψ([u](x), νu (x)) dH N−1 (x) Q∩Su



 W (B) + inf



ψ([u](x), νu (x)) dH

N−1

(x) : u ∈

Q∩Su

Cpbulk(A, B)

,

and, therefore, also yield the lower bound Hp (A, B)  W (B)  

. ψ([u](x), νu (x)) dH N−1 (x) : u ∈ Cpbulk(A, B) . + inf

(3.58)

Q∩Su

To obtain an upper bound for Hp (A, B) we note from (3.12) that

.

 Cpbulk(A, B) ⊃ u ∈ SBV (Q; RN ) : u|∂Q (x) = aA (x), ∇u = B L N -a.e. on Q =: C (A, B),

so that 

Hp. (A, B) = inf

W (∇u(x)) dx

Q



+

ψ([u](x), νu (x)) dH

N−1

(x) : u ∈

Q∩Su



 inf

Cpbulk(A, B)

W (∇u(x)) dx Q





+

ψ([u](x), νu (x)) dH

N−1

(x) : u ∈ C (A, B)

Q∩Su





= W (B) + inf

ψ([u](x), νu (x)) dH

N−1

(x) : u ∈ C (A, B)

Q∩Su



= W (B) + inf

ψ([u](x), νu (x)) dH N−1 (x) : Q∩Su

 u ∈ SBV (Q, RN ), u|∂Q (x) = aA−B (x), ∇u = 0 a.e.

62

3 Energetic Relaxation to First-Order Structured Deformations



= W (B) + inf

ψ([u](x), νu (x)) dH N−1 (x) : u ∈ SBV (Q, RN ), Q∩Su



u|∂Q (x) = aA−B (x),

∇u = 0, |∇u| ∈ L (Ω; R p

N×N

 ) ,

Q

(3.59) where the last equality follows from the equivalence of (iii) and (iv) in Theorem 3.9 and the inclusions S C∗bulk(A − B) ⊂ Cpbulk(A − B, 0) ⊂ C∗bulk(A − B). It is easy to see that the upper bound (3.59) and lower bound (3.58) just obtained for Hp (A, B) are the same. The relation (3.57) follows from (3.56) and from item (i) in Theorem 3.9. In formulae (3.56) and (3.57), B plays the role of G(x) and A the role of ∇g(x), so that A−B in (3.57) plays the role of M(x). As pointed out in [46], Proposition 3.6 thus provides sufficient conditions under which the relaxed bulk density has the form H\ (x, G(x)) + Hd (x, M(x)),

.

an energy density without disarrangements plus an energy density due to disarrangements. An analogous additive decomposition appears in Theorem 3.4, (3.39) in the context of strong structured deformations. Decompositions of this type have been used in studies of plasticity in various contexts [19, 27, 46]. Proposition 3.10 below offers another result (in the context of periodic homogenization) stemming from the convexity of the initial energy densities.

3.2.2.2 Motivation for Non-local Models The assumption that the relaxed bulk energy density admits an additive decomposition of the form H (∇g, G) = H (1)(∇g − G) + H (2)(G)

.

(3.60)

was shown by Larsen [43] not to hold in general for the relaxation process (3.11). Proposition 3.6 provides sufficient conditions on the initial energy for (3.60) to hold, but analyses in one-dimensional settings [26], [30, Part Two, Proposition 2.2] indicate that the relaxation scheme in the SBV setting, detailed in Sect. 3.1, leads to relaxed densities H (1) of linear type, a form too restrictive for many applications involving yielding and hysteresis. The determination of an upscaling of energy that justifies a more robust decomposition (3.60) was studied in [30] in a onedimensional setting by adding to the energy E in (3.1) a term with a non-local

3.2 Applications

63

dependence on the jumps [u] (see also [29, Section 2.6]). The additional non-local term

1 

[u](z) . Ψ (3.61) dx 2r 0 z∈Su ∩(x−r,x+r)

introduced in [30] involves the values of an energy density Ψ at the average

.

z∈Su ∩(x−r,x+r)

[u](z) 2r

of the jumps [u](z) of the deformation u at all points z ∈ Su that lie within the interval (x − r, x + r); the energy density then is integrated over the interval (0, 1) representing the one-dimensional body under consideration. The main result obtained there for a particular class of energy densities Ψ is the formula

.

lim lim

r→0 n→∞ 0



1

Ψ

z∈Sun ∩(x−r,x+r)



1 [un ](z) dx = Ψ (M(x)) dx 2r 0

(3.62)

that holds for sequences of the deformation un that converge to (g, G) in the sense (1.2) and whose jumps are all of the same sign. The iterated limit on the left-hand side of this formula is interpreted as the operation of upscaling, “ lim ”, n→∞ followed by the operation of spatial localization, “ lim ”, and is seen to result in a r→0 1 bulk energy 0 Ψ (M(x)) dx in which the original energy density Ψ is evaluated at the disarrangement tensor M = ∇g − G for the target structured deformation (g, G). In particular, properties of the energy density Ψ such as periodicity persist under the operations of upscaling and localization. Moreover, sufficient conditions were provided in [30] in order that the total energy obtained by addition of the nonlocal term (3.61) to the energy E(un ) satisfies  lim lim

r→0 n→∞



.



1

E(un ) +

Ψ 0

= I (g, G) +

z∈Sun ∩(x−r,x+r)

 [un ](z) dx 2r

1

Ψ (M(x)) dx 0

where I (g, G) is given by Theorem 3.2 for the considered structured deformation (we are not stressing here the dependence on the summability exponent p). The analysis of upscaling and spatial localization of non-local energies in multidimensional setting [46] will be discussed in Sect. 3.2.5.

64

3 Energetic Relaxation to First-Order Structured Deformations

3.2.3 Dimension Reduction in the Context of Structured Deformations In this section we present two approaches to dimension reduction results [17, 49] in the context of structured deformations. They both concern dimension reduction from three to two dimensions, as is customary in plate or membrane theory. The first one by Matias and Santos [49] relies on the model for first-order structured deformations presented in Sect. 3.1.2. A key aspect of the model developed in [49] is avoidance of the formation of macroscopic voids (or “holes”) in the direction normal to the two-dimensional target domain as a consequence of the dimension-reduction process. This is achieved by imposing that all jumps in the approximating sequences be properly aligned as is illustrated in Example 3.3 below, and it necessarily leads to a control on the second derivatives; as a matter of fact, this was the main motivation for the derivation of the mathematical formulation of first-order structured deformations in [8] and presented in Sect. 3.1.2. We refer the reader to, e.g., [16, 44, 45] for alternative variational approaches and to, e.g., [32] for an approach via Taylor expansion. We fix the dimensions N = d = 3 and we define the energy of a three dimensional structure with thickness  > 0 as follows



E (w) := W (∇w(y), ∇ 2 w(y)) dy + ψ (1) ([w](y), νw (y)) dH 2 (y) .

Sw ∩Ωε

Ω



+

S∇w ∩Ωε

ψ (2) ([∇w](y), ν∇w (y)) dH 2 (y), (3.63)

for w ∈ SBV 2 (Ω ; R3 ), where Ω = ω × (− 2ε , 2ε ) and ω ⊂ R2 is an open bounded set. Regarding the energy densities in (3.63), W , ψ (1) and ψ (2) satisfy Assumptions 3.2; moreover, ψ (2) is required to satisfy the following additional hypothesis

.

 for every A ∈ R3×3 and for every ν = (να , ν3 ) ∈ S2 (for α = 1, 2) ψ (2) (A, (να , ν3 )) = ψ (2) (A, (να , −ν3 )).

(3.64)

As usual in dimensional reduction problems, a change of variables allows one to consider a fixed domain at the expenses of rescaling some terms in the energy functional. Precisely, let y = (yα , y3 ) ∈ Ω and define x = (xα , x3 ) ∈ Ω := Ω1 = ω × (− 12 , 12 ) through xα = yα (for α = 1, 2) and x3 = y3 /. Then u(xα , x3 ) := w(xα , x3 )

.

3.2 Applications

65

is a function in SBV 2 (Ω; R3 ) and the integral in (3.63) becomes !

E (u) =  .

 2 2 u ∇ 2 u ∇3β u ∇33 ∇3 u 2 , ∇α,β u, α3 , , 2 W ∇α u, dx     Ω 

(νu )3 ψ (1) [u], (νu )α , dH 2 +  Su  "

[∇3 u] (ν∇u )3 , (ν∇u )α , dH 2 . + ψ (2) [∇α u],   S∇u

Studying the dimension reduction now consists in studying the asymptotic behavior as  → 0+ , in the sense of Γ -convergence (see Sect. 2.3), of the -scaled energies J (u) :=

.

E (u) . 

(3.65)

One seeks at this point an integral representation result for the functional   I (g, G, b) := inf lim inf Jn (un ) : εn → 0+ , {un } ⊂ R MS (g, G, b; {εn }) ,

.

n→∞

(3.66) defined for (g, G, b) ∈ GSD(ω; R3 × R3×2 ) × BV (ω; R3 ), where the class of admissible sequences is, for each sequence {εn } with limn→∞ εn = 0,   ∇3 un L1 R MS (g, G, b; {εn }) := {un } ∈ SBV 2 (Ω; R3) : un −→ (g, G), →b . SSD εn (3.67)

.

Notice that the generalized structured deformation (g, G) on which the functional I of (3.66) is defined is a pair of functions on the two-dimensional cross section ω ⊂ R2 of the body (which is also the limit object when the thickness ε → 0+ ). In particular, they only depend on the in-plane variables xα = (x1 , x2 ), and not on the out-of-plane variable x3 . Nonetheless, the effects of the limiting thinning procedure are still present in the fact that (g, G) takes values in R3 × R3×2 ; moreover, the functional I depends on the vector field b : ω → R3 that tracks the limiting orientation and stretchings of line segments in Ωε initially perpendicular to the cross-section ω, as seen in (3.67). We refer the reader to Remark 3.9 below, for further comments. The main result proved by Matias and Santos in [49] is Theorem 3.10 below, which holds under the above-mentioned hypotheses on the energy densities W , ψ (1) , and ψ (2) in (3.63). Before stating the theorem, we define the following objects: for

66

3 Energetic Relaxation to First-Order Structured Deformations

A ∈ R3×2 , B ∈ R3×3×2 , λ ∈ R3 , Γ1 , Γ2 , L ∈ R3×3 , and ν ∈ S1 , let

C

.

C

C

bulk,(1)

(A; Q ) :={u ∈ SBV (Q ; R3 ) : u|∂Q = 0, ∇u = A a.e. in Q },

C

bulk,(2)

(B; Q ) :={u ∈ SBV (Q ; R3×3 ) : u|∂Q (x) = aB (x)},

surf,(1)

surf,(2)

(λ, ν; Q ν ) :={u ∈ SBV (Q ; R3 ) : u|∂Q = sλ,ν , ∇u = 0 a.e. in Q ν },

(Γ1 , Γ2 , ν; Q ν ) :={u ∈ SBV (Q ν ; R3×3 ) : u|∂Q = sΓ1 ,Γ2 ,ν , ∇u = 0 a.e. in Q ν },

and  W (A, B) := inf W (A, B, B3 ) : B3 ∈ R3×3×1 ,  (1) . ψ (λ, να ) := inf ψ1 (λ, να , t) : t ∈ R ,  (2) ψ (Γ, να ) := inf ψ2 (Γ, να , t) : t ∈ R . Finally, for U ∈ A (ω), u ∈ SBV (ω; R3 ), and w ∈ SBV (ω; R3×3 ), let E

surf,(1)

(u; U ) :=

(2)

E L (w; U ) := E

rec,(2)

(1)

([u](x), νu (xα )) dH 1 (xα ),

W (L, ∇w(x)) dx +

.

Su ∩U

ψ

U

Sw ∩U

ψ

(2)

([w](x), νw (xα )) dH 1 (x),

∞

(w; U ) :=

W (w(x), ∇w(x)) dx U



+

Sw ∩U

ψ

(2)

([w](x), νw (x)) dH 1 (x)

Theorem 3.10 Let ω ⊂ R2 be a bounded, open set, let W , ψ (1) , and ψ (2) satisfy Assumptions 3.2 and (3.64), and let (g, G, b) ∈ GSD(ω; R3 ×R3×2 )×BV (ω; R3 ). The functional I defined in (3.66) does not depend on the sequence n → 0+ and admits an integral representation of the form I (g, G, b) = I (1) (g, G) + I (2) (G, b),

.

(3.68)

where



I (1) (g, G) :=

h(1) ([g](xα ), νg (xα )) dH 1 (xα )

H (1)((G − ∇g)(xα )) dxα +

.

ω

+



Sg

 dD c g (x H (1) − ) d|D c g|(xα ), α d|D c g| ω

3.2 Applications

67

I (2) (G, b) :=

H (2)(G(xα ), b(xα ), ∇G(xα ), ∇b(xα )) dxα ω



+

S(G,b)

+

(H ω

h(2) ((G, b)+ (xα ), (G, b)− (xα ), ν(G,b) (xα )) dH 1 (xα )  dD c (G, b) (xα ) d|D c (G, b)|(xα ). ) G(xα ), b(xα ), d|D c (G, b)|

(2) ∞

For A ∈ R3×2 , λ ∈ R3 , and ν ∈ S1 ,   bulk,(1) surf,(1) H (1)(A) := inf E (u; Q ) : u ∈ C (A; Q ) .  surf,(1)  surf,(1) (u; Qν ) : u ∈ C (λ, ν; Qν ) , h(1) (λ, ν) := inf E and for Γ1 , Γ2 , L ∈ R3×3 , B ∈ R3×3×2 , and ν ∈ S1 ,   bulk,(2) (2) H (2)(L, B) := inf E L (u; Q ) : u ∈ C (B; Q ) , .  rec,(2)  surf,(2) (u; Q ν ) : u ∈ C (Γ1 , Γ2 , ν; Q ν ) h(2) (Γ1 , Γ2 , ν) := inf E Proof (Sketch of the Proof of Theorem 3.10) The proof relies on the standard technique of proving upper and lower bounds for the two functionals in the righthand side of (3.68). The results provided in [12] (and in [49, Theorem 1.4]) yield integral representations for each of these two functionals. The lower bound for the relaxed energy densities follows then from Theorem 3.4, while assumption (3.64) is crucial for the proof of the upper bound estimates, where it is used to construct the admissible sequences for the functionals I (1) and I (2) . Remark 3.9 We observe the following. • Assumption (3.64) is a property of invariance under the particular reflection associated with the plane in the reference configuration occupied by the twodimensional “reduced” continuum. Most lattices and submacroscopic geometries indeed have such a plane of symmetry. • The term I (1) captures the energy due to the disarrangements given by the deformation at the submacroscopic scale. The growth condition Assumption 3.25 on ψ (1) implies that H (1) is homogeneous of degree 1 and, therefore, it coincides with its recession function: as a consequence, the term that takes the contribution of the Cantor part of the derivative into account is still H (1). Moreover, from the same assumption, it follows that H (1)(0) = 0. In particular, in the regions where ∇g = G there is a diffusion of jumps in the mid-surface that may appear macroscopically as a smooth deformation with a bulk contribution through the density H (1).

68

3 Energetic Relaxation to First-Order Structured Deformations

• The term I (2) captures the limiting lattice distortion in the mid-surface and in the transversal section, along with an interfacial penalization on jumps in the fields G and b. Example 3.3 To illustrate the geometrical meaning of the vector b as a tracker of the limiting deformation of out-of-plane segments, we consider the following example. Given a sequence of deformations clamped at the boundary and with finite total energy, for a given sequence {n }, one can consider a sequence {wn } ⊂ SBV 2 (Ωn ; R3 ) such that wn (x) = x in a neighborhood of ∂ω × (− 2n , 2n ), which, after rescaling, gives rise to a new sequence {un } ⊂ SBV 2 (Ω; R3) such that un (x) = (xα , n x3 ) and .

sup Jn (un ) < ∞. n

The coercivity conditions considered for W , ψ (1) , and ψ (2) then imply that .

|[un ]|(x) dH 2 (x) < +∞,



sup |D(∇un )|(Ω) + n

Su n

which, together with the boundary condition, implies in turn the boundedness of un and ∇un in the BV norm. Thus, up to a subsequence, un → g in L1 and ∇un  G in L1 . Defining bn := ∇3nun , the hypotheses considered imply in turn that .

sup |D(bn )|(Ω) < +∞, n

which, together with the boundary condition bn = (0, 0, 1) implies the boundedness of bn in the BV norm and consequently the existence of a subsequence such that bn → b in L1 . The field g represents the deformation of the mid-surface and the field b represents the rotation and compression of the normal sections. On the other hand, we have that # $

.

sup |D3 (∇un )|(Ω) + n

|[un ]e3 | + |D3 (bn )|(Ω) < Cn , Su n

which, together with boundary conditions, implies that the limit fields g, G, and b do not depend on x3 . The second approach we present is that provided in the work [17] where the procedures of dimension reduction and the incorporation of structured deformations to a thinning domain are both applied, one after the other, in either order. In this way, we obtain both a relaxed bulk and a relaxed interfacial energy at each stage of these two sequential applications of these procedures. General integral representations are obtained and, for a specific choice of an initial energy including only the surface term, the final energy densities can be made explicit and turn out to be the same,

3.2 Applications

69 3d-body (i)

(ii)

3d-body with disarrangements

2d-body (i)

(ii)

2d-body with disarrangements

2d-body with disarrangements

Fig. 3.1 The two paths for refinements of classical continuum theories: (i) structured deformations (SD) and (ii) dimension reduction (DR)

independent of the order of the relaxation processes. These explicit results can be compared with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time, as just presented in Theorem 3.10. The conceptual scheme of the sequential relaxation processes studied in [17] is presented in Fig. 3.1.The right-hand path in the graph below begins with the incorporation of disarrangements (i) and then applies dimension reduction (ii), while the left-hand path reverses the order. We depart as in Sect. 3.1 from bulk and interfacial tridimensional energy densities W3d : R3×3 → [0, +∞) and ψ3d : R3 × S2 → [0, +∞), satisfying Assumption 3.1-1 for p > 1 and Assumptions 3.14, 5, and 6, respectively, (with no x-dependence) as well as the following coercivity condition on W3d : there exists C > 0 such that 1 |A|p  W3d (A) (3.69) C   for all A ∈ R3×3 . Letting Ω := ω × − 2 , 2 , for u ∈ SBV (Ωε ; R3 ), we consider the initial energy .





Eε (u) :=

W3d (∇u(x)) dx +

.

Ωε

Ωε ∩Su

  ψ3d [u](x), ν(u)(x) dH 2 (x).

(3.70)

The Left-Hand Path To implement the first step “DR” in the left-hand path, we rescale as in (3.65):    ∇3 u(x) dx W3d ∇α u(x) ε Ω  

 (νu )3 (x)  + ψ3d [u](x), (νu )α (x) dH 2 (x). ε Ω∩Su

Fε (u) = .

70

3 Energetic Relaxation to First-Order Structured Deformations

For (u, d) ∈ SBV (ω; R3 ) × Lp (ω; R3 ) and εn → 0+ , define the relaxed functional   IpDR (u, d) := inf lim inf Fεn (un ) : {un } ∈ RpDR (u, d; {εn }) ,

.

n→∞

where  L1 RpDR (u, d; {εn }) := {un } ⊂ SBV (Ω; R3) : un → u, νun · e3 = 0,

.

I

 ∇3 un dx3  d in Lp (ω; R3 ) . εn

(3.71)

Here, I = (−1/2, 1/2) and, in writing the convergence un → u in L1 (Ω; R3 ) in formula (3.71), it is understood that u is extended to a function on Ω which is independent of x3 . The requirement νun · e3 = 0 in (3.71) rules out the occurrence of slips and separations on surfaces with normals parallel to the thinning directions e3 as part of the dimension reduction. Remark 3.10 Following the model in [13], we consider the weak convergence of the average with respect to the third variable to a field d(xα ) depending only on the coordinates in the cross-section ω. The dimension reduction scheme results in the replacement of the vector field u by a pair (u, d) of vector fields defined on a two-dimensional body, where u places the two-dimensional body into the threedimensional space and d is a “director field” on the two-dimensional body that is a geometrical residue of the passage from a three-dimensional body to a twodimensional body. To present the statement of the representation theorem for the relaxed energy obtained as a consequence of the dimension reduction process, we introduce the following objects: for A ∈ R3×2 , d, λ ∈ R3 , and η ∈ S1 let  p Cpbulk,DR (A, d; Q ) := (u, z) ∈ SBV (Q ; R3 ) × LQ −per (R2 ; R3 ) :

.

u|∂Q (xα ) = aA (xα );  Cpsurf,DR (λ, η; Q η ) := u ∈ SBV (Q ν ; R3 ) :

Q

 z(xα ) dxα = d ,

u|∂Q ν (xα ) = sλ,η (xα ); ∇u = 0 a.e. in Q η ,

3.2 Applications

71

where, by LQ −per (R2 ; R3 ) we mean the set of p-integrable functions z : Q → R3 extended by Q -periodicity to all of R2 ; for U ⊂ A (ω), u ∈ SBV (U ; R3 ), and p z ∈ LQ −per (R2 ; R3 ), let p

E surf,3d (u; U ) :=

.

E

bulk,3d

Su ∩U

(u, z; U ) :=

ψ 3d ([u](xα ), νu (xα )) dH 1 (xα )

W 3d (∇α u(xα )|z(xα )) dxα + E surf,3d (u; U ). U

The following is the main result on dimension reduction in the present setting [17, Theorem 11 and Remark 1]. Theorem 3.11 Let (u, d) ∈ SBV (ω; R3 ) × Lp (ω; R3 ). Then, under Assumptions 3.1-1, 4, 5, and 6 and (3.69), every sequence εn → 0+ admits a subsequence such that

IpDR (u, d) = HpDR (∇u(xα ), d(xα )) dxα + .

ω



(3.72)

+ ω∩Su

1 hDR p ([u](xα ), νu (xα )) dH (xα ),

3 1 where HpDR : R3×2 ×R3 → [0, +∞) and hDR p : R ×S → [0, +∞) are the relaxed bulk and interfacial energy densities and are defined as follows. For A ∈ R3×2 , d, λ ∈ R3 , and η ∈ S1 ,

  HpDR (A, d) := inf E bulk,3d (u, z; Q ) : (u, z) ∈ Cpbulk,DR (A, d; Q ) , . (3.73a)   surf,3d (u; Q η ) : u ∈ Cpsurf,DR (λ, η; Q η ) . (3.73b) hDR p (λ, η) := inf E

.

On the left-hand path in Fig. 3.1, the dimension reduction process is followed by the relaxation to structured deformations. To this end, given a triple (g, G, d) ∈ SDp (ω; R3 × R3×2 ) × Lp (ω; R3 ) (see Definition 3.1), we define the relaxed energy   IpDR,SD (g, G, d) := inf lim inf IpDR (un , d) : {un } ∈ RpDR,SD (g, G) ,

.

n→∞

(3.74)

where   RpDR,SD (g, G) := {un } ⊂ SBV (ω; R3 ) : un − (g, G)

.

SDp

(3.75)

72

3 Energetic Relaxation to First-Order Structured Deformations

where the convergence − is the one of in (3.7) with Ω replaced by ω. We now SDp

define the localizations of the terms of the functional defined in (3.72), namely we let, for U ∈ A (ω),

1 E surf,DR (u; U ) := hDR p ([u](xα ), νu (xα )) dH (xα ); U ∩Su



.

E bulk,DR (u, d; U ) := U

HpDR (∇u(xα ), d(xα )) dxα + E surf,DR (u; U );

moreover, for A, B ∈ R3×2 , for d, λ ∈ R3 , and for η ∈ S1 , we set  Cpbulk,DR,SD (A, B; Q ) := u ∈ SBV (Q ; R3 ) : u|∂Q (xα ) = aA (xα ),

.

Q





∇u(xα ) dxα = B, |∇u| ∈ L (Q ) ; p

 Cpsurf,DR,SD (λ, η; Q η ) := u ∈ SBV (Q η ; R3 ) : u|∂Q η = sλ,η , ∇u = 0 a.e. . The next theorem [17, Theorem 15] provides the desired integral representation for the energy IpDR,SD in (3.74), obtained at the end of the left-hand path in Fig. 3.1. The novel features of its proof are highlighted in [17, Remark 14, Proposition 12]. Theorem 3.12 Under Assumptions 3.1-1, 4, 5, 6, and (3.69), for each (g, G, d) ∈ SDp (ω; R3 × R3×2 ) × Lp (ω; R3 ), the energy IpDR,SD (g, G, d) defined in (3.74) admits an integral representation of the form

IpDR,SD (g, G, d) =

ω

.

HpDR,SD (∇g(xα ), G(xα ), d(xα )) dxα

(3.76)

+ ω∩Sg

hDR,SD ([g](xα ), νg (xα )) dH 1 (xα ), p

where, for A, B ∈ R3×2 , d, λ ∈ R3 , and η ∈ S1 ,  bulk,DR HpDR,SD . (A, B, d) := inf E (u, d; Q ) : (u, d)  ∈ Cpbulk,DR,SD (A, B; Q ) , . (3.77a)   (λ, η) := inf E surf,DR (u; Q η ) : u ∈ Cpsurf,DR,SD (λ, η; Q η ) . (3.77b) hDR,SD p

3.2 Applications

73

The Right-Hand Path To perform this sequence of relaxation to structured deformations followed by dimension reduction, we follow the analysis in [17] by considering a target structured deformation which is classical in the third component of the gradient. The structured deformations (g, (G\3 |∇3 g)) that we consider are determined from pairs (g, G\3 ) ∈ SBV (Ωε ; R3 ) × Lp (Ωε ; R3×2 ),

.

(3.78)

where the matrix field G\3 has only two columns, by adjoining the column vector field ∇3 g. Starting from the initial energy (3.70), we consider a target pair (g, G\3 ) as in (3.78) and we define the functional   IpSD (g, G\3 ) := inf lim inf E(un ) : {un } ∈ RpSD (g, G\3 ) ,

(3.79)

   RpSD (g, G\3 ) := {un } ⊂ SBV (Ωε ; R3 ) : un − g, (G\3 |∇3 g) ,

(3.80)

.

n→∞

where .

SDp

the convergence − being the one in (3.7) with Ω replaced by Ωε . SDp

Under the hypotheses of Theorem 3.12 on W3d and ψ3d , the integral representation result for IpSD follows immediately from Corollary 3.1 and Theorem 3.2. That is, we have that



SD \3 2 .Ip (g, G ) = HpSD (∇g(x), G\3 (x)) dx+ hSD p ([g](x), νg (x)) dH (x), Ω ∩Sg

Ω

with HpSD and hSD p given by the following instances of (3.17) and (3.18), respectively: for all A ∈ R3×3 , B ∈ R3×2 , λ ∈ R3 , ν ∈ S2 , HpSD (A, B \3 ) = inf





W3d (∇u(x)) dx + Q

ψ3d ([u](x), νu (x)) dH 2 (x) : Q∩Su

u ∈ SBV (Q; R3 ), u|∂Q (x) = aA (x), |∇u| ∈ Lp (Q; R3 ), 

∇u(x) dx = (B|Ae3 ) ;

.

Q



hSD p (λ, ν) = inf .

ψ3d ([u](x), νu (x)) dH 2 (x) : u ∈ SBV (Qν ; R3 ), Qν

 ∇u = 0 a.e., u|∂Qν (x) = sλ,ν (x) .

74

3 Energetic Relaxation to First-Order Structured Deformations

In order to apply the dimension reduction process, we rescale the variables by (xα , x3 ) → (xα , x3 /ε), thereby replacing the domain of integration Ωε by Ω = Ω1 , and we rescale the energy IpSD by dividing it by ε, obtaining SD Fp,ε (g, G\3 ) .

   ∇3 g(x) \3  , G (x) dx ∇α g(x) = ε Ω   

 (νg )3 (x)  + hSD ) (x) [g](x), (ν dH 2 (x). g α p  ε Ω∩Sg

HpSD

Also in this case, we expect the emergence of the vector d, measuring the residual out-of-plane deformation as an effect of the dimension reduction. Therefore, given (g, G, d) ∈ SDp (ω; R3 × R3×2 ) × Lp (ω; R3 ) and εn → 0+ , we define the energy SD,DR

.Ip

  SD (g , G) : {g } ∈ R DR (g, d; {ε }) , (g, G, d) := inf lim inf Fp,ε n n p n n

(3.81)

n→∞

where RpDR (g, d; {εn }) is defined in (3.71). We now define the localizations of the terms of the functional defined in (3.81), namely we let, for U ∈ A (ω), g ∈ SBV (U ; R3 ), z ∈ Lp (U ; R3 ), and G\3 ∈ Lp (U ; R3×2 ),

  2 E surf,SD (g; U ) := hSD p [g](xα ), ((νg )α (xα )|0) dH (xα ); .

E bulk,SD (g, z, G\3 ; U ) :=



U ∩Sg

U

  HpSD (∇α g(xα )|z(xα )), G\3 (xα ) dxα

+ E surf,SD (g; U ); moreover, for A, B ∈ R3×2 , for d, λ ∈ R3 , and for η ∈ S1 , we let  p Cpbulk,SD,DR (A, d; Q ) := (u, z) ∈ SBV (Q ; R3 ) × LQ −per (R2 ; R3 ) : |∇u| ∈ Lp (Q ), .

u|∂Q (xα ) = aA (xα ),



Q

z(xα ) dxα = d ;

 Cpsurf,SD,DR (λ, η; Q η ) := u ∈ SBV (Q η ; R3 ) : u|∂Q η (xα ) = sλ,η (xα ), ∇u = 0 a.e. .

3.2 Applications

75

The following integral representation theorem [17, Theorem 18] for the energy IpSD,DR in(3.81) on the right-hand path is the analog of Theorem 3.12 for the energy IpDR,SD in (3.76) on the left-hand path. Theorem 3.13 Under Assumptions 3.1-1, 4, 5, 6, and (3.69), given (g, G, d) ∈ SDp (ω; R3 × R3×2 ) × Lp (ω; R3 ), every sequence εn → 0+ admits a subsequence such that the relaxed energy IpSD,DR defined in (3.81) admits the integral representation

IpSD,DR (g, G, d) = HpSD,DR (∇g(xα ), G(xα ), d(xα )) dxα ω

.



(3.82)

+ ω∩Sg

hSD,DR ([g](xα ), νg (xα )) dH 1 (xα ), p

where, for A, B ∈ R3×2 , d, λ ∈ R3 , and η ∈ S1 ,  H. pSD,DR (A, B, d) := inf E bulk,SD (g, z, B; Q ) : (g, z)  ∈ Cpbulk,SD,DR (A, d; Q ) , . (3.83)   (λ, η) := inf E surf,SD (g; Q η ) : g ∈ Cpsurf,DR,SD (λ, η; Q η ) . (3.84) hSD,DR p Proof (Idea of the Proof of the Results) The most challenging results are the dimension reduction Theorems 3.11 and 3.13. The proofs follow arguments in [13]. The integral representation for IpSD is immediate by application of Theorem 3.2, taking into account Corollary 3.1. Theorem 3.13 also follows from Theorem 3.2 after proving that HpDR and hDR p satisfy Assumptions 3.1-1, 4, 5, and 6. Remark 3.11 We invite the reader to compare the settings for relaxation to structured deformations of (3.74) and (3.79) to the setting provided in (3.11): in the former cases, the classes RpDR,SD (g, G) and RpSD (g, G\3 ) (defined in (3.75)  and (3.80), respectively) do not contain the request that supn∈N ∇un Lp +  un BV < +∞, because this is granted by the coercivity conditions (3.69) and Assumption 3.1-4. As noted in points 3. and 4. of Remark 3.5, the coercivity hypotheses on W3d and ψ3d can be lifted, provided the relaxation is performed under the stricter condition just mentioned. A similar modification must be carried out for the class RpDR (u, d; {εn }) defined in (3.71), namely by requesting that   supn∈N ∇α un Lp + un BV < +∞. Remark 3.12 In general, it is not clear if the two processes lead to the same relaxed energy densities, that is, if IpDR,SD of formula (3.76) and IpSD,DR of formula (3.82) coincide. Some more details in this direction can be given for particular choices of the initial energy densities W3d and ψ3d in (3.70), as reported next.

76

3 Energetic Relaxation to First-Order Structured Deformations

In light of Remark 3.11, we now study the following particular case. Assume that [17, Section 5] W3d ≡ 0

and

.

ψ3d (λ, ν) = |λ · ν|,

(3.85)

for which the results proved in Sect. 3.2.1 will be used. In the computation of the relaxed energy densities going through the left-hand path, we obtain HpDR (A, d) = 0 in (3.73a) and hDR p (λ, η) = ψ3d (λ, (η|0)) in (3.73b), by applying Theorem 3.8. It is important to notice here that in the dimension reduction process of an initial energy deprived of the bulk contribution the dependence on the vector d is lost. In the second step, when relaxing to structured deformations, we obtain   HpDR,SD (A, B, d) =  tr((A|0) − (B|0)) = |A11 + A22 − B11 − B22 |

.

(3.86)

in (3.77a) and hDR,SD (λ, η) = hDR p p (λ, η) = ψ3d (λ, (η|0)) in (3.77b), again thanks to Theorem 3.8. Thus, the energy (3.76) has the expression

IpDR,SD (g, G, d) =

ω

    tr (∇g|0) − (G|0)  dxα

  [g] · (νg |0) dH 1 (xα )

+

.

= ω

ω∩Sg

  ∂1 g 1 + ∂2 g 2 − G11 − G22  dxα

+ ω∩Sg

(3.87)

  [g 1 ](νg )1 + [g 2 ](νg )2  dH 1 (xα ).

The computations for the right-hand path yield HpSD (A, B \3 ) = | tr(A − (B \3 |A3 ))| \3 3 and hSD p (λ, ν) = ψ3d (λ, ν) = |λ · ν|, by Theorem 3.8, where (B |A ) denoted the \3 3×3 matrix whose first two columns are B and whose third column is A3 , the third column of A. In (3.83) one can prove that HpSD,DR (A, B, d) = HpDR,SD (A, B, d) (λ, η) = hSD that we found in (3.86) and, once again, that hSD,DR p p (λ, (η|0)) = ψ3d (λ, (η|0)) in (3.84), so that the energy (3.82) has the same expression as the energy in (3.87). The result is therefore that, with the choice (3.85), the relaxed energies (3.76) and (3.82) coincide and are given by the formula in (3.87), see [17, Proposition 19]. This result can be summarized in Fig. 3.2. Remark 3.13 In [17, Section 6] it is also proved that, in the special case (3.85), the energy in (3.87) is strictly larger than that obtained in Theorem 3.10. Indeed, it is possible to show that the energy in formula (3.68) is zero. In fact, setting ψ (2) = 0 makes the energy I (2) vanish, while an ad hoc construction given in [17] shows that I (1) also vanishes. Therefore, we obtain that the simultaneous relaxation process described in Theorem 3.10 yields an energy which is strictly smaller than

3.2 Applications

77 W3d ≡ 0

(i)

(ii)

H pSD (A, B\3 ) = | tr(A − B\3 )|

H pDR = 0

(i)

(ii) H pDR,SD (A, B, d) = H pSD,DR (A, B, d)

Fig. 3.2 The two paths for refinements of classical continuum theories under the assumption (3.85): we see that the two paths give the same final relaxed energy

those given by Theorems 3.12 and 3.13, confirming the heuristic idea that letting the material adjust through disarrangements while performing a dimension reduction process is energetically more favorable than a succession of the two mechanisms.

3.2.4 Optimal Design of Fractured Media In [48] an optimal design problem is studied which can incorporate elements of plasticity in a way that it is suited to treat both composite materials (made of components with different mechanical properties) and polycrystals (where the same material develops different types of slips and separations at the microscopic level, see [54, 55]). The pioneering variational approach for these problems is due to Kohn and Strang [39–42] (see also [53]). In order to do so, the framework introduced by Ambrosio and Buttazzo in [4] (see also [38]) is extended by considering a material with two components each of which undergoes an independent (first-order) structured deformation. The generalization of our model to account for materials with more than two components, or to polycrystals, is straightforward. The energy functional considered (see (3.88) below) features: 1. different bulk densities associated with each of two components; 2. surface energy densities to account for the jumps in the deformations inside each component; 3. a perimeter penalization (which measures the boundary between the two components, independently of any discontinuities in the deformation); and finally 4. a surface energy term that accounts for the interaction between neighboring components (where both discontinuities in the deformation and in the components are counted).

78

3 Energetic Relaxation to First-Order Structured Deformations

More precisely, in order to take the presence of two components into account, consider a set of finite perimeter U ⊂ Ω describing one of them, and let χ ∈ BV (Ω; {0, 1}) be its characteristic function. Denoting by {χ = 1} the set of points in Ω with density 1, by {χ = 0} the set of points in Ω with density 0 (see [5, Definition 3.60]), and by letting u ∈ SBV (Ω; Rd ), the following functional E : BV (Ω; {0, 1})×SBV (Ω; Rd ) → [0, +∞[ is considered

  (1 − χ(x))W 0 (∇u(x)) + χ(x)W 1 (∇u(x)) dx E(χ, u) := Ω



+

ψ10 ([u](x), νu (x)) dH N−1 (x)

.

Ω∩{χ=0}∩Su

ψ11 ([u](x), νu (x)) dH N−1 (x)

+

+

Ω∩{χ=1}∩Su

ψ2 (χ + (x), χ − (x), u+ (x), u− (x), νu (x)) dH N−1 (x)

Ω∩Sχ ∩Su

+ |Dχ|(Ω). (3.88) For i ∈ {0, 1}, W i : Rd×N → [0, +∞[ is the bulk energy density associated with the ith component, ψ1i : Rd × SN−1 → [0, +∞[ is the surface energy density associated with jumps in the deformation in the ith component, and ψ2 : {0, 1}2 × (Rd )2 × SN−1 → R is the surface energy density associated with the jumps in the deformation at the interface between the two components. The energy contribution of the interface ∂U , discounting the discontinuities of the deformation u, is carried by |Dχ|(Ω), the total variation of Dχ in Ω. In (3.88) the jump set S(χ,u) of the pair (χ, u) is split into the disjoint union S(χ,u) = (Sχ ∩ Su ) ∪ (Su \ Sχ ) ∪ (Sχ \ Su ). In this way, the energy contribution of the jumps in u occurring in {χ = 0} ∩ (Su \ Sχ ) and {χ = 1} ∩ (Su \ Sχ ) is carried by ψ10 and ψ11 , respectively. Finally, the energy contribution of the jumps of u across ∂U is carried by ψ2 in Sχ ∩ Su . The main goal in [48] was to find an integral representation for the functional I : BV (Ω; {0, 1}) × SD(Ω; Rd × Rd×N ) → [0, +∞[ defined as the relaxation of the functional E in (3.88) by   I (χ, . g, G) := inf lim inf E(χn , un ) : {(χn , un )} ∈ RpOD (χ, g, G; Ω) , n→∞

(3.89)

3.2 Applications

79

where, for p > 1, χ ∈ BV (Ω; {0, 1}), (g, G) ∈ SD(Ω; Rd × Rd×N ), and U ∈ A (Ω) the class of admissible sequences is defined as  RpOD (χ, g, G; U ) := {(χn , un )} ⊂ BV (U ; {0, 1})×SBV (U ; Rd ) : .  p ∗ χn  χ in BV (U ; {0, 1}), un − (g, G) SD

p

and where here by un − (g, G) we mean SD

un → g

.

in L1 (Ω; Rd )

and

∇un  G

in Lp (Ω; Rd×N ).

In addition to Assumptions 3.1-1, 4, 5, and 6 (with no x-dependence) on W i and ψ1i (for i ∈ {0, 1}), we will need the following hypotheses on ψ2 . Assumptions 3.3 Let ψ2 : {0, 1}2 × (Rd )2 × SN−1 → [0, +∞) be a continuous function satisfying the following conditions 1. there exists C > 0 such that for all a, b ∈ {0, 1}, c, d ∈ Rd , and ν ∈ SN−1 , 0  ψ2 (a, b, c, d, ν)  C(1 + |a − b| + |c − d|);

.

2. (mechanical consistency of the surface energy density) for all a, b ∈ {0, 1}, c, d ∈ Rd , and ν ∈ SN−1 , ψ2 (a, b, c, d, ν) = ψ2 (b, a, d, c, −ν);

.

3. there exists C > 0 such that for all a, b ∈ {0, 1}, ci , di ∈ Rd , i = 1, 2, and ν ∈ SN−1 ,   |ψ2 (a, b, c1, d1 , ν) − ψ2 (a, b, c2 , d2 , ν)|  C |c1 − c2 | − |d1 − d2 | .

 C|(c1 − d1 ) − (c2 − d2 )|.

Remark 3.14 We notice that Assumption 3.3-2 is a symmetry condition on the density ψ2 analogous to those stated in (3.53) and (3.55). The main result [48, Theorem 3.3] reads as follows. Theorem 3.14 Let χ ∈ BV (Ω; {0, 1}) and (g, G) ∈ SD(Ω; Rd × Rd×N ). Let I be the functional defined by (3.89) as the relaxation of the functional E defined in (3.88), for density functions W i , ψ1i , i ∈ {0, 1} satisfying Assumptions 3.1-

80

3 Energetic Relaxation to First-Order Structured Deformations

1, 4, 5, and 6 (with no x-dependence) for some p ∈ (1, +∞), and ψ2 satisfying Assumptions 3.3. Then I (χ, g, G) admits an integral representation of the form

I (χ, g, G) =

Hp (χ(x), ∇g(x), G(x)) dx Ω

.



hp (χ + (x), χ − (x), g + (x), g − (x), νg (x)) dH N−1 (x),

+ Ω∩S(χ,g)

where, for i ∈ {0, 1} and A, B ∈ Rd×N ,  Hp (i, A, B) := inf E1i (u, Q) : u ∈ Cpbulk(A, B)

.

(3.90)

(E1i is the analogue of (3.15b) with densities W i and ψ1i instead of W and ψ) and, for a, b ∈ {0, 1}, c, d ∈ Rd , and ν ∈ SN−1 ,  hp (a, b, c, d, ν) := inf E1surf,0 (u, Qν ∩ {χ = 0}) + E1surf,1 (u, Qν ∩ {χ = 1}) + E2surf (χ, u, Qν ) + |Dχ|(Qν ) : (χ, u) ∈ Cpsurf (a, b, c, d, ν)

.

(3.91) where E1surf,i

.

is the analogue of (3.15a) with density ψ1i instead of ψ,

E2surf (χ, u, Qν ) :=

.

ψ2 (χ + (x), χ − (x), u+ (x), u− (x), νu (x)) dH N−1 (x),

Qν ∩Sχ ∩Su

and (recalling (3.14))

.

Cpsurf (a, b, c, d, ν) := {(χ, u) ∈ BV (Qν ; {0, 1})×SBV (Qν ; Rd ) : χ|∂Qν = sa,b,ν , u|∂Qν = sc,d,ν , ∇u = 0 L N -a.e.}.

Remark 3.15 We observe that the relaxed bulk energy density Hp : BV (Ω; {0, 1})× (L1 (Ω; Rd×N ))2 → [0, +∞[ defined in (3.90) depends on the structured deformation on {χ = 0} or {χ = 1}, and that the relaxed interfacial energy density hp : {0, 1}2 ×(Rd )2 ×SN−1 → [0, +∞[ defined in (3.91) can be further specialized on the various pieces of the decomposition of S(χ,u) , as detailed below. Note also that in the classical deformation setting, that is no jumps in u and G = ∇u, the optimal design problem studied by Carita and Zappale [18] is recovered; moreover, considering just one material leads to the results in [20]. The formula for H only sees each component separately, so for i ∈ {0, 1} and A, B ∈ Rd×N , we can write

3.2 Applications

81

Hp (i, A, B) = Hpi (A, B), where the latter is the bulk energy density given by formula (3.17). The formula for hp can be specialized giving rise to three cases: • in Sχ ∩ Su we have in fact the formula in (3.91) fully reflects both the optimal design and structured deformation effects. • in Su \ Sχ the formula for hp reads as (3.18) for i ∈ {0, 1}, as it only takes into account the structured deformation effect in each component; • in Sχ \ Su the formula for hp reads   hp (a, b, ν) := inf |Dχ|(Qν ) : χ ∈ Cpsurf (a, b, ν) = |(b − a) ⊗ ν| = 1

.

for a, b ∈ {0, 1} and ν ∈ SN−1 , with  Cpsurf (a, b, ν) := χ ∈ BV (Qν ; {0, 1}) : χ|∂Qν = sa,b,ν .

.

and only reflects the optimal design setting of [18]. Proof (Sketch of the Proof of Theorem 3.14) The proof follows the usual procedure in the previous results (see, e.g., Theorem 3.2), that is: • prove that the functional I (χ, g, G) defined in (3.89) is the restriction of a suitable Radon measure to open subsets of Ω; • prove a lower bound and an upper bound estimate in terms of its integral representation when the target field g is in L∞ (Ω; Rd ); • prove the general case via a truncation argument. Some extra care is needed when following this procedure, since the convergence of χn also needs to be taken into account. To do this, arguments in [18] are successfully combined with those in [20]; we refer the reader to [48] for the details.

3.2.5 Relaxation of Non-local Energies In [46] a multidimensional generalization of the one-dimensional model discussed in Sect. 3.2.2.2 is presented. In particular, multiscale geometrical changes are described via structured deformations (g, G) ∈ SD(Ω; Rd × Rd×N ) and the nonlocal energetic response at a point x ∈ Ω via a function Ψ of the weighted averages ∗ of the jumps [un ](y) of admissible deformations un − (g, G) at points y within SD

a distance r of x. These weighted averages account for non-linear contributions of [un ] to the relaxed bulk energy density Hp through the disarrangement tensor M. (See the discussion in Sect. 3.2.2.2). Under broader conditions on Ψ , the upscaling “n → ∞” results in a macroscale energy that depends through Ψ on (1) the jumps [g] of g and the disarrangement field M = ∇g − G, (2) the “horizon” r, and (3) the weighting function αr for

82

3 Energetic Relaxation to First-Order Structured Deformations

microlevel averaging of [un ](y). The upscaling “n → ∞” is followed by spatial localization “r → 0” and this succession of processes results in a purely local macroscale energy I (g, G) that depends through Ψ upon the jumps [g] of g and upon the disarrangement field M = ∇g − G, alone. The convolution kernels that will be responsible for the averaging process in the non-local term of the energy play a role analogous to the operation “division by 2r” in the functional (3.61). Let αr (x) :=

.

1 x  , α rN r

(3.92a)

where α ∈ Cc∞ (RN ), spt α ⊂ B1 , α  0, α(−x) = α(x), and

α(x) dx = 1.

.

B1

(3.92b) It is easily verified that, for every x ∈ RN , the convolution

μ ∗ αr (x) =

αr (x − y)dμ(y) = r

.

U

−N





x−y α r Br (x)∩U

dμ(y).

is well defined and it is a regular function. The convolution of a measure with a kernel of the type in (3.92) has good convergence properties with respect to the weak-* and ·-strict convergences of measures. We now introduce and describe two classes of continuous functions from which we are going to take the energy density of the non-local energy. We consider a continuous function Ψ : Ω × Rd×N → [0, +∞) belonging to the class (E) or (L), which are characterized as follows: (E) Ψ can be extended to a function (still denoted by Ψ ) belonging to C(Ω×Rd×N ) with the property that limt →+∞ Ψ (x, tξ )/t exists uniformly in x ∈ Ω and ξ with |ξ | = 1. Such functions Ψ form the class E(Ω × Rd×N ). In particular, this entails that (i) Ψ has at most linear growth at infinity with respect to the second variable, namely there exists CΨ > 0 such that |Ψ (x, ξ )|  CΨ (1 + |ξ |)

.

for all x ∈ Ω and ξ ∈ Rd×N ;

(ii) for all x ∈ Ω and ξ ∈ Rd×N there exists the limit .

lim

x →x ξ →ξ t →+∞

Ψ (x , tξ ) . t

(3.93)

3.2 Applications

83

(L) (i) Ψ is Lipschitz with respect to the second variable, i.e., there exists LΨ > 0 such that |Ψ (x, ξ ) − Ψ (x, ξ )|  LΨ |ξ − ξ |, .

for all x ∈ Ω and ξ, ξ ∈ Rd×N ;

(3.94)

(ii) there exists a continuous function ω : [0, +∞) → [0, +∞), with ω(s) → 0+ as s → 0+ , such that |Ψ (x, ξ )−Ψ (x , ξ )|  ω(|x −x |)(1+|ξ |),

.

for all x, x ∈ Ω, ξ ∈ Rd×N .

Notice that, by fixing ξ ∈ Rd×N , (3.94) implies that there exists CΨ > 0 such that (3.93) holds. For Ψ : Ω × Rd×N → [0, +∞) in (E) or (L), set Ωr := {x ∈ Ω : dist(x, ∂Ω) > r}, and for u ∈ SBV (Ω; Rd ), define the averaged interfacial energy

E (u) := αr

Ωr



.

= Ωr

  Ψ x, (D s u ∗ αr )(x) dx 

Ψ x,

αr (x − y)[u](y) ⊗ νu (y) dH N−1 (y) dx. Br (x)∩Su

(3.95) In order to pass to the limit in n (keeping r fixed) in (3.95) with u replaced by ∗ un − (g, G), the following condition on the admissible sequences is needed: SD

∗

D s un  (∇g − G)L N + D s g

.

in M (Ω; Rd×N ).

(3.96)

Given (g, G) ∈ SD(Ω; Rd × Rd×N ), we let   ∗ Ad(g, G) := un ∈ SBV (Ω; Rd ) : un − (g, G) in and (3.96) holds .

.

SD

The goal is to find an integral representation for the functional     Jp (g, G) := lim inf lim inf E(un ) + E αr (un ) , {un } ∈ Ad(g, G) ,

.

r→0

n→∞

(3.97)

where E(u) is given by (3.10). The limiting procedure of the non-local term (3.95) is addressed independently and then coupled with the local term, that is, with the relaxed energy in the context of Theorem 3.2. Remark 3.16 We note that given (g, G) ∈ SD(Ω; Rd × Rd×N ), any sequence of functions un ∈ SBV (Ω; Rd ) admissible for Theorem 3.2, such that lim infn→∞ EL (un ) is finite, belongs to Ad(g, G). In fact, by (3.11) and (3.9),

84

3 Energetic Relaxation to First-Order Structured Deformations

un converges to (g, G) in the sense of (3.4), thus providing a uniform bound on the L1 norm of un and on the Lp norm of ∇un . Then, the coercivity of ψ in Assumption 3.1-4 ensures that {|D s un |} is also a uniformly bounded sequence of measures, so that {Dun } is bounded in total variation and therefore has a weakly-* ∗ converging subsequence (not relabelled) such that D s un  (∇g − G)L N + D s g, that is, (3.96) holds true. Moreover, by the metrizability on compact sets of the weak-* convergence, (see [5, Remark 1.57, Theorem 1.59, and subsequent comments]) and Urysohn’s principle, the whole sequence un belongs to Ad(g, G). Moreover, the collection of such sequences un is non-empty, and the infimum over such un of lim infn→∞ EL (un ) equals IL (g, G). The result concerning the innermost limit in (3.97) for the non-local energy E αr introduced in (3.95) is the following. Theorem 3.15 ([46, Theorem 3.1]) Let Ω ⊂ RN be a bounded Lipschitz domain, let Ψ : Ω × Rd×N → [0, +∞) be a continuous function, for r > 0 let αr be as in (3.92), and let E αr be as in (3.95). Then for every (g, G) ∈ SD(Ω; Rd × Rd×N ) and for every admissible sequence un ∈ Ad(g, G) 

Ψ x,

lim E (un ) = αr

n→∞

Ωr

.

αr (y − x)(∇g − G)(y) dy Br (x)



αr (y − x)[g](y) ⊗ νg (y) dH

+

N−1

(y) dx

Br (x)∩Sg

=: I αr (g, G; Ωr ). (3.98) Proof (Sketch of the Proof of Theorem 3.15) Set μn := D s un and μ := (∇g − G)L N + D s g. Since un ∈ Ad(g, G), and by the properties of convolutions, we have that μn ∗ αr (x) → μ ∗ αr (x) for L N -a.e. x ∈ Ωr .

.

The result then follows upon proving an uniform bound on |Ψ (x, μn ∗αr (x))|,which together with the continuity of Ψ , imply that Ψ (x, μn ∗ αr )(x) → Ψ (x, μ ∗ αr )(x) for L N a.e. x ∈ Ωr .

.

The result follows then immediately by application of Lebesgue’s Dominated Convergence Theorem.

3.2 Applications

85

In order to address the limit in r of the non-local term, we need to extend appropriately (g, G) ∈ SD(Ω; Rd × Rd×N ) to a larger domain. In particular, we ¯ ∈ BV (RN ; Rd ) × L1 (RN ; Rd×N ) satisfying provide a pair (g, ¯ G) ¯ Ω = (g, G); (e1) (g, ¯ G)| (e2) |D g|(R ¯ N )  CgBV (Ω;Rd ) , for some constant C > 0; (e3) |D s g|(∂Ω) ¯ = 0. Because ∂Ω is Lipschitz and g ∈ BV (Ω; Rd ), a function g¯ ∈ BV (RN ; Rd ) satisfying g| ¯ Ω = g, (e2), and (e3) is provided by Gerhardt [36, Theorem 1.4]. Any ¯ ∈ L1 (RN ; Rd×N ) satisfying G| ¯ Ω = G provides the second element function G ¯ ¯ satisfying (e1-3) and for αr as of the pair (g, ¯ G) satisfying (e1-3). For any (g, ¯ G) in (3.92), we define N ¯ μ¯ := (∇ g¯ − G)L + D s g¯

and

.

μ¯ r := (μ¯ ∗ αr )L N ,

where we observe that the latter expression is well defined for every x ∈ RN . N + |D s g|. ¯ Moreover, |μ| ¯ = |∇ g¯ − G|L ¯ Finally, we define the functional αr ¯ I (g, ¯ G; Ω) by .

αr ¯ Ω) := ¯ G; I (g,

Ψ (x, (μ¯ ∗ αr )(x)) dx. Ω

and note that it can be written as ¯ Ω) = I αr (g, G; Ωr ) + .I (g, ¯ G; αr

Ψ (x, (μ¯ ∗ αr )(x)) dx. Ω\Ωr

We are now ready to state and sketch the proof of the result concerning the outermost limit in (3.97) for the non-local energy E αr introduced in (3.95). Theorem 3.16 ([46, Theorem 3.2]) Let Ω ⊂ RN be a bounded Lipschitz domain, let Ψ : Ω × Rd×N → [0, +∞) be a continuous function belonging to (E) or (L), and let αr be as in (3.92). Then for any (g, G) ∈ SD(Ω; Rd × Rd×N ) and for I αr (g, G; Ωr ) defined in (3.98) INL (g, G) := lim I αr (g, G; Ωr ),

.

r→0+

(3.99)

is given by

  Ψ x, ∇g(x)−G(x) dx +

INL (g, G) =

.

Ω

Ω∩Sg

 dD s g (x) d|D s g|(x), Ψ ∞ x, d|D s g|

with Ψ ∞ defined by (2.2). Proof (Sketch of the Proof of Theorem 3.16) Let (g, G) ∈ BV (RN ; Rd ) × L1 (RN ; Rd×N ) satisfying (e1-3) above. By the properties of convolutions, (e3), and

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3 Energetic Relaxation to First-Order Structured Deformations

the linear growth of Ψ , it is easily shown that

.

lim

r→0+ Ω\Ωr

Ψ (x, (μ¯ ∗ αr )(x)) dx = 0

and that INL (g, G) in (3.99) is well defined. The proof of the integral representation of INL (g, G) follows from Theorem 2.1. In particular, for Ψ belonging to the class (E), we conclude by applying the continuity result stated in Theorem 2.1(ii), while for Ψ belonging to the class (L), we conclude by applying the upper semicontinuity result stated in Theorem 2.1(i), using arguments similar to the ones in [7]. Finally, the following result shows that the local term captured through Theorem 3.2 and the non-local effect captured by the previous theorems can be added to obtain an integral representation for Jp (g, G). Theorem 3.17 ([46, Theorem 5.5]) Let Ω ⊂ RN be a bounded Lipschitz domain, let Ψ : Ω × Rd×N → [0, +∞) be a continuous function belonging to (E) or (L), and let αr be as in (3.92). Let W, ψ be as in Theorem 3.2. Then Jp (g, G) defined in (3.97) is given by Ip (g, G) in Theorem 3.2 plus INL (g, G) in Theorem 3.16. Proof The result follows immediately from the superadditivity of liminf together with Remark 3.16. Remark 3.17 In special settings, such macroscale energies Jp (g, G) have been shown to support the phenomena of yielding and hysteresis, and the results in this section provide a broader setting for studying such yielding and hysteresis. As an illustration, in [46] the results are applied in the context of the plasticity of single crystals.

3.2.6 Periodic Homogenization of Structured Deformations In this section, which summarizes the results contained in [2], we treat a periodic homogenization problem in the context of structured deformations; more precisely, we focus on heterogeneous, hyperelastic, defective materials featuring a fine periodic microstructure, with the goal of providing an asymptotic analysis of the energies associated with these materials, as the fineness of their microstructure vanishes and as the disarrangements of their deformations diffuse throughout the materials. Both the bulk and the surface initial energy densities W and ψ are going to depend explicitly on the spatial variable in a periodic fashion, namely the energy associated with a deformation u ∈ SBV (Ω; Rd ) has the expression

Eε (u) :=

W

.

Ω

x ε

 , ∇u(x) dx + Ω∩Su

ψ

x ε

 , [u](x), νu (x) dH N−1 (x), (3.100)

3.2 Applications

87

where W : RN × Rd×N → [0, +∞) and ψ : RN × Rd × SN−1 → [0, +∞) are Qperiodic in the first variable (namely, W (x + q, ξ ) = W (x, ξ ) and ψ(x + q, λ, ν) = ψ(x, λ, ν) for every q ∈ ZN ), and ε > 0 is the length scale of the microscopic heterogeneities. The standing assumptions on W and ψ are collected here: some of them are essentially the same already introduced in Assumptions 3.1 and the symmetry condition (3.55), with the only difference that now W and ψ are defined on the whole of RN as far as the x-dependence is concerned. Assumptions 3.4 Let p > 1 and let W : RN × Rd×N → [0, +∞) and ψ : RN × Rd × SN−1 → [0, +∞) be continuous functions such that 1. for every A ∈ Rd×N , λ ∈ Rd , and ν ∈ SN−1 , the functions x → W (x, A) and x → ψ(x, λ, ν) are Q-periodic; 2. there exists CW > 0 such that, for every x ∈ RN and for every A1 , A2 ∈ Rd×N , |W (x, A1 ) − W (x, A2 )|  CW |A1 − A2 |(1 + A1 |p−1 + |A2 |p−1 );

.

3. there exists a function ωW : [0, +∞) → [0, +∞) such that ωW (s) → 0 as s → 0+ such that for every x1 , x2 ∈ RN and A ∈ Rd×N |W (x1 , A) − W (x2 , A)|  ωW (|x1 − x2 |)(1 + |A|p );

.

> 0, and c > 0 such that W (x, A)  C |A|p − c for every 4. there exist CW W W W A ∈ Rd×N and a.e. x ∈ Ω. 5. there exists cψ , Cψ > 0 such that, for every x ∈ RN , λ ∈ Rd , and ν ∈ SN−1 ,

cψ |λ|  ψ(x, λ, ν)  Cψ |λ|;

.

6. there exists a function ωψ : [0, +∞) → [0, +∞) such that ωψ (s) → 0 as s → 0+ and such that for every x1 , x2 ∈ RN and (λ, ν) ∈ Rd × SN−1 |ψ(x1 , λ, ν) − ψ(x2 , λ, ν)|  ωψ (|x1 − x2 |)|λ|;

.

7. for every x ∈ RN and ν ∈ SN−1 , the function λ → ψ(x, λ, ν) is positively homogeneous of degree one, i.e., for every λ ∈ Rd and t > 0, ψ(x, tλ, ν) = tψ(x, λ, ν);

.

8. for every x ∈ RN and ν ∈ SN−1 , the function λ → ψ(x, λ, ν) is subadditive, i.e., for every λ1 , λ2 ∈ Rd , ψ(x, λ1 + λ2 , ν)  ψ(x, λ1 , ν) + ψ(x, λ2 , ν);

.

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3 Energetic Relaxation to First-Order Structured Deformations

9. for every x ∈ RN , the function (λ, ν) → ψ(x, λ, ν) is symmetric, i.e., for every λ ∈ Rd and ν ∈ SN−1 , ψ(x, λ, ν) = ψ(x, −λ, −ν);

.

Remark 3.18 We make the following observations. 1. The p-Lipschitz continuity in Assumption 3.4-2 jointly with Assumptions 3.4-1 and 3 imply that W has p-growth from above in the second variable, namely that there exists CW > 0 such that for every (x, A) ∈ RN × Rd×N W (x, A)  CW (1 + |A|p ).

.

(3.101)

On the contrary, p-growth from above jointly with the quasiconvexity of the bulk energy density in the gradient variable (which is the natural assumption in equilibrium problems in elasticity) returns the p-Lipschitz continuity. 2. Assumptions 3.4-5 and 8 imply Lipschitz continuity of the function λ → ψ(x, λ, ν), i.e., for every (x, ν) ∈ RN × SN−1 and for every λ1 , λ2 ∈ Rd , |ψ(x, λ1 , ν) − ψ(x, λ2 , ν)|  Cψ |λ1 − λ2 |

.

(3.102)

(compare with Remark 3.2). 3. Assumption 3.4-9 (compare with (3.55)) allows one to identify ψ(x, λ, ν) with (x, λ ⊗ ν), for a suitable function ψ  : RN × Rd×N → [0, +∞). Compare with ψ Theorem 3.9, Corollary 3.2, and Assumption 3.3-2. We will perform a relaxation analogous to that in (3.11) for the ε-dependent initial energy (3.100). For every sequence εn → 0, recalling (3.9), we define   {εn } Ihom (g, G) := inf lim inf Eεn (un ) : {un } ∈ RpCF (g, G; Ω) .

.

n→∞

(3.103)

The main result in the periodic homogenization of structured deformations is the following theorem, which provides a representation result analogous to that in (3.3), {εn } is independent of the and shows by means of (3.104), (3.105), and (3.106) that Ihom choice of the sequence εn → 0. Theorem 3.18 Let p > 1; let W : RN × Rd×N → [0, +∞) and ψ : RN × Rd × SN−1 → [0, +∞) satisfy Assumptions 3.4; let u ∈ SBV (Ω; Rd ) and let Eε (u) be the energy defined by (3.100). Then, for every (g, G) ∈ SDp (Ω; Rd × Rd×N ), {εn } the homogenized functional Ihom (g, G) defined in (3.103) is independent of the

3.2 Applications

89

sequence εn → 0 and admits the integral representation

Ihom (g, G) =

Hhom (∇g(x), G(x)) dx Ω

.



(3.104)

+

hhom ([g](x), νg (x)) dH N−1 (x) Ω∩Sg

The relaxed energy densities Hhom : Rd×N × Rd×N → [0, +∞) and hhom : Rd × SN−1 → [0, +∞) are given by the formulae 1 Hhom (A, B) := inf N inf k∈N k



W (x, A + ∇u(x)) dx kQ

ψ(x, [u](x), νu (x)) dH N−1 (x) :

+

.

kQ∩Su

u∈

(3.105)



bulk (A, B; kQ) C#,p

for every A, B ∈ Rd×N , and hhom (λ, ν) := inf

1



inf k∈N k N−1

.

(kQν )∩Su

ψ(x, [u](x), νu (x)) dH N−1 (x) : 

(3.106)

u ∈ C surf (λ, ν; kQν )

for every (λ, ν) ∈ Rd × SN−1 . In (3.105) and (3.106), we have defined, for A, B ∈ Rd×N , (λ, ν) ∈ Rd × SN−1 , and R, Rν ⊂ RN cubes, bulk C#,p (A, B; R)

 := u ∈ SBV# (R; Rd ) : |∇u| ∈ Lp (R),  ∇u(x) dx = B − A ,



.

(3.107a)

R

.

 C surf (λ, ν; Rν ) := u ∈ SBV (Rν ; Rd ) : u|∂Rν (x) = sλ,ν (x),

∇u(x) = 0 a.e. in Rν .

(3.107b)

In formula (3.107a), we denote by SBV# (R; Rd ) the set of Rd -valued SBV functions with equal traces on opposite faces of the cube R. Moreover, we notice that the class C surf (λ, ν; Rν ) is like the class Cpsurf (λ, ν) defined in (3.13) for p > 1,

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3 Energetic Relaxation to First-Order Structured Deformations

where Rν is a generic cube with two faces perpendicular to ν ∈ SN−1 , instead of the unit cube Qν . We notice that (3.105) and (3.106) are asymptotic cell formulae, as is expected in the context of homogenization when no convexity assumptions are made on the initial energy densities (see, e.g., [14]). In the special case of functions W and ψ which are convex in the gradient and jump variable, respectively, we are able to show that (3.105) reduces to a cell problem in the unit cell (see Proposition 3.10 below); whether the same result holds for hhom is still unknown. Before presenting a sketch of the proof of Theorem 3.18, we collect some properties of the homogenized densities Hhom and hhom defined in (3.105) and (3.106), respectively. Proposition 3.7 (Translation Invariance) For A, B ∈ Rd×N , let Hhom (A, B) be τ (A, B), defined by (3.105). Then for every τ ∈ Q, we have Hhom (A, B) = Hhom where 

1 τ Hhom (A, B) := inf N inf W (x + τ, A + ∇u(x)) dx k∈N k kQ

+ ψ(x + τ, [u](x), νu (x)) dH N−1 (x) : . kQ∩Su

 bulk u ∈ C#,p (A, B; kQ) . (3.108) bulk (A, B; kQ) is defined in (3.107a). where C#,p For (λ, ν) ∈ Rd × SN−1 , let hhom (λ, ν) be defined by (3.106). Then for every τ ∈ Q, we have hhom (λ, ν) = hτhom (λ, ν), where

hτhom (λ, ν) .

:= inf

k∈N

1 k N−1



ψ(x + τ, [u](x), νu (x)) dH N−1 (x) :

inf

(kQν )∩Su

u∈C

surf

 (λ, ν; kQν ) , (3.109)

where C surf (λ, ν; kQν ) is defined in (3.107b). Proof The proof of both (3.108) and (3.109) is a straightforward adaptation of the proof of [47, Proposition 2.15].

3.2 Applications

91

The next proposition contains further properties of hhom . Proposition 3.8 Let ψ satisfy Assumptions 3.4 and let % hhom : Rd × SN−1 → [0, +∞) be the function defined by % hhom (λ, ν) := lim sup .

T →+∞

1 T N−1



ψ(x, [u](x), νu (x)) dH N−1 (x) :

inf

(T Qν )∩Su

u∈C

surf

 (λ, ν; T Qν ) . (3.110)

Then the following properties hold true: 1. the function % hhom is a limit which is independent of the choice of the cube Qν , once ν ∈ SN−1 is fixed; 2. the function % hhom is continuous on Rd × SN−1 and for every (λ, ν) ∈ Rd × SN−1 cψ |λ|  % hhom (λ, ν)  Cψ |λ|,

.

(3.111)

where cψ and Cψ are the constants in Assumptions 3.4-5; 3. for every λ ∈ Rd and ν ∈ SN−1 , we have hhom (λ, ν) = % hhom (λ, ν), where hhom is the function defined in (3.106). Proof The proof of items 1 and 2 is essentially the same as that of [15, Proposition 2.2], upon observing that our density ψ satisfies (3.102), which is a stronger continuity assumption than condition [15, (iii) page 304]. Conclusion 1 is obtained verbatim as in [15, proof of Proposition 2.2, Steps 1–4]; we sketch here a proof of conclusion 2 for the reader’s convenience. The continuity of % hhom can be obtained by arguing in the following way: (a) one shows that the function % hhom (λ, ·) is continuous on SN−1 , uniformly with respect to λ, when λ varies on bounded sets; (b) one shows that for every ν ∈ SN−1 , the function % hhom (·, ν) is continuous on Rd ; (c) one shows that % hhom is continuous in the pair (λ, ν). The proof of point (a) above relies on the fact that for every fixed λ ∈ Rd , formula (3.110) does not depend on the cube Qν once the direction ν is prescribed, by 1. For the proof of point (b), we can argue as in [15, proof of Proposition 2.2, Step 6] (here we exploit Assumptions 3.4-5 and the Lipschitz continuity of ψ, see (3.102)). Point (c) can be obtained by arguing as in [25, proving (ii) from (i) in Theorem 2.8]. To conclude the proof of 2, we need to prove (3.111). The estimate from above can be easily obtained from the very definition of % hhom in (3.110), by using Assumptions 3.4-5. Concerning the estimate from below, it is sufficient to  observe that the functional SBV (Ω; Rd )  u → Su |[u](x)| dH N−1 (x) is lower

92

3 Energetic Relaxation to First-Order Structured Deformations

semicontinuous with respect to the convergence un − (g, 0), with g a pure jump SDp

function, as it follows from the lower semicontinuity of the total variation with respect to the weak-* convergence and, again, from Assumptions 3.4-5. To prove 3, we take inspiration from the proof of [22, Lemma 2.1]: we show that % hhom is an infimum over the integers. Together with 1, we will conclude that % hhom = hhom , as desired. Let gT : Rd × SN−1 → [0, +∞) be defined by gT (λ, ν) :=



1

ψ(x, [u](x), νu (x)) dH N−1 (x) :

inf

T N−1

(T Qν )∩Su

.

u∈C

surf

 (λ, ν; T Qν ) ,

(3.112)

so that we can write (3.110) as % hhom (λ, ν) = lim supT →+∞ gT (λ, ν). We start by proving a monotonicity property of gT over multiples of integer values of T , namely we prove that, for every (λ, ν) ∈ Rd × SN−1 , ghk (λ, ν)  gk (λ, ν)

.

for every h, k ∈ N.

(3.113)

To this aim, let u ∈ C surf (λ, ν; kQν ) be a competitor for gk (λ, ν) and consider the function u¯ : hkQν → Rd defined by ⎧ ⎪ ⎪ ⎨0 := u(k x/k) .u(x) ¯ ⎪ ⎪ ⎩λ

if x · ν < −k/2, if |x · ν| < k/2, if x · ν > k/2,

obtained by replicating u by periodicity in the (N − 1)-dimensional strip perpendicular to ν and extending it to 0 and λ appropriately. It is immediate to see that u¯ ∈ C surf (λ, ν; hkQν ), so that (3.113) follows. We now consider two integers 0 < m < n and a function u ∈ C surf (λ, ν, mQν ); we define u˜ : nQν → Rd by := u(x) ˜

.

 u(x)

if x ∈ mQν ,

sλ,ν (x) if x ∈ nQν \ mQν

and notice that u˜ ∈ C surf (λ, ν; nQν ). Then, invoking Assumptions 3.4-5,

ψ(x, [u](x), ˜ νu˜ (x)) dH N−1 (x)

.

=

ψ(x, [u](x), νu (x)) dH N−1 (x) +



(nQν )∩Su˜

(mQν )∩Su

ψ(x, [u](x), ˜ νu˜ (x)) dH N−1 (x)

(nQν \mQν )∩Su˜

  ψ(x, [u](x), νu (x)) dH N−1 (x) + Cψ |λ| nN−1 − mN−1 , (mQν )∩Su

3.2 Applications

93

so that, by infimizing first over u˜ ∈ C surf (λ, ν; nQν ) and then over u ∈ C surf (λ, ν; mQν ), we obtain   Cψ |λ| nN−1 − mN−1 .gn (λ, ν)  gm (λ, ν) + . nN−1 Using [n/m]m in place of m and (3.113), we get   * n +N−1 mN−1 Cψ |λ| nN−1 − m .gn (λ, ν)  gm (λ, ν) + . nN−1 By 1, % hhom (λ, ν) = limT →+∞ gT (λ, ν), so that, by taking the limit as n → ∞ in the inequality above, we can write % .hhom (λ, ν) = lim gT (λ, ν) = lim gn (λ, ν)  gm (λ, ν), T →+∞

n→∞

for every m ∈ N;

this yields, by taking the infimum over the integers, .

inf gn (λ, ν)  lim gn (λ, ν)  inf gm (λ, ν),

n∈N

n→∞

m∈N

(3.114)

the first inequality being obvious. Recalling the definition (3.106) of hhom (λ, ν), this gives the equality % hhom = hhom of 3 and concludes the proof. The proof of (3.104) is obtained by computing the Γ -limit in (3.103) by combining blow-up techniques à la Fonseca-Müller [33, 34] already used for Theorem 3.2 with rescaling techniques typically used in homogenization problems [14, 15]. This interplay is especially visible in the construction of the recovery sequences for proving that the densities in (3.104) are indeed given by (3.105) and (3.106) (see [2, Section 4]). To deduce the upper bound for the homogenized surface energy density hhom , comparison results in a Γ -convergence setting (see [15, 23]) are also used. We start by showing that, for every U ∈ A (Ω) and for every (g, G) ∈ SDp (U ; Rd × Rd×N ), the localization U → Ihom (g, G; U ) of the functional {εn } Ihom (g, G) of (3.103), defined as   Ihom (g, G; U ) := inf lim inf Eεn (un ) : {un } ∈ RpCF (g, G; U ) ,

.

n→∞

(3.115)

is the trace of a Radon measure which is absolutely continuous with respect to L N + |D s g|. The following proposition, whose proof relies on Theorem 2.2 and is omitted, is in the spirit of [20, Proposition 2.22] and will allow us to proceed with the same technique used to prove Theorem 3.2.

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3 Energetic Relaxation to First-Order Structured Deformations

Proposition 3.9 Assume that Assumptions 3.4 hold and let (g, G) ∈ SDp (Ω; Rd × Rd×N ). Then the localized functional A (Ω)  U → Ihom (g, G; U ) defined in (3.115) is the trace on A (Ω) of a finite Radon measure on B(Ω). Proof (Sketch of the Proof of Theorem 3.18) The proof is achieved by obtaining upper and lower bounds for the Radon–Nikodým derivatives of the functional Ihom defined in (3.103) with respect to the Lebesgue measure L N and to the Hausdorff measure H N−1 in terms of the homogenized bulk and surface energy densities Hhom and hhom defined in (3.105) and (3.106), respectively. Step 1—The Bulk Energy Density Hhom Formula (3.105) is obtained by combining the blow-up method used by Choksi and Fonseca [20] with the scaling techniques of homogenization [14]. In doing this, the initial bulk and surface energy densities W and ψ are computed at points of the form (x0 + rk y)/εn , where x0 ∈ Ω \ Sg is the center of the blow-up, rk is a vanishing sequence of radii (those for computing the Radon–Nikodým derivative), εn → 0, and y is the integration variable. The periodicity Assumption 3.4-1, the continuity of the initial densities in the first variable (Assumptions 3.4-3 and 6), and the translation invariance property from Proposition 3.7 are then used to prove (3.105). Step 2—The Surface Energy Density hhom From now on, we consider a point x0 ∈ Sg . Recalling Proposition 3.9, for every U ∈ A (Ω) and for every (g, G) ∈ SDp (U ; Rd × Rd×N ), the functional U → Ihom (g, G; U ) in (3.103) is a measure. In particular, there exists C > 0 such that   Ihom (g, G; U )  C L N (U ) + |D s g|(U ) .

(3.116)

.

Observe that (3.116) guarantees that, for every g ∈ SBV (Ω; Rd ), the computation dIhom (g, G) of the Radon–Nikodým derivative (x0 ) does not depend on G. Indeed, d|D s g| let us consider {uk } ∈ RpCF (g, G; U ) a recovery sequence for Ihom (g, G; U ) and, by Theorem 2.4 and Lemma 2.1, let us consider v ∈ SBV (U ; Rd ) such that ∇v = −G and piece-wise constant functions vk ∈ SBV (U ; Rd ) such that vk → v in L1 (U ; Rd ). Finally, let us define wk := uk + v − vk , so that wk − (g, 0) and SDp

therefore 

Ihom (g, 0; U )  lim inf k→∞

W U

.

+



x εk

U ∩Swk

 , ∇wk (x) dx ψ

x εk

  N−1 , [wk ](x), νwk (x) dH (x) .

3.2 Applications

95

Thus, by invoking Assumption 3.4-2 and Hölder’s inequality for the volume integrals, and first the sub-additivity of ψ (see Assumption 3.4-8) then the linear growth of ψ (see Assumption 3.4-5) for the surface integrals, we can estimate Ihom (g, 0; U )− Ihom (g, G; U )     x  x W , ∇wk (x) − W , ∇uk (x) dx  lim inf k→∞ εk εk U

x  + ψ , [wk ](x), νwk (x) dH N−1 (x) εk U ∩Swk 

x  . − ψ , [uk ](x), νuk (x) dH N−1 (x) εk U ∩Suk 

(1 + |G|p (x)) dx + |[v](x)| dH N−1 (x)  C lim inf k→∞

U ∩Sv

U



+

U ∩Svk

 |[vk ](x)| dH N−1 (x) ,

where C > 0 is a suitable constant. By virtue of the estimate in (2.16) and by (2.17), the two surface integrals in the last line above are bounded by the volume integral, so that, by exchanging the roles of Ihom (g, G; U ) and Ihom (g, 0; U ), we arrive at the conclusion that

.|Ihom (g, 0; U ) − Ihom (g, G; U )|  C (1 + |G|p (x)) dx, U

for every U ∈ A (Ω). In turn, this guarantees that, for H N−1 -a.e. x0 ∈ Sg , .

dIhom (g, 0) dIhom (g, G) (x0 ) = (x0 ). s d|D g| d|D s g|

In view of this, without loss of generality, we will consider G = 0 for the rest of the proof. Step 2.1—The Surface Energy Density hhom —Lower Bound The lower bound .

dIhom (g, 0) (x0 )  hhom ([g](x0 ), νg (x0 )), d|D s g|

is obtained considering g of the type sλ,ν = sλ,0,ν in (3.14), with (λ, ν) ∈ (Rd \ {0}) × SN−1 , essentially by following the proof of [15, Proposition 6.2].

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3 Energetic Relaxation to First-Order Structured Deformations

Step 2.2—The Surface Energy Density hhom —Upper Bound The upper bound .

dIhom (g, 0) (x0 )  hhom ([g](x0 ), νg (x0 )), d|D s g|

(3.117)

can be obtained by following the lines of [15, Proposition 6.2]. First we consider functions g taking finitely many values, that is g ∈ BV (Ω; L) where L ⊂ Rd is a set with finite cardinality, for which (3.117) is proved by using the abstract representation result contained in [3, Theorem 3.1]. In particular, the upper bound holds for functions of the type g = sλ,ν . To conclude, the general case can be obtained via standard approximation results (relying on the continuity properties of hhom , see Proposition 3.8) as in [20, Theorem 4.4, Step 2] (stemming from the ideas [6, Proposition 4.8]). Remark 3.19 We would like to underline that a crucial passage in the proof of Theorem 3.2 is the possibility of providing a sequential characterization of the relaxed bulk and energy densities, see [20, Proposition 3.1] for H and [20, Propositions 4.1 and 4.2] for h. From the proof of Proposition 3.8 (see (3.114)), a sequential characterization holds for hhom , but whether the same holds for Hhom is not clear at the moment. The next proposition shows that if the initial bulk and surface energy densities W and ψ are convex in the gradient and jump variable, respectively, then the homogenized bulk energy density reduces to a cell formula in the unit cube. Proposition 3.10 Let W and ψ satisfy Assumptions 3.4, let us assume that the functions ξ → W (x, ξ ) and λ → ψ(x, λ, ν) are convex for every x ∈ Rd and every ν ∈ SN−1 , and let 

cell Hhom (A, B) := inf

W (x, A + ∇u(x)) dx Q



.

+

ψ(x, [u](x), νu (x)) dH N−1 (x) : Q∩Su

u∈

bulk (A, B; Q) C#,p

 .

cell (A, B) for every A, B ∈ Rd×N . Then Hhom(A, B) = Hhom

Proof Let A, B ∈ Rd×N be given, and let us denote by mk (A, B) the inner infimization problem in the definition of Hhom (A, B), so that (3.105) reads cell (A, B) = Hhom (A, B) = infk∈N mk (A, B). With this notation, we also have Hhom m1 (A, B). We obtain the desired result if we prove that mk (A, B) = m1 (A, B). bulk (A, B; Q) be an admissible function for m (A, B). By To this aim, let u ∈ C#,p 1 bulk extending u by Q-periodicity on kQ, we obtain a function in C#,p (A, B; kQ) which is a competitor for mk (A, B), whence mk (A, B)  m1 (A, B). To show the

3.2 Applications

97

bulk reverse inequality, we consider u ∈ C#,p (A, B; kQ) a competitor for mk (A, B), and we use the standard method of averaging its translates along with Jensen’s bulk inequality to produce a competitor v ∈ C#,p (A, B; Q) for m1 (A, B), see [14, proof of Theorem 14.7]. By letting J := {0, 1, . . . , k − 1}N , it is easy to see that the function v : RN → Rd defined by

Q  x → v(x) :=

.

1 u(x + j ) kN j ∈J

and extended by periodicity is Q-periodic and satisfies

∇v(x) dx = Q



1 1 ∇u(x + j ) dx = ∇u(y) dy kN kN Q Q−j j ∈J

.

=

1 kN



,

j∈J (Q−j )



j ∈J

∇u(y) dy =

∇u(y) dy = B − A, kQ

bulk (A, B; Q) and therefore m (A, B)  m (A, B), yielding the so that v ∈ C#,p 1 k sought-after equality mk (A, B) = m1 (A, B) and the independence of the size of the cube. The thesis follows.

3.2.7 Hierarchical First-Order Structured Deformations In [31], Deseri and Owen extended the theory of [28] to hierarchical structured deformations in order to include the effects of disarrangements at more than one sub-macroscopic level. This extension is based on the fact that many natural and man-made materials exhibit different levels of disarrangements. Muscles, cartilage, bone, plants, and biomedical materials are just some of the materials whose mechanical behavior can be addressed within this generalized field theory. The energetic relaxation to first order structured deformations of [20] is extended to hierarchical structured deformations in [11]. We start by defining the set of hierarchical structured deformations and by extending the notion of convergence (3.4) to this context. We let H SDK (Ω; Rd × Rd×N ) := SBV (Ω; Rd ) × L1 (Ω, Rd×N ) × · · · × L1 (Ω; Rd×N ) ./ 0 -

.

K-times

denote the set of K-level (first-order) structured deformations on Ω.

98

3 Energetic Relaxation to First-Order Structured Deformations

Definition 3.5 Let (g, G1 , . . . , GK ) ∈ H SDK (Ω; Rd × Rd×N ) and let the sequence NK  (n1 , . . . , nK ) → un1 ,...,nK ∈ SBV (Ω; Rd ) be given. We say that un1 ,...,nK converges to (g, G1 , . . . , GK ) if (i)

lim · · · lim un1 ,...,nK = g, with each of the iterated limits in the sense of

n1 →∞ nK →∞ L1 (Ω; Rd );

(ii) for all k = 1, . . . , K − 1, the (K − k)-fold iterated limit gn1 ,...,nk :=

.

lim

nk+1 →∞

· · · lim un1 ,...,nK nK →∞

is in SBV (Ω; Rd ) and lim · · · lim ∇gn1 ,...,nk = Gk , with each of the n1 →∞

nk →∞

second iterated limits in the sense of the weak-* convergence in M (Ω; Rd×N ); (iii) lim · · · lim ∇un1 ,...,nK = GK with each of the iterated limits in the sense n1 →∞

nK →∞

of weak-* convergence in M (Ω; Rd×N ). ∗

In this case, we write un1 ,...,nK  (g, G1 , . . . , GK ). H

For convenience, it will be useful to write G0 := ∇g. Notice that in the case K = 1 we recover the convergence to (g, G) in the sense of (3.4). With this definition in mind, we can state the following result, which generalizes the Approximation Theorem [20, Theorem 2.12] for hierarchical structured deformations. Theorem 3.19 (Approximation Theorem) For every hierarchical structured deformation (g, G1 , . . . , GK ) ∈ H SDK (Ω; Rd × Rd×N ) there exists a sequence ∗ of functions (n1 , . . . , nK ) → un1 ,...,nK ∈ SBV (Ω; Rd ) such that un1 ,...,nK  H

(g, G1 , . . . , GK ) in the sense of Definition 3.5. Proof For every k = 1, . . . , K, let hk ∈ SBV (Ω; Rd ) such that ∇hk = Gk−1 − Gk be the functions provided by Theorem 2.4 and let nk → h¯ nk be K piecewise constant sequences approximating hk in L1 (Ω; Rd ) provided by Theorem 2.1. Define now un1 ,...,nK := g +

.

K

(h¯ nk − hk ); k=1

it is easily verified that this sequence approximates the given (g, G1 , . . . , GK ) in the sense of Definition 3.5. Indeed, lim · · · lim un1 ,...,nK = lim · · · lim

n1 →∞

nK →∞

.

= lim · · · n1 →∞

lim

nK−1 →∞

n1 →∞

nK →∞

 K

g+ (h¯ nk − hk ) k=1

 K−1

¯ g+ (hnk − hk ) = · · · = g,

in L1 (Ω; Rd ),

k=1

(3.118)

3.2 Applications

99

proving (i). Using (3.118), we have that gn1 ,...,nk := =

lim

· · · lim un1 ,...,nK

lim

· · · lim

nk+1 →∞

nk+1 →∞

=g +

nK →∞

nK →∞

 K

(h¯ nj − hj ) g+ j =1

k

(h¯ nj − hj ) ∈ SBV (Ω; Rd ); j =1

.

lim · · · lim ∇gn1 ,...,nk

n1 →∞

nk →∞

 k

¯ = lim · · · lim ∇ g + (hnj − hj ) n1 →∞

nk →∞

j =1

 k

= lim · · · lim ∇g + (Gj − Gj −1 ) n1 →∞

=G0 +

nk →∞

j =1

k

(Gj − Gj −1 ) = Gk , j =1

which is condition(ii). Finally, condition (iii) follows upon observing that ∇un1 ,...,nK = ∇g + K k=1 (Gk − Gk−1 ) = GK . Let us now consider an initial energy as in (3.10), suppressing, for the moment, the dependence on x. For a function u ∈ SBV (Ω; Rd ), we consider

.E

(0) (u) := Ω

W (0) (∇u(x)) dx +

Su ∩Ω

ψ (0) ([u](x), νu (x)) dH N−1 (x),

(3.119)

where W (0) : Rd×N → [0, +∞) and ψ (0) : Rd × SN−1 → [0, +∞) satisfy Assumptions 3.1-1, 3, 4, 5, and 6. We provide here a recursive relaxation scenario for recovering relaxed bulk and interfacial energy densities that can determine the energy of a K-level hierarchical structured deformation. This special procedure corresponds to performing K relaxations à la Choksi–Fonseca, starting at the finest microscopic level, level K, and ending at the macroscopic level, level 0. In the relaxation from level k to level k − 1 (for every k ∈ {1, . . . , K}) the kinematical quantities associated with levels k + 1 to K are held fixed, and we can think of the process as a partial relaxation. We notice that each index k ∈ {0, 1, · · · , K} in the recursion below would correspond to fields associated with submacroscopic level K − k.

100

3 Energetic Relaxation to First-Order Structured Deformations

Given the standing hypotheses on W (0) and ψ (0) , and recalling (3.9), we can relax the energy E (0) in (3.119) to an energy E (1) according to (3.11), which, by Theorem 3.2, has the integral representation



E. (1) (g, G) =

W (1) (∇g(x), G(x)) dx + Ω

ψ (1) ([g](x), νg (x)) dH N−1 (x), Ω∩Sg

(3.120) where W (1) and ψ (1) are provided by the cell formulas (3.17) and (3.18). Moreover, we note that E (1) : SBV (Ω, Rd ) × L1 (Ω, Rd×N ) → [0, +∞), while in (3.119) E (0) : SBV (Ω, Rd ) → [0, +∞). In particular, if we replace the field G in (3.120) by a constant mapping with value B1 ∈ Rd×N , we find that E (1) (·, B1 ) : SBV (Ω, Rd ) → [0, +∞) has the same form as E (0) , with W (0) replaced by W (1) (·, B1 ) and with ψ (0) replaced by ψ (1) . We suppose that, for each B1 ∈ Rd×N , the densities W (1) (·, B1 ) : Rd×N → [0, +∞) and ψ (1) : Rd × SN−1 → [0, +∞) also satisfy Assumptions 3.1-1, 3, 4, 5, and 6. Then the appeal to Theorem 3.2 and the argument in the previous paragraph yield a relaxed energy E (2) as well as the following counterpart of (3.120):

E (2) (g, G, B1 ) = .

W (2) (∇g(x), G(x), B1 ) dx Ω



(3.121)

+

ψ

(2)

([g](x), νg (x), B1 ) dH

N−1

(x).

Ω∩Sg

The second relaxed energy E (2) (·, ·, B1 ) : SBV (Ω, Rd ) × L1 (Ω, Rd×N ) → [0, +∞) in (3.121), the bulk second relaxed density W (2) (·, ·, B1 ) : Rd×N × Rd×N → [0, +∞), and the interfacial second relaxed density ψ (2) (·, ·, B1 ) : Rd × SN−1 → [0, +∞) are determined by the cell formulas (3.17) and (3.18) with W replaced by W (1) (·, B1 ) and with ψ replaced by ψ (1) . For the case p = 1, the presence of the recession function of W (1) (·, B1 ) in the cell formula (3.18) tells us that the values of ψ (2) may depend upon B1 , so that the notation ψ (2) (·, ·, B1 ) must be used to describe the interfacial second relaxed density; for the case p > 1, the recession function is absent from the cell formula (3.18), so that the dependence on B1 in the notation ψ (2) (·, ·, B1 ) is illusory. The steps described in the previous paragraphs are preparative for a recursive specification: given bulk and interfacial densities W (0) : Rd×N → [0, +∞) and ψ (0) : Rd × SN−1 → [0, +∞) satisfying Assumptions 3.1-1, 3, 4, 5, and 6, specify pairs (W (1) , ψ (1) ), . . . , (W (k) , ψ (k) ), . . . , (W (K) , ψ (K) ), with the stopping point m ∈ {1, 2, . . . , K} and the index k ∈ {0, 1, 2, · · · , m}, and with k+1  W (k) : Rd×N → [0, +∞) . k−1  → [0, +∞) ψ (k) : Rd × SN−1 × Rd×N

(3.122)

3.2 Applications

101

satisfying the following counterparts of the cell formulas (3.17) and (3.18) (for k  notational convenience we write Bk := (Bk , . . . , B1 ) ∈ Rd×N for k  1): for all Ak , B1 , . . . , Bk ∈ Rd×N , λ ∈ Rd , and ν ∈ SN−1 , 

W (Ak , Bk ) = inf (k)

W (k−1) (∇u(x), Bk−1 ) dx Q



+

.

ψ (k−1) ([u](x), νu (x), Bk−2 ) dH N−1 (x) : Q∩Su

 u ∈ Cpbulk(A, B)  ψ

(k)



.

(W (k−1) )∞ (∇u(x), Bk−1 ) dx

(λ, ν, Bk−1 ) = inf δ1 (p) +

Qν

Qν ∩Su

ψ (k−1) ([u](x), νu (x), Bk−2 ) dH N−1 (x) : 

u ∈ Cpsurf (λ, ν) (where it is intended that both B0 and B−1 signify no dependence on B). In addition to the cell formulas, W (k) (·, Bk ) must satisfy Assumptions 3.1-1 and 3 and ψ (k) (·, ·, Bk−1 ) must satisfy Assumptions 3.1-4, 5, and 6 for all Bk and Bk−1 , respectively. We note that in the passage from the bulk energy W (k−1) to W (k) , the expression (k−1) W (Ak−1 , Bk−1 , . . . , B1 ) is replaced by W (k) (Ak , Bk , Bk−1 , . . . , B1 ), so that the single matrix Ak−1 in the arguments of W (k−1) is replaced by the pair of matrices (Ak , Bk ) in the arguments of W (k) . Remark 3.20 The discussion presented so far can be easily extended to the case where the energy densities show an explicit spatial dependence, namely the case where both W (0) and ψ (0) depend on the x variable, as in Sect. 3.1.1. In that case, the same arguments carry through provided we ask that the bulk energy densities satisfy Assumption 3.1-2 and that the surface energy densities satisfy Assumption 3.1-7 whenever requested. The designation of the integer m as the stopping point corresponds to the assertion that at least one of the required Assumptions 3.1-1 or 3 on W (m) (·, Bm ) or one of Assumptions 3.1-4, 5, or 6 on ψ (m) (·, ·, Bm−1 ) is not satisfied. There need not exist a stopping point, as the following example shows.

102

3 Energetic Relaxation to First-Order Structured Deformations

Example 3.4 Let p > 1 be given, let W (0) be convex and satisfy Assumption 3.1-1, and define ψ (0) : Rd ×SN−1 → [0, +∞) by ψ (0) (λ, ν) := |λ·ν|. From Theorem 3.9, Proposition 3.6, and (3.46), the densities in (3.120) are given by W (1) (A1 , B1 ) = W (0) (B1 ) + | tr(A1 − B1 )| .

ψ (1) (λ, ν) = |λ · ν| = ψ (0) (λ, ν).

(3.123)

(The recession function of W (0) does not appear in the formula for ψ (1) because p > 1.) We note that W (1) (·, B1 ) also is convex and satisfies Assumption 3.1-1, and, noting that Cpbulk(A2 , B2 ) is used in the bulk density cell formula for k = 2, we may then perform a second relaxation via the argument that led to (3.123) to obtain the relaxed densities in (3.121): W (2) (A2 , B2, B1 ) = W (1) (B2 , B1 ) + | tr(A2 − B2 )| = W (0) (B1 ) + | tr(B2 − B1 )| + | tr(A2 − B2 )| . ψ (2) (λ, ν) = |λ · ν|. It is straightforward to show by induction that the recursive relaxation determined by W (0) and ψ (0) in this example has no stopping point and that for k  1 the relaxed densities at the k-th recursive stage are given by W

.

(k)

(Ak , Bk ) = | tr(Ak − Bk )| +

k

| tr(Bj − Bj −1 | + W (0) (B1 ), (3.124)

j =2

ψ (k) (λ, ν) = |λ · ν|. In this example, the K-fold relaxed bulk and interfacial relaxed energy densities W (K) and ψ (K) just obtained can be used to assign an energy E (K) to a hierarchical structured deformation (g, G1 , . . . , GK ) ∈ H SDK (Ω; Rd × Rd×N ):

E (K) (g, G1 , . . . , GK ) :=

W (K) (∇g(x), G1 (x), . . . , GK (x)) dx Ω

.



(3.125)

+

|[g](x) · νg (x)| dH N−1 (x). Ω∩Sg

If we put G0 := ∇g and use (3.124) with k = K, the integrand W (K) (∇g(x), G1 (x), . . . , GK (x)) in the volume integral above can be written K

.

i=1

| tr(Gi (x) − Gi−1 (x))| + W (0) (GK (x)).

(3.126)

References

103

Assumption 3.1-1 and the expression (3.126) imply that x → W (K) (∇g(x), G1 (x), . . . , GK (x)) is in L1 (Ω) whenever (g, G1 , . . . , GK ) ∈ H SDK (Ω; Rd × Rd×N ) and GK ∈ Lp (Ω; Rd×N ), so that the definition (3.125) is meaningful. The matrices Gi−1 (x) − Gi (x) in (3.126) are the contributions to the macroscopic deformation gradient ∇g from the disarrangements arising in upscaling from submacroscopic level i to submacroscopic level i −1, so that the sum over i represents a bulk energy density arising from hierarchical disarrangements. The remaining term W (0) (GK (x)) represents a bulk energy density arising from the contributions at the macrolevel of deformation without disarrangements at submacroscopic level K.

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Chapter 4

Energetic Relaxation to Second-Order Structured Deformations

4.1 Spaces of Second-Order Structured Deformations, Approximation Theorems, and Representation of Relaxed Energies The setting of elasticity with disarrangements in the context of first-order structured deformations precludes capturing directly the effects of “gradient disarrangements”, i.e., of jumps in the gradients of deformations that approximate geometrical changes at the submacroscopic level. In particular, jumps in ∇un , with un converging to g and ∇un converging to G, cannot be accounted for in the theory developed in Chap. 3. As described in Sect. 1.2, the multiscale geometry of structured deformations was broadened in [13] to provide additional fields capable of describing effects at the macrolevel of gradient disarrangements. These effects play a crucial role, for instance, in the mechanics of phase transitions in metals, see, e.g., [1, 2, 12]. The general form of a second-order structured deformation is that of a triple (g, G, Γ ) in which (g, G) is a first-order structured deformation (possibly with additional smoothness granted to g and G) and Γ : Ω → Rd×N×N is a field intended to describe the contributions at the macrolevel of smooth bending and of curving at submacroscopic levels. In [14], various versions of approximation theorems are obtained that provide sequences of approximations un with un converging to g, ∇un converging to G, and ∇ 2 un converging to Γ (see Theorems 4.1 and 4.6 below for the detailed statements). There are two approaches to the variational theory of second-order structured deformations available in the literature. In [5], Barroso et al. extended the variational setting proposed by Choksi and Fonseca in [6] to triples (g, G, Γ ) with g : Ω → Rd , G : Ω → Rd×N , and Γ : Ω → Rd×N×N . In [8], Fonseca et al. considered structured deformations of the form (g, Γ ) ∼ (g, ∇g, Γ ) with g : Ω → Rd and

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. Matias et al., Energetic Relaxation to Structured Deformations, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-19-8800-4_4

107

108

4 Energetic Relaxation to Second-Order Structured Deformations

Γ : Ω → Rd×N×N , the space of tensors with the symmetry property sym Γij k = Γikj ,

for every i = 1, . . . , d and for every j, k = 1, . . . , N.

.

(4.1)

The two different approaches are determined by the choice of the function spaces in which second-order structured deformations are studied, SBV 2 and SBH (see (2.11) and (2.14)), respectively, as will be clear in Sects. 4.1.1 and 4.1.2 below. Recalling the definitions of the fields M and Ξ given in the introduction, we can notice that in the SBV 2 case we have M = ∇g − G ∈ Rd×N

and

.

Ξ = ∇G − Γ ∈ Rd×N×N ,

(4.2a)

whereas, in the SBH case, we have M = 0 ∈ Rd×N

and

.

Ξ = ∇ 2 g − Γ ∈ Rd×N×N , sym

(4.2b)

so that the SBH case is more constrained than the SBV 2 case and requires different analytical techniques to be tackled. Notice that, in particular, in the SBH case there are only gradient disarrangements. As we did in Chap. 3, we now consider an initial energy E(u) defined on functions in u ∈ SBV 2 (Ω; Rd ) or u ∈ SBH (Ω; Rd ), and we relax it to secondorder structured deformations (g, G, Γ ) or (g, Γ ). Relevant initial energies should feature a bulk contribution, depending on ∇u and ∇ 2 u in both settings, and feature surface contributions depending on [u] and [∇u] in the SBV 2 setting, with only one surface contribution depending on [∇u] in the SBH setting. For this reason, we postpone specifying the exact form of the initial energies that we consider. We also point out that only the case in which the bulk energy density has growth of order p = 1 is considered. In the ensuing sections, we provide integral representation results for the relaxed energy   ∗ I (g, G, Γ ) := inf lim inf E(un ) : un − (g, G, Γ ) ,

.

n→∞

SD 2

(4.3)

∗

where the convergence un − (g, G, Γ ) is defined as SD 2

un → g .

∗

in L1 (Ω; Rd ),

∇ 2 un  Γ

∇un → G in L1 (Ω; Rd×N ),

in M (Ω; Rd×N×N ),

(4.4)

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

109

and is the one that provides the approximation of (g, G, Γ ) in the sense of Theorem 4.1, in the SBV 2 setting. The situation is slightly different in the SBH setting. Here, the analogue would be to consider the relaxed energy   ∗ I (g, Γ ) := inf lim inf E(un ) : un − (g, Γ ) ,

.

n→∞

SD2

(4.5)

∗

where the convergence − is defined as SD2

un → g

.

in W 1,1 (Ω; Rd ),

∗

∇ 2 un  Γ

in M (Ω; Rd×N×N ),

(4.6)

and is the one that provides the approximation of (g, Γ ) in the sense of Theorem 4.6, in the SBH setting. Introducing the class of admissible sequences, for (g, Γ ) ∈ SD2 (Ω) and U ∈ A (Ω),   ∗ R SD2 (g, Γ ; U ) := {un } ⊂ SBH (U ; Rd ) : un − (g, Γ ) ,

.

SD2

(4.7)

the definition (4.5) can be written as   I (g, Γ ) := inf lim inf E(un ) : {un } ∈ R SD2 (g, Γ ; Ω) .

.

n→∞

(4.8)

In Sect. 4.1.2, reporting on the research in [8], the relaxation is considered in a weaker sense, so that Theorem 4.9 provides an integral representation for the relaxed energy   n ) : un → g in L1 (Ω; Rd ), I˜(g, Γ ) := inf lim inf E(u .

n→∞

 ∗ ∇ 2 un  Γ in M (Ω; Rd×N×N ) ,

(4.9)

 is a suitable extension of the initial energy E (see formula (4.47) below where E for the precise definition). Formulae structurally analogous to (3.3), with due modifications, will be specified later, since they strongly depend on the functional setting considered.

4.1.1 Relaxation in SBV 2 We begin by defining the space of second-order structured deformations in the SBV 2 setting.

110

4 Energetic Relaxation to Second-Order Structured Deformations

Definition 4.1 The space of (Rd -valued) second-order structured deformations on a domain Ω ⊂ RN in the SBV 2 setting is SD 2 (Ω) = SD 2 (Ω; Rd × Rd×N × Rd×N×N ) .

:= SBV 2 (Ω; Rd ) × SBV (Ω; Rd×N ) × L1 (Ω; Rd×N×N ).

It is endowed with the natural norm induced by the product structure (g, G, Γ )SD 2 (Ω) := gBV 2 (Ω;Rd ) + GBV (Ω;Rd×N ) + Γ L1 (Ω;Rd×N×N ) ,

.

which is going to be denoted by (g, G, Γ )SD 2 when no domain specification is needed. We now state and prove the approximation theorem for second-order structured deformations in the SBV 2 setting. Theorem 4.1 (Approximation Theorem [5, Remark 4.2], [14, Theorem 3.2]) Let Ω ⊂ RN and (g, G, Γ ) ∈ SD 2 (Ω). Then there exists {un } ⊂ SBV 2 (Ω; Rd ) ∗ such that un − (g, G, Γ ) (see (4.4)) and such that, for all sufficiently large n ∈ N, SD 2

and for a constant C > 0, depending only on N,   |Dun |(Ω) + |D(∇un )|(Ω)  C 1 + (g, G, Γ )SD 2 (Ω) .

.

(4.10)

Proof The proof can be easily obtained by applying Theorem 2.4 and Lemma 2.1 twice. Given (g, G, Γ ) ∈ SD 2 (Ω), by Theorem 2.4 there exist H ∈ SBV (Ω; Rd×N ) such that ∇H = ∇G − Γ and such that, by (2.16), |D s H |(Ω)  CN ∇G − Γ L1 (Ω;Rd×N×N ) .

.

(4.11)

By Lemma 2.1, there exist a sequence of piecewise constant functions Hn such that Hn → H in L1 (Ω; Rd×N ), and such that |DHn |(Ω) → |DH |(Ω) (see (2.17)). For every n ∈ N, let Wn := G − H + Hn ; we have n→∞

Wn −→ G

.

in L1 (Ω; Rd×N )

and

∇Wn = ∇G − ∇H = Γ.

(4.12)

We now repeat the construction above, for every n ∈ N. By Theorem 2.4, there exists hn ∈ SBV (Ω; Rd ) such that ∇hn = ∇g − Wn and such that, by (2.16), |D s hn |(Ω)  C N ∇g − Wn L1 (Ω;Rd×N ) .

.

(4.13)

By Lemma 2.1, there exists a sequence of piecewise constant functions hn,m such that (by (2.17)) m→∞

hn,m −→ hn

.

in L1 (Ω; Rd )

and

m→∞

|Dhn,m |(Ω) −→ |Dhn |(Ω). (4.14)

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

111

For every n, m ∈ N, let wn,m := g − hn + hn,m ; we have m→∞

wn,m −→ g

.

in L1 (Ω; Rd )

∇wn,m = ∇g − ∇hn = Wn .

(4.15)

  1 and |Dhn,m(n) |(Ω) − |Dhn |(Ω)  n

(4.16)

and

For every n ∈ N let m(n) ∈ N be such that hn,m(n) − hn L1 (Ω;Rd ) 

.

1 n

(which is granted by the convergences (4.14)) and define un := wn,m(n) . We claim that the sequence un converges to (g, G, Γ ) in the sense of (4.4). Indeed, since un is defined by diagonalization, we have .

lim un = lim lim wn,m

n→∞

n→∞ m→∞

so that the desired convergence follows by (4.12) and (4.15). To prove (4.10) we estimate each summand in the left-hand side. We have |Dun |(Ω) = ∇un L1 (Ω;Rd×N ) + |D s un |(Ω)  Wn L1 (Ω;Rd×N ) + |D s g|(Ω) + |D s hn |(Ω) + |D s hn,m(n) |(Ω)  |D s g|(Ω) + 2C N ∇g − Wn L1 (Ω;Rd×N ) + Wn L1 (Ω;Rd×N ) +

.

1 n

 max{1, 2C N }|Dg|(Ω) + (1 + 2C N )GL1 (Ω;Rd×N ) + (1 + 2C N )G − Wn L1 (Ω;Rd×N ) +

1 , n

where we have used the convergence in (4.12), (4.13), the equality in (4.15), and the second inequality in (4.16). Now, for n sufficiently large, we obtain   |Dun |(Ω)  (1 + 2C N ) (g, G, Γ )SD 2 + 1 .

.

(4.17)

Upon noticing that, as in the statement of [14, Theorem 3.2], ∇ 2 un = ∇Wn = Γ , and using (4.11) and the equality in (4.15) again, we can estimate, for n sufficiently large,   |D(∇un )|(Ω)  (1 + 2CN ) (g, G, Γ )SD 2 + 1 ,

.

(4.18)

so that we conclude (4.10) with the constant C = max{1 + 2CN , 1 + 2C N }. Example 4.1 (Example 1.3 Revisited) Let d = N = 2 and, Ω = (0, 1)2 , and let (g, G, Γ ) = (g, ∇g, 0) be given by g(x, y) = (x + h(y), y), for an unspecified function h ∈ C 2 (0, 1) ∩ W 2,1 (0, 1). This second-order structured deformation describes bending of the domain Ω through submacroscopic simple shearing. This is achieved by introducing discontinuities in the gradient in the following way.

112

4 Energetic Relaxation to Second-Order Structured Deformations

For n ∈ N \ {0} and for i = 0, . . . , n, let yi := i/n, define mi := (h(yi+1 ) − h(yi ))/(yi+1 − yi ) for i = 0, . . . , n − 1 and un : Ω → R2 by un (x, y) = (x +hn (y), y),

where hn (y) =

.

n−1

(mi (y −yi )+h(yi ))χ(yi ,yi+1 ) (y). i=0

It is not difficult to see that un → g in L1 (Ω; R2), ⎛ ⎜1 ∇un = ⎜ ⎝ 0

n−1

.

i=1

⎞ mi χ(yi ,yi+1 ) ⎟ ⎟ → ∇g ⎠ 1

in L1 (Ω; R2×2 ),

and that ∇ 2 un ≡ 0, so that (4.4) holds. Let us now look at the distributional derivatives. Since un is a continuous functions, we have Dun = ∇un L 2 Ω, and D 2 un is the 2×2×2-tensor with the following characteristics: its only non-vanishing component is the one indexed by 122 and it consists only of the singular measure of jump type given by   (D 2 un )122 = [∇un ] ⊗ ν∇un 122 H 2

.

S∇un .

The jump set of the gradient is the union of horizontal lines S∇un =

n−1 

.

(0, 1) × {yi };

i=1

it is easy to see that H 1 (S∇un ) = L 1 ((0, 1)) · H 0 as n → ∞, whereas

 ,n−1

i=1 {yi }



= n − 1 diverges

   n−2  |D s (∇un )|(Ω) = |(D 2 un )122 |(Ω) =  (mi+1 − mi ) i=0

   n−2 h(yi+2 ) − h(yi+1 ) h(yi+1 ) − h(yi )   =  −  yi+2 − yi+1 yi+1 − yi .

i=0

   n−2 h(yi+2 ) − 2h(yi+1 ) + h(yi )   =   1/n i=0

   n−2  1   1 h(yi+2 ) − 2h(yi+1 ) + h(yi )      h (y) dy  = →  2 n 1/n 0 i=0

as n → ∞. Notice that this limit is finite since h ∈ C 2 (0, 1) ∩ W 1,2 (0, 1).

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

113

We now specify the class of initial energies to be relaxed to second-order structured deformations in the SBV 2 setting. Given u ∈ SBV 2 (Ω; Rd ) we consider the following energy functional (of the type as in (3.32) but including explicit dependence on the spatial variable)

E(u) :=

W (x, ∇u(x), ∇ 2 u(x)) dx Ω



+

.

+

Su ∩Ω

ψ (1) (x, [u](x), νu (x)) dH N−1 (x)

S∇u ∩Ω

(4.19)

ψ (2) (x, [∇u](x), ν∇u (x)) dH N−1 (x),

where the bulk energy density W and the surface energy densities ψ (1) and ψ (2) satisfy the following hypotheses. Assumptions 4.1 The functions W : Ω×Rd×N ×Rd×N×N → [0, +∞), ψ (1) : Ω× Rd × SN−1 → [0, +∞) and ψ (2) : Ω × Rd×N × SN−1 → [0, +∞) are continuous functions that satisfy the following conditions 1. there exists C > 0 such that, for all x ∈ Ω, A ∈ Rd×N , and Γ ∈ Rd×N×N , .

1 (|A| + |Γ |) − C  W (x, A, Γ )  C(1 + |A| + |Γ |); C

2. there exists C > 0 such that, for all x ∈ Ω, A1 , A2 ∈ Rd×N , and Γ1 , Γ2 ∈ Rd×N×N |W (x, A1 , Γ1 ) − W (x, A2 , Γ2 )|  C(|A1 − A2 | + |Γ1 − Γ2 |)

.

3. there exists a continuous function ωW : [0, +∞) → [0, +∞) with ωW (s) → 0 as s → 0+ such that, for every x, x0 ∈ Ω, A ∈ Rd×N , and Γ ∈ Rd×N×N , |W (x, A, Γ ) − W (x0 , A, Γ )|  ωW (|x − x0 |)(1 + |A| + |Γ |);

.

4. there exist C, T > 0 and 0 < α < 1 such that, for all x ∈ Ω, A ∈ Rd×N , and Γ ∈ Rd×N×N with |Γ | = 1     ∞ . W (x, A, Γ ) − W (x, A, tΓ )   C , for all t > T ,  tα  t with W ∞ denoting the recession function at infinity of W with respect to Γ , see (2.2); 5. there exist c1 , C1 > 0 such that, for all x ∈ Ω, λ ∈ Rd , and ν ∈ SN−1 , c1 |λ|  ψ (1) (x, λ, ν)  C1 |λ|;

.

114

4 Energetic Relaxation to Second-Order Structured Deformations

there exist c2 , C2 > 0 such that, for all x ∈ Ω, Λ ∈ Rd×N and ν ∈ SN−1 , c2 |Λ|  ψ (2) (x, Λ, ν)  C2 |Λ|;

.

6. (positive 1-homogeneity) for all x ∈ Ω, λ ∈ Rd , Λ ∈ Rd×N , ν ∈ SN−1 , and t >0 ψ (1) (x, tλ, ν) = tψ (1) (x, λ, ν)

.

and

ψ (2) (x, tΛ, ν) = tψ (2) (x, Λ, ν);

7. (sub-additivity) for all x ∈ Ω, λ1 , λ2 ∈ Rd , Λ1 , Λ2 ∈ Rd×N , and ν ∈ SN−1 , ψ (1) (x, λ1 + λ2 , ν)  ψ (1) (x, λ1 , ν) + ψ (1) (x, λ2 , ν),

.

ψ (2) (x, Λ1 + Λ2 , ν)  ψ (2) (x, Λ1 , ν) + ψ (2) (x, Λ2 , ν);

.

8. there exist continuous functions ωψ (i) : [0, +∞) → [0, +∞), i = 1, 2 with ωψ (i) (s) → 0 as s → 0+ such that, for every x, x0 ∈ Ω, λ ∈ Rd , Λ ∈ Rd×N , and ν ∈ SN−1 , |ψ (1) (x, λ, ν) − ψ (1) (x0 , λ, ν)|  ωψ (1) (|x − x0 |)|λ|,

.

|ψ (2) (x, Λ, ν) − ψ (2) (x0 , Λ, ν)|  ωψ (2) (|x − x0 |)|Λ|.

.

For (g, G, Γ ) ∈ SD 2 (Ω) and U ∈ A (Ω), define the class of admissible sequences   2 ∗ R SD (g, G, Γ ; U ) := {un } ⊂ SBV 2 (U ; Rd ) : un − (g, G, Γ ) ,

.

SD 2

(4.20)

∗

with − as defined in (4.4). The energy (4.3) associated with a second-order SD 2

structured deformation in the SBV 2 setting can then be written as the relaxation ∗ of the initial energy (4.19) with respect to the − convergence, namely SD 2

  2 I (g, G, Γ ) = inf lim inf E(un ) : {un } ∈ R SD (g, G, Γ ; Ω) .

.

n→∞

(4.21)

Remark 4.1 We notice that by (4.10) and the upper bounds in Assumptions 4.1-1 and 5, for every (g, G, Γ ) ∈ SD 2 (Ω), the energy I (g, G, Γ ) < +∞ (see also [5, Lemma 4.1]). Moreover, Assumptions 4.1-2 and 4 imply that the values of the recession function (x, A, Γ ) → W ∞ (x, A, Γ ) do not depend upon the matrix variable A ∈ Rd×N . This observation was missed in [5] and permits one to ignore the spurious appearance of G(x) as an argument of h(2) in the counterparts in [5] of our (4.33) and (4.35) below.

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

115

The first step in proving the integral representation Theorem 4.5 below is the following decomposition Theorem 4.2 that states that it is possible to decompose, in an additive fashion, the relaxed energy (4.21) into the contribution of an energy I (1) (g, G, Γ ) (see (4.27a) below) that depends on the first-order part of second-order structured deformation (see Remark 4.2 below, pointing out that the dependence on Γ here is apparent) and of an energy I (2) (G, Γ ) (see (4.27b) below) that does not depend on the deformation g. This second energy captures the “structuredness” of the deformation at the level of the first-order and secondorder gradients. In view of this, it is convenient to introduce the analogue of ∗ the convergence −, defined in (3.4) for deformation functions, for deformation SD

gradients. Given (G, Γ ) ∈ SBV (Ω; Rd×N ) × L1 (Ω; Rd×N×N ), we write that ∗ Vn − (G, Γ ) if the sequence of functions Vn ∈ SBV (Ω; Rd×N ) is such that g−SD

Vn → G

.

in L1 (Ω; Rd×N )

∗

and ∇Vn  Γ

in M (Ω; Rd×N×N ).

(4.22)

Given such a pair (G, Γ ) and a set U ∈ A (Ω), we also define the following class of admissible sequences   ∗ R g−SD (G, Γ ; U ) := {un } ⊂ SBV (U ; Rd×N ) : un − (G, Γ ) .

.

g−SD

For a function u ∈ SBV (Ω; Rd ), we let

(1) .E (u) := ψ (1) (x, [u](x), νu (x)) dH N−1 (x); Su ∩Ω

(4.23)

(4.24)

for a function V ∈ SBV (Ω; Rd×N ), we let

E. (2) (V ) := W (x, V (x), ∇V (x)) dx

+

Ω

(4.25) ψ

(2)

SV ∩Ω

(x, [V ](x), νV (x)) dH

N−1

(x),

where the bulk and surface energy densities W , ψ (1) , and ψ (2) are those defining the initial energy E in (4.19), which now can also be written as E(u) = E (1)(u) + E (2) (∇u), for u ∈ SBV 2 (Ω; Rd ). Theorem 4.2 (Decomposition Theorem [5, Theorem 4.3]) For every (g, G, Γ ) ∈ SD 2 (Ω) the relaxed energy I (g, G, Γ ) defined in (4.21) can be decomposed as I (g, G, Γ ) = I (1) (g, G, Γ ) + I (2) (G, Γ ),

.

(4.26)

116

4 Energetic Relaxation to Second-Order Structured Deformations

where   2 I (1) (g, G, Γ ) := inf lim inf E (1)(un ) : {un } ∈ R SD (g, G, Γ ; Ω) , (4.27a) . n→∞   I (2) (G, Γ ) := inf lim inf E (2)(Vn ) : {Vn } ∈ R g−SD (G, Γ ; Ω) . (4.27b)

.

n→∞

Proof The inequality  in (4.26) is an immediate consequence of the properties of the infimum. To prove the  inequality, two admissible sequences {un } and {Vn } for the relaxations in (4.27) are combined using Assumptions 4.1-5 and 8, and Lemma 2.1 to obtain an admissible sequence for (4.21). Inspecting the expression of the energy E (2) defined in (4.25) and the definition of its relaxation I (2) in (4.27b) with respect to the convergence (4.22), we can notice a structure which is very similar to the relaxation of energies to first-order structured deformations dealt with in Chap. 3 by means of Theorem 3.2—only in the present case, the relaxation machinery takes place at the level of gradients. The only difference is that the initial energy E in (3.10) does not depend on the deformation function u itself, whereas the energy E (2) (V ) depends on both V and ∇V . Thanks to the following result, we will be able to apply Theorem 3.2 directly to relax the (2) energy EG defined in (4.29) below and to obtain an integral representation for the (2) energy I . Proposition 4.1 ([5, Proposition 4.6]) For every pair (G, Γ ) ∈ SBV (Ω; Rd×N )× L1 (Ω; Rd×N×N ) we have I (2) (G, Γ ) = I˜(2) (G, Γ ), where   (2) I˜(2) (G, Γ ) := inf lim inf EG (Vn ) : {Vn } ∈ R g−SD (G, Γ ; Ω)

.

n→∞

(4.28)

and, for V ∈ SBV (Ω; Rd×N ),

(2)

E. G (V ) :=

W (x, G(x), ∇V (x)) dx

+

Ω

(4.29) ψ

(2)

SV ∩Ω

(x, [V ](x), νV (x)) dH

N−1

(x). (2)

Proof Notice that the difference between E (2)(Vn ) and EG (Vn ) arises because V (x) in (4.25) is replaced by G(x) in (4.24). The desired equality then follows easily by using the Lipschitz continuity of W granted by Assumption 4.1-2, together with the fact Vn → G in L1 (Ω; Rd×N ). From the Decomposition Theorem 4.2 and Proposition 4.1, it is clear that an integral representation for the functional I holds if we can prove that both functionals I (1) in (4.27a) and I˜(2) in (4.28) admit an integral representation (see Theorems 4.3 and 4.4 below, respectively). To this aim, we introduce the following classes of competitors for the cell formulae for the relaxed bulk and surface energy densities.

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

117

For all λ ∈ Rd , A, B ∈ Rd×N , L1 , L2 ∈ Rd×N×N , ν ∈ SN−1 , let  C bulk,(1) (A) := u ∈ SBV 2 (Q; Rd ) : u|∂Q = 0, ∇u = A for a.e. x ∈ Q ;  C bulk,(2) (L1 , L2 ) := V ∈ SBV (Q; Rd×N ) : V |∂Q (x) = aL1 (x), .

 ∇V (x) dx = L2 ;

Q



C surf,(1) (λ, ν) := u ∈ SBV 2 (Qν ; Rd ) : u|∂Qν (x) = sλ,ν (x),

∇u = 0 for a.e. x ∈ Qν ;

C

surf,(2)

 (B, ν) := V ∈ SBV (Qν ; Rd×N ) : V |∂Qν (x) = sB,ν (x),  ∇V (x) dx = 0 .

Qν

For x0 ∈ Ω, U ∈ A (Ω) and W ∞ the recession function of W with respect to its last entry (see (B4) and (2.2)), let

Exsurf,(1) (u; U ) := 0

.

(2)



Ex0 ,A (V ; U ) :=

Su ∩U

ψ (1) (x0 , [u](x), νu (x)) dH N−1 (x);

W (x0 , A, ∇V (x)) dx U



+

SV ∩U

(V ; U ) := Exrec,(2) 0

ψ (2) (x0 , [V ](x), νV (x)) dH N−1 (x);

W ∞ (x0 , ∇V (x)) dx U



+

SV ∩U

ψ (2) (x0 , [V ](x), νV (x)) dH N−1 (x).

(4.30)

We state now the integral representation theorems. Theorem 4.3 (Integral Representation of I (1) [5, Theorem 5.1]) Let ψ (1) : Ω × Rd ×SN−1 → [0, +∞) be a continuous function satisfying Assumptions 4.1-5, 6, 7, and 8; let (g, G, Γ ) ∈ SD 2 (Ω) and let I (1) (g, G, Γ ) be given by (4.27a). Then there exist H (1) : Ω × Rd×N → [0, +∞) and h(1) : Ω × Rd × SN−1 → [0, +∞) such that  I (1) (g, G, Γ ) = Ω H (1) (x, ∇g(x) − G(x)) dx  . (4.31) + Sg ∩Ω h(1) (x, [g](x), νg (x)) dH N−1 (x).

118

4 Energetic Relaxation to Second-Order Structured Deformations

For every x0 ∈ Ω, A ∈ Rd×N , λ ∈ Rd , and ν ∈ SN−1 ,   bulk,(1) H (1)(x0 , A) = inf Exsurf,(1) (u; Q) : u ∈ C (A) ;. 0   surf,(1) (u; Q ) : u ∈ C (λ, ν) . h(1) (x0 , λ, ν) = inf Exsurf,(1) ν 0 .

(4.32a) (4.32b)

Theorem 4.4 (Integral Representation of I˜(2) [5, Theorem 5.7]) Let W : Ω × Rd×N × Rd×N×N → [0, +∞) and ψ (2) : Ω × Rd×N × SN−1 → [0, +∞) satisfy Assumptions 4.1; let (G, Γ ) ∈ SBV (Ω; Rd×N ) × L1 (Ω; Rd×N×N ) and let I˜(2) (G, Γ ) be given by (4.28). Then there exist H (2) : Ω × Rd×N × Rd×N×N × Rd×N×N → [0, +∞) and h(2) : Ω × Rd×N × SN−1 → [0, +∞) such that

H (2)(x, G(x), ∇G(x), Γ (x)) dx I˜(2) (G, Γ ) = .

Ω



+

(4.33) h (x, [G](x), νG (x)) dH (2)

SG ∩Ω

N−1

(x).

For every x0 ∈ Ω, A, B ∈ Rd×N , L1 , L2 ∈ Rd×N×N , λ ∈ Rd , and ν ∈ SN−1 ,   (2) H (2)(x, A, L1 , L2 ) = inf Ex0 ,A (V ; Q) : V ∈ C bulk,(2)(L1 , L2 ) ;   h(2) (x, B, ν) = inf Exrec,(2) (V ; Qν ) : V ∈ C surf,(2)(B, ν) . 0

.

(4.34a) . (4.34b) surf,(1)

Remark 4.2 We note that formulas (4.31), (4.32), and the definitions of Ex0 , C bulk,(1) , and C surf,(1) show that I (1) does not depend upon the field Γ . Consequently, the dependence of I on Γ arises only from the bulk density H (2) in Theorem 4.4. The proofs of Theorems 4.3 and 4.4 rely on the following properties of the relaxed bulk and energy densities defined in (4.32) and (4.34). Proposition 4.2 (Properties of the Energy Densities (4.32) and (4.34) [5, Propositions 4.8, 4.9, 4.11, and 4.12]) Let H (1), h(1) and H (2), h(2) be defined by (4.32) and (4.34), respectively. Then (1) H (1)(x0 , 0) = 0 for every x0 ∈ Ω; (2) H (2)(x0 , A, 0, 0)  W (x, A, 0) for every x0 ∈ Ω and A ∈ Rd×N ; (3) there exists C > 0 such that |H (1)(x0 , A1 ) − H (1)(x0 , A2 )|  C|A1 − A1 | for every x0 ∈ Ω and A1 , A2 ∈ Rd×N ; (4) there exists C > 0 such that |H (2)(x0 , A1 , L, M1 ) − H (2)(x0 , A2 , L, M2 )|  C(|A1 − A2 | + |M1 − M − 2|) for every x0 ∈ Ω, A1 , A2 ∈ Rd×N , and L, M1 , M2 ∈ Rd×N×N ; (5) there exists C > 0 such that h(1) (x0 , λ, ν)  C|λ|, for every x0 ∈ Ω, λ ∈ Rd , and ν ∈ SN−1 ;

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

119

(6) there exists C > 0 such that h(2) (x0 , B, ν)  C|Λ|, for every x0 ∈ Ω, B ∈ Rd×N , and ν ∈ SN−1 ; (7) there exist continuous functions ωh(i) : [0, +∞) → [0, +∞), i = 1, 2 with ωh(i) (s) → 0 as s → 0+ such that |h(1)(x, λ, ν) − h(1) (x0 , λ, ν)|  ωh(1) (|x − x0 |)|λ|,

.

|h(2) (x, B, ν) − h(2) (x0 , B, ν)|  ωh(2) (|x − x0 |)(1 + |B|), for every x, x0 ∈ Ω, λ ∈ Rd , B ∈ Rd×N , and ν ∈ SN−1 ; (8) there exists C > 0 such that |h(1)(x0 , λ1 , ν) − h(1) (x0 , λ2 , ν)|  C|λ1 − λ2 |,

.

|h(2) (x0 , B1 , ν) − h(2) (x0 , B2 , ν)|  C|B1 − B2 |, for every x0 ∈ Ω, λ1 , λ2 ∈ Rd , B1 , B2 ∈ Rd×N , and ν ∈ SN−1 ; (9) h(i) are upper semicontinuous in their domains. Proof All properties are proved by means of the very definition of the energy densities in (4.32) and (4.34) as infima and of Assumptions 4.1 of the initial energy densities W , ψ (1) , and ψ (2) . Proof (Sketch of the Proof of Theorem 4.3) The proof is similar, in spirit, to that of Theorem 3.2 for first-order structured deformations: localization of the energy and the blow-up method are used to prove upper and lower bounds. One can follow the lines of the proof of the representation theorem in [3], together with [4] to address the explicit dependence on the space variable. The properties of the energy densities listed in Proposition 4.2 are used to prove the upper bounds. Proof (Sketch of the Proof of Theorem 4.4) In view of the comments made before the statement of Proposition 4.1, the representation is obtained by Theorem 3.2 (2) applied to EG . To show that its hypotheses are satisfied, it is necessary to prove that the energy density (x, L) → WG (x, L(x)) := W (x, G(x), L(x)) satisfies Assumptions 3.11 (with p = 1), 2, and 3: this is ensured by Proposition 4.2. Theorems 4.3 and 4.4 together provide the proof of the following result. Theorem 4.5 ([5, Theorem 3.2]) Let W : Ω × Rd×N × Rd×N×N → [0, +∞), ψ (1) : Ω × Rd × SN−1 → [0, +∞), and ψ (2) : Ω × Rd×N × SN−1 → [0, +∞) satisfy Assumptions 4.1; let (g, G, Γ ) ∈ SD 2 (Ω) and let I (g, G, Γ ) be given

120

4 Energetic Relaxation to Second-Order Structured Deformations

by (4.21). Then there exist H (1) : Ω × Rd×N → [0, +∞), H (2) : Ω × Rd×N × Rd×N×N → [0, +∞), h(1) : Ω × Rd × SN−1 → [0, +∞), and h(2) : Ω × Rd×N × SN−1 → [0, +∞) such that

 (1)  H (x, ∇g(x) − G(x)) + H (2)(x, G(x), ∇G(x), Γ (x)) dx I (g, G, Γ ) = Ω



+

.

+

Sg ∩Ω

SG ∩Ω

h(1) (x, [g](x), νg (x)) dH N−1 (x) h(2) (x, [G](x), νG (x)) dH N−1 (x), (4.35)

where the relaxed bulk and surface energy densities H (1), H (2), h(1) , and h(2) are defined in (4.32) and (4.34) and have the properties (1)–(9) in Proposition 4.2.

4.1.2 Relaxation in SBH The second approach to relaxation in the context of second-order structured deformations is the one carried out in [8] and is set in the context of SBH functions (see (2.14)), using some results concerning the relaxation of functionals in BH [11]. As observed at the beginning of this chapter, second-order structured deformations in this setting are identified by pairs (g, Γ ), in which g is the deformation function and Γ is an Rd×N×N -valued field describing the second-order gradient without sym disarrangements. We define now the space of second-order structured deformations in the SBH setting (see [7]). Definition 4.2 The space of (Rd -valued) second-order structured deformations on a domain Ω ⊂ RN in the SBH setting is SD2 (Ω) = SD2 (Ω; Rd × Rd×N×N ) := SBH (Ω; Rd ) × L1 (Ω; Rd×N×N ). sym sym

.

It is endowed with the natural norm induced by the product structure (g, Γ )SD2 (Ω) := gBH (Ω;Rd ) + Γ L1 (Ω;Rd×N×N ) ,

.

which is going to be denoted by (g, Γ )SD2 when no domain specification is needed.

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

121

Remark 4.3 In view of Remark 2.1, the inclusion SD2 (Ω)  SD 2 (Ω) is immediate, so that the SBH setting appears more constrained than the SBV 2 setting. The restrictions appear at two levels: • asking that g ∈ SBH (Ω; Rd ) so that, in particular g ∈ W 1,1 (Ω; Rd ) does not allow for disarrangements to appear though jumps in g itself; see (4.2b), where it is highlighted that the disarrangements tensor M = 0; • asking that Γ takes values in Rd×N×N restricts the second-order gradients sym without disarrangements to be symmetric in the sense of (4.1), whereas this constraint is visibly absent in the SBV 2 case. The implications of the observations of Remark 4.3 are so deep that, despite the SBH setting being a particular case of the SBV 2 setting, the approximation techniques presented in Sect. 4.1.1 cannot be directly applied. In particular, they do not guarantee that elements (g, Γ ) ∈ SD2 (Ω) can be approximated by functions un ∈ SBH (Ω; Rd ) (and therefore in W 1,1 (Ω; Rd )). Instead, an approximation theorem for second-order structured deformation in SD2 (Ω) is proved using Theorem 2.5, an extension of Alberti’s result to second-order gradients, and Lemma 2.3. Theorem 4.6 (Approximation Theorem [8, Theorem 3.2]) Let (g, Γ ) ∈ ∗ SD2 (Ω). Then there exists a sequence {un } ⊂ SBH (Ω; Rd ) such that un − SD2

(g, Γ ) (see (4.6)) and such that, for a constant C > 0 depending only on N, .

sup un BH  C(g, Γ )SD2 .

(4.36)

n∈N

Proof (Sketch of the Proof of Theorem 4.6) Let us suppose that ⎧ ⎨for every Δ ∈ L1 (Ω; Rd×N×N ) there exists fε ∈ SBH (Ω; Rd ) such that sym ∗

.

⎩fε − (0, Δ) and sup |D 2 fε |(Ω)  CΔL1 (Ω;Rd×N×N ) , SD2

ε

(4.37) for a certain C > 0. It follows that the sequence of functions un := g + fεn , with fε chosen as in (4.37) for Δ = Γ − ∇ 2 g, converges to (g, Γ ) in the sense of (4.6). Indeed, by (4.37) we have that ∗

un → g+0 in W 1,1 (Ω; Rd ) and ∇ 2 un  ∇ 2 g+Γ −∇ 2 g in M (Ω; Rd×N×N ). sym

.

In particular,   sup un BH  sup gBH + fεn BH n∈N .

n∈N

 gBH + sup |D 2 fεn |(Ω) + sup fεn W 1,1 (Ω;Rd ) , n∈N

so that (4.36) follows by the second line in (4.37).

n→∞

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4 Energetic Relaxation to Second-Order Structured Deformations

The proof of (4.37) is technical and is achieved by explicitly constructing the sequence {fε } in the following way: upon tessellating the space with cubes {Qε, } centered on the lattice (εZ)N and of side-length ε, it is requested that



∇ 2 fε (x) dx =

.

Qε,

Δ(x) dx.

(4.38)

Qε,

(Essentially, fε are suitable truncations of localized quadratic forms determined by tensors Aε, ∈ Rd×N×N .) We refer the reader interested in the details to the proof of sym [8, Theorem 3.2], where the construction is explicit for N = 2 and the convergence fε → 0 in W 1,1 (Ω; Rd ) is obtained via the estimates

|fε (x)| dx  Cε2 ΔL1 (Ω;Rd×N×N ) and sym

.

Ω

Ω

|∇fε (x)| dx  CεΔL1 (Ω;Rd×N×N ) ; sym

moreover, the estimates

.

Ω

S∇fε ∩Ω

|∇ 2 fε (x)| dx  CΔL1 (Ω;Rd×N×N ) sym

|[∇fε (x)]| dH N−1 (x)  CΔL1 (Ω;Rd×N×N ) sym

imply the last condition in (4.37); finally, (4.38) and Lemma 2.3 provide the ∗ convergence ∇ 2 fε  Δ in M (Ω; Rd×N×N ), which concludes the proof of (4.37) and of the theorem. The Approximation Theorem 4.6 just obtained will be crucial to show that the class of competitors R SD2 defined in (4.7) for the relaxation problem (4.8) defining the energy I (g, Γ ) of a second-order structured deformation in the SBH setting is non-empty. Theorem 4.9 below will provide the desired integral representation result for the functional I(g, Γ ) defined in (4.9). We refer the reader to Remark 4.6 for a comparison between I (g, Γ ) and I(g, Γ ). We anticipate that the proof of the integral representation Theorem 4.9 will be a mere corollary of the integral representation result contained in Theorem 4.8 proved in the setting of the global method for relaxation synthesized in Sect. 2.4. To this end, we consider an abstract functional F : SD2 (Ω) × A (Ω) → [0, +∞]

.

satisfying the following hypotheses.

(4.39)

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

123

Assumptions 4.2 We require that the abstract functional F in (4.39) satisfies: 1. (g, Γ ) → F (g, Γ ; ·) is the restriction to A (Ω) of a Radon measure; 2. U → F (·, ·; U ) is SD2 (Ω)-lower semicontinuous, namely if (g, Γ ) ∈ SD2 (Ω) ∗ and {(gn , Γn )} ⊂ SD2 (Ω) are such that gn → g in W 1,1 (Ω; Rd ) and Γn  Γ in M (Ω; Rd×N×N ), then F (g, Γ ; U )  lim inf F (gn , Γn ; U );

.

n→∞

(4.40)

3. there exists C > 0 such that for all (g, Γ ) ∈ SD2 (Ω) and U ∈ A (Ω),   2 C −1 Γ L1 (U ;Rd×N×N ) + |D g|(U )  F (g, Γ ; U ) sym .   2  C L N (U ) + Γ L1 (U ;Rd×N×N + |D g|(U ) ; ) sym 4. F is local, that is for all U ∈ A (Ω), (g1 , Γ1 ), (g2 , Γ2 ) ∈ SD2 (Ω), if g1 (x) = g2 (x) and Γ1 (x) = Γ2 (x) for L N -a.e. x ∈ U , then F (g1 , Γ1 ; U ) = F (g2 , Γ2 ; U ). Remark 4.4 We notice that the locality property was a consequence of the (strong) L1 -lower semicontinuity Assumption 2.1-(2) in the general setting for the global method for relaxation in BV , see Remark 2.3. In the case of the SD2 (Ω)-lower ∗ semicontinuity of Assumption 4.2-2, the weak-* convergence Γn  Γ in the sense of measures is not enough to guarantee locality; consequently, locality must be imposed separately in Assumption 4.2-4. In order to continue with the program of the global method for relaxation, we need to define the class of admissible functions for the definition of the set function m(g, Γ ; ·) (which will be the analogue of the m(u; ·) defined in (2.22)). Given (g, Γ ; U ) ∈ SD2 (Ω) × A (Ω), let  

    A(g, . Γ ; U ) := (g, ˜ Γ ) ∈ SD2 (Ω) : spt(g˜ − g)  U, Γ (x) − Γ (x) dx = 0 , U

(4.41) so that we can define m : SD2 (Ω) × A (Ω) → [0, +∞] by   m(g, Γ ; U ) := inf F (g, ˜ Γ; U ) : (g, ˜ Γ) ∈ A(g, Γ ; U ) .

.

(4.42)

The first result is that the abstract functional F of (4.39) and the functional m defined in (4.42) have the same Radon-Nikodým derivative with respect to the measure μ := L N

.

Ω + |(D 2 )s g|.

(4.43)

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4 Energetic Relaxation to Second-Order Structured Deformations

We observe that this particular measure μ is intended to capture the singularities at the level of the second distributional derivative of the function g: it is composed of the Lebesgue measure on Ω and of the singular part of the second-order distributional derivative of g. This is consistent with the fact that g ∈ SBH (Ω; Rd ) is such that Dg = ∇g, so that any singularities with respect to the Lebesgue measure can only be seen at the level of (D 2 )s g. Theorem 4.7 ([8, Theorem 4.3]) Let Assumptions 4.2 hold. Then for every (g, Γ ) ∈ SD2 (Ω), for every ν ∈ SN−1 , and for μ-a.e. x0 ∈ Ω we have .

lim

ε→0

F (g, Γ ; Qν (x0 , ε)) m(g, Γ ; Qν (x0 , ε)) = lim . ε→0 μ(Qν (x0 , ε)) μ(Qν (x0 , ε))

With this result, the analogue of Theorem 2.6 can be obtained for F , and the corresponding densities H and h(2) (see (4.45) below) will be obtained with formulae analogous to (2.23), where the functional m involved is that defined in (4.42) above. The integral representation result for the abstract functional F of (4.39) is the following. Theorem 4.8 (Integral Representation [8, Theorem 4.6]) Let Assumptions 4.2 hold. Then for every (g, Γ ) ∈ SD2 (Ω; Rd ) and for every U ∈ A (Ω) we have

F (g, Γ ; U ) = H (x, g(x), ∇g(x), ∇ 2 g(x), Γ (x)) dx U

.

+

S∇g ∩U

h(2) (x, g(x), ∇g + (x), ∇g − (x), ν∇g (x)) dH N−1 (x), (4.44)

where, for all x0 ∈ Ω, b ∈ Rd , A, B ∈ Rd×N , Φ, Λ ∈ Rd×N×N , and ν ∈ SN−1 , sym H (x0 , b, A, Φ, Λ) := lim

.

ε→0+

h(2) (x0 , b, A, B, ν) := lim

ε→0+

m(qb,A,Φ (· − x0 ), Λ; Q(x0 , ε)) ,. εN

(4.45a)

m(b + ΣA,B,ν (· − x0 ), O; Qν (x0 , ε)) , εN−1

(4.45b)

where O ∈ Rd×N×N is the zero element, qb,A,Φ : RN → Rd is the function defined by 1 qb,A,Φ (x) := b + Ax + Φx, x, 2

.

4.1 Spaces of Second-Order Structured Deformations, Approximation. . .

125

and ΣA,B,ν : RN → Rd is the function defined by ΣA,B,ν (x) :=

.

 Ax

if x · ν  0,

Bx

if x · ν < 0.

Proof (Sketch of the Proof of Theorem 4.8) The proof is obtained by using approximate first- and second-order differentiability results for functions in BH (see [8, Theorem 2.1]) and a continuity property (see [8, Corollary 4.5]) for the functional G (g, Γ ) := lim

.

ε→0

m(g, Γ ; Qν (x0 , ε)) , λ(Qν (x0 , ε))

where λ is a non-negative Radon measure. The representation result follows by Theorem 4.7 and by applying a blow-up argument à la Fonseca-Müller [9, 10] to G . The general integral representation Theorem 4.8 can be used to deduce a formula like (4.44) if the abstract functional F of (4.39) is indeed obtained from an integral functional E : SBH (Ω; Rd ) → [0, +∞) determined by suitable bulk and surface energy densities W and ψ (2) . In this case, we will be able to tackle problem (4.5) of defining an energy I (g, Γ ) for a second-order structured deformation (g, Γ ) ∈ SD2 (Ω) as the relaxation of the initial energy E. The process in this situation is analogous to that of Sect. 4.1.1 but the initial energy needs to be both extended and localized from the beginning in such a way to apply the machinery of the global method for relaxation. For u ∈ SBH (Ω; Rd ) and U ∈ A (Ω), let

E(u; U ) := W (x, u(x), ∇u(x), ∇ 2 u(x)) dx .

U

+

S∇u ∩U

ψ (2) (x, u(x), ∇u+ (x), ∇u− (x), ν∇u (x)) dH N−1 (x), (4.46)

where the bulk energy density W : Ω × Rd × Rd×N × Rd×N×N → [0, +∞) and the surface energy density ψ (2) : Ω × Rd × Rd×N × Rd×N × SN−1 → [0, +∞) are functions that satisfy the following hypotheses. Assumptions 4.3 We require that the functions W and ψ (2) satisfy: 1. W is measurable in x and continuous in all other variables, and there exists C > 0 such that, for all x ∈ Ω, b ∈ Rd , A ∈ Rd×N , and Λ ∈ Rd×N×N , .

1 |Λ|  W (x, λ, A, Λ)  C(1 + |Λ|); C

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4 Energetic Relaxation to Second-Order Structured Deformations

2. ψ (2) is continuous and there exists C > 0 such that for all x ∈ Ω, b ∈ Rd , A, B ∈ Rd×N , and ν ∈ SN−1 , .

1 |A − B|  ψ(x, b, A, B, ν)  C(1 + |A − B|). C

We now define the following energy, for U ∈ A (Ω) and u ∈ SBV 2 (Ω; Rd ),  U ) := E(u;

.

 E(u; U ) +∞

if u ∈ SBH (Ω; Rd ), otherwise.

(4.47)

For (g, Γ ) ∈ SD2 (Ω) and U ∈ A (Ω), define the class of admissible sequences  SD2 (g, Γ ; U ) := {un } ⊂ SBH (U ; Rd ) : un → g in L1 (U ; Rd ), R .  ∗ ∇ 2 un  Γ in M (U ; Rd×N×N ) .

(4.48)

The energy (4.9) associated with a second-order structured deformation in the SBH setting can then be written as the relaxation of the (extended) initial energy (4.47) SD2 , namely, for every U ∈ A (Ω) in the class R   SD2 (g, Γ ; U )  n ; U ) : {un } ∈ R I(g, Γ ; U ) = inf lim inf E(u

.

n→∞

(4.49)

The integral representation result in the SBH setting is the following. Theorem 4.9 Let W : Ω ×Rd ×Rd×N ×Rd×N×N → [0, +∞) and ψ (2) : Ω ×Rd × Rd×N × Rd×N × SN−1 → [0, +∞) satisfy Assumptions 4.3; let (g, Γ ) ∈ SD2 (Ω) and let, for every U ∈ A (Ω), the energy I(g, Γ ; U ) be given by (4.49). Then there exist functions H : Ω × Rd × Rd×N × Rd×N×N × Rd×N×N → [0, +∞) and h(2) : Ω × Rd × Rd×N × Rd×N × SN−1 → [0, +∞) such that

 H (x, g(x), ∇g(x), ∇ 2 g(x), Γ (x)) dx I (g, Γ ; U ) = U .

+

S∇g ∩U

h(2) (x, g(x), ∇g + (x), ∇g − (x), ν∇g (x)) dH N−1 (x), (4.50)

where the relaxed bulk and surface energy densities H and h(2) are defined in (4.45), where the functional m is given by (4.42), with F = I.

4.2 Outlook for Applications

127

Proof Assumptions 4.3 allow one to prove that the Assumptions 4.2 required to apply the global method for relaxation in the SD2 context are satisfied, so the proof of the theorem follows by the abstract result of Theorem 4.8. Remark 4.5 A technical part of the proof of Theorem 4.9, [8, Lemma 5.1], shows that the very definition (4.49) of  I implies that it is lower semicontinuous in the following sense: for every U ∈ A (Ω), (g, Γ ) ∈ SD2 (Ω), and {(gn , Γn )} ⊂ ∗ SD2 (Ω) such that un → u in L1 (Ω; Rd ) and Γn  Γ in M (Ω; Rd×N×N ), I(g, Γ ; U )  lim inf I(gn , Γn ; U ).

.

n→∞

(4.51)

SD2 and it guarantees This lower semicontinuity motivates the definition of R that the SD2 -lower semicontinuity Assumption 4.2-2 can be fulfilled. Indeed, if {(gn , Γn )} ⊂ SD2 (Ω) is such that gn → g in W 1,1 (Ω; Rd ) and Γn → Γ in M (Ω; Rd×N×N ), for a certain (g, Γ ) ∈ SD2 (Ω), then, in particular gn → g in L1 (Ω; Rd ), so that (4.51) implies that (4.40) is satisfied for I. Remark 4.6 A comparison of (4.6) and (4.7), on the one hand, with (4.48) on the other shows that the difference between the functionals I (g, Γ ) in (4.5) and I(g, Γ ; Ω) in (4.49) resides in the different topologies with respect to which the relaxation is performed. The topology of the class R SD2 is stronger than that of the SD2 , so that one naturally has that class R I(g, Γ ; Ω)  I (g, Γ ).

.

Theorem 4.9 thus provides an integral representation of the functional  I in (4.49) assigning to a second-order structured deformation (g, Γ ) ∈ SD2 (Ω) a lower energy than does the functional I in (4.5).

4.2 Outlook for Applications The principal variational results for second-order structured deformations, Theorems 4.2 and 4.3 in the SBV 2 -theory and Theorems 4.8 and 4.9 in the SBH -theory, are recent enough that few examples and applications have appeared in the literature at the time of this writing (see [5, Section 6] and [8, Section 6]). Beyond these instances, all of the applications of the variational theory of first-order structured deformations presented here in Sect. 3.2 are, in principle, open for reconsideration in the context of second-order structured deformations. In addition, the application of second-order structured deformations to multiscale differential geometry presented here in Sect. 5.2 provides a setting in which the relaxation of energies assigned to classical hypersurfaces would lead to a rational method for the assignment of an energy to structured hypersurfaces and, thereby, would suggest many further applications.

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4 Energetic Relaxation to Second-Order Structured Deformations

References 1. A. Acharya and C. Fressengeas: Continuum mechanics of the interaction of phase boundaries and dislocations in solids. Differential Geometry and Continuum Mechanics, Ed: G. Q. Chen, M. Grinfeld, R.J. Knops. Proc. in Mathematics & Statistics Springer 137 (2015), 125–168. 2. V. Agrawal and K. Dayal: Dynamic Phase-field Model for Structural Transformations and Twinning: Regularized Interfaces with Transparent Prescription of Complex Kinetics and Nucleation. Part I: Formulation and One-Dimensional Characterization. J. Mech. Phys. Solids, 85 (2015), 270–290. 3. M. Baía, J. Matias, and P. M. Santos: A relaxation result in the framework of structured deformations in the bounded variation setting. Proc. Royal Soc. Edinburgh A, 142 (2012), 239–271. 4. A. C. Barroso, G. Bouchitté, G. Buttazzo, and I. Fonseca: Relaxation of bulk and interfacial energies. Arch. Rational Mech. Anal., 135 (1996), 107–173. 5. A. C. Barroso, J. Matias, M. Morandotti and D. R. Owen: Second-order structured deformations: relaxation, integral representation and examples. Arch. Rational Mech. Anal., 225 (2017), 1025–1072. 6. R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103. 7. G. Del Piero and D. R. Owen: Multiscale Modeling in Continuum Mechanics and Structured Deformations. CISM Courses and Lecture Notes 447, Springer, 2004. 8. I. Fonseca, A. Hagerty, and R. Paroni: Second-order structured deformations in the space of functions of bounded hessian. J. Nonlinear Sci., 29(6) (2019), 2699–2734. 9. I. Fonseca and S. Müller: Quasi-convex integrands and lower semicontinuity in L1 . SIAM J. Math. Anal., 23(5)(1992), 1081–1098. 10. I. Fonseca and S. Müller: Relaxation of quasiconvex functionals in BV(Ω, R p ) for integrands f (x, u, ∇u). Arch. Rational Mech. Anal., 123(1) (1993), 1–49. 11. A. Hagerty: Relaxation of functionals in the space of vector-valued functions of bounded Hessian. Calc. Var. PDE, 58(1) (2019), Paper No. 4, 38 pages. 12. R. D. James and K. F. Hane: Martensitic transformations and shape-memory materials. Acta Mater., 48 (2000), 197–222. 13. D. R. Owen and R. Paroni: Second-order structured deformations. Arch. Rational Mech. Anal. 155 (2000), 215–235. 14. R. Paroni: Second-order structured deformations: approximation theorems and energetics. Multiscale Modeling in Continuum Mechanics and Structured Deformations, edited by G. Del Piero and D. R. Owen, CISM 447 2004.

Chapter 5

Outlook for Future Research

5.1 Microdegree and Micromixing The following description of concepts and preliminary results addresses issues that arose in discussions at various times among the authors and our colleagues Ana Cristina Barroso, Gianni Dal Maso, and Elvira Zappale. Although the description below mainly concerns structured deformations .(g, G) in the original .L∞ -setting [4], the principal concepts introduced here in that setting suggest counterparts in the variational settings for structured deformations described in Chaps. 3 and 4 of this book as well as in descriptions of mixtures in the .L∞ -setting [4, 6, 7]. The present motivation for this line of investigation is the emergence of submacroscopic voids during multiscale deformations of continua and the need to provide precise measures of the “degree of occupancy” arising from such deformations. These considerations point to a notion of “microdegree” whose density is a local measure of the degree of occupancy and is multiplicative under composition of structured deformations. We briefly explore here possible counterparts in the SBV -setting from Chap. 3 that may be useful in the description of “micromixing” of a continuum. Let A and E be bounded, non-empty open subsets of .RN , let .f : A → RN , and define ⎧

⎨ H 0 (f −1 ({y})) dy if y → H 0 (f −1 ({y})) is measurable, degA,E f := E ⎩ +∞ otherwise. (5.1)   Here, E := (L N (E))−1 E , and .H 0 is the counting measure, so that the integrand .H 0 (f −1 ({y})) is the cardinality of the preimage of the point .y ∈ E under the given mapping f . The function .y → H 0 (f −1 ({y}) is defined on all of N and is called the multiplicity function for f . We call .deg .R A,E f the mean degree of occupancy for f within E or, briefly, the occupancy degree of f in E. In the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. Matias et al., Energetic Relaxation to Structured Deformations, SpringerBriefs on PDEs and Data Science, https://doi.org/10.1007/978-981-19-8800-4_5

129

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5 Outlook for Future Research

particular case where the open set E equals .f (A) and where .y → H 0 (f −1 ({y}) is measurable, we have the following useful formula for the integral in (5.1):

.

H 0 (f −1 ({y})) dy =

f (A)

RN

H 0 (f −1 ({y}) ∩ A) dy.

(5.2)

This formula follows from the fact that, in the formula above, the integrand on the right, .y → H 0 (f −1 ({y})∩A) with .y ∈ RN , is the extension to .RN of the integrand 0 −1 ({y})) with .y ∈ f (A) by the zero function 0 on .RN \ f (A). .y → H (f In the case where f is injective and has minimal regularity, the following result shows that .degA,E f measures the fraction of the volume of E that is occupied by the range of f . Remark 5.1 If f is injective and .f (A) is .L N -measurable, then .degA,E f = L N (E ∩ f (A))/L N (E) ∈ [0, 1]. In this case (i) .degA,E f = 0 if and only if N .E ∩ f (A) is an .L -null set, and (ii) .degA,E f = 1 if and only if .E \ f (A) is an N .L -null set. In particular, .degA,f (A) f = 1. Proof By the injectivity of f and properties of the counting measure .H 0 , we have  H 0 (f −1 ({y})) =

.

+1 if y ∈ E ∩ f (A), 0

if y ∈ E \ f (A),

so that .H 0 (f −1 ({y})) = χE∩f (A) (y) for all .y ∈ E. The conclusion then follows from the definition of .degA,E f . The hypothesis of injectivity when applied to the macroscopic deformations of a continuous body is synonymous with the notion that the matter composing one part of the body cannot penetrate the matter composing another part initially disjoint from the first. This hypothesis of “noninterpenetrability of matter” may also be called the “absence of mixing”. The following example in the one-dimensional case .N = 1 provides the values of the occupancy degree .degA,f (A) f when A differs from .(0, 1) by an .L 1 -null set and when f represents a uniform stretching of A by a factor .γ followed by a complicated macroscopic mixing of smaller and smaller subintervals of .(0, 1). Example 5.1 (Stretching and Macromixing) Put .N = 1, let .γ > 0 be given, define sequences .m → ym , and .m → xm for .m ∈ N \ {0} by ym =

.

γ 4m

and

xm =

1 2m−1

,

(5.3)

5.1 Microdegree and Micromixing

131

and define .f : A → R (with A to be specified below) as follows: for .m ∈ N \ {0}, for .j = 0, 1, . . . , 2m − 1, and for . 21m (1 + 2jm ) < x < 21m (1 + j2+1 m ),  j  1  := .f (x) γ x− m 1+ m . 2 2

(5.4)

Let .m ∈ N \ {0} be given. , 2m  − 1, the function f restricted  1For eachj .j =1 0, 1, . .j .+1 to the interval .Im,j := 2m (1 + 2m ), 2m (1 + 2m ) is thus a linear function with ,m derivative .γ and range .(0, ym ). Because . 2j =0−1 I¯m,j = [xm , xm−1 ], it follows that ,m  f restricted to .Am := (xm+1 , xm ) \ 2j =1−2 21m (1 + 2jm ) is a piecewise linear function with derivative .γ , and with range .(0, ym ). Moreover, an induction on m shows easily that for all .y ∈ (ym+1 , ym ) H (f 0

.

−1

({y})) =

m

2k = 2(2m − 1)

(5.5)

k=1

which reflects the fact that, for each .y ∈ (ym+1 , ym ), there are in total .2(2m − 1) k points .xk,j ∈ Ikj for ,∞.k = 1, 2, . . . , m and .j = 0, 1, . . . , 2 − 1 that map via f onto y. With .A := m=1 Am , the definition (5.1) of the occupancy degree .degA,E f m then yields ,∞ immediately .degA,Em f = 2(2 − 1). In addition, for the target set .E∪ := m=1 (ym+1 , ym ), which differs from the interval .(0, y1 ) = (0, γ /4) by an 1 .L -null set, we can calculate .degA,E f as follows ∪

degA,E∪ f =

H 0 (f −1 ({y})) dy = E∪

4 γ



γ /4

H 0 (f −1 ({y})) dy

0

∞

∞ 4 4 1 0 −1 = H (f ({y})) dy = L (Em ) degA,Em f γ γ Em m=1

=

4 γ

∞

m=1

γ (4−m − 4−(m+1) )2(2m − 1) = 4.

m=1

The conclusion that the occupancy degree of f in its entire range .E∪ is finite (with value 4) reflects the fact that the sets .Em , while having occupancy degree that increases with m at the asymptotic rate .2m , have measure that decreases at the rate .4−m and shows that the definition of occupancy degree supports complicated macromixings of a continuum.

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5 Outlook for Future Research

In order to address the issue of submacroscopic mixing (or, more briefly, micromixing), we consider first the particular case where • A and E are piecewise-fit regions [4]: each of A and E is a finite union of (possibly overlapping) fit regions i.e., of bounded, regularly open sets of finite perimeter whose boundaries have zero .L N -measure, and • f is a simple deformation from A in the sense of Del Piero and Owen [4]: f is injective, and the restriction of f to each of the fit regions whose union is A extends to a .C 1 -diffeomorphism of .RN with .det ∇f (x) > 0 for all .x ∈ A. (We take the set .κ in the original definition of simple deformation to be the empty set.) The injectivity and piecewise-regularity of simple deformations permit the explicit evaluation of their occupancy degrees by application of Remark 5.1. Because a simple deformation f is “piecewise Lipschitzian” (albeit without f itself possessing a continuous extension to the closure of A), an additional formula for .degA,E f for simple deformations arises again in the case where .E = f (A):

det ∇f (x) dx .

degA,f (A) f = 1 =

A

L N (f (A))

.

(5.6)

The first equality follows from Remark 5.1, while the area formula [1, Theorem 2.71, Remark 2.72], and [4, Lemma 3.9] imply the equality of the numerator and denominator of the fraction in (5.6). If A is piecewise fit and non-empty and if .(g, G) is a structured deformation from A in the sense of Del Piero and Owen, then the Approximation Theorem [4, Theorem 5.8] guarantees the existence of a sequence of simple deformations .n → fn such that (see (1.2)) fn − gL∞ (A;RN ) → 0

.

and

∇fn − GL∞ (A;RN×N ) → 0.

(5.7)

In the spirit of obtaining a measure of the degree to which the ranges of the approximating simple deformations .fn occupy the range of the simple deformation g, we define .

  L N (fn (A)) degA (g, G) := inf lim inf , n→∞ L N (g(A))

(5.8)

the occupancy degree of .(g, G) in .g(A), where the infimum is taken over the sequences .n → fn of simple deformations satisfying (5.7). Because such sequences carry geometrical information from submacroscopic geometrical changes, such as slipping and the formation of voids, we also use the term the microdegree of .(g, G) in .g(A) for .degA (g, G). The next result shows that the infimum .degA (g, G) is attained on every sequence satisfying (5.7).

5.1 Microdegree and Micromixing

133

Theorem 5.1 (Formula for the Degree) For each piecewise-fit region A, each structured deformation .(g, G) from A, and each sequence of simple deformations .n → fn satisfying (5.7) there holds

lim L N (fn (A)) n→∞

. degA (g, G) = lim = n→∞ L N (g(A))

det ∇fn (x) dx A

det ∇g(x) dx

A

det G(x) dx = A . det ∇g(x) dx A

(5.9) Moreover, for each .x ∈ A there holds .

lim degBε (x) (g, G) =

ε→0

det G(x) , det ∇g(x)

(5.10)

and both of the numbers .degA (g, G) and .limε→0 degBε (x)(g, G) lie in the interval (0, 1].

.

Proof The formula (5.9) follows from the fact that .fn and g are simple deformations, from the reasoning following the formula (5.6), and from the nature of the convergence of .∇fn to G in (5.7). The formula (5.10) results when one puts .A := Bε (x), divides both numerator and denominator of the last fraction in (5.9) by .L N (Bε (x)), and takes the limit .ε → 0. The last assertion follows immediately from the accommodation inequality (1.1). For each structured deformation .(g, G) from A we call the scalar field mA (g, G) :=

.

det G : A → (0, 1], det ∇g

(5.11)

the microdegree density of .(g, G). The value of the microdegree density .mA (g, G) at each .x ∈ A gives the degree of occupancy of .(g, G) at .g(x), i.e., the fraction of the portion of the range .g(A) of g near .g(x) that is occupied under the action of the structured deformation .(g, G) on its domain A. The counterpart of .mA (g, G) in the mechanics literature typically is called the volume fraction field. The difference .1 − mA (g, G) : A → [0, 1) evaluated at .x ∈ A gives the degree of vacancy of the range of g near .g(x), and the counterpart of .1−mA (g, G) in the mechanics literature is called the void fraction field. The behavior of the microdegree density .mA (g, G) in (5.11) under the composition operation .♦ of structured deformations [4], (h, H )♦(g, G) := (h ◦ g, (H ◦ g)G),

.

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5 Outlook for Future Research

is revealed in the calculation det(H ◦ g) det G det H det G det((H ◦ g)G) = =( ◦ g) , det ∇(h ◦ g) det((∇h ◦ g)∇g) det ∇h det ∇g

.

which shows that mA ((h, H )♦(g, G)) = [mg(A) (h, H ) ◦ g]mA (g, G),

.

(5.12)

i.e., the microdegree density of the composition is the product of the microdegree densities (adjusted by composition with g). For example, following [4], if we put −1 G and .H := G(∇g)−1 then we obtain the two factorizations .K := (∇g) (g, G) = (i, H ◦ g −1 )♦(g, ∇g) = (g, ∇g)♦(i, K)

.

(5.13)

of .(g, G) in which the classical deformation .(g, ∇g) precedes or follows one of the two purely submacroscopic deformations .(i, K) and .(i, H ◦ g −1 ). The relations −1 ) ◦ g = det(H ◦ g −1 ) ◦ g = det H , .mA (g, G) = det K = det H , .mg(A) (i, H ◦ g .mA (g, ∇g) = 1, and .mA (i, K) = det K confirm the validity of the composition rule (5.12) for the factorizations (5.13) that turn out to play a central role in many of the applications of structured deformations (see, e.g., [2, 3, 9, 10]). The notions of microdegree and of microdegree density for structured deformations .(g, G) in the .L∞ -setting rest heavily on both the piecewise-smoothness and the injectivity of g, as well as of its approximations .fn , properties that are not guaranteed in the BV - and SBV -variational settings. The availability of piecewise-smooth, injective approximations .fn satisfying (5.7) for .(g, G) rests on the accommodation inequality 0 < c < det G(x)  det ∇g(x) for all x ∈ A,

.

(5.14)

[4, Theorems 4.17 and 5.8]. In fact, the cited theorems show that the accommodation inequality is essentially a necessary and sufficient condition for the availability of injective approximations satisfying (5.7). Of course, .det G(x) and .det ∇g(x), in general, have no meaning in the variational settings, so that the accommodation inequality, itself, is meaningless in those settings. Accordingly, it is natural to seek counterparts of (5.9), (5.10), and (5.14) that are meaningful in variational settings. To this end, we note from Theorem 5.1 and (5.9) that, in the .L∞ -setting, .

L N (fn (A)) , n→∞ L N (g(A))

degA (g, G) = lim

(5.15)

5.1 Microdegree and Micromixing

135

while (5.2) and the proof of Remark 5.1 show that

L N (fn (A)) f (A) = n . L N (g(A))

H 0 (fn−1 ({y})) dy H (g 0

−1

RN

=

({y})) dy

g(A)

RN

H 0 (fn−1 ({y}) ∩ A) dy H 0 (g −1 ({y}) ∩ A) dy

.

(5.16) Therefore, the content of (5.15) can be expressed via (5.16) in terms of the multiplicity functions for .fn and for g. Moreover, in the special case where .G = ∇h with h a simple deformation, we can write the last expression for .degA (g, G) in (5.9) as

H 0 (h−1 ({y}) ∩ A) dy N R , . degA (g, G) =

(5.17) 0 −1 H (g ({y}) ∩ A) dy RN

a relation that expresses the occupancy degree of .(g, G) in terms of the multiplicity functions of g and of h, a simple deformation whose gradient is G. The relation (5.17) in the .L∞ -setting for the case .G = ∇h, with .∇h the classical derivative of a smooth function, points us immediately to the SBV -setting for structured deformations. There, Alberti’s Theorem 2.4 tells us that, for every 1 N×N ), there exists .h ∈ SBV (Ω; RN ) such that .G = ∇h, with .G ∈ L (Ω; R .∇h now denoting the matrix field associated with the absolutely continuous part N of the distributional derivative of h. This result leads us in the SBV -setting .∇hL to consider .(g, G) ∈ SD(Ω; RN × RN×N ) as in Definition 3.1 and .A ⊂ Ω for which .y → H 0 (g −1 ({y}) ∩ A) is measurable, and then to define (provisionally)

 degA (g, G) := inf

lim inf n→∞

.

N

R

RN

H 0 (h−1 n ({y}) ∩ A) dy H 0 (g −1 ({y}) ∩ A) dy

:

 hn ∈ SBV (A; RN ), ∇hn = G, hn → g in L1 (A; RN ) , (5.18) the occupancy degree of .(g, G) in the SBV -setting for structured deformations. Motivated by (5.10) we define (again provisionally) for .L N -almost every .x ∈ A mA (g, G)(x) := lim degBε(x) (g, G).

.

ε→0

to be the microdegree density at x for .(g, G).

(5.19)

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5 Outlook for Future Research

At the time of writing, we have not determined which, if any, additional restrictions on .(g, G) ∈ SD(Ω; RN × RN×N ) are required to render the provisional definitions (5.18) and (5.19) meaningful and useful. Should this issue be settled satisfactorily with these definitions substantially unchanged, it is natural to offer further a counterpart of the accommodation inequality (5.14) that may be meaningful and significant in the SBV -setting: for .(g, G) ∈ SD(Ω; RN × RN×N ) there exists a sequence .{hn } admissible for (5.18) such that for .L N -almost every .y ∈ RN and for every .n ∈ N 0 −1 H 0 (h−1 n ({y}) ∩ A)  H (g ({y}) ∩ A).

.

(5.20)

This “multiplicity inequality” would imply in non-trivial cases that .degA (g, G) lies in the interval .(0, 1]. However, when this restriction on .degA (g, G) is not satisfied, e.g., if in (5.18) .degA (g, G) > 1, then this would signal a degree of micromixing of a continuum under the structured deformation .(g, G).

5.2 Multiscale Differential Geometry In this section we present a preliminary investigation of the use of structured deformations to broaden the scope of differential geometry through a notion of structured manifolds, i.e., manifolds that possess not only classical differential structures with associated tangent spaces and curvatures, but also that reflect similar geometrical properties at a submacroscopic level. We have in mind the idea that approximations of structured deformations would translate into approximations of structured manifolds and would then provide a rich setting in which the relaxation techniques presented in Chaps. 3 and 4 could be applied. Our initial attempt below to explore this idea is limited to a study of structured hypersurfaces in .RN+1 . Nevertheless, this narrow aspect of differential geometry allows us to bring to bear the .L∞ -, .SBV 2 -, and SBH -settings of second-order structured deformations treated in Chaps. 1 and 4 and to apply results of those treatments to a different branch of mathematics. Moreover, the interdisciplinary research [3] points to contexts in biophysics and bioengineering in which structured hypersurfaces in .R3 may have fruitful applications.

5.2.1 Toward a Multiscale Enrichment of Classical Hypersurfaces in RN +1 The content of this subsection provides a specific, classical differential-geometric setting that is the basis for our definition of “structured hypersurfaces”, a multiscale counterpart of classical hypersurfaces to be explored in subsequent subsections

5.2 Multiscale Differential Geometry

137

of Sect. 5.2.2. We here follow [11, Chapter 5] for the definition of classical hypersurfaces and for a characterization of classical hypersurfaces in terms of local parametrizations. In the following we take .N  1, and the case .N = 1 treats 2 2 .C -curves in .R . We provide here a detailed treatment of the well-known result on the symmetry of the Weingarten map, a map that at a given point of a classical hypersurface is the restriction of the gradient of the unit normal vector field to the tangent space at that point. The arguments in our treatment are needed to explore a refined notion of curvature for structured hypersurfaces. Definition 5.1 ([11, p. 109]) A subset .S of .RN+1 is called a classical hypersurface if, for every .x ∈ S , there is an open set .U ⊂ RN+1 containing x, an open set N+1 , and a .C 2 - diffeomorphism .h : U → V such that .V ⊂ R h(U ∩ S ) = V ∩ (RN × {0}) = {v ∈ V : v = (y, 0) with y ∈ RN }.

.

(5.21)

The intuition underlying this definition is: .S is locally, up to a diffeomorphism, a portion of the hyperplane .RN × {0} in .RN+1 . In other terms, for each .x ∈ S , the mapping f given by f (y) := h−1 (y, 0),

.

(5.22)

that maps the subset .{y ∈ RN : (y, 0) ∈ V } of .RN onto .U ∩ S , “lifts” to the .C 2 -diffeomorphism .h−1 of V onto U . The following theorem provides an equivalent description of .C 2 -hypersurfaces, employing counterparts of the mappings f in (5.22) as building blocks. Theorem 5.2 ([11, Theorem 5-2]) A subset .S of .RN+1 is a classical hypersurface if and only if, for each .x ∈ S there is an open set .U ⊂ RN+1 containing x, an open set .W ⊂ RN and an injective .C 2 -mapping .f : W → RN+1 such that 1. .f (W ) = U ∩ S , 2. .∇f (y) has rank N for each .y ∈ W , and 3. .f −1 : f (W ) → W is continuous. Spivak uses the term “coordinate system around x” to describe such mappings f . Alternatively, in order to make distinctions appropriate for structured hypersurfaces, we shall use the term classical parameterization around x for the mappings f in the theorem. The proof of the above theorem in [11] shows that (i) the Conditions 1–3 are sufficient for the existence of a “lifting” of f to a .C 2 diffeomorphism . of open subsets of .RN+1 that is a counterpart of .h−1 in (5.22), so that .f (y) = (y, 0) for all .y ∈ W , and (ii) the lifting . can be chosen so that its gradient .∇ and its second gradient .∇ 2 , when restricted to .W × {0}, are determined by the corresponding derivatives 2 .∇f and .∇ f of the given classical parametrization f . It is helpful here to note that the values of both . and f lie in .RN+1 , that the values of .∇ and of .∇ 2 lie in .R(N+1)×(N+1) and .R(N+1)×(N+1)×(N+1) ,

138

5 Outlook for Future Research

respectively, while the values of .∇f and of .∇ 2 f lie in .R(N+1)×N and .R(N+1)×N×N , respectively. The phrase “are determined by” in (ii) amounts to the assertion that, for each .y ∈ W , the matrix .∇f (y) is a submatrix of the corresponding matrix 2 .∇ (y, 0) and that the matrix .∇ f (y) is a submatrix of the corresponding matrix 2 2 .∇ (y, 0). Moreover, the enlargements of .∇f (y) and .∇ f (y) to .∇ (y, 0) and to 2 .∇ (y, 0), respectively, can be achieved by addition of matrix entries chosen from the set .{0, 1}. Given a classical parameterization .f : W → S ∩ U of .S around x, Condition 2 in Theorem 5.2 implies that the collection of all vectors . dtd f (c(t))|t =0 generated by smooth curves c in .RN+1 with .c(0) = f −1 (x) forms an N-dimensional subspace .Sx of .RN+1 , the tangent space of .S at x, which turns out to be independent of the particular classical parameterization considered. For each basis N −1 (x))v : p = 1, · · · , N} is a basis of .S , .{v1 , · · · , vN } of .R , the set .{∇f (f p x 1N so that the wedge-product . p=1 ∇f (f −1 (x))vp ∈ RN+1 provides a non-zero vector in .Sx⊥ , the one-dimensional orthogonal complement in .RN+1 of the tangent space .Sx . 1 We recall here that, for all .u1 , . . . , uN ∈ RN+1 , the wedge product . N p=1 up ∈ RN+1 is defined by 2 N .

up · v = det(u1 , . . . , uN , v)

for all v ∈ RN+1 .

(5.23)

p=1

(Note that in (5.23) we regard .det as a skew-symmetric, multilinear function of the N + 1 column vectors of an .(N + 1) × (N + 1) matrix.) As x varies through .S ∩ U , the unit vectors

.

 −1 2  2 N  N  −1  .nx :=  ∇f (f (x))v ∇f (f −1 (x))vp p   p=1  p=1

(5.24)

determine the continuous vector field .x → nx on .S ∩ U with values in the unit sphere in .RN+1 . (The choice .x → −nx is a second such vector field and may be used in instances where prescribed conventions on signs of curvatures need be followed.) The vector field .x → nx , to within a factor of .−1, is independent of the choice of basis .{v1 , . . . , vN } of .RN and classical parameterization f . For definiteness, we fix from now on the basis .{v1 , . . . , vN } to be the standard basis of 1N N .R , i.e., .(vp )k = δpk , with .δ the Kronecker symbol, so that . p=1 vp = vN+1 . If . is a lifting of f as described in (i) and (ii) above, then (5.24) reads −1   2 2 N   N  .nx =  ∇ (y, 0)| v ∇y (y, 0)|y=ΠN −1 (x)vp , −1 y y=ΠN (x) p    p=1 p=1

(5.25)

5.2 Multiscale Differential Geometry

139

where .ΠN : RN+1 → RN is the mapping .(x1 , . . . , xN , xN+1 ) → (x1 , . . . , xN ) and, for .p = 1, . . . , N, the pth column of .∇y (y, 0) ∈ R(N+1)×N is the partial derivative of . with respect to .yp evaluated at .(y, 0). Because .ΠN , . −1 , .∇ , and .det 1 are continuously differentiable and . N p=1 ∇y (y, 0)|y=ΠN −1 (x) vp = 0, the righthand side of (5.25) is a continuously differentiable function of x and, therefore, provides a continuously differentiable extension .x → nex of .x → nx from .S ∩ U onto an open subset of .RN+1 containing .S ∩ U . In the relation (5.24) we may replace .nx by .nex and evaluate both sides at .x = f (y) with .y ∈ W to obtain nef (y)

.

−1   2 2 N   N =  ∇f (y)vp  ∇f (y)vp  p=1 p=1

for all y ∈ W,

(5.26)

and, after differentiating both  sides with respect to y and multiplying both sides by   1N  .μ(y) :=   p=1 ∇f (y)vp  = 0 (the “area” magnification factor at y), we obtain 

μ(y)∇nef (y) ∇f (y) = I − nef (y) ⊗ nef (y)

N 

⎛ ⎝

j =1 .

∧∇ 2 f (y)[·]vj ∧

 2 N

 j2 −1

∇f (y)vp ∧

p=1

⎞ ∇f (y)vp ⎠ .

(5.27)

p=j +1

Note that the linear mapping .I − nef (y) ⊗ nef (y) is the perpendicular projection of .RN+1 onto the tangent space .Sf (y) = Rng∇f (y), and (5.27) along with the relation .Rng∇f (y) = Sf (y) then tell us that .∇nef (y) Sf (y) ⊂ Sf (y) . We now wish to show that the restriction of .∇nef (y) to .Sf (y) , the Weingarten map, is a symmetric linear mapping on .Sf (y) . For .k, l = 1, . . . , N, if we apply both sides of (5.27) to the vector .vk , take the inner product of both of the resulting sides with .∇f (y)v , use the symmetry of the linear mapping .I − nef (y) ⊗ nef (y) and the fact that .nef (y) · .∇f (y)v = 0, the relation (5.27) takes the form ⎛  j2 −1 N

  e ⎝ ∇f (y)vp ∧ ∇ 2 f (y)[vk ]vj ∧ μ(y)∇nf (y)∇f (y)vk · ∇f (y)vl = j =1 .

p=1

⎞  2 N ∇f (y)vp ⎠ · ∇f (y)vl . ∧ p=j +1

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5 Outlook for Future Research

Each of the N terms in the sum on the right-hand side with .j = l has .∇f (y)vl repeated twice and therefore vanishes, so that the quantity above reduces to ⎛ ⎝

l−1 2

p=1

.

∇f (y)vp ∧ ∇ 2 f (y)[vk ]vl ∧

N 2

⎞ ∇f (y)vp ⎠ · ∇f (y)vl

p=l+1

⎧ 2 ⎪ ⎨ det(∇f (y)v1 , . . . , ∇f (y)vl−1 , ∇ f (y)[vk ]vl , ∇f (y)vl+1 , . . . , ∇f (y)vN , ∇f (y)vl ) = ⎪ ⎩det(∇ 2 f (y)[v ]v , ∇f (y)v ) k l l

(5.28) if N  2 if N = 1.

For .N = 1, .k = l = 1 so that .det(∇ 2 f (y)[vk ]vl , ∇f (y)vl ) is unchanged when .vk and .vl are interchanged. In the expression involving the determinant for the case .N  2, the vector .∇f (y)vk occurs in the kth argument of .det, the vector .∇ 2 f (y)[vk ]vl occurs in the lth argument, and .∇f (y)vl occurs in the st argument. Switching the vectors in the kth and .(N + 1)st arguments .(N + 1) and, then, the vectors in the kth and lth arguments does not alter the value of that expression, since each of the two switches introduces the factor of .−1. Moreover, the symmetry of the second gradient .∇ 2 f (y) allows us to switch .vk and .vl in the vector .∇ 2 f (y)[vk ]vl , appearing now in the kth argument, without altering the vector in the kth argument. Thus, these three switches leave the value of the right-hand side of (5.28) unchanged. However, the derivation of (5.28) shows that expression obtained on the right-hand side after the three switches is precisely e .(μ(y)∇n f (y) ∇f (y)vl ) · ∇f (y)vk ; because .∇f (y)v1 , . . . , ∇f (y)vN are a basis of e .Sf (y) and .μ(y) = 0, this establishes the symmetry of .∇n f (y) |Sf (y) , the restriction e e of .∇nf (y) to .Sf (y). Equivalently, .∇nx |Sx is symmetric for every .x ∈ S ∩ U . The eigenvalues of .∇nex |Sx are called the principal curvatures of the hypersurface .S at x, while .det(∇nex |Sx ) and .tr(∇nex |Sx ) are called the Gaussian curvature and the mean curvature, respectively. In subsequent subsections we provide a multiscale setting in which classical hypersurfaces are enriched by means of structured hypersurfaces and in which the classical notions of curvature just identified are refined to capture curvature at a submacroscopic level.

5.2.2 Structured Hypersurfaces in RN +1 in an L∞ -Setting We employ here the characterization of classical hypersurfaces in Theorem 5.2 in Sect. 5.2.1 as a starting point for the introduction of the concept “structured hypersurfaces in .RN+1 ”. In this development it is convenient to employ the notion of a fit region in .RN (see [4, 5]), that is, a bounded, regularly open set having finite perimeter and whose boundary has .L N -measure zero. Fit regions permit us to identify a special collection of classical hypersurfaces whose differential structure admits “piecewise-classical parametrizations”, i.e., globally defined mappings that

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141

are piecewise smooth and that still carry the classical differential structure of the hypersurface. Definition 5.2 A subset .S ⊂ RN+1 is a finitely generated classical hypersurface if there is a positive integer M, open subsets .U1 , . . . , UM of .RN+1 with .S ⊂ U := M , Uj , fit regions .W1 , . . . , WM of .RN , and an injective, .C 2 -mapping .f : W = j =1 M , j =1

Wj → RN+1 such that, for each .j = 1, . . . , M, the restriction .f |Wj of f to

Wj satisfies

.

1. .f |Wj is a classical parametrization of .S ∩ Uj around every .x ∈ S ∩ Uj , and 2. .f |Wj , .∇f |Wj , and .∇ 2 f |Wj extend as continuous mappings to .W j . Condition 1 means that .f |Wj satisfies Conditions 1–3 in Theorem 5.2 with f there replaced by .f |Wj , with W replaced by .Wj , and with .S replaced by .S ∩ Uj . This permits one to view .S as being composed of the m classical hypersurfaces .S ∩ Uj (.j = 1, . . . , M). In Condition 2 the existence of continuous extensions to .W j of .f |Wj and its first two derivatives implies that f and its first M , two derivatives are essentially bounded functions on .W = Wj . Conditions 1 j =1

and 2 provide justification for calling f a piecewise classical parametrization of the finitely generated, classical hypersurface .S . Thus, in the context of finitely generated classical hypersurfaces, we obtain global parametrizations f , but we may sacrifice some smoothness properties. Indeed, because W , itself, need not be a fit region, the mappings f , .∇f , and .∇ 2 f may not extend to .W as continuous mappings across internal boundaries of W . Nevertheless, the presence of discontinuities in f and in .∇f allows us to invoke aspects of the .L∞ -theory of structured deformations that lead as described below to a notion of “structured hypersurface”. We note that the requirement that a piecewise classical parametrization f be injective rules out self-intersection of a finitely generated classical hypersurface and is imposed to draw a close parallel to the requirement that simple deformations be injective in the .L∞ -theory of structured deformations. Before defining structured hypersurfaces, we note that finitely generated classical hypersurfaces are, in fact, classical hypersurfaces. Consequently, around a given M , point .x ∈ S (or, equivalently, around a point .y ∈ Wj ), the local analysis j =1

of unit normals and curvatures presented above applies for each f . In particular, in the context of finitely generated classical hypersurfaces we may write for each

142

y∈

.

5 Outlook for Future Research M , j =1

Wj the following instance of the formula (5.28) for the components of the

Weingarten map .∇nef (y) 

 μ(y)∇nef (y)∇f (y)vk · ∇f (y)vl ⎧ 2 ⎪ ⎨ det(∇f (y)v1 , . . . , ∇f (y)vl−1 , . . . , ∇ f (y)[vk ]vl , if N  2, . ∇f (y)vl+1 , . . . , ∇f (y)vN , ∇f (y)vl ) = ⎪ ⎩det(∇ 2 f (y)[v ]v , ∇f (y)v ) if N = 1, k l l

(5.29)

  2  M  N  , where .μ(y) :=  ∇f (y)vp  = 0. (Here, because .y ∈ Wj , we are justified in j =1 p=1  writing f in place of an appropriate restriction .f |Wj .) Now, in the spirit of second-order structured deformations in the .L∞ -setting as sketched in the Introduction, we suppose that there is a sequence of piecewise classical parametrizations f for finitely generated classical hypersurfaces .Sm ⊂ M ,m RN+1 such that the domain .W = Wj of .fm is, to within an .L N -null set, j =1

independent of m and such that each of the sequences .m → fm , .m → ∇fm , and 2 ∞ mappings appropriate for its codomain, as .m → ∇ fm converges in a space of .L indicated in the next display. If we write g := lim fm ∈ L∞ (W, RN+1 ) m→∞

.

G := lim ∇fm ∈ L∞ (W, R(N+1)×N ) m→∞

(5.30)

Γ := lim ∇ 2 fm ∈ L∞ (W, R(N+1)×N×N ) m→∞

for the limits of the three sequences, the fact that (5.29) is valid at every point of W and the continuity of .det implies that the right-hand side of (5.29) converges for .N  2 to the essentially bounded scalar field y → det(G(y)v1 , . . . , G(y)vk , . . . , Γ (y)[vk ]vl , . . . , G(y)vN , G(y)vl )

.

on W and for .N = 1 to the essentially bounded scalar field y → det(Γ (y)[vk ]vl , G(y)vl ).

.

Moreover, the sequence of positive-valued, essentially bounded functions .m → μm   1  N   converges to the essentially bounded scalar field .y →  G(y)vp   0; for each p=1  .p ∈ {1, . . . , N}, .m → ∇fm vp converges to the essentially bounded vector field

5.2 Multiscale Differential Geometry

143

y → G(y)vp , and, almost everywhere in W , the field .Γ retains the symmetry of second gradients enjoyed by each field .∇ 2 fm . If the rank of .G(y) is N at almost 1   N  every point .y ∈ W , then . G(y)vp  is positive almost everywhere, and the p=1  vectors .G(y)vp , .p ∈ {1, . . . , N} are linearly independent for almost every y in W. These observations can be used to show that, if the rank of .G(y) is N at almost every point .y ∈ W , then the Weingarten maps .∇nefm (y) |Sfm (y) , as y varies over W , determine the sequence .m → ∇nefm (·) |Sfm (·) of essentially bounded mappings on W with values in .RN×N , and that sequence converges in .L∞ to the symmetric matrix-valued, essentially bounded mapping .W\ on W whose entries are given, for almost every .y ∈ W , by the formula

.

(W\ )kl (y) =  −1 ⎧  N  ⎨ det(G(y)v1 , . . . , G(y)vl−1 , Γ (y)[vk ]vl , . 2  for N  2,   G(y)vl+1 , . . . , G(y)vN , G(y)vl ) G(y)v p   ⎩ p=1  for N = 1. det(Γ (y)[vk ]vl , G(y)vl ) (5.31) Aside from its dependence on the basis vectors .v1 , . . . , vN that are fixed throughout this development, the limiting Weingarten matrix-valued field .W\ is determined solely by the fields G and .Γ , the fields without disarrangements that capture only smooth features of the piecewise classical parametrizations .fm . Accordingly, we call .W\ the Weingarten field without disarrangements for the pair of fields G and .Γ , and, for almost every .y ∈ W , the eigenvalues of the symmetric matrix .W\ (y) will be called here the principal curvatures at y without disarrangements. Passing to the limit .m → ∞ involves, in practice, partitioning the common domain W into ever more numerous and smaller pieces, and we therefore interpret the curvatures at y without disarrangements as reflecting geometrical changes at a submacroscopic level. The shortcoming of the limiting procedure described above is the absence in the limit of a definite classical hypersurface that captures curvatures and other geometrical changes at the macroscopic level. The only candidate to emerge from the limiting procedure as a possible parametrization of a classical hypersurface, the limit .g = limn→∞ fm , fails to qualify as such because it is only guaranteed to be an essentially bounded mapping and need not be locally injective. The following definition of structured hypersurfaces is intended to provide a limiting procedure that captures geometric changes at a submacroscopic level through fields of the type G and .Γ , as well as a limiting object g that does qualify as a piecewise classical parameterization of a finitely generated classical hypersurface. The path toward this goal is provided by the .L∞ -theory of second-order structured deformations described in the Introduction. In that context, a secondorder structured deformation .(g, G, Γ ) has the mapping g not only essentially bounded but also is piecewise .C 2 and injective, while the fields .∇g and G are

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5 Outlook for Future Research

square matrix-valued and are required to satisfy the accommodation inequality, in particular the relation .det G  det ∇g. The accommodation inequality then assures through the approximation theorem the existence of injective, piecewise 2 .C -mappings .um satisfying (1.6). In the present context of hypersurfaces, the fields .∇g and G are .R(N+1)×N -valued and so that the accommodation inequality is not meaningful. However, the geometrical interpretation of the accommodation inequality, namely, that submacroscopic volume changes should not exceed macroscopic volume changes, does have an analogue in the context of hypersurfaces: submacroscopic area changes should not exceed macroscopic area changes. This observation provides an analogue of the accommodation inequality for second-order structured deformations that underlies our definition of structured hypersurfaces. Definition 5.3 A structured hypersurface in .RN+1 is specified by giving a finitelygenerated classical hypersurface .S ⊂ RN+1 , a positive integer M, together with M , Wj → RN+1 is a .C 2 -piecewise classical a triple .(g, G, Γ ) in which .g : W = j =1

parametrization of .S (as in Definition 5.2), in which .G : W → R(N+1)×N is a 1 (N+1)×N×N is a continuous mapping such .C -mapping, and in which .Γ : W → R that 1. for each .j = 1, . . . , M, G, .∇G, and .Γ have continuous extensions to .W j , 2. for each .y ∈ W , the matrix .Γ (y) ∈ R(N+1)×N×N has the symmetry of a second gradient, i.e., for all .i = 1, . . . , N + 1 and .k, l = 1, . . . , N, .Γikl (y) = Γilk (y), 3. .∇g and G satisfy the area-accommodation inequality : there exists .c > 0 such that for every .y ∈ W ,      2  2   N   N    .c < G(y)vp    ∇g(y)vp  .   p=1  p=1

(5.32)

This definition mirrors that of a second-order structured deformation given in [8, Definition 3]: there, the “N”’s appearing in the above relations are replaced everywhere by “.N + 1”’s, item 3 above is replaced by the (volume)-accommodation inequality .0 < c < det G(y)  det ∇g(y), and the term “piecewise classical parametrization of .S ”, used here in describing g, is replaced by the term “simple deformation from W ”. Item 3 in the present definition implies that the N vectors .G(y)vp , for .p = 1, . . . , N, are linearly independent, so that the matrix .G(y) has rank N.

5.2.3 L∞ -Approximation of Structured Insertions As was noted in the introductory chapter, the approximation theorem [8, Theorem 2] is the assertion that, for every second-order structured deformation, there is a

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145

sequence of simple deformations that determines the structured deformation in the sense of (1.6). Here, we investigate the following analogous assertion: for each structured hypersurface .S with accompanying triple .(g, G, Γ ) there exists a sequence .m → Sm of finitely-generated, classical hypersurfaces whose piecewise classical parametrizations .fm determine the given structured hypersurface .S : g = lim fm ,

.

m→∞

G = lim ∇fm , m→∞

and Γ = lim ∇ 2 fm , m→∞

(5.33)

with convergence in the sense of .L∞ . We limit ourselves initially to a class of structured hypersurfaces .S for which the approximation property by finitely-generated hypersurfaces (5.33) follows quite directly from the .L∞ -approximation theorem for second-order structured deformations [8, Theorem 2]. The elements of this class will be generated from structured hypersurfaces .S with associated triple .(g, G, Γ ) in which the mapping g is the following insertion mapping .ι of W into .RN : ι(y) = (y, 0)

.

for y ∈ W,

(5.34)

which has the following lifting .ιe : W × (−1, 1) → RN+1 ιe (y, z) = (y, z) for (y, z) ∈ W × (−1, 1),

.

(5.35)

namely, .ιe = i |W ×(−1,1) , the identity mapping on .W × (−1, 1). By (5.34), the hypersurface .S associated with the triple .(ι, G, Γ ) is the set .ι(W ) = W × {0}. N 1 ∇ι(y)vp equals .vN+1 and that the area-accommodation Note that the product . p=1

inequality (5.32) becomes in this case   2  2 2    N N      .0 < c < G(y)vp   1 =  ∇ι(y)vp  .  p=1 p=1  

(5.36)

We call a structured hypersurface .W × {0} with structured parametrization .(ι, G, Γ ) a structured insertion of W into .RN+1 . Given a structured insertion of W into .RN+1 , we define .Ge : W × (−1, 1) → R(N+1)×(N+1) and .Γ e : W × (−1, 1) → R(N+1)×(N+1)×(N+1) by ⎧ ⎪ ⎪ ⎨G(y)vk N e 2 .G (y, z)vk = ⎪ G(y)vp ⎪ ⎩ p=1

for k = 1, . . . , N, for k = N + 1,

(5.37)

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5 Outlook for Future Research

and e Γikl (y, z) =

.

 Γikl (y) for i = 1, . . . , N + 1, and for k, l = 1, . . . , N, 0

for i = 1, . . . , N + 1, and k = N + 1 or l = N + 1. (5.38)

Thus, the mappings .Ge and .Γ e are independent of the variable .z ∈ (−1, 1), and we have 2    2 N N 2   e  . det G (y, z) = det G(y)v1 , . . . , G(y)vN , G(y)vp =  G(y)vp   p=1 p=1 and .

det ∇ιe (y, z) = det I = 1.

The last two relations permit us to write (5.36) in the form 0 < c2 < det Ge (y, z)  det ∇ιe (y, z),

.

for (y, z) ∈ W × (−1, 1),

which is the accommodation inequality (5.32) for a second-ordered structured deformation on .W × (−1, 1). In fact, items 1–3 of the definition of structured hypersurface assure that the quadruple .(Ø, ιe , Ge , Γ e ) is a second-order structured deformation from the piecewise-fit region .W × (−1, 1) [8, Definition 3]. By the approximation theorem [8, Theorem 2] and its proof, we may choose a sequence 2 .m → (κm , fm ) of simple deformations such that (1) .fm , .∇fm , and .∇ fm converge ∞ e e e in .L to .ι , .G , and .Γ , respectively, on .W × (−1, 1), (2) the disarrangement site .κm for .fm converges to the empty set .Ø, (3) .W \ κm is piecewise fit, and (4) N (κ ∩ (W × {0})) = 0. These conclusions permit the use of [4, Lemma 4.11] .H m and yield the following approximation theorem for structured insertions. Theorem 5.3 Let the structured insertion .W × {0} with structured parametrization (ι, G, Γ ) be given, and let .m → (κm , fm ) be a sequence of simple deformations such that .fm , .∇fm , and .∇ 2 fm converge in .L∞ to .ιe , .Ge , and .Γ e , respectively, on .W × (−1, 1) and such that the disarrangement sites .κm for .fm converge to the empty set .Ø and satisfy .H N (κm ∩ (W × {0})) = 0. The sequences .m → fm (·, 0), 2 .m → ∇fm (·, 0), and .m → ∇ fm (·, 0), obtained first by restricting .fm to .(W \ κm ) × {0} and second by appropriate differentiation, then converge uniformly to .ι, G, and .Γ , respectively, on a subset of W with full N-dimensional Lebesgue measure N .L . Moreover, each .fm (·, 0) is a piecewise classical parametrization of the finitely generated hypersurface .Sm := fm ((W \ κm ) × {0}). .

Given a structured insertion .W × {0} with structured parametrization .(ι, G, Γ ) and a determining sequence .m → fm (·, 0) as in the approximation theorem, the

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147

reasoning that led to the relation (5.31) now permits us to call the matrix-valued field .W\ on W determined by (5.31) the Weingarten field without disarrangements and to call the associated field of eigenvalues the principal curvature field without disarrangements for the structured insertion .W × {0}. Because .ι, itself, is a classical parametrization of the hypersurface .W × {0} that provides .W × {0} with the Weingarten field 0, we have the following remark. Remark 5.2 The hypersurface .W × {0} with classical parametrization .ι carries only zero curvatures via its macroscopic Weingarten field 0, while the structured insertion .W × {0} with structured parametrization .(ι, G, Γ ) carries (generally) nonzero curvatures via its submacroscopic Weingarten field .W\ in (5.31). In the same vein, .W × {0}, as a classical hypersurface, carries the constant field of tangent spaces .Lsp{vp : p = 1, . . . , N} ⊂ RN+1 from the geometry at the macrolevel, and, as a structured emersion, .W × {0} carries the (generally) variable field .y → Lsp{G(y)vp : p = 1, . . . , N} ⊂ RN+1 of tangent spaces without disarrangements from the geometry at the submacroscopic level. As an example in which .W ×{0} is a line segment in .R2 , for .N = 1 and .y ∈ W := (0, 1), we have .ι(y) = (y, 0) and specify .G(y) ∈ R2×1 and .Γ (y) ∈ R2×1×1  R2×1 by G(y) =

.

 λ , 0

Γ (y) =

 0 . κ

(5.39)

with .λ, .κ real numbers and .0 < λ  1. Items 1 and 2 in the definition of structured hypersurface are easily verified, and the area-accommodation inequality in Item 3 follows from the formulas 1 2 .

p=1

 0 ∇ι(y)vp = 1

and

1 2 p=1

 0 G(y)vp = . λ

For this example, the Weingarten matrix .W\ (y) ∈ R1×1 without disarrangements is calculated from (5.31) (with .vp = vk = vl = 1) as follows:  −1  1   2  0λ −1   .W\ (y) = G(y) det(Γ (y), G(y)) = λ det = −κ.  κ 0 p=1 

(5.40)

For the structured insertion .(ι, G, Γ ) in this example, it is2 easy to show that the → R of piecewise quadratic following sequence .m → hm : (0, 1) \ m1 , . . . , m−1 m parametrizations  hm (y) =

.

   i i i i+1 κ i 2 +λ y − , for y ∈ , y− , i = 0, . . . , m−1. m m 2 m m m

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5 Outlook for Future Research

of approximating hypercurves .Sm has the convergence properties of .m → fm (·, 0) in the approximation theorem. The parabolic curve in the .x1 -.x2 plane traced out by .hm on the interval .(i/m, (i + 1)/m) of length .1/m projects onto the interval .(i/m, (i + λ)/m) of length .λ/m on the .x1 -axis; the same parabolic curve projects onto the interval .(0, κ/2m2 ) of length .|κ|/2m2 on the .x2 -axis. Thus, the structured insertion of .(0, 1) into .R2 in the example is approximated via .hm by m mutually congruent parabolic curves emanating from .(0, 1) × {0}, with adjacent parabolic segments separated in the .x1 -direction by a gap of length .(1 − λ)/m and with each segment separating from the .x1 -axis by a maximum distance .|κ|/2m2. The structured insertion .(ι, G, Γ ) captures through G the shrinking by the factor .λ and through .Γ the curving by amount .κ in the approximations .hm . (We note that, with the choice of opposite unit normal field .−nx in (5.24), the Weingarten matrices both for the curves associated with .hm and with .(ι, G, Γ ) would have the single entry .κ, agreeing with the standard convention on curvature for planar curves.)

5.2.4 L∞ -Approximation of Other Structured Hypersurfaces In order to expand the scope of the above approximation theorem for structured insertions to a broader class of structured hypersurfaces, we explore the idea of composing the approximations in that theorem, .hm := fm (·, 0) : W \ κm → RN+1 , with a fixed, .C 2 -diffeomorphism h of .RN+1 onto itself. Thus, we consider .h ◦ hm and its derivatives ∇(h ◦ hm ) = (∇h ◦ hm )∇hm

.

∇ 2 (h ◦ hm ) = ∇{(∇h ◦ hm )∇hm } = ∇(∇h ◦ hm )∇hm + (∇h ◦ hm )∇ 2 hm = (∇ 2 h ◦ hm )[∇hm ]∇hm + (∇h ◦ hm )∇ 2 hm . The use of [4, Lemma 4.11] in the construction of .m → hm guarantees that this sequence, along with the sequences .m → ∇hm and .m → ∇ 2 hm , converge N uniformly on the set .W \ ∪∞ m=1 κm , a subset of W of full .L -measure. It follows 2 from the smoothness of h that .m → ∇h ◦ hm , .m → (∇ h ◦ hm )[∇hm ]∇hm , and ∞ κ and, therefore, 2 .m → (∇h ◦ hm )∇ hm also converge uniformly on .W \ ∪ m=1 m ∞ converge in the sense of .L on W . Passing to the limit in the above formulas, using the continuity of h and its derivatives as well as the approximation properties of .hm and its derivatives, yields lim h ◦ hm = h ◦ ι,

m→∞ .

lim ∇(h ◦ hm ) = (∇h ◦ ι)G,

m→∞

lim ∇ 2 (h ◦ hm ) = (∇ 2 h ◦ ι)[G]G + (∇h ◦ ι)Γ.

m→∞

(5.41)

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149

Thus, starting from the structured parameterization .(ι, G, Γ ) of .W × {0} that is approximated by .m → hm , we have obtained the limiting triple h

.

(ι, G, Γ ) := (h ◦ ι, (∇h ◦ ι)G, (∇ 2 h ◦ ι)[G]G + (∇h ◦ ι)Γ )

(5.42)

that can be approximated by .m → h ◦ hm . Moreover, each .h ◦ hm is a piecewise classical parametrization of the finitely generated classical hypersurface .h(hm (W )). To check whether or not .h (ι, G, Γ ) is a structured parametrization of the limiting, classical hypersurface .h(W × {0}), we note the correspondences arising from (5.42) for the quantities g, G, .Γ in Definition 5.3 g ↔ h ◦ ι.

.

G ↔ (∇h ◦ ι)G,

Γ ↔ (∇ 2 h ◦ ι)[G]G + (∇h ◦ ι)Γ,

and it is easy to verify that items 1 and 2 are satisfied for .h relationship N 2 .

p=1

Avp = (det A)A−T

N 2

vp = (det A)A−T vN+1 ,

(5.43)

(ι, G, Γ ). The

(5.44)

p=1

which holds for every invertible .A ∈ R(N+1)×(N+1) , implies that the areaaccommodation inequality for .h (ι, G, Γ ) in item 3 is equivalent to the assertion that the inequality      N  2     −T −T  ◦ ι)  .c < (∇h ◦ ι) Gv v (∇h p N+1     p=1

(5.45)

hold throughout W . Two cases in which it is easy to verify that (5.45) is satisfied are (i) .∇(h ◦ ι) = (∇h ◦ ι)G, in which case .G = ∇ι, and (ii) for the structured insertion .W × {0}, the tangent space without disarrangements .Lsp{G(y)vp : p = 1, · · · , N} coincides with the tangent space .Lsp{vp : p = 1, · · · , N}. In case (i) equality holds in (5.45), so that micromagnifications coincide, and   and macro-area  1 N N 1   in case (ii) we have . Gvp = ±  Gvp  vN+1 , so that (5.45) follows from the   p=1 p=1 area-accommodation inequality for the structured insertion .(ι, G, Γ ). In case (i) we say that .h (ι, G, Γ ) causes no area disarrangements, and in case (ii) we say that .h (ι, G, Γ ) causes only tangential disarrangements. For the example just treated, where .N = 1 and where G and .Γ are given by (5.39) (and in which actually the term “no arc-length disarrangements” is more appropriate), case (i) applies if and only if .λ = 1 and case (ii) applies for every choice of .λ and .κ, whatever the diffeomorphism h may be. In conclusion, we have identified through (5.45) a necessary and sufficient condition that the composition .h (ι, G, Γ ) is a structured parametrization of

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5 Outlook for Future Research

h(W ×{0}), and the cases (i) of no area disarrangements and (ii) of purely tangential disarrangements provide alternative sufficient conditions on h, G, and .Γ for the same conclusion. The relations (5.41) then tell us that the area-accommodation inequality (5.45) provides a sufficient condition that the hypersurface .h(W × {0}) with parametrization .h (ι, G, Γ ) in (5.42) is both a structured hypersurface and a limit of finitely-generated classical hypersurfaces. In this manner, we have broadened the class of structured hypersurfaces for which the approximation theorem is valid. The question of whether or not an approximation theorem holds for arbitrary structured hypersurfaces in this .L∞ -setting remains open at the time of this writing.

.

5.2.5 SBV 2 - and SBH -Approximation of Structured Hypersurfaces in RN +1 The challenges encountered above in obtaining an approximation theorem that holds for arbitrary structured hypersurfaces in the .L∞ -setting stem from the fact that, in the underlying .L∞ -theory of second-order structured deformations .(g, G, Γ ) and their approximating functions .um , (i) the mappings g and .um are required to have domains and ranges in the same Euclidean space and to have enough smoothness that .det ∇g and .det ∇um are defined, and (ii) g and the approximations .um are required to be injective mappings. The requirement on the domains and ranges in (i) led to the use of structured insertions in the approximation of structured hypersurfaces in the .L∞ -setting, and the requirement of injectivity in (ii) led to the imposition of the area-accommodation inequality in the definition of structured hypersurfaces. In the .SBV 2 - and SBH -theories of second-order structured deformations the restrictions on domains and ranges as well as the requirement of injectivity are absent. Consequently, the adaptation of approximation theorems in those theories to the problem of approximation of structured hypersurfaces is more straightforward, and the problem of assigning an energy to structured hypersurfaces then can be attacked via the relaxation results in Chap. 4. However, the most general analogues of structured hypersurfaces in .SBV 2 - and SBH -settings would not have enough smoothness to permit straightforward access to notions of tangent space and curvature, so that the objects to be approximated as well as the approximating objects would be farther removed from the classical theory of hypersurfaces than is the case in the .L∞ -setting. In a first attempt to make available notions of tangent space and curvature for structured hypersurfaces in .SBV 2 - and SBH -settings, while retaining the availability of approximations and relaxation, we require that structured hypersurfaces have the piecewise smoothness properties given in Definition 5.3 for the .L∞ -setting while only requiring that approximating hypersurfaces have .SBV 2 - or SBH -regularity.

5.2 Multiscale Differential Geometry

151

In the .SBV 2 -setting we note that, for a given piecewise-fit region .W ⊂ RN , if .(g, G, Γ ) is a structured parametrization in the sense of Definition 5.3, then   (g, G, Γ ) ∈ SD 2 W, RN+1 × R(N+1)×N × R(N+1)×N×N

.

(5.46)

as in Definition 4.1. The properties of .(g, G, Γ ) in Definition 5.3 permit us to use the formula (5.31) to define the components of the Weingarten matrix without disarrangements that reflect submacroscopic curvatures of .g(W ), as well as to use the derived formula for the Weingarten matrix at the macrolevel (5.28) to assign curvatures to .g(W ) at the macrolevel, with f replaced everywhere in (5.28) by g. Moreover, the properties of .(g, G, Γ ) by virtue of (5.46) permit us to invoke Theorem 4.1 to obtain approximations .um ∈ SBV 2 (W, RN+1 ) that converge to .(g, G, Γ ) in the sense of (4.4) (with .d = N + 1). The structured hypersurface N+1 is thereby endowed with not only macro- and micro-notions of .g(W ) in .R curvature, but also with approximating parametrizations .um that can support the results on relaxation of energies provided in Chap. 4. However, in contrast to the approximations .h ◦ hm of structured parametrizations of the form .h (ι, G, Γ ) in (5.41), where the finitely-generated classical hypersurfaces .(h ◦ hm )(W ) can be assigned curvatures via (5.28) with f replaced by .h ◦ hm , the approximations .um need not be smooth enough to permit a similar assignment of curvatures. In the SBH -setting, we specialize to structured parametrizations for which .G = ∇g, i.e., to triples .(g, ∇g, Γ ) that are structured parametrizations in the sense of Definition 5.3. In this case we have   (g, Γ ) ∈ SD2 W, RN+1 × R(N+1)×N×N sym

.

(5.47)

(see Definition 4.2). Moreover, the properties of .(g, Γ ) in Definition 5.3 again permit us to use the formula (5.31), with G replaced by .∇g throughout, to define the components of the Weingarten matrix without disarrangements reflecting submacroscopic curvatures. The derived formula for the Weingarten matrix at the macrolevel (5.28), with f there replaced everywhere by g, allow us as in the 2 .SBV -setting to identify curvatures at the macrolevel for .g(W ). The fact that   N+1 × R(N+1)×N×N permits us now to invoke Theorem 4.6 .(g, Γ ) ∈ SD2 W, R sym to obtain approximations .um ∈ SBH (W, RN+1 ) that converge to .(g, Γ ) in the sense of (4.6) (with .d = N + 1). The structured hypersurface .g(W ) in .RN+1 is thereby endowed with not only macro- and micro-notions of curvature, but also with approximating parametrizations .um that can support the results on relaxation of energies provided in Chap. 4. Although the approximations .um ∈ SBH (W, RN+1 ) are more regular than in the .SBV 2 -setting, they still lack the regularity needed to assign curvatures via (5.28) to the associated hypersurfaces .um (W ). Another approach to the incorporation of a notion of structured hypersurface into the .SBV 2 - and SBH -settings is to employ the notion of “approximate tangent space” from geometric measure theory [1, Section 2.1] that is an extension of the notion of tangent space employed in differential geometry. This opens the possibility

152

5 Outlook for Future Research

that both “approximate tangent spaces” and “approximate tangent spaces without disarrangements” could be meaningful for an appropriate class of subsets in .RN+1 associated with .SBV 2 - and/or SBH -mappings.

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