Calculus of Variations on Thin Prestressed Films: Asymptotic Methods in Elasticity 3031174941, 9783031174940

Citation preview

Progress in Nonlinear Differential Equations and Their Applications 101

Marta Lewicka

Calculus of Variations on Thin Prestressed Films Asymptotic Methods in Elasticity

Progress in Nonlinear Differential Equations and Their Applications Volume 101 Series Editor Haïm Brezis, Rutgers University, New Brunswick, NJ, USA Editorial Board Members Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy Henri Berestycki, École des Hautes Études en Sciences Sociales, Paris, France Luis Caffarelli, University of Texas at Austin, Austin, TX, USA Sun-Yung Alice Chang, Princeton University, Princeton, NJ, USA Jean-Michel Coron, Université Marie et Pierre Curie, Paris, France Manuel Del Pino, University of Bath, Bath, UK Lawrence C. Evans, University of California, Berkeley, Berkeley, CA, USA Alessio Figalli, ETH Zürich, Zürich, Switzerland Rupert Frank, Mathematisches Institut, LMU Munich, München, Germany Nicola Fusco, Università di Napoli “Federico II”, Naples, Italy Sergiu Klainerman, Princeton University, Princeton, NJ, USA Robert Kohn, Courant Institute of Mathematical Sciences, New York, NY, USA Pierre-Louis Lions, Collège de France, Paris, France Andrea Malchiodi, Scuola Normale Superiore, Pisa, Italy Jean Mawhin, Université Catholique de Louvain, Louvain-la-Neuve, Belgium Frank Merle, Université de Cergy-Pontoise, Cergy, France Giuseppe Mingione, Università di Parma, Parma, Italy Felix Otto, MPI MIS Leipzig, Leipzig, Germany Paul Rabinowitz, University of Wisconsin, Madison, WI, USA John Toland, University of Bath, Bath, UK Michael Vogelius, Rutgers University, Piscataway, NJ, USA

Marta Lewicka

Calculus of Variations on Thin Prestressed Films Asymptotic Methods in Elasticity

Marta Lewicka Mathematics Department University of Pittsburgh Pittsburgh, PA, USA

ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications ISBN 978-3-031-17495-7 (eBook) ISBN 978-3-031-17494-0 https://doi.org/10.1007/978-3-031-17495-7 Mathematics Subject Classification (2020): 35B40, 35C20, 49-00, 49-02, 53A35, 53Z30, 74B10, 74B20, 74G05, 74G22, 74G35, 74G40, 74G55, 74G65, 74G75 © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved 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 book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part I Tools in mathematical analysis 2

Γ -convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Definition, examples and fundamental properties . . . . . . . . . . . . . . . . 9 2.2 Example of Γ -convergence in linearised elasticity . . . . . . . . . . . . . . . 12 2.3 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3

Korn’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Korn’s inequality and First Korn’s inequality . . . . . . . . . . . . . . . . . . . . 3.2 Variants of Korn’s inequality with different boundary conditions . . . 3.3 Proof of Korn’s inequality: preliminary estimates . . . . . . . . . . . . . . . . 3.4 Proof of Korn’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Korn’s constant under tangential boundary conditions . . . . . . . . . . . . 3.6 Approximation theorem and Korn’s constant in thin shells . . . . . . . . 3.7 Killing vector fields and Korn’s inequality on surfaces . . . . . . . . . . . . 3.8 Blowup of Korn’s constant in thin shells . . . . . . . . . . . . . . . . . . . . . . . 3.9 Uniformity of Korn’s constant under tangential boundary conditions in thin shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Proofs of uniformity of Korn’s constant under tangential boundary conditions in thin shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 26 30 36 39 42 46 51

¨ Friesecke-James-Muller’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Liouville’s theorem and quantitative rigidity estimate . . . . . . . . . . . . 4.2 First order truncation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Local rigidity estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Proof of Friesecke-James-M¨uller’s inequality . . . . . . . . . . . . . . . . . . . 4.5 Approximation theorem and Friesecke-James-M¨uller’s constant in thin shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 68 73 77

4

53 59 62

80 v

vi

Contents

4.6 Friesecke-James-M¨uller’s inequality in the plane . . . . . . . . . . . . . . . . 83 4.7 Rigidity estimates in conformal setting . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.8 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Part II Dimension reduction in classical elasticity 5

Limiting theories for elastic plates and shells: nonlinear bending . . . . 95 5.1 Set-up of three dimensional nonlinear elasticity . . . . . . . . . . . . . . . . . 96 5.2 Elasticity on shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3 Kirchhoff’s theory for shells: compactness and lower bound . . . . . . . 104 5.4 Second order truncation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.5 Proof of second order truncation result . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.6 Kirchhoff’s theory for shells: recovery family . . . . . . . . . . . . . . . . . . . 119 5.7 Kirchhoff’s theory for shells: Γ -limit and convergence of minimizers123 5.8 Non-compactness by wrinkling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.9 Compactness beyond Kirchhoff’s scaling . . . . . . . . . . . . . . . . . . . . . . . 127 5.10 Lower bound beyond Kirchhoff’s scaling . . . . . . . . . . . . . . . . . . . . . . . 132 5.11 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6

Limiting theories for elastic plates and shells: sublinear and linear . . . 139 6.1 Von K´arm´an’s theory for shells: recovery family . . . . . . . . . . . . . . . . . 140 6.2 Von K´arm´an’s theory for shells: Γ -limit and convergence of minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.3 Linear elasticity for shells: recovery family, Γ -limit and convergence of minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4 Shells with variable thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.5 Convergence of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.6 Von K´arm´an’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.7 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7

Limiting theories for elastic plates: linearised bending . . . . . . . . . . . . . . 173 7.1 Linearised Kirchhoff’s theory for shells . . . . . . . . . . . . . . . . . . . . . . . . 174 7.2 Linearised Kirchhoff’s theory for plates . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3 Matching infinitesimal to exact isometries on plates . . . . . . . . . . . . . . 182 7.4 Linearised Kirchhoff’s theory for plates: recovery family for Lipschitz displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.5 Density result on plates and linearised Kirchhoff’s theory: recovery family, Γ -limit and convergence of minimizers . . . . . . . . . . 192 7.6 Elastic shallow shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.7 Linearised Kirchhoff’s theory for shallow shells . . . . . . . . . . . . . . . . . 201 7.8 Matching isometries on shallow shells . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.9 Convexity of weakly regular displacements . . . . . . . . . . . . . . . . . . . . . 210 7.10 Density result and recovery family on shallow shells . . . . . . . . . . . . . 216 7.11 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Contents

vii

8

Infinite hierarchy of elastic shell models . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.1 Heuristics on isometry matching and collapse of theories . . . . . . . . . 223 8.2 Matching infinitesimal to second order isometries on surfaces of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.3 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

9

Limiting theories on elastic elliptic shells . . . . . . . . . . . . . . . . . . . . . . . . . . 233 9.1 Linear problem sym∇w = B on elliptic surfaces . . . . . . . . . . . . . . . . . 233 9.2 Matching infinitesimal to exact isometries on elliptic surfaces . . . . . 241 9.3 Density result on elliptic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.4 Collapse of theories beyond Kirchhoff’s scaling for elliptic shells: recovery family, Γ -limit and convergence of minimizers . . . . . . . . . . 247 9.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

10

Limiting theories on elastic developable shells . . . . . . . . . . . . . . . . . . . . . 251 10.1 Developable surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 10.2 Matching properties on developable surfaces . . . . . . . . . . . . . . . . . . . . 256 10.3 Density result on developable surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 260 10.4 Collapse of theories beyond Kirchhoffs scaling for developable shells: recovery family, Γ -limit and convergence of minimizers . . . . 263 10.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Part III Dimension reduction in prestressed elasticity 11

Limiting theories for prestressed films: nonlinear bending . . . . . . . . . . 269 11.1 Three dimensional non-Euclidean elasticity . . . . . . . . . . . . . . . . . . . . . 270 11.2 Prestressed thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.3 Kirchhoff-like theory for prestressed films: compactness and lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 11.4 Kirchhoff-like theory for prestressed films: recovery family . . . . . . . 285 11.5 Identification of Kirchhoff’s scaling regime . . . . . . . . . . . . . . . . . . . . . 290 11.6 Coercivity of Kirchhoff-like energy for prestressed films . . . . . . . . . . 295 11.7 Effective energy density under isotropy condition . . . . . . . . . . . . . . . . 299 11.8 Application to liquid crystal glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.9 More examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 11.10 Connection to experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 11.11 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

12

Limiting theories for prestressed films: von K´arm´an-like theory . . . . . 317 12.1 Energy quantisation and approximation lemmas . . . . . . . . . . . . . . . . . 318 12.2 Von K´arm´an-like theory for prestressed films: compactness and lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 12.3 Von K´arm´an-like theory for prestressed films: recovery family . . . . . 332 12.4 Identification of von K´arm´an’s scaling regime and coercivity of von K´arm´an-like energy for prestressed films . . . . . . . . . . . . . . . . . . . 337 12.5 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

viii

Contents

12.6 Beyond von K´arm´an’s regime: an example . . . . . . . . . . . . . . . . . . . . . 346 12.7 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 13

Infinite hierarchy of limiting theories for prestressed films . . . . . . . . . . 349 13.1 Energy quantisation and identification of scaling regimes for prestressed films beyond von K´arm´an’s regime . . . . . . . . . . . . . . . . . . 350 13.2 Higher order theories for prestressed films: compactness and preliminary lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 13.3 Identification of lower bound’s curvature term . . . . . . . . . . . . . . . . . . 359 13.4 Higher order theories for prestressed films: lower bound . . . . . . . . . . 365 13.5 Higher order theories for prestressed films: recovery family . . . . . . . 367 13.6 Convergence of minima and coercivity of linear elasticity-like energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 13.7 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

14

Limiting theories for weakly prestressed films . . . . . . . . . . . . . . . . . . . . . 375 14.1 Weakly prestressed films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 14.2 Von K´arm´an and linear elasticity-like theories for weakly prestressed films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 14.3 Compactness and lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 14.4 Von K´arm´an and linear elasticity-like theories for weakly prestressed films: recovery family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 14.5 Elimination of out-of-plane displacements . . . . . . . . . . . . . . . . . . . . . . 393 14.6 Identification of scaling regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 14.7 Linearised Kirchhoff-like theory for weakly prestressed films . . . . . . 405 14.8 Matching isometries on weakly prestressed films . . . . . . . . . . . . . . . . 408 14.9 Uniqueness of minimizers to linearised Kirchhoff-like energy for prestressed films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 14.10 Critical points in radially symmetric case . . . . . . . . . . . . . . . . . . . . . . 418 14.11 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 425 Terminology and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

To my Friends

Chapter 1

Introduction

This monograph concerns the analytical and geometrical questions emerging from the study of thin elastic films exhibiting residual stress at free equilibria. Prestressed thin films are present in many contexts and applications, ranging from growing tissues, through plastically strained sheets, engineered swelling or shrinking gels, to petals and leaves of flowers, atomically thin graphene layers, etc. While the related questions about the physical basis for shape formation (morphogenesis) lie at the intersection of biology, chemistry and physics, fundamentally they have analytical and geometrical character. Indeed, they may be seen as a variation on the classical themes: in differential geometry - that of isometrically embedding a shape with a given metric in an ambient space of possibly different dimension; and in calculus of variations - that of minimizing non-convex energy functionals parametrised by a quantity in whose limit the functionals become in some sense degenerate. Motivation from differential geometry. The field of differential geometry began with the study of curves and surfaces in R3 . The abstract concept of a Riemannian manifold, formulated in the XIXth century, and the natural question of whether each such object coincides with a subset (submanifold) of some Euclidean space RM , quickly assumed a position of fundamental conceptual importance. This problem, called the isometric immersion problem, can be formulated as the question of solvability of the following system of partial differential equations: (∇u)T ∇u = g

for u : RN ⊃ Ω → RM ,

(1.1)

where g : Ω → RN×N is a given symmetric, positive definite matrix field. A remarkable positive resolution due to Nash states that any such smooth Riemannian metric g admits a smooth isometric immersion u in RM (i.e. the solution to (1.1)) for some suitably large dimension M = M(N). On the other hand, according to the celebrated Nash-Kuiper theorem, any C 1 short immersion u0 of a C 1 metric g into RN+1 (i.e. u0 ∈ C 1 (Ω , RN+1 ) satisfying 0 < (∇u0 )T ∇u0 < g in the sense of matrix inequalities, at each point of Ω ), can be uniformly approximated by C 1 solutions to (1.1). This regularity has been recently improved to H¨older continuous u ∈ C 1,α (Ω , RN+1 ), with the optimal exponent α being the subject of vigorous research. © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_1

1

2

1 Introduction

For N = M, the isometric immersion problem is linked with the satisfaction or failure of orientation preservation by u : Ω → RN , expressed as: det ∇u > 0

in Ω .

(1.2)

Without the above restriction, there always exists a Lipschitz u : Ω → RN solving (1.1). However, under (1.2) the same problem becomes rigid: a sufficient and necessary condition for the (local) solvability of (1.1) (1.2) is the vanishing of the Riemannian curvature of g at each point of Ω . In this latter context, it is natural to pose the quantitative question: what is the infimum of the averaged pointwise deficit of a map u from being an orientation-preserving isometric immersion of g on Ω ? This deficit is measured by the following non-Euclidean energy: Eg (u) =

Z


dist2 (∇u)g−1/2 , SO(N) dx.

(1.3)

Ω

Indeed, u satisfies (1.1) (1.2) if and only if ∇u(x) ∈ SO(N)g(x)1/2 for almost every x ∈ Ω , which is precisely when Eg (u) = 0. In this monograph, we will be largely concerned with the following questions: (i) Can one quantify the infimum of Eg in relation to g and Ω ? (ii) What is the structure of minimizers to (1.3), if they exist? (iii) In the limit of Ω becoming (N − 1)-dimensional, what can be said about asymptotic properties of non-Euclidean energies and their minimizers in relation to the Riemann curvatures of g? Motivation from calculus of variations. The field of calculus of variations originally centered around minimization problems for integral functionals of the form: E (u) =

Z


W x, u(x), ∇u(x) dx

for u : RN ⊃ Ω → RM ,

(1.4)

Ω

where W : Ω ×RM ×RM×N → R is given energy density, and where u may be subject to various constraints, for example the boundary conditions. The systematic study of existence of minimizers to (1.4), their uniqueness and qualitative properties, began with Euler and Bernoulli in the XVIIth century and progressed due to seminal contributions by Tonelli, Morrey and De Giorgi in the XXth century. These and other related questions are strongly tied to the convexity (with respect to ∇u) properties of W , which in turn imply the so-called sequential lower semicontinuity of E ; a condition necessary to conclude that the minimizing sequences to (1.4) accumulate at the minimizers. This is the celebrated direct method of calculus of variations which bypasses solving the potentially complicated Euler-Lagrange equations of (1.4), and allows one to treat the aforementioned minimization problem directly. However, this technique does not apply to the functional in (1.3), due to its lack of convexity. For a class of domains Ω that are thin films, whose diameter in a chosen direction is much smaller than in other directions, one may however consider a family of energies parametrised by the small thickness h of Ω h = ω × (−h/2, h/2):

1 Introduction

3

Egh (uh ) =

Z Ωh


dist2 (∇uh )g−1/2 , SO(N) dx

for u : Ω h → RN .

(1.5)

The task is now to determine the asymptotic limit of the above minimization problems as h → 0, rather than to minimize Egh for each particular h. This can be achieved using the method of Γ -convergence, which allows one to identify the “singular limit” energy functional Ig , characterized by the property that the minimizers and minimum values of (1.5) converge to the minimizers and the minimum values of Ig . In this monograph, we will be concerned with the following related questions: (i) What is the optimal mode of convergence in the above approach, allowing to recover the most information for the original minimization problems? (ii) Can one expect that inf Egh ' hγ as h → 0 and can one determine the optimal scaling exponent γ directly from the metric tensor g? (iii) Which curvatures or components of g are involved in this dimension reduction process and how do they contribute to the residual energy Ig ? Motivation from solid mechanics and elasticity. The theory of elasticity is one of the most important subfields of continuum mechanics. Elasticity studies materials which are capable of undergoing large deformations, due to the distribution of local stresses and displacements, and resulting from the application of mechanical or thermal loads. The basic variational model investigated in this monograph pertains to the non-Euclidean version of the nonlinear elastic energy of deformations: Eg (u) =

Z


W (∇u)g−1/2 dx

for u : R3 ⊃ Ω → R3 ,

(1.6)

Ω

where W : R3×3 → R is the given energy density, satisfying the necessary physicallyrelevant conditions. These include frame invariance W (RF) = W (F) valid for all rotations R ∈ SO(3) and all F ∈ R3×3 , and the zero-penalty for all rigid motions: W (R) = 0. The model (1.6) postulates formation of a target Riemannian metric g and the induced multiplicative decomposition of the deformation gradient ∇u into the elastic part (∇u)g1/2 , and the inelastic part g1/2 responsible for the morphogenesis. Equivalently, Eg (u) quantifies the total pointwise deviation of ∇u from the prestress g1/2 , modulo rotations that do not cost any energy. The functional in (1.6) corresponds to a range of hyperelastic energies approximating the behavior of a large class of elastomeric materials, and it is consistent with the microscopic derivations based on statistical mechanics. We note that it has the general form (1.4) and also that it reduces (via a change of variables) to the classical nonlinear three-dimensional elasticity, for g with vanishing Riemannian curvature, which is precisely when Eg (u) = 0 for some u. In the opposite case i.e. for a non-Euclidean g, the infimum of Eg in the absence of any forces or boundary conditions remains strictly positive, pointing to the existence of residual strain. The energy (1.6) reduces also to (a version of) classical linear elasticity when g ' Id3 . The goal is now to answer the following questions:

4

1 Introduction

(i) How to determine the minimizing shapes u of a tissue attaining an orientationpreserving configuration closest to being an isometric immersion of g, in terms of an appropriate mechanical theory? (ii) Is it possible to quantify the separation of scales arising in slender structures from the prescription of growth laws? (iii) How to pose and resolve geometric design problems involving prestress as inverse problems to the minimization of (1.6)? Overview of the monograph. This monograph consists of three parts. Part I introduces three key tools in the mathematical analysis, that we will rely on while investigating the minimization problems of the energy functionals (1.3) (1.6). These are: the Γ -convergence discussed in chapter 2, Korn’s inequality introduced in chapter 3, and Korn’s inequality nonlinear counterpart which is FrieseckeJames-M¨uller’s inequality in chapter 4. Our treatment is self-contained and the proofs are complete, including Hardy’s inequality and the Lusin-type truncationapproximation result in the Sobolev spaces, which are of independent interest. We also derive various other estimates, including the properties of constants appearing in both inequalities in relation to the dimension of the problem or to the geometry of the domain, such as the star-shaped domains and thin films. In Part II we formulate the modern description of nonlinear elasticity of plates and shells, where the analysis of the scaling of the energy infima in terms of the film’s thickness leads to the rigorous derivation of a hierarchy of limiting variational theories. These theories are differentiated by shells’ responses to external forces: in chapter 5 we derive the Kirchhoff theory (fully nonlinear bending) of elastic plates and shells, while in chapter 6 we turn to the von K´arm´an (nonlinear) theory and the linear elasticity. In chapter 7 we discuss the linearised Kirchhoff theory (linearised bending) for plates. The aforementioned four plate theories were rigorously derived from the nonlinear elasticity in the fundamental work by Friesecke, James and M¨uller, relying on their nonlinear rigidity estimate studied in chapter 4. Chapter 8 provides a heuristic derivation of the infinite hierarchy of elastic shell models and explains how the infinite hierarchy reduces to the finite hierarchy of plates due to the matching and density properties of infinitesimal isometries on twodimensional domains. Other matching and density results lead to an even larger collapse of theories: for elliptic shells as derived in chapter 9, and for developable shells in chapter 10. In each particular theory, we give the complete details of the Γ -convergence results, including the compactness analysis, the lower bound estimates, constructions of the recovery family, and the convergence of minimizers. We frequently take a detour and present generalizations: to shells with variable thickness, and to shallow shells where the curvature of the midsurface competes with the shell’s thickness in the order of vanishing. Part III is the central part of this monograph, wherein we continue the discussion of the dimension reduction, but now in the context of the prestress-driven response. Here, the obtained Γ -limiting theories are differentiated by the embeddability properties of the target metrics (rather than by the magnitude of applied forces) and,

1 Introduction

5

a-posteriori, by the emergence of isometry constraints on deformations with low regularity. In chapter 11 we derive the Kirchhoff-like theory for prestressed thin films, and in chapter 12 we turn to the von K´arm´an-like theory. In chapter 13 we show the energy quantisation result, in the sense that only the even powers of films’ thickness are viable as the scaling of the energy at minimizers, and all of them are also attained. This leads to the remaining family of the linear elasticity-like theories in the infinite hierarchy of prestressed films’ models. Along the way, we provide many examples, including applications to liquid crystal elastomers and the relation to experimental observations. There are still unresolved dichotomies between theory and experiments, which call for a thorough understanding: the insufficient understanding of the role of curvature in determining the mechanical properties of the material, and of the effects of the symmetry and the symmetry breaking solutions. The final chapter 14 analyzes the case of the weak prestress, using constructions and arguments similar to those in chapter 7 treating the case of shallow shells. There, many problems regarding multiplicity, singularities and regularity of the critical points of the obtained models remain open and are hard to tackle. On the other hand, our analysis leads to further questions of rigidity and flexibility of solutions to the weak formulations of the related partial differential equations, including the weak Sobolev or H¨older solutions to the Monge-Amp`ere equation. Overview of the back matter. The bibliography contains the main references to the material treated in this monograph, updated to 2022. Each chapter ends with the bibliographical notes, while the attributions are kept to a minimum in the body of the text. We also provide an index of terminology, containing both the classical notation in analysis and differential geometry, the standard notation in elasticity theory, and the remaining terminology consistently used throughout the text. Finally, we include the alphabetical index of topics, concepts, and named theorems. Prerequisites. The monograph is self-contained and suitable for graduate students in analysis, with some prior exposure to concepts in differential geometry. No a priori knowledge of elasticity theory is assumed. This monograph aims at the systematic and comprehensive treatment of the theory of dimension reduction for thin elastic (prestressed or lacking prestress) plates and shells. Starting from the view that shape is the consequence of metric frustration in an ambient space, we explore many surprising connections between the classical Nash embedding problem, its quantitative version via the variational description, the Monge-Amp´ere equation and the biological morphogenesis. We hope that our text will serve as a friendly introduction to this beautiful and multifaceted topic, as well as suggest and encourage new research directions. Marta Lewicka Pittsburgh, March 2022.

Part I

Tools in mathematical analysis

Chapter 2

Γ -convergence

In this chapter, we briefly discuss the notion of the Gamma-convergence, which will play the key role in the dimension reduction analysis in Parts II and III of this monograph. Indeed, the results on convergence of minimizers to families of parametrised variational problems can be conveniently expressed using this general set-up. In section 2.1 we present the definition and a handful of examples distinguishing the introduced notion from other modes of convergence in the mathematical analysis. We then explain how Γ -convergence when combined with a compactness assumption yields a one-to-one correspondence between the almost-minimizers of the given sequence of functionals, and the minimizers of this sequence’s Γ -limit. In section 2.2 we discuss an example of Γ -convergence in linearised elasticity. We introduce the notion of a displacement on a thin plate, and its linear elastic and total energies. Using the simplest form of Korn’s inequality, we then show how the family of energies given in terms of the symmetrized displacement gradients, Γ converges in the limit of the vanishing thickness to the energy determined by the Hessian of a scalar displacement on the (two-dimensional) midplate. This limiting energy yields then the biharmonic equation as its Euler-Lagrange equation.

2.1 Definition, examples and fundamental properties The following important tool can be used to rigorously justify the passage to the limit in the minimizers to a sequence of functionals: ¯ ∞ be a sequence of functions on a metric space Definition 2.1. Let {Fn : X → R} n=1 ¯ provided (X, d). We say that this sequence Γ -converges to a function F : X → R, that the following two conditions hold: (i) for every x ∈ X and for every sequence {xn ∈ X}∞ n=1 , if limn→∞ xn = x then F(x) ≤ lim infn→∞ Fn (xn ). (ii) for every x ∈ X there exists a sequence {xn ∈ X}∞ n=1 (called a recovery sequence for x) such that limn→∞ xn = x and F(x) = limn→∞ Fn (xn ). © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_2

9

10

2 Γ -convergence Γ

We then write Fn −→ F, or Γ - lim Fn = F. It is not hard to observe that the Γ -limit, when exists, is unique. The examples below show that the notion of Γ -convergence is quite different from other modes of convergence encountered in analysis: Γ

˜ Example 2.2. For a constant sequence of functions {Fn = F}∞ n=1 , we have: Fn −→ F where F˜ is the lower-semicontinuous envelope of F, given by:  ˜ F(x) = min lim inf F(xn ); {xn ∈ X}∞ n=1 such that lim xn = x . n→∞

n→∞

Γ

In particular, Fn −→ F if and only if F is lower-semicontinuous, namely F(x) ≤ lim infn→∞ F(xn ) whenever limn→∞ xn = x. Similarly, the Γ -limit is always lowersemicontinuous. Example 2.3. The following sequence of continuous functions converges to 0, pointwise in X = R and locally uniformly in R \ {0}: Fn (x) = F1 (nx),

where F1 (x) =

√ −(2x2 −1)/2 2xe .

Γ

However, Fn −→ F with F(0) = −1 and F|R\{0} = 0. A possible recovery sequence at x = 0 is given by xn = −1/n. We also observe the same convergence of the Γ

negated sequence: −Fn −→ F. Thus, in general Γ - lim(−Fn ) , −Γ - lim Fn , and also Γ - lim(Fn + F¯n )) , Γ - lim Fn + Γ - lim F¯n . On the other hand, when F¯ is continuous, ¯ = Γ - lim Fn + F. ¯ there holds: Γ - lim(Fn + F)) Example 2.4. The sequence Fn (x) = − cos(nx) has no pointwise converging subse ∞ quence on X = R, but it Γ -converges to F ≡ −1. One can take xn = bnx/(2π)c2π n n=1 as a recovery sequence at a given x ∈ R. Example 2.5. For each x ∈ R and each n, define:  0 if x < Q or if x = Fn (x) = (−1)n otherwise.

k m

with 1 ≤ m ≤ n,

Then the sequence {Fn }∞ n=1 converges pointwise to 0 on X = R, but its Γ -limit does not exist at any point. The motivation behind Definition 2.1 is revealed in the simple result below. Informally speaking, any limit of a sequence of approximate minimizers to {Fn }∞ n=1 must be a minimizer of that sequence’s Γ -limit F. Note that this statement cannot be reversed: not every minimizer is the limit of approximate minimizers, as shown Γ by taking Fn (x) = x2 /n where Fn −→ 0 on [0, 1]. There, the only minimizer of F = 0 that is also the limit of approximate minimizers, is x = 0.

2.1 Definition, examples and fundamental properties

11

Γ

Theorem 2.6. Assume that Fn −→ F in a metric space (X, d) and also assume that for a compact set K ⊂ X there holds: inf Fn = inf Fn X

for all n.

K

Then, the limit function F has at least one minimizer in X. Moreover: (i) if a sequence {xn ∈ K}∞ n=1 satisfies: lim |Fn (xn ) − inf Fn | = 0,

n→∞

X

(2.1)

then {xn } has a subsequence that converges to a minimizer of F. (ii) for every minimizer x of F in X, there holds x = limn→∞ xn for some sequence of approximate minimizers {xn } , i.e. satisfying (2.1). Proof. To show existence of a minimizer of F, take any {xn ∈ K}∞ n=1 that satisfies (2.1). By the compactness assumption and passing to a subsequence if necessary, limn→∞ xn = x ∈ K. Let now x¯ ∈ X. We now use Definition 2.1 (i) and (ii) with a recovery sequence {x¯n }∞ ¯ to obtain: n=1 corresponding to x, ¯ F(x) ≤ lim inf Fn (xn ) ≤ lim inf inf Fn ) ≤ lim Fn (x¯n ) = F(x). n→∞

n→∞

n→∞

X

Consequently, x is a minimizer of F and we have also achieved the proof of (i). Taking x¯ = x, there follows additionally:
min F = F(x) = lim inf Fn . X

n→∞

X

To show (ii), let x ∈ X be a minimizer of F and take its recovery sequence {xn }∞ n=1 . Then limn→∞ Fn (xn ) = F(x) = limn→∞ (infX Fn ), implying (2.1). Before our next example of Γ -convergence, presented in section 2.2, and leading the way to the more complex discussion of dimension reduction in the nonlinear and prestressed elasticity, as addressed in the following chapters, we point out: Remark 2.7. When X is only a topological space, the definition of Γ -convergence involves systems of neighborhoods rather than sequences. However, when the functionals {Fn }∞ n=1 are equi-coercive and X is a reflexive Banach space equipped with its weak topology, one can still use (i) and (ii) in Definition 2.1 for weakly converging sequences, as an equivalent version.

12

2 Γ -convergence

2.2 Example of Γ -convergence in linearised elasticity In this section, we develop a more involved example of Γ -convergence than the four introductory examples in section 2.1. This will also be our first dimension reduction example, of which we will see many throughout this monograph. Let ω ⊂ R2 be an open, bounded, connected domain with Lipschitz boundary. For each h  1 we define the following thin plate with middle surface ω: h h h
 = x = (z,t) ∈ R3 ; z ∈ ω, |t| < . Ωh = ω × − , 2 2 2 For a vector field vh ∈ H 1 (Ω h , R3 ), we consider its elastic and total energies: E h (vh ) =

1 h

Z

J h (vh ) = E h (vh ) +

|sym∇vh |2 dx,

Ωh

1 h

Z Ωh

h f h , vh i dx,

with respect to the given f h ∈ L2 (Ω h , R3 ). The family {vh }h→0 can be interpreted as displacement fields on thin plates {Ω h }h→0 , while { f h }h→0 are external forces, each acting on the corresponding plate Ω h . Lemma 2.8. In the above context, assume that for each (z,t) ∈ Ω h there holds: f h (z,t) = h3 f (z) where f = ( f 1 , f 2 , f 3 ) ∈ L2 (ω, R3 ). Suppose that: Z

Z

f dz = 0, ω

Z

f 3 (z)z dz = 0,

ω

(2.2)

hAz, ( f 1 , f 2 )(z)i dz = 0

for all A ∈ so(2).

ω

Then we have, with a positive constant C that depends only on ω and f :  −Ch4 ≤ inf J h (vh ); vh ∈ H 1 (Ω h , R3 ) ≤ 0.

(2.3)

Proof. 1. Note first that conditions (2.2) imply (they are, in fact, equivalent to): Z Ωh

h f h , Ax + bi dx = 0

for all A ∈ so(3), b ∈ R3 .

(2.4)

Indeed, the first condition in (2.2) yields the above for A = 0, while for b = 0 and A ∈ so(3) we use the other two conditions to conclude that: Z Ωh

h f h , Axi dx = h3 4

=h

Z

h

ZΩ

 ( f 1 , f 2 )(z), A2×2 z + t(A13 , A23 ) − f 3 (z)hz, (A13 , A23 )i dx

 ( f 1 , f 2 )(z), A2×2 z dz = 0.

ω

In particular, (2.4) gives: J h (vh ) = J h (vh + (Ax + b))

for all vh ∈ H 1 (Ω h , R3 ), A ∈ so(3), b ∈ R3 ,

2.2 Example of Γ -convergence in linearised elasticity

13

and similarly we see that (2.2) is necessary for having: inf J h > −∞. 2. We now prove (2.3). The right hand side inequality follows trivially, upon taking vh = 0. To prove the left hand side, we introduce the change of variables in: 1  h 1 h 1 h v v v (z, ht) for all x = (z,t) ∈ Ω 1 , (2.5) , , h2 1 h2 2 h 3

which to a given vh = vh1 , vh2 , vh3 ∈ H 1 (Ω h , R3 ) associates v˜h = v˜h1 , v˜h2 , v˜h3 ∈ H 1 (Ω 1 , R3 ). The value of the energy J h on vh can be computed in terms of v˜h : v˜h (x) =

1 J h (vh ) = h4

" # ∇z (v˜h1 , v˜h2 ) 1h ∂t (v˜h1 , v˜h2 ) 2 sym dx 1 1 h Ω1 ∂ v˜h h ∇z v˜3 h2 t 3 Z 

  + h ( f 1 , f 2 ), (v˜h1 , v˜h2 ) + f 3 v˜h3 dx

Z

Ω1

≥

Z Ω1

Z sym∇v˜h 2 dx +

Ω1



1 (h f , h f 2 , f 3 ), v˜h dx

where the last bound holds for h < 1. We now invoke the Korn-Poincar´e inequality on the domain Ω 1 , asserting existence of Ah ∈ so(3), bh ∈ R3 such that: kv˜h − (Ah x + bh )kH 1 (Ω 1 ) ≤ Cksym∇v˜h kL2 (Ω 1 ) with a constant C that depends only on ω. Korn’s inequality (and the above KornPoincar´e inequality, see Corollary 3.4) will be discussed from different points of view and proved in chapter 3. Consequently, and recalling (2.4) we get: Z

 1 h h h h h 2 J (v ) ≥ ck v ˜ − (A x + b )k + (h f 1 , h f 2 , f 3 ), v˜h − (Ah x + bh ) dx 1 (Ω 1 ) H 4 h Ω1 h h h 2 ≥ ckv˜ − (A x + b )kL2 (Ω 1 ) − k f kL2 (ω) kv˜h − (Ah x + bh )kL2 (Ω 1 ) .

Since the quadratic polynomial R 3 s 7→ cs2 − sk f kL2 (ω) is bounded from below whenever c > 0, the claimed bound (2.3) has been achieved. Remark 2.9. We observe that for every open bounded ω ⊂ R2 , there exists a S nonzero f ∈ L2 (ω, R3 ) satisfying (2.2). Indeed, take any partition ω = 4i=1 ωi into four measurable sets with non-zero areas. Define f 1 = f 2 = 0 and f 3 = ∑4i=1 ai 1ωi with some {ai ∈ R}4i=1 . Then (2.2) is equivalent to: 4

∑ ai |ωi | = 0

i=1

4

and

∑ ai

i=1

Z

z dz = 0. ωi

The existence of a nonzero (a1 , . . . , a4 ) ∈ R4 as above, immediately follows from
4  R the linear dependence of the four vectors |ωi |, ωi z dz ∈ R3 i=1 .

14

2 Γ -convergence

Remark 2.10. By a reasoning entirely similar as in the proof of Lemma 2.8, via using Korn’s inequality on a single domain Ω h , one can easily deduce that each functional J h has a minimizer on H 1 (Ω , R3 ). Thus “inf” in (2.3) can be replaced by “min”. Moreover, this minimum is unique up to the additive modification by affine functions with skew-symmetric gradients. Towards stating a Γ -limit result for the energies J h in Corollary 2.13, we first derive both a compactness statement to replace the assumption of Theorem 2.6, and a lower bound statement, corresponding to Definition 2.1 (i): Theorem 2.11. Assume that f h (x) = h3 f (z) for all x = (z,t) ∈ Ω h , where f = ( f 1 , f 2 , f 3 ) ∈ L2 (ω, R3 ) satisfies (2.2). Let {vh ∈ H 1 (Ω h , R3 )}h→0 satisfy: |J h (vh )| ≤ Ch4 .

(2.6)

Then, possibly modifying each vh to vh − (Ah x + bh ) with some Ah ∈ so(3) and bh ∈ R3 , the following holds for the rescalings v˜h in (2.5): (i) up to a subsequence that we do not relabel, {v˜h } converges as h → 0 weakly in H 1 (Ω 1 , R3 ), to some limit v˜ = (w1 , w2 , v), (ii) the normal component field v of v˜ satisfies: ∂3 v = 0 in Ω 1 and so v is only a function of z ∈ ω; moreover v3 ∈ H 2 (ω, R), (iii) the tangent component field w = (w1 , w2 ) ∈ H 1 (Ω 1 , R2 ) of v˜ satisfies: w(x) = −t∇z v(z) + h(z)

for all x = (z,t) ∈ Ω 1

with some vector field h ∈ H 1 (ω, R2 ), (iv) we have the lower bound: 1 1 lim inf 4 J h (vh ) ≥ J (v)  h→0 h 12

Z

2

2

|∇ v| dz +

ω

Z

f 3 v dz.

(2.7)

ω

Proof. 1. As in the proof of Lemma 2.8, we choose Ah ∈ so(3), bh ∈ R3 so that: kv˜h kH 1 (Ω 1 ) ≤ Cksym∇v˜h kL2 (Ω 1 ) where C is a Korn-Poincar´e constant on Ω 1 . Then in view of (2.6) we have: C≥ ≥

1 1 h h J (v ) ≥ 4 E h (vh ) − k f kL2 (ω) kv˜h kL2 (Ω 1 ) 4 h h Z Ω1

|sym∇v˜h |2 dx − k f kL2 (ω) kv˜h kL2 (Ω 1 ) ≥ ckv˜h k2H 1 (Ω 1 ) − k f kL2 (ω) kv˜h kH 1 (Ω 1 ) .

Consequently kv˜h kH 1 (Ω 1 ) ≤ C, which yields (i). Recalling the expression of ∇vh in terms of v˜h , it further follows that: # " Z ∇z (v˜h1 , v˜h2 ) 1h ∂t (v˜h1 , v˜h2 ) 2 1 h h (2.8) E (v ) = dx ≤ C. sym 1 1 h h h4 Ω1 ∂ v ˜ t 2 3 h ∇z v˜3 h

2.2 Example of Γ -convergence in linearised elasticity

15

In particular, the above implies that k∂t vkL2 (Ω 1 ) = k∇z v + ∂t wkL2 (Ω 1 ) = 0, because: k∂t v˜h3 kL2 (Ω 1 ) + k∇z v˜h3 + ∂t (v˜h1 , v˜h2 )kL2 (Ω 1 ) → 0

as h → 0.

We obtain that v = v(z) as claimed in (ii), and also ∂t w = −∇z v proving that w is affine in t. Since w ∈ H 1 (Ω 1 , R2 ) we may now choose t1 ,t2 ∈ (− 12 , 21 ) such that both traces w(·,t1 ) and w(·,t2 ) belong to H 1 (ω, R2 ). Then: 1

2

H (ω, R ) 3 w(z,t1 ) − w(z,t2 ) =

Z t2 t1

∂t w(z,t) dt = −(t2 − t1 )∇z v(z)

which yields v ∈ H 2 (ω, R) and hence proves (ii) and (iii). 2. To show the lower bound in (iv), we observe the following weak convergence in L2 (Ω 1 , R2×2 ) as h → 0, where we also use (iii): sym∇z (v˜h1 , v˜h2 ) * sym∇z w = −t∇2 v + sym∇z h. Recalling (2.8), the weak lower-semicontinuity of the L2 (Ω 1 ) norm gives: lim inf h→0

1 h h E (v ) ≥ lim inf h→0 h4

Z Ω1

Z Z 1/2

=

Z sym∇z (v˜h , v˜h ) 2 dx ≥ 1

2

t 2 |∇2 v|2 dt dz +

ω −1/2

Ω1

Z

− t∇2 v + sym∇z h 2 dx

|sym∇z h|2 dz ≥

ω

1 12

Z

|∇2 v|2 dz.

ω

Finally, we get: lim inf h→0

Z

1   1 h h h h J (v ) = lim inf E (v ) + (h f 1 , h f 2 , f 3 ), v˜h dx 4 4 h→0 h h Ω1 Z Z 1 |∇2 v|2 dz + f 3 v dx. ≥ 12 ω Ω1

The proof is done. In the next result we show the upper bound corresponding to condition (ii) in Definition 2.1 of Γ -convergence: Theorem 2.12. Let f h (x) = h3 f (z) for all x = (z,t) ∈ Ω h , where f ∈ L2 (ω, R3 ) satisfies (2.2). Then, for every v ∈ H 2 (ω, R) there is {vh ∈ H 1 (Ω h , R3 )}h→0 so that: (i) there holds: |J h (vh )| ≤ Ch4 , (ii) the corresponding rescalings {v˜h } defined in (2.5) converge as h → 0, weakly in H 1 (Ω 1 , R3 ), to v˜ = (−t∇z v, v), (iii) recalling the definition of J in (2.7), there holds: 1 J h (vh ) = J (v). h→0 h4 lim

(2.9)

16

2 Γ -convergence

Proof. For every z ∈ ω and |t| < 2h , we put:
vh (z,t) = − th∂1 v(z), −th∂2 v(z), hv(z) ,
so that: v˜h (x) = − t∂1 v(z), −t∂2 (z), v for all x = (z,t) ∈ Ω 1 . Further, by (2.8): 1 h h E (v ) = h4 = 1 h5

Z Ωh

# " Z −t∇2 v − h1 ∇z v 2 dx = t 2 |∇2 v|2 dx sym 1 1 Ω Ω1 0 h ∇z v

Z

1 12

Z

|∇2 v|2 dz,

ω

h f h , vh i dx =

Z

f 3 v dz.

ω

This ends the proof.

Fig. 2.1 The Kirchhoff-Love extension relative to the deformation φ h : ω → R3 .

The interpretation of the recovery family {vh }h→0 construction in Theorem 2.12 is as follows. The midplate ω ⊂ R2 is deformed by the map ω 3 z 7→ φ h (z) = (z, hv(z)) ∈ R3 , with the displacement given by vh|ω . A non-vanishing normal vector field on the image surface φ h (ω) may be calculated as: ∂1 φ h × ∂2 φ h = (e1 + h∂1 ve3 ) × (e2 + h∂2 ve3 ) = e3 − h∇z v + O(h2 ). The unit normal has then the same expansion: N(z, hv(z)) =

∂1 φ h × ∂2 φ h = e3 − h∇z v(z) + O(h2 ), |∂1 φ h × ∂2 φ h |

and we see that the deformation of the thin plate Ω h , given by: Ω h 3 (z,t) 7→ (z,t) + vh (z,t) = (z, hv(z)) + tN(z, hv(z)) + O(h2 ) ∈ R3

(2.10)

2.2 Example of Γ -convergence in linearised elasticity

17

coincides, at the leading order terms, with the so-called Kirchhoff-Love ansatz. Namely, points along the unit normal vector e3 to the undeformed midplate are mapped onto points along the unit normal vector to the deformed midsurface. Concluding, Theorems 2.11 and 2.12 yield: Corollary 2.13. Let ω ⊂ R2 be an open, bounded, Lipschitz domain. Assume that f h (x) = h3 f (z) for all x = (z,t) ∈ Ω h where f ∈ L2 (ω, R3 ) satisfies (2.2). Given v, ¯ u¯ ∈ H 1 (Ω 1 , R3 ) and h  1, we set vh (z,t)  v(z,t/h) ¯ ∈ H 1 (Ω h , R3 ) and define:  Z Z 1 1  h h |sym∇vh |2 dx + h f h , vh i dx if u¯ = v/h, ¯ J (v ) = h F (v, ¯ u) ¯ = h Ωh h Ωh ∞ otherwise. Then

1 h F h4

Γ

−→ F with respect to the weak convergence in H 1 (Ω , R3 )2 , where:

Z Z  1 2 3 2 3  |∇ u ¯ | dz + f 3 u¯3 dz J ( u ¯ ) =    12 ω ω if v¯ = 0 and u¯ ∈ H 2 (ω, R3 ) and (u¯1 , u¯2 )(z) = Az + b F(v, ¯ u) ¯ =   for some A ∈ so(2), b ∈ R2 ,   ∞ otherwise.

Proof. 1. We need to validate the two conditions in Definition 2.1. For (i), we take a sequence in {(v¯h , u¯h )}h→0 converging weakly in H 1 (Ω , R3 )2 to (v, ¯ u). ¯ It suffices to assume that lim infh→0 h14 F h (v¯h , u¯h ) < ∞, for otherwise the inequality in (i) holds trivially. In particular, we see that v¯h = hu¯h converges (strongly) to v¯ = 0. Also, by Theorem 2.11 we get the following convergence (weakly in H 1 (Ω 1 , R3 ), up to a subsequence that we do not relabel): 1

1 1  v¯h1 , 2 v¯h2 , v¯h3 − Ah x + bh * h1 (z) − t∂1 v(z), h2 (z) − t∂2 v(z), v(z) 2 h h h for some v ∈ H 2 (ω, R), h = (h1 , h2 ) ∈ H 1 (ω, R2 ), and {Ah ∈ so(3), bh ∈ R3 }h→0 . Consequently, recalling that u¯h = h1 v¯h , we get:   u¯h (x) − hAh2×3 x + h(bh1 , bh2 ), h(Ah31 , Ah32 ), zi + bh3 * (0, 0, v(z)), weakly in H 1 (Ω 1 , R3 ) as h → 0. But u¯h * u, ¯ so it follows by direct inspection that:

u(x) ¯ = 0, 0, v(z) + Az, 0 + b, for some A ∈ so(2), b ∈ R3 . Finally, the bound and definition (2.7) of the limit functional J in Theorem 2.11 (iv) yield, in view of (2.2):

18

2 Γ -convergence

lim inf h→0

1 h h h F (v¯ , u¯ ) ≥ J (v) = J (u¯3 ). h4

2. To validate condition (ii) of Definition 2.1, let (v, ¯ u) ¯ ∈ H 1 (Ω 1 , R3 )2 . If F(v, ¯ u) ¯ = h h h ∞, then we take (v¯ , u¯ ) = (v, ¯ u) ¯
so that limh→0 F (v¯h , u¯h ) = ∞. Otherwise, when v¯ = 0 and u(z,t) ¯ = Az + b, u¯3 (z) for some A ∈ so(2), b ∈ R2 and u¯3 ∈ H 2 (ω, R), we define accordingly with Theorem 2.12:  
v¯h (z,t) = − th2 ∇z u¯3 + h Az + b , hu¯3 (z) ,

1 u¯h = v¯h . h

Then, (v¯h , u¯h ) converge as h → 0 to (v, ¯ u), ¯ strongly in H 1 (Ω 1 , R3 ) and moreover: 1 h h h F (v¯ , u¯ ) = J h (u¯3 ), h→0 h4 lim

by a direct calculation. The proof is done. In the statement of Corollary 2.13 we have abused the notion of Γ -convergence because the weak topology is not metrizable, whereas in Definition 2.1 we required the domain of F h , F to be a metric space. Nonetheless, the same definition can be formulated with respect to weak convergence (see Remark 2.7), and also, in virtue of the compactness statements in Theorem 2.11, it suffices to consider F h restricted to bounded subsets of H 1 (Ω 1 , R3 ), where the weak topology is metrizable. Combining Corollary 2.13 and recalling the proof of Theorem 2.6 we finally note: Remark 2.14. In the context of Corollary 2.13, the minimizers vh of J h converge, after a possible modification by affine maps with skew-symmetric gradients, and after the rescaling of their variables to v˜h according to (2.5), to the limiting vector field (w, v) which is completely determined by its out-of-plane component v. This component has regularity v ∈ H 2 (ω, R) and it is the unique, up to linear functions, minimizer of the functional J defined in (2.7). There also holds: 1 min J h = J (v) = min J . h→0 h4 H 1 (Ω h ,R3 ) H 2 (ω,R) lim

By a direct inspection, it follows that v satisfies the Euler-Lagrange equations together with the natural boundary conditions: 1 2 ∆ v + f 3 = 0 in ω, 6 h∇2 v : η ⊗ ηi = 0, ∂τ h∇2 v : η ⊗ τi + ∂η ∆ v = 0

on ∂ ω,

where η is the outward unit normal normal and τ is the unit tangent vector along the midplate’s boundary ∂ ω.

2.3 Bibliographical notes

19

2.3 Bibliographical notes Definition 2.1 is due to De Giorgi and Franzoni [1975]. Γ -convergence is linked to previous notions of convergence such as Mosco’s convergence or Kuratowski’s convergence of sets. We refer to the book by Dal Maso [1993] for an in-depth discussion of the various concepts and applications of Γ -convergence. The book by Rindler [2018] is a friendly introduction to the general field of calculus of variations, in which the notion of Γ -convergence plays a prominent role. Examples in section 2.1 are taken from Braides [2002]. Theorem 2.6 and results in section 2.2 are “folklore”. Literature using the notion of Γ -convergence is abundant. Since this monograph focuses on the dimension reduction in nonlinear elasticity, we list below only one reference per topic in which Γ -convergence has been displayed in an active field of research in the past decades: phase transitions in Conti et al. [2002], various homogenization problems in Braides and Defrancheschi [1999] , Ginzburg-Landau type functionals in Alberti [2001], elasticity in Friesecke et al. [2006], free-discontunuity problems in Braides et al. [1996], image segmentation problems in Ambrosio and Tortorelli [1990], signal transmission in Amar and Braides [1995], discrete problems in Piatnitski and Remy [2001], finite difference schemes in Bartels et al. [2017], passage from atomistic to continuum models in Alicandro and Cicalese [2006]. Many more references may be given for each topic.

Chapter 3

Korn’s inequality

Among the most important inequalities in mathematical analysis is Korn’s inequality. Discovered in the early XXth century in the context of the boundary value problem of linear electrostatics, it is a basic tool to prove existence of solutions of the linearised displacement-traction equations in elasticity. Another area of application is fluid dynamics in the presence of Navier’s boundary conditions, where Korn’s inequality replaces Poincar´e’s inequality under the Dirichlet boundary conditions. The outline of this chapter is as follows. In section 3.1 we introduce Korn’s inequality (sometimes called the second Korn’s inequality) as a rigidity estimate, quantifying the simple observation that a gradient field that is skew-symmetric must be constant. We show how through an argument by contradiction, it can be deduced from First Korn’s inequality which involves the norm of the vector field itself, in addition to the norm of its gradient. In section 3.2 we digress to deduce a few useful variants of Korn’s inequality, in presence of boundary conditions: the homogeneous inequality where Korn’s constant equals 2 regardless of the domain (the same result is true on polyhedra under tangential boundary conditions), and Korn’s inequality under mixed boundary conditions (Dirichlet and tangential), in which it suffices to look for the approximating skew-symmetric matrix among those matrices that are gradients of affine functions satisfying the same boundary conditions. In particular, if the portion of the boundary corresponding to the Dirichlet condition is nonempty or when the domain has no rotational symmetry, this set is trivial leading to the homogeneous Korn’s inequality (albeit with a constant possibly different than 2). In sections 3.3 and 3.4 we give a proof of First Korn’s inequality. The argument relies on the weighted (through the distance from the boundary) reverse Poincar´einequality and on estimates in star-shaped domains. Since star-shaped domains are the building blocks of Lipschitz domains (as proved in section 3.3), the final result is obtained by decomposition. In section 3.5 we prove that under tangential boundary conditions, Korn’s constant is at least 2. There is however no upper bound, because any Killing field on a given midsurface gives rise to a family of displacements on thin shells that are tangential on the shells’ boundaries and whose Korn’s constants blow up as the inverse square of shell’s thickness. This statement is established in section 3.5 for the case © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_3

21

22

3 Korn’s inequality

of a curve in R2 , and for the general case of a hypersurface in RN in section 3.8. In section 3.6 we deduce an approximation result in which the displacement gradient on a thin shell is approximated by a field of skew-symmetric matrices, rather than a constant matrix, while the corresponding Korn’s constants are uniform with respect to thickness. The construction is a combination of the local application of Korn’s inequality and a mollification argument, and it can be repeated in other contexts. In particular, it is of central importance regarding the dimension reduction analysis in nonlinear prestressed elasticity, discussed in Parts II and III of this monograph. In section 3.7 we prove the counterparts of the two Korn’s inequalities on hypersurfaces. Rather than redeveloping previous arguments in the Riemannian geometry setting, we first apply First Korn’s inequality on thin shells and subsequently pass to the limit with the vanishing thickness. Then, an argument by contradiction naturally identifies gradients of Killing fields as approximations of the covariant derivatives of arbitrary tangent fields, up to an error quantifying the symmetrized gradients. Killing fields are the infinitesimal generators of isometries on the surface and as such they serve as the replacements of affine maps with skew-symmetric gradients, which are the “linearised rigid motions” on open domains in RN . In sections 3.9 and 3.10 we prove that the presence of Killing fields is the only obstruction from the uniformity of Korn’s constant on thin shells under tangential boundary conditions. Indeed, for vector fields within any cone that is separated from the displacements derived from Killing fields, Korn’s constant is uniform.

3.1 Korn’s inequality and First Korn’s inequality In this section we introduce the two versions of Korn’s inequality and show how to deduce one, namely the rigidity estimate in Theorem 3.1, from the other which is First Korn’s inequality in Theorem 3.5. Theorem 3.1. [Korn’s inequality] Let Ω ⊂ RN be an open, bounded, connected, Lipschitz domain. Then, for every vector field v ∈ H 1 (Ω , RN ) there exists a matrix A ∈ so(N) satisfying: Z Ω

|∇v − A|2 dx ≤ C

Z

|sym ∇v|2 dx.

(3.1)

Ω

The above constant C depends only on Ω , but not on v. The inequality (3.1) is an example of a rigidity estimate, quantifying the rigidity statement below. Namely, the vanishing of the right hand side in (3.1) i.e. having sym∇v ≡ 0 in Ω , implies the vanishing of its left hand side i.e. having ∇v constant and skew-symmetric. This remarkable observation may be proved directly:

3.1 Korn’s inequality and First Korn’s inequality

23

Lemma 3.2. Let Ω ⊂ RN be an open, connected domain. If v ∈ H 1 (Ω , RN ) satisfies ∇v ∈ so(N) a.e. in Ω , then: v(x) = Ax + b

for some A ∈ so(N), b ∈ RN .

Proof. The following useful formula, valid in the sense of distributions: 
 1 (3.2) ∆ v = 2 div sym∇v − tr sym∇v IdN 2
1 1 1 1 T may be obtained by noting
div sym∇v = 2 div ∇v+ 2 div (∇v) = 2 ∆ v+ 2 ∇(div v), and that div (div v)IdN = ∇(div v). Consequently, if sym∇v = 0 then ∆ v = 0 in Ω , so v is harmonic and automatically smooth. By assumption, one has ∇v = −(∇v)T . To each row {∇vi = −∂i v}i=1...N of this identity, we apply the N-dimensional curl operator: curl u  {∂k u j − ∂ j uk } j,k=1...N . Since curl ∇vi = 0 it follows that ∇curl v = 0, and so curl v must be constant within the connected domain Ω . It now suffices to note that coefficients of curl v coincide with the entries of the matrix ∇v = skew ∇v. The proof is done. Recall that every matrix in RN×N is the orthogonal sum of its symmetric and skewsymmetric parts, so in particular: |sym∇v(x)| = dist(∇v(x), so(N))

for all x ∈ Ω .

The inequality (3.1) is thus the quantitative version of Lemma 3.2, in the sense that the total pointwise distance of ∇v from so(N), measured in the L2 (Ω ) norm, yields the deviation of ∇v from being constant skew-symmetric, again measured in L2 (Ω ):
distL2 (Ω ) ∇v, so(N) 

Z

inf A∈so(N)

|∇v − A|2 dx


1/2

Ω

(3.3)

≤ Ck dist(∇v, so(N))kL2 (Ω ) . It is obvious that a reverse inequality also holds, namely:
k dist(∇v, so(N))kL2 (Ω ) ≤ distL2 (Ω ) ∇v, so(N) . Hence, Korn’s inequality guarantees the equivalence of commuting the operations of taking the distance from so(N) and integrating. One has to be careful though: the right hand side computes the L2 norm of the pointwise distance in RN×N , whereas the distance in the left hand side is in the function space L2 (Ω , RN×N ). Further: Remark 3.3. The infimum in (3.3) is uniquely attained by the minimizing matrix A: ? A = skew ∇v dx. Ω

24

3 Korn’s inequality

Indeed, let A ∈ so(N) be such that Ω |∇v − A|2 dx ≤ Ω |∇v − (A + εB)|2 dx for all ε ∈ R and B ∈ so(N). Expanding the right hand side as: R

Z

|∇v − A|2 dx + 2ε

Ω

R

Z

h∇v − A : Bi dx + ε 2

Ω

Z

|B|2 dx,

Ω

and passing to the limit with ε → 0, we obtain the Euler-Lagrange equations:

Z

 Z ∇v − A dx : B = h∇v − A : Bi dx = 0

Ω

The matrix

>

Ω

for all B ∈ so(N).

Ω

∇v − A is thus orthogonal to so(N), hence symmetric: ?

?
∇v dx − A = 0.

∇v dx − A = skew

skew Ω

Ω

This yields the claimed result. By combining (3.1) with the Poincar´e-Wirtinger inequality, one gets another useful bound (that we already applied in section 2.2): Corollary 3.4. [Korn-Poincar´e inequality] Let Ω ⊂ RN be an open, bounded, connected, Lipschitz domain. Then, for every v ∈ H 1 (Ω , RN ) there exists A ∈ so(N) and b ∈ RN such that: kv − (Ax + b)kH 1 (Ω ) ≤ Cksym ∇vkL2 (Ω ) . The constant C depends only on Ω , but not on v. Korn’s inequality in Theorem 3.1 classically follows via an argument by contradiction, from a stronger result called First Korn’s inequality: Theorem 3.5. [First Korn’s inequality] Let Ω ⊂ RN be an open, bounded, Lipschitz domain. There holds: Z Ω

2

|∇v| dx ≤ C

Z

|v|2 + |sym ∇v|2 dx,

(3.4)

Ω

for every v ∈ H 1 (Ω , RN ), where the constant C depends on Ω , but not on v. We postpone the proof of (3.4) to section 3.3, and deduce (3.1) right away: Proof of Theorem 3.1. Assume that there is no universal constant C for (3.1) to hold. In view of Remark 3.3, this implies existence of a sequence {vn ∈ H 1 (Ω , RN )}∞ n=1 , such that:

3.1 Korn’s inequality and First Korn’s inequality

?

Z

2 ∇vn dx > n

∇vn − skew

Ω

Denoting wn (x)  vn (x) − skew ?

>

25

Z

|sym∇vn |2 dx.

(3.5)

Ω

Ω

>
>

v (y) − skew ∇v x − ∇vn y dy we get: n n Ω Ω Ω ?

wn dx = 0,

∇wn dx = 0.

skew

Ω

(3.6)

Ω

Multiplying each vn by an appropriate constant, we may also ensure that: Z

|∇wn |2 dx = 1,

1 |sym∇wn |2 dx < , n Ω

Z

Ω

(3.7)

where the second assertion follows by (3.5). By virtue of Poincar´e’s inequality, the 1 sequence {wn }∞ n=1 is thus bounded in H (Ω ), and so it has a subsequence (that we do not relabel), converging weakly to some w ∈ H 1 (Ω , RN ). In particular, the 2 N×N ) to sym∇w. sequence {sym∇wn }∞ n=1 converges weakly in L (Ω , R By (3.7) we see that sym∇w = 0 a.s. in Ω , so Lemma 3.2 implies that w(x) = N Ax + b for some A ∈ > so(N) and b ∈ R> . On the other hand, passing to the limit in (3.6) implies: skew Ω ∇w dx = 0 and Ω w dx = 0. In conclusion, A = 0, b = 0 and so w = 0, yielding the following convergences: Z

|wn |2 dx → 0,

Ω

Z

|sym∇wn |2 dx → 0

as n → ∞.

Ω

We now apply (3.4) to get: Z

|∇wn |2 dx → 0

as n → ∞.

Ω

This contradicts the first assertion in (3.7). The proof is done. As an anticipating point of comparison, we remark that a nonlinear version of (3.1) will be discussed in chapter 4. This celebrated nonlinear geometric rigidity estimate (see Theorem 4.1), due to Friesecke, James and M¨uller, is “nonlinear” in the sense that in both of its sides it quantifies the appropriate distances of the gradient ∇v from the compact manifold SO(N), rather than from a linear subspace so(N) ⊂ RN×N as in (3.1). Another connection is that so(N) is the tangent space to SO(N) at IdN , so (3.1) formally follows by collecting the lowest order terms in the expansion of the nonlinear estimate (4.2) close to ∇u = IdN . These observations are also inherently related to the concepts of deformations and displacements in the mathematical description of, respectively, the nonlinear and linear elasticity. We have used Corollary 3.5 in section 2.2 to prove a Γ -convergence result in the dimension reduction of linearly elastic plates. The aforementioned FrieseckeJames-M¨uller inequality will naturally be applied in dimension reduction analysis of the nonlinear prestressed plates and shells in Parts II and III of this monograph.

26

3 Korn’s inequality

3.2 Variants of Korn’s inequality with different boundary conditions In this section we prove variants of Korn’s inequality valid under specific boundary conditions. The rule of thumb is that, without violating the uniformity of the constant C in (3.1), it suffices to seek the skew-symmetric matrix A in its left hand side, only within constant skew-symmetric gradients of those linear maps that obey the same boundary conditions. This space is, in general, a proper subspace of so(N) and may even be trivial. Before we make this observation more precise, we prove a classical variant of Korn’s inequality where the constant C can be made specific. Theorem 3.6. [Homogeneous Korn’s inequality] For every open domain Ω ⊂ RN and every v ∈ H01 (Ω , RN ), there holds: Z

|∇v|2 dx ≤ 2

Z

Ω

|sym ∇v|2 dx.

(3.8)

Ω

Moreover, the constant 2 above is optimal, for any Ω . Proof. Without loss of generality, we assume that v ∈ C0∞ (Ω , RN ). Recall the identity (3.2), multiply both sides by v and then integrate by parts on Ω . Thus we arrive at the equality below, implying the claimed bound (3.8): Z

|∇v|2 dx = 2

Z

Ω

Ω

|sym∇v|2 dx −

Z

|div v|2 dx ≤ 2

Ω

Z

|sym∇v|2 dx.

Ω

For the optimality of Korn’s constant C = 2, it suffices to take any divergence-free, compactly supported v, for which the above formula yields equality in (3.8). The same arguments as in the proof of Theorem 3.6 work also under tangential boundary conditions when Ω is a polyhedron: Example 3.7. Let Ω be a (bounded) polyhedron in RN . We will show that (3.8) holds for v ∈ H 1 (Ω , RN ) satisfying hv, ni = 0 on ∂ Ω . Indeed, (3.2) results in: Z

|sym∇v|2 dx =

2 Ω

Z

|∇v|2 + |div v|2 dx +

Z

Ω




(∇v)T − (div v)IdN n, v dσ (x),

∂Ω

where we applied integration by parts. The boundary integral equates to: Z ∂Ω

Z

 (∇v)v, n − (div v)hv, ni dσ (x) =

∂v hv, ni − (div v)hv, ni dσ (x) = 0.

∂Ω

The last equality above follows as n is locally constant and v is tangent on ∂ Ω . To go beyond the particular geometry in Example 3.7 and also to deal with mixed boundary conditions, let us denote the following space of affine maps with skewsymmetric gradients on Ω :

3.2 Variants of Korn’s inequality with different boundary conditions

27

 I (Ω ) = Ax + b; A ∈ so(N), b ∈ RN .

(3.9)

In section 3.7 we will extend the notion of I (Ω ) to encompass the space of socalled Killing vector fields I (S), a concept that is central to Korn’s inequality on surfaces S and thin shells having S as their midsurface. We anticipate that Killing fields are precisely the generators of one-parameter paths of isometries, which is why they appear in Korn’s inequality when seen as a linear counterpart of the nonlinear geometric rigidity estimate (4.2) in chapter 4. Theorem 3.8. Let Ω ⊂ RN be an open, bounded, connected, Lipschitz domain. Given two open, disjoint (possibly empty) subsets Γ0 ,Γ1 ⊂ ∂ Ω , denote:  VΓ0 ,Γ1 (Ω ) = v ∈ H 1 (Ω , RN ); v = 0 on Γ0 , hv, ni = 0 on Γ1 , IΓ0 ,Γ1 (Ω ) = I (Ω ) ∩ VΓ0 ,Γ1 (Ω ). Then, for every v ∈ VΓ0 ,Γ1 (Ω ) there exists w ∈ IΓ0 ,Γ1 (Ω ) with: Z

|∇v − ∇w|2 dx ≤ C

Z

Ω

|sym ∇v|2 dx,

(3.10)

Ω

where the constant C above depends only on Ω , Γ0 and Γ1 . Proof. 1. We first claim that for all vector fields v ∈ VΓ0 ,Γ1 (Ω ) which are L2 (Ω )orthogonal to the subspace IΓ0 ,Γ1 (Ω ), we have the Poincar´e inequality: Z

|v|2 dx ≤ C

Z

Ω

|∇v|2 dx,

(3.11)

Ω

with C depending only on Ω , Γ0 , Γ1 , but not on v. For, suppose that we had a sequence {vn ∈ VΓ0 ,Γ1 (Ω )}∞ n=1 satisfying the orthogonality condition and such that: Z

|vn |2 dx = 1,

Ω

Z

|∇vn |2 dx → 0

as n → ∞.

(3.12)

Ω

Passing to a subsequence (that we do not relabel), we obtain that vn → b0 in H 1 (Ω , RN ), where b0 ∈ RN is a constant. It follows that b0 ∈ VΓ0 ,Γ1 (Ω ), and we claim that actually b0 = 0. In the case when Γ0 , 0/ or if the space W  span{n(z); z ∈ Γ1 } equals RN , then this is obvious. In the remaining case of Γ0 = 0/ and W being ⊥ so in particular b ∈ I a proper subspace of RN , we get that b0 ∈ W 0 Γ0 ,Γ1 (Ω ). R R The orthogonality condition implies then 0 = Ω hvn , b0 i dx → Ω |b0 |2 dx, so indeed b0 = 0, contradicting the first identity in (3.12) and thus proving (3.11). 2. For all v as in step 1 there also holds the homogeneous Korn inequality: Z Ω

|∇v|2 dx ≤ C

Z Ω

|sym∇v|2 dx,

(3.13)

28

3 Korn’s inequality

with a universal constant C as in the statement of the result. To show (3.13), we argue by contradiction as in the proof of Theorem 3.1. Assume the existence of a sequence {vn ∈ VΓ0 ,Γ1 (Ω )}∞ n=1 satisfying the orthogonality condition and such that: Z

Z

|∇vn |2 dx = 1,

Ω

|sym∇vn |2 dx → 0

as n → ∞.

(3.14)

Ω

1 N By virtue of (3.11), the sequence {vn }∞ n=1 is bounded in H (Ω , R ), so it converges (up to a subsequence that we do not relabel), weakly to some limit vector field v. Consequently, sym∇v = 0 and so v ∈ I (Ω ) in virtue of Lemma 3.2. Since weak convergence implies strong convergence of traces in RL2 (∂ Ω , RN ), there also holds v ∈ IΓ0 ,Γ1 (Ω ). Finally, passing to the limit yields: Ω hv, wi dx = 0 for all w ∈ IΓ0 ,Γ1 (Ω ), and we thereby conclude that v = 0. Hence:

Z

|vn |2 dx → 0

as n → ∞.

Ω

Together with the second assertion in (3.14), the estimate in Theorem 3.5 implies R |∇v | dx → 0. This contradicts the first assertion in (3.14) and proves (3.13). n Ω 3. Now, given any v ∈ VΓ0 ,Γ1 (Ω ), let w be its orthogonal projection on the finitedimensional subspace IΓ0 ,Γ1 (Ω ) ⊂ L2 (Ω , RN ). Condition (3.13) implies then: Z

|∇v − ∇w|2 dx =

Ω

Z

|∇(v − w)|2 dx ≤ C

Ω

Z

|sym∇v|2 dx.

Ω

The proof is done. Remark 3.9. With the same analysis as in Remark 3.3, one can identify the unique optimal vector field w ∈ IΓ0 ,Γ1 (Ω ) in (3.10). Given v ∈ VΓ0 ,Γ1 (Ω ), let: ? A  P{∇w; w∈IΓ ,Γ (Ω )} ∇v dx, 0 1

Ω

where P denotes the orthogonal projection onto the indicated linear subspace of RN×N . Then there exists b ∈ RN so that w(x) = Ax + b belongs to IΓ0 ,Γ1 (Ω ) and hence is the desired vector field. Theorem 3.8 implies Theorem 3.6 (with some uniform constant C rather than the optimal constant 2) in view of the observation below. In fact, (3.13) holds with any nonempty Γ0 , because: Corollary 3.10. Let Ω ⊂ RN be an open, bounded, connected, Lipschitz domain. In the context of Theorem 3.8 we have IΓ0 ,Γ1 (Ω ) = {0}, in any of the two cases: (i) Γ0 , 0, / (ii) Γ0 = 0, / Γ1 = ∂ Ω and Ω has no rotational symmetry. Then, for every v ∈ VΓ0 ,Γ1 (Ω ) and with C that depends only on Ω ,Γ0 ,Γ1 , there holds:

3.2 Variants of Korn’s inequality with different boundary conditions

Z

|∇v|2 dx ≤ C

Ω

Z

29

|sym∇v|2 dx.

Ω

Proof. 1. To prove (i), let x0 ∈ Γ0 be such that the normal vector n(x0 ) is well defined. Consider the linear subspace: M = span{x − x0 ; x ∈ Γ0 } ⊂ RN . Then, M contains the (well defined) tangent space to ∂ Ω at x0 , because every y ∈ −x0 ∈ M}∞ Tx0 ∂ Ω is generated by some sequence { |xxnn −x n=1 in the limit of Γ0 3 xn → x0 . 0| Let now w = Ax + b ∈ IΓ0 ,Γ1 (Ω ). It is easy to note that M ⊂ ker A, since: A(x − x0 ) = w(x) − w(x0 ) = 0

for all x ∈ Γ0 .

Thus dim ker A ≥ dim M ≥ N − 1. We further claim that A = 0. Indeed, if y ∈ RN was a unit vector in (ker A)⊥ , then not only hAy, yi = 0 by skew-symmetry, but also: hAy, zi = −hy, Azi = 0

for all z ∈ ker A.

(3.15)

We see that Ay is orthogonal to RN and hence y ∈ ker A, which is a contradiction. Since A = 0, it easily follows that b = 0 as well, proving the claim w = 0. 2. To show (ii), we take w = Ax + b ∈ IΓ0 ,Γ1 (Ω ) and prove below that the flow generated by the tangent vector field w|∂ Ω must be a rotation. Then, the lack of nontrivial rotations of Ω will directly yield w = 0. Since A ∈ so(n) it follows that ker A ⊕ im A is an orthogonal decomposition of RN , in virtue of (3.15). Accordingly, we write b = bker + Ab0 with bker ∈ ker A and some b0 ∈ RN . Consider the translated domain Ω0 = Ω + b0 . Since: w(x) = A(x + b0 ) + bker

for all x ∈ Ω ,

it follows that y 7→ Ay + bker is a tangent vector field on ∂ Ω0 . The flow α which this field generates, is given by: α 0 (t) = Aα(t) + bker ,

α(0) ∈ ∂ Ω0 ,

where we may also write α(t) = β (t) + δ (t), with:  0 β (t) = Aβ (t), β (0) ∈ im A δ 0 (t) = bker , δ (0) ∈ ker A, β (0) + δ (0) = α(0). Notice that β (t) remains bounded, because: d |β (t)|2 = 2hβ (t), Aβ (t)i = 0, dt while δ (t) = δ (0) + tbker is unbounded for bker , 0. Since α(t) ∈ ∂ Ω0 for all t ≥ 0, it must be that bker = 0. Hence the flow α(t) = etA β (0) + δ (0) indeed is a rotation generated by A ∈ so(N) on ∂ Ω0 . The proof is done.

30

3 Korn’s inequality

Remark 3.11. (i) From the proof of Corollary 3.10 (ii) it follows that each w ∈ N I0,∂ / Ω (Ω ) has the form w(x) = A(x + b0 ) with A ∈ so(N), b0 ∈ R . We thus obtain 3 the following characterisation when Ω ⊂ R :  if Ω has no rotational symmetry  {0} a 1-parameter family if Ω has one rotational symmetry I0,∂ (Ω ) = / Ω  a 3-parameter family if Ω = Br (x). (ii) Let Ω = B1 (0) ⊂ R3 . Each A ∈ so(3) can be written as Ax = a × x for some 3 a ∈ R3 , and hence we obtain: I0,∂ / Ω (B1 (0)) = {a×x; a ∈ R }. The perpendicularity condition in the proof of Theorem 3.8 then reads: Z

Z ha × x, v(x)i dx = a,

0= B1 (0)

x × v(x) dx



B1 (0)

for all a ∈ R3 .

Consequently, for the class of vector fields v in: Z  1 3 v ∈ H (B1 (0), R ); hv, ni = 0 on ∂ B1 (0) and

x × v(x) dx = 0 ,

B1 (0)

there holds the uniform homogeneous estimate (3.13).

3.3 Proof of Korn’s inequality: preliminary estimates In this and the next sections, we prove the First Korn’s inequality (3.1). We start with two preliminary results: Lemma 3.12. Let Ω be an open, bounded, connected, Lipschitz domain, and let g ∈ L2 (Ω , R) satisfy ∆ g = 0 in Ω . Then: Z

|∇g|2 dist2 (x, ∂ Ω ) dx ≤ 4

Ω

Z

|g|2 dx.

Ω

Proof. For each small ε > 0, consider the open, Lipschitz domain: Ωε = {x ∈ Ω ; dist(x, ∂ Ω ) > ε}. Integrating by parts, it follows that: Z

Z
2 dist(x, ∂ Ω ) − ε |∇g|2 dx =

 
2 div dist(x, ∂ Ω ) − ε g∇g dx

Ωε

Ωε

−2

Z




 dist(x, ∂ Ω ) − ε g ∇x dist(x, ∂ Ω ), ∇g dx

Ωε

= −2

Z Ωε




 dist(x, ∂ Ω ) − ε g ∇x dist(x, ∂ Ω ), ∇g dx.

3.3 Proof of Korn’s inequality: preliminary estimates

31

√ Upon applying −2ab ≤ a2 + b2 with a = 2g(x) and b = √12 dist(x, ∂ Ω ) −


 ε ∇x dist(x, ∂ Ω ), ∇g(x) , at each x ∈ Ωε under the integration sign, we obtain: Z


2 dist(x, ∂ Ω ) − ε |∇g|2 dx

Ωε

≤2

Z

|g|2 dx +

Ωε

1 2

Z

dist(x, ∂ Ω ) − ε


2

2 ∇x dist(x, ∂ Ω ), ∇g dx.

(3.16)

Ωε

Since the function x 7→ dist(x, ∂ Ω ) has Lipschitz constant 1, we further get:

 ∇x dist(x, ∂ Ω ), ∇g ≤ ∇x dist(x, ∂ Ω ) · ∇g(x) ≤ |∇g(x)|. It now follows that the last term in the right hand side of (3.16) can be bounded by half of the exact term in the left hand side, leading to: Z

Z
2 dist(x, ∂ Ω ) − ε |∇g|2 dx ≤ 4

Ωε

|g|2 dx.

Ωε

Passing with ε → 0 while applying Fatou’s lemma to the left hand side and the monotone convergence theorem to the right hand side, yields the result. 1 ((0, T ), R) satisfy lim Lemma 3.13. Let g ∈ Hloc t→0 g(t) = 0. Then there holds:

Z T

|g|2 dt ≤ 4

Z T

|g0 |2 |T − t|2 dt.

0

0

1 ((0, ∞), R), given by: h(t) = g(T −t)1 Proof. Consider the function h ∈ Hloc (0,T ) (t). This function is absolutely continuous on (0, ∞) and identically equal to zero beyond T . For each t ∈ (0, T ) we have:

Z ∞ 2 2 
1 h0 (s)s3/2 s−3/2 ds h0 (s) ds = 4t −1 R ∞ −3/2 ds t t t s Z ∞ Z ∞
1 2 h0 (s)s3/2 s−3/2 ds = 2t −1/2 h0 (s)2 s3/2 ds, ≤ 4t −1 R ∞ −3/2 s ds t t t

h(t)2 =

Z

∞

where the inequality above follows by applying Jensen’s inequality to the convex function x 7→ x2 , with the probability measure obtained as the normalisation of the measure s−3/2 ds on the interval (t, ∞). Integrating on (0, ∞) yields: Z T

h(t)2 dt =

Z ∞

h(t)2 dt ≤ 2

0

0

0

=

Z ∞ Z s 0

Z ∞

0

t −1/2

Z ∞

h0 (s)2 s3/2 ds dt

t

Z  t −1/2 dt h0 (s)2 s3/2 ds = 4 0

∞

h0 (s)2 s2 ds = 4

Z T

|h0 |2 s2 ds,

0

in virtue of Fubini’s theorem and changing the integration order. Applying the reflection of variables t 7→ (T − t) results in the claimed bound for g.

32

3 Korn’s inequality

The main arguments in the proof of Theorem 3.5 will be given in star-shaped domains, which by Lemma 3.15 can be seen as building blocks of Lipschitz domains. Definition 3.14. We say that an open domain Ω ⊂ RN is star-shaped with respect to its interior ball Br (z) b Ω , when: {tx + (1 − t)x; ¯ t ∈ [0, 1]} ⊂ Ω

for all x ∈ Ω , x¯ ∈ Br (z).

(3.17)

Fig. 3.1 A domain that is star-shaped with respect to an interior ball, see Definition 3.14.

Lemma 3.15. Let Ω ⊂ RN be an open, bounded domain. (i) if Ω is star-shaped with respect to an interior ball, then it is Lipschitz, (ii) if Ω is Lipschitz then it can be written as a finite union Ω = {Ωi }ni=1 of open domains Ωi that are each star-shaped with respect to some interior ball. Proof. 1. To prove (i), we may without loss of generality assume that Ω ⊂ BR (0) and that it is star-shaped with respect to Br (0) b Ω . For every z ∈ ∂ Br (0), define:  z f (z) = sup t > 0; (t + r) ∈ Ω . r By (3.17), the graph of f over ∂ Br (0) coincides with ∂ Ω . We will show that the function f is Lipschitz, which by a simple (local) change of variable will imply the Lipschitz condition on Ω . For any z ∈ ∂ Br (0) there holds f (z) ∈ [amin r, amax r] with some uniform constant r amin > 0 and amax = Rr − 1. Further, define the angle γ(z) so that: cos γ(z) = f (z)+r . It is clear that γ(z) ≥ γ0 , where cos γ0 = amin1+1 . Take now any pair z, z¯ ∈ ∂ Br (0) r satisfying: γ  ∠(¯z, z) ≤ 21 γ0 . In virtue of (3.17) we have: f (z) + r > cos(γ(¯ z)−γ) . Consequently, by the mean value theorem, we get:

f (z) − f (¯z) = f (z) + r − f (¯z) + r ≥

r rγ r − ≥− 2 , cos(γ(¯z) − γ) cos γ(¯z) cos θ


for some angle θ ∈ γ(¯z) − γ, γ(¯z) for which: cos2 θ > cos2 γ(¯z) ≥ (a 1+1)2 . In max conclusion and by a symmetric argument, there follows the final bound:

3.3 Proof of Korn’s inequality: preliminary estimates

33

| f (¯z) − f (z)| ≤ r(amax + 1)2 γ, justifying the Lipschitz continuity of f . 2. To show (ii), it suffices to construct the claimed finite decomposition by covS ering only a thin boundary layer z∈∂ Ω Bε (z) ∩ Ω , for some ε  1. The remaining interior set can then be covered by finitely many balls, which are clearly star-shaped with respect to their respective concentric balls with halved radii. Let f : B¯ N−1 3r (0) → R be a strictly positive, Lipschitz function, whose graph coincides with a portion of ∂ Ω and whose subgraph is contained in Ω . We denote some upper bound of the Lipschitz constant of f by L.Without loss of generality we may assume that L ≤ 1 and min f ≥ 2Lr. Observe that for each z0 ∈ BN−1 (0), the “upside down” cone with its tip at the r point (z0, f (z0 )), its height f (z0 ) and the radius of its base 2r, is contained in the domain (z,t) ∈ RN ; z ∈ BN−1 3r (0), 0 < t < f (z) . Consequently, the open set:  (0), 0 < t < f (z) Ωi = Br (0) ∪ (z,t) ∈ RN ; z ∈ BN−1 r is star-shaped with respect to its interior ball Br/2 (0) b Ωi . By construction and compactness of Ω¯ , finitely many of the domains of the type Ωi cover the sufficiently narrow boundary layer of Ω , as claimed. From Lemma 3.13 there follow the key estimates on star-shaped domains: Theorem 3.16. Let Ω ⊂ BR (0) ⊂ RN be an open domain that is star-shaped with respect to Br (0), for some 0 < r < R. Then, for every g ∈ H 1 (Ω , R) there holds: Z Z  Z |g|2 dx + |∇g|2 dist2 (x, ∂ Ω ) dx , (i) |g|2 dx ≤ C Br (0)

Ω

Ω

(ii) there exists a ∈ R such that:

Z

|g − a|2 dx ≤ C

Ω

Z

|∇g|2 dist2 (x, ∂ Ω ) dx,

Ω

with constants C depending only on N and R/r. Proof. 1. By a simple scaling argument, it suffices to assume that R = 1. To prove R (i), we first derive an estimate on Ω \Br (0) |g|2 dx. Choose a smooth cut-off function θ : (0, ∞) → [0, 1] with the following properties: θ|(0,r/2) ≡ 0,

θ|(r,∞) ≡ 1,

4 kθ 0 kL∞ ≤ . r

For every p ∈ ∂ Ω we apply Lemma 3.13 to the function θ g on the segment [0, p]: Z |p|

|θ g|2 dt ≤ 4

Z |p|

(θ g)0 2 |p| − t 2 dt

0

0

8 ≤ 2 r

Z |p| 0



p |θ 0 g|2 + |θ |2 |∇g|2 dist2 t , ∂ Ω dt, |p|

where we used Ω being start-shaped with respect to Br (0) to conclude that:

34

3 Korn’s inequality

|p| − |x| |p| 1 ≤ ≤ dist(x, ∂ Ω ) r r

for all x ∈ [0, p].

Consequently, it follows that: Z |p|

|g|2 dt ≤ C

r

Z

|g|2 dt +

Z |p| r/2

r/2

r

|∇g|2 dist2 t


 p , ∂ Ω dt , |p|

and further, with the constant C that again depends only on r: Z |p|

t N−1 |g|2 dt ≤ C

Z

r

r

t N−1 |g|2 dt +

Z |p|

t N−1 |∇g|2 dist2 t

r/2

r/2


 p , ∂ Ω dt . |p|

Integrating in spherical coordinates, we finally arrive at an inequality that yields (i): Z Z Z
 |g|2 dx ≤ C |g|2 dx + |∇g|2 dist2 x, ∂ Ω dx . Ω \Br (0)

Br (0)

Ω

2. To prove (ii), we invoke the Poincar´e-Wirtinger inequality to get: ? Z Z 2 g − g dx ≤ Cr2 |∇g|2 dx Br/2 (0)

Br/2 (0)

Br/2 (0)

≤C

Z

(3.18)

|∇g|2 dist2 (x, ∂ Ω ) dx,

Br/2 (0)

where C depends only on N. Apply now the> statement in (i) with the interior ball Br/2 (0) replacing Br (0), to the function g − B (0) g on Ω . It follows that: r/2

Z

?

g −

Ω

2 g dx ≤ C

Br/2 (0)

Br/2 (0)

≤C

?

Z Z

g −

Z  2 g dx + |∇g|2 dist2 (x, ∂ Ω ) dx

Br/2 (0)

Ω

|∇g|2 dist2 (x, ∂ Ω ) dx,

Ω

where in the last inequality we used (3.18). Taking a =

> Br/2 (0)

g dx achieves (ii).

Theorem 3.17. Let Ω ⊂ RN be an open, bounded, connected, Lipschitz domain. Then, for every g ∈ H 1 (Ω , R) there holds: Z Z
(i) |g|2 dx ≤ C |g|2 + |∇g|2 dist2 (x, ∂ Ω ) dx, Ω

Ω

(ii) there exists a ∈ R such that:

Z Ω

|g − a|2 dx ≤ C

Z

|∇g|2 dist2 (x, ∂ Ω ) dx,

Ω

with constants C depending only on Ω . Moreover, these constants can be chosen uniformly for a family of domains Ω which are bilipschitz equivalent with controlled Lipschitz constants.

3.3 Proof of Korn’s inequality: preliminary estimates

35

Proof. 1. The uniform estimate in (i) follows directly from Theorem 3.16 in view of the finite decomposition statement in Lemma 3.15 (ii). To show the weighted Poincar´e-type inequality (ii), for each ε > 0 we consider the domain Ωε = {x ∈ Ω ; dist(x, ∂ Ω ) > ε}, > and apply (i) to the function g − Ω g: ε

?

Z

2 g dx

g −

Ω

?

Ωε



≤ C ε2

Z Ω \Ωε

g −

Z 2 g dx +

?

Ωε

Ωε

Z  2 g dx + |∇g|2 dist2 (x, ∂ Ω ) dx .

g −

Ω

Ωε

The constant C above depends only on Ω , so for ε  1 sufficiently small to have Cε 2 ≤ 12 , the first term in the right hand side can be absorbed by the left hand side: ? ? Z Z Z  2 2 g − g dx ≤ 2C g− g dx + |∇g|2 dist2 (x, ∂ Ω ) dx . Ω

Ωε

Ωε

Ω

Ωε

We now apply the Poincar`e-Wirtinger inequality on Ωε to get: ? Z Z Z 2 Cε g − |∇g|2 dist2 (x, ∂ Ω ) dx, g dx ≤ Cε |∇g|2 dx ≤ 2 ε Ω Ωε Ω Ωε ε ε where by writing Cε we indicate the dependence of the constant on ε. Combining the two displayed inequalities, we arrive at (ii). 2. To prove the uniformity of constants, assume that ϕ : Ω → Ω 0 is a bi-Lipschitz map with Lipschitz constants for ϕ and ϕ −1 bounded by some L > 0. Given h ∈ H 1 (Ω 0 , R), define g  h ◦ ϕ ∈ H 1 (Ω , R). Then: Z Ω0

Z

2

h dx ≤ k det ∇ϕkL∞ (Ω )

Z

2

g dx ≤ CL

Ω

N

Z

g2 dx,

Ω


g2 + |∇g|2 dist2 (x, ∂ Ω ) dx

Ω

≤ k det ∇(ϕ −1 )kL∞ (Ω 0 ) ≤ CLN+2 (1 + L2 )

Z Ω0

Z Ω0


h2 + L2 |∇h|2 dist2 (ϕ −1 (x), ϕ −1 (∂ Ω 0 )) dx


h2 + |∇h|2 dist2 (x, ∂ Ω 0 ) dx,

where C above depends only on the dimension N. This implies the estimate in (i) on Ω 0 with the constant that only depends on the constant in (i) on Ω and on L. The argument for (ii) is the same. This ends the proof of the theorem. The above result is an important corollary from Theorem 3.16. We now have all the ingredients towards proving Korn’s inequality (Theorem 3.5) in the next section.

36

3 Korn’s inequality

3.4 Proof of Korn’s inequality In this section, we complete the proof of Theorem 3.5. In the first step, we decompose the given vector field v as the sum of its harmonic part w and the correction u that equals zero on ∂ Ω . A simple integration by part yields a desired bound for u. To deal with w, one applies Lemma 3.12 to g = sym∇w and uses the fact that ∇2 w ' ∇(sym∇w), in combination with Theorem 3.16 (i). The argument is then lifted from star-shaped domains to arbitrary Lipschitz domains, by Lemma 3.15 (ii). Theorem 3.18. Let Ω ⊂ BR (0) ⊂ RN be open and star-shaped with respect to Br (0), for some 0 < r < R. Then, there holds: Z Z Z  |∇v|2 dx ≤ C |∇v|2 dx + |sym ∇v|2 dx , (3.19) Br (0)

Ω

Ω

for every v ∈ H 1 (Ω , RN ), where C depends only on N and R/r. Proof. 1. We decompose v as the sum: v = u + w, where u ∈ H 1 (Ω , RN ) satisfies: ∆ u = ∆ v in Ω ,

u = 0 on ∂ Ω .

Using identity (3.2), it follows that: 
 1 ∆ u = 2 div sym∇v − tr sym∇v IdN . 2 Integration by parts against u then leads to: Z

 1 ∇u : sym∇v − tr(sym∇v)IdN dx 2 Ω ≤ C k∇ukL2 (Ω ) ksym∇vkL2 (Ω ) ,

|∇u|2 dx = 2

Ω

Z

and further, where C as also above depends only on N: Z

|∇u|2 dx ≤ C

Ω

Z

|sym∇v|2 dx.

(3.20)

Ω

2. Consider now the harmonic corrector w = v − u ∈ H 1 (Ω , RN ), which satisfies: ∆ w = 0 in Ω ,

w = v on ∂ Ω .

The application of Lemma 3.12 to each of the N 2 components of the harmonic matrix field sym∇w ∈ L2 (Ω , RN×N ) yields, upon noting Lemma 3.15 (i): Z

Z ∇(sym∇w) 2 dist2 (x, ∂ Ω ) dx ≤ C |sym∇w|2 dx.

Ω

Ω

3.4 Proof of Korn’s inequality

37

We now observe the useful fact that the second partial derivatives are always a linear combination of the first derivatives of the components of the symmetric gradient: [∇2 wi ] jk = ∂ j [sym∇w]ik + ∂k [sym∇w]i j − ∂i [sym∇w] jk for all i, j, k = 1 . . . N.

(3.21)

The above identity can be checked by a direct calculation. Consequently, the previous bound becomes: Z Z ∇2 w 2 dist2 (x, ∂ Ω ) dx ≤ C |sym∇w|2 dx, Ω

(3.22)

Ω

again with a constant C that depends only on N. 3. Finally, we apply Theorem 3.16 (i) to each component g = ∂i w j of ∇w to get: Z Z Z
 (3.23) |∇w|2 dx ≤ C |∇w|2 dx + |∇2 w|2 dist2 x, ∂ Ω dx , Br (0)

Ω

Ω

where C depends on N and R/r. Combining (3.20), (3.22) and (3.23) results in: Z Z Z  |∇v|2 dx ≤ C |∇u|2 dx + |∇w|2 dx Ω Ω Ω Z Z Z
 2 ≤C |sym∇v| dx + |∇w|2 dx + |∇2 w|2 dist2 x, ∂ Ω dx Br (0)

Ω

≤C

Z

|sym∇v|2 dx +

Z

≤C

|∇w|2 dx +

Br (0)

Ω

Z

Ω

2

|sym∇v| dx +

Z

Z

|sym∇w|2 dx



Ω 2



|∇v| dx .

Br (0)

Ω

The proof is done. We remark that the same splitting technique used above will also be present in the proof of Friesecke-James-M¨uller’s inequality (4.2) in the next chapter, as well as in the dimension reduction analysis in the presence of prestress, in Part III of this monograph. Two corollaries are now in order. In the first one, we already may derive the estimate in Theorem 3.5 on star-shaped domains: Corollary 3.19. [First Korn’s inequality on star-shaped domains] In the setting of Theorem 3.18, for every v ∈ H 1 (Ω , RN ) there holds: Z Z Z  |∇v|2 dx ≤ C |sym∇v|2 dx + |v|2 dx , Ω

Ω

Ω

where the constant C depends only on N, R/r and dist(Br (0), ∂ Ω ).

(3.24)

38

3 Korn’s inequality

Proof. Let φ ∈ C0∞ (Ω , [0, 1]) be some smooth test function satisfying φ|Br (0) ≡ 1. By Theorem 3.6 applied to φ v ∈ H01 (Ω , RN ), we obtain: Z

2

|∇v| dx ≤

Br (0)

Z

2

|∇(φ v)| dx ≤ 2

Ω

Z

|sym∇(φ v)|2 dx

Ω

Z Z  ≤4 |sym∇v|2 dx + |∇φ |2 |v|2 dx . Ω

Ω

Since φ can be taken radially symmetric with k∇φ kL∞ depending only on the quantity dist(Br (0), ∂ Ω ), the estimate (3.19) yields the result. Proof of Theorem 3.5 The same estimate (3.24) is evidently valid on any Ω that is a finite union of domains satisfying assumptions of Theorem 3.18. Hence, the proof of Theorem 3.5 is achieved in virtue of Lemma 3.15 (ii). Recall that the argument by contradiction that appears in the proof of Theorem 3.1, also yields Korn’s inequality in the form of a rigidity estimate on those domains for which First Korn’s inequality has been established. Below we observe a direct proof of the same statement, which on star-shaped domains gives information on the dependence of Korn’s constant on Ω : Corollary 3.20. [Korn’s inequality on star-shaped domains] In the setting of Theorem 3.18, for every v ∈ H 1 (Ω , RN ) there holds: ? Z Z |∇v − skew ∇v|2 dx ≤ C |sym∇v|2 dx Ω

(3.25)

Ω

Ω

where C depends only on N and R/r. Proof. Let v = u + w be the decomposition as in the proof of Theorem 3.18. We apply Theorem 3.16 (ii) to the components g = ∂i w j of the harmonic matrix field ∇w, to get B ∈ RN×N so that, recalling (3.22): Z

|∇w − B|2 dx ≤ C

Ω

Z

|∇2 w|2 dist2 (x, ∂ Ω ) dx ≤ C

Ω

Z

|sym∇w|2 dx.

(3.26)

Ω

Since for every x ∈ Ω there holds: |sym B| ≤ |sym(B − ∇w(x))| + |sym ∇w(x)| ≤ |B − ∇w(x)| + |sym∇w(x)|, it follows that Ω |B − skew B|2 ≤ C be improved to: R

Z Ω

R Ω

|sym∇w|2 dx, and consequently (3.26) may

|∇w − skew B|2 dx ≤ C

Z Ω

|sym∇w|2 dx.

(3.27)

3.5 Korn’s constant under tangential boundary conditions

39

In conclusion, by (3.20) and (3.27) we get: k∇v − skew BkL2 (Ω ) ≤ k∇ukL2 (Ω ) + k∇w − BkL2 (Ω ) ≤ Cksym∇vkL2 (Ω ) .

(3.28)

Invoking Remark 3.3 completes the proof.

3.5 Korn’s constant under tangential boundary conditions We have seen in Theorem 3.6 that under Dirichlet boundary conditions, Korn’s constant is universal and equals 2 for all domains Ω . In this section, we deduce that under tangential boundary conditions Korn’s constant is always at least 2. As shown in the example below there is however no upper bound on it. Example 3.21. Consider any smooth, closed, nonintersecting curve γ in R2 , which is not rotationally symmetric. Denote by γ 3 z 7→ n(z), τ(z) the smooth unit normal and unit tangent vector fields on γ, and recall that ∂τ n = κτ and ∂τ τ = −κn, where κ is the scalar curvature field on γ. For each z ∈ γ and a small |t|  1, let: v(z + tn(z)) = (1 + tκ(z))τ(z). Then v|Ω¯ h ∈ C ∞ (Ω¯ h , R2 ) where for each small h > 0 we define a thin strip Ω h ⊂ R2 around γ, by setting:  h h
Ω h = x = z + tn(z); z ∈ γ, t ∈ − , . 2 2 It is clear that hv, ni = 0 on ∂ Ω h , and also we calculate directly: ∂n v(z + tn) = κτ(z),  1  t∂τ κ (1 + tκ)∂τ τ(z) + t∂τ κ(z)τ(z) = −κn(z) + τ(z). 1 + tκ 1 + tκ







 t∂τ κ Consequently: ∂n v, n = 0, ∂τ v, n + ∂n v, τ = 0 and ∂τ v, τ = 1+tκ , yielding: ∂τ v(z + tn) =

Z Ωh

|sym∇v|2 dx ≤ Ch3 ,

Z Ωh

|∇v|2 dx ≥ ch

as h → 0.

(3.29)

By Corollary 3.10, the uniform bound (3.13) holds for all v ∈ H 1 (Ω h , R2 ) tangential on the boundary, with some constant C = Ch that depends only on Ω h . However, (3.29) implies that Ch ≥ hc2 as h → 0. We point out that Example 3.21 can be carried out in higher dimensions N > 2 as well, upon replacing the tangent vector field τ along a curve γ, by a Killing vector field on a surface S. We refer to Example 3.34 for the related construction.

40

3 Korn’s inequality

Fig. 3.2 Thin two-dimensional domains in Example 3.21.

The following is the main result of this section: Theorem 3.22. Let Ω ⊂ RN be an open, bounded, connected, Lipschitz domain. In the context of Theorem 3.8, let Γ0 = 0/ and Γ1 = ∂ Ω . Then, there holds C ≥ 2 for the constant C in (3.10). Equivalently: Z n 2 ≤ sup min |∇v − ∇w|2 dx; v ∈ H 1 (Ω , RN ), hv, ni = 0 on ∂ Ω , w∈I0,∂ / Ω (Ω ) Ω

Z

and

o |sym∇v|2 dx = 1

Ω

Proof. 1. Without loss of generality, we may assume that 0 ∈ Ω . Fix some vector R field v¯ ∈ H 1 (RN , RN ) satisfying Ω |sym∇v| ¯ 2 dx = 1 and define the sequence {v¯n ∈ 1 N N ∞ N/2−1 H (R , R }n=1 by v¯n (x) = n v(nx). ¯ Then, there holds: Z RN

|∇v¯n |2 dx =

Z RN

|∇v| ¯ 2 dx,

Z RN

|sym∇v¯n |2 dx =

Z RN

|sym∇v| ¯ 2 dx = 1.

Let φ ∈ Cc∞ (Ω , [0, 1]) be a test function, equal identically to 1 in a neighborhood of 0 in Ω . For the modified sequence {vn  φ v¯n ∈ H01 (Ω , RN )}∞ n=1 , we have: ∇vn = φ ∇v¯n + v¯n ⊗ ∇φ , where the second term satisfies: lim kv¯n ⊗ ∇φ kL2 (Ω ) ≤ lim k∇φ kL∞ (Ω ) n−1 kvk ¯ L2 (RN ) = 0.

n→∞

n→∞

We now make the following claims: lim k∇vn kL1 (Ω ) = 0,

(3.30)

n→∞

lim k∇vn kL2 (Ω ) = k∇vk ¯ L2 (RN ) ,

n→∞

lim ksym∇vn kL2 (Ω ) = 1.

n→∞

The first convergence in (3.31) follows by noting that:

(3.31)

3.5 Korn’s constant under tangential boundary conditions

2 lim φ ∇v¯n L2 (Ω ) = lim

n→∞

n→∞

41

2 x
φ ∇v(x) ¯ dx = k∇vk ¯ 2L2 (Ω ) . n RN

Z

The second convergence follows similarly. For the remaining assertion (3.30), we use the result (3.32) in step 2 below to conclude the last equality in: lim kφ ∇v¯n kL1 (Ω ) ≤ lim kφ kL∞ n−N/2 k∇vk ¯ L1 (nΩ ) = 0.

n→∞

n→∞

2. We now prove the following statement, valid for any g ∈ L2 (RN , R): lim R−N/2 kgkL1 (BR (0)) = 0.

(3.32)

R→∞

Fix ε > 0 and let R > m be two constants, sufficiently large to ensure that: kgkL2 (RN \Bm (0)) < ε

 m N/2

and

R

kgkL2 (RN ) ≤ ε.

Then we have, as claimed: Z

R−N/2 kgkL1 (BR ) = R−N/2 −N/2

≤R

|g| dx +

Z

BR (0)\Bm (0)

|BR (0)|

1/2

≤ |B1 (0)|1/2 ε +




Bm (0) −N/2

ε +R  m N/2 R

|g| dx

|Bm (0)|1/2 kgkL2 (RN )

|B1 (0)|1/2 kgkL2 (RN ) ≤ 2|B1 (0)|1/2 ε.

3. Note that, in virtue of Remark 3.9 we obtain: ?

min k∇vn − ∇wkL2 (Ω ) = ∇vn − P ∇vn dx L2 (Ω )

w∈IΓ0 ,Γ1 (Ω )

?

≥ k∇vn kL2 (Ω ) − P Ω

Ω

∇vn dx L2 (Ω ) ≥ k∇vn kL2 (Ω ) − |Ω |−1/2 k∇vn kL1 (Ω ) .

By (3.30) and (3.31), the right hand side above converges to k∇vk ¯ L2 (RN ) as n → ∞, while the left hand side is bounded by C1/2 ksym∇vn kL2 (Ω ) , with C being the Korn constant to be estimated. Passing to the limit and using (3.31) again, we conclude: k∇vk ¯ L2 (RN ) ≤ C1/2 ksym∇vk ¯ L2 (RN ) = C1/2 , which yields: C ≥ sup

nZ RN

|∇v| ¯ 2 dx; v¯ ∈ H 1 (RN , RN ),

Z RN

o |sym∇v| ¯ 2 dx = 1 .

The proof is done by recalling homogeneous Korn’s inequality in Theorem 3.6. The bound in Theorem 3.22 is actually achieved, as shown in Example 3.7. The final observation of this section is that, in general, Korn’s constant under tangen-

42

3 Korn’s inequality

tial boundary condition blows up (at the rate h−2 ) when the thickness h of a 2dimensional domain goes to 0. Thin domains are of primary importance in this monograph and will be studied in the context of elasticity, in Parts II and III.

3.6 Approximation theorem and Korn’s constant in thin shells In this section we show how to approximate a displacement gradient ∇v on a thin shell by a field of skew-symmetric matrices rather than by a constant matrix. Now Korn’s constant in the latter approximation generally blows up like h12 along the vanishing shell’s thickness h (as we have already seen in Example 3.21), while in the former approximation the corresponding constants are independent of h → 0. Let S be a smooth, closed hypersurface (i.e. a compact boundaryless manifold of co-dimension 1) in RN . We will use the following notation: n for the outward unit normal to S (seen as the boundary of some bounded domain in RN ), Tz S for the tangent space to S at z ∈ S, and π for the projection onto S along n: π(z + tn(z)) = z

for all z ∈ S, |t|  1.

For two families of positive, C 1 functions {gh1 , gh2 : S → R}h>0 , we will consider a family {Sh }h>0 of thin shells around S, itself viewed as their midsurfaces: Sh = {x = z + tn(z); z ∈ S, − gh1 (z) < t < gh2 (z)}.

(3.33)

We have the following uniform estimates: Theorem 3.23. Let S ⊂ RN be a smooth, closed hypersurface. Assume that {Sh }h>0 is given by (3.33) where {gh1 , gh2 ∈ C 1 (S, R)}h>0 satisfy: C1 h ≤ ghi (z) ≤ C2 h,

|∇ghi (z)| ≤ C3 h

for all z ∈ S, h  1,

(3.34)

with some positive constants C1 ,C2 ,C3 independent of h. Then, for every vector field v ∈ H 1 (Sh , RN ) there exists a map A ∈ H 1 (S, so(N)) such that: Z

(i)

ZSh

|∇v − A ◦ π|2 dx ≤ C

|∇A|2 dσ (z) ≤

(ii) S

C h3

Z Sh

Z Sh

|sym∇v|2 dx,

|sym∇v|2 dx.

The constants C above depend on S and {Ci }3i=1 , but not on v or h  1. Proof. 1. For each z ∈ S, define the following sets: Dz,h = Bh (z) ∩ S,

Bz,h = π −1 (Dz,h ) ∩ Sh ,

3.6 Approximation theorem and Korn’s constant in thin shells

43

where Bh (z) denotes the ball in RN . We observe that Bz,h is contained in a ball of radius (C2 +1)h and it is star-shaped with respect to a ball of radius r(C1 ,C2 ,C3 , S)h, for h sufficiently small. Hence, an application of Corollary 3.20 on Bz,h yields a skew-symmetric matrix Az,h ∈ so(N) such that: Z Bz,h

|∇v(x) − Az,h |2 dx ≤ C

Z

|sym∇v|2 dx,

(3.35)

Bz,h

where C depends only on the quantities indicated in the statement of the result. The goal now is to replace Az,h by A(z) which depends smoothly on z, and thus ultimately replace Az,h by A(πz) in (3.35). The desired estimate on Sh will then follow by summing over a finite family of cylinders Dz,h . To this end, let θ : [0, 1) → [0, 2] be a smooth cut-off function that is compactly R supported, constant in a neighbourhood of 0, and satisfies 01 θ = 1. Define: ηz (x) = R

Sh

θ (|πx − z|/h) θ (|πx − z|/h) dx

for all z ∈ S, x ∈ Sh .

Then ηz is supported on Bz,h and, moreover, we have: Z Sh

|ηz | ≤

ηz (x) dx = 1,

C , hN

|∇z ηz | ≤

C . hN+1

(3.36)

Finally, we define the skew-symmetric matrix field A as the average: Z

A(z) =

Sh

ηz (x) skew∇v(x) dx.

R

2. Since A(z) − Az,h = Sh ηz (x) skew(∇v(x) − Az,h ) dx, the Cauchy-Schwarz inequality together with (3.35) and (3.36) yield: |A(z) − Az,h |2 ≤

Z Sh

ηz (x)|∇v(x) − Az,h | dx


2

R

≤

C hN

Z

|sym∇v|2 dx.

(3.37)

Bz,h

R

To estimate ∇A we use that: Sh ∇z ηz (x) dx = ∇z ( Sh ηz (x) dx) = 0, to get: ∇A(z) = R R (∇ η ) skew∇v dx = (∇ z z z ηz ) skew(∇v − Az,h ) dx. Consequently: Sh Sh |∇A(z)|2 ≤

Z Bz,h

|∇z ηz |2 dx ·

Z ∇v − Az,h 2 dx ≤ C |sym∇v|2 dx, (3.38) hN+2 Bz,h Bz,h

Z

in virtue of (3.35) and (3.36). Similarly, for all z0 ∈ Dz,h there holds: |∇A(z0 )|2 ≤

C hN+2

Z Bz0 ,h

|sym∇v|2 dx ≤

C hN+2

Z

|sym∇v|2 dx,

(3.39)

2Bz,h

where 2Bz,h = π −1 (Dz,2h ) ∩ Sh . From this, by the fundamental theorem of calculus:

44

3 Korn’s inequality

|A(z00 ) − A(z)|2 ≤

C hN

Z

for all z00 ∈ Dz,h .

|sym∇v|2 dx

2Bz,h

In combination with (3.35) and (3.37) the above yields: Z

|∇v(x) − A(πx)|2 dx ≤ C

Z

|sym∇v|2 dx.

(3.40)

2Bz,h

Bz,h

n(h)

3. We now cover Sh with a family of sets {Bzi ,h }i=1 with the property that the n(h)

covering number of the derived family {2Bzi ,h }i=1 is independent of h. An argument for the existence of such a covering goes as follows. The surface S is contained Sn(h) in the finite union of balls i=1 Bh/2 (ki ) where ki ∈ ( 2h Z)N . Fix a one-to-one map S ki 7→ zi ∈ S ∩ Bh/2 (ki ), so that Sh = i Bzi ,h . Then, if x ∈ 2Bzi ,h then it must be that π(x) ∈ B2h (zi ), so that |ki − π(x)| ≤ |ki − xi | + |π(x) − zi | ≤ 5h/2. Therefore ki ∈ B5h/2 (x) ∩ ( h2 Z)n . The cardinality of this last set is bounded by 10N , which serves as n(h)

an upper bound on the covering number for the family {2Bzi ,h }i=1 . Summing (3.40) over i = 1 . . . n proves (i). Integrating (3.39) on Dz,h we get: Z

|∇A(z0 )|2 dσ (z0 ) ≤

Dz,h

C h3

Z

|sym∇v|2 dx,

2Bz,h

which in turn yields (ii) by using the same covering argument above. A similar construction as above, combining a mollification argument with Korn’s inequality (3.1) used locally, will be carried out in section 4.5 to obtain the SO(N)valued approximations of a deformation gradient on thin shells. This approximation, based on the Friesecke-James-M¨uller nonlinear version of (3.1), will be of key importance in the dimension reduction analysis in Parts II and III of this monograph. As a corollary to Theorem 3.23, we readily deduce: Theorem 3.24. Let S ⊂ RN be a smooth, closed hypersurface and assume (3.34). Then, for every vector field v ∈ H 1 (Sh , RN ) defined on Sh in (3.33) with h  1, there exists A0 ∈ so(N) such that: Z Sh

|∇v − A0 |2 dx ≤

C h2

Z Sh

|sym∇v|2 dx.

The constant C above depends only on S and constants {Ci }3i=1 in (3.34). Proof. Let A : S → so(N) be as in Theorem 3.23 and define: ? A0  A(z) dσ (z) ∈ so(N). S

Applying additionally the Poincar´e inequality on S, we obtain:

3.6 Approximation theorem and Korn’s constant in thin shells

Z Sh

|∇v − A0 |2 dx ≤ C ≤C

Z Sh

Z Sh

|∇v − Aπ|2 dx + h

Z S

|sym∇v|2 dx + h

45


|A(z) − A0 |2 dσ (z)


CZ |∇A|2 dσ (z) ≤ 2 |sym∇v|2 dx. h Sh S

Z

This ends the proof. The decomposition and mollification argument as in Theorem 3.23 can be also applied towards a useful uniform Poincar´e-Wirtinger type inequality in thin shells: Theorem 3.25. Let S ⊂ RN be a smooth, closed hypersurface and assume (3.34). Then, for every g ∈ H 1 (Sh , R), there exists a ∈ R so that: Z Sh

|g − a|2 dx ≤ C

Z Sh

|∇g|2 dx.

The constant C above depends only on S and constants {Ci }3i=1 in (3.34). Proof. Let Dz,h , Bz,h ,> ηx be as in the proof of Theorem 3.23. We will show the result for the constant a = S a(z) ˜ dσ (z), where we define: Z

a(z) ˜ =

Sh

for all z ∈ S.

ηz (x)g(x) dx

First, by Lemma 3.16 (ii), the local estimate (3.35) can be replaced by: Z Bz,h

|g − az,h |2 dx ≤ Ch2

Z

|∇u|2 dx.

Bz,h

Repeating the calculations leading to (3.37) and (3.38), it follows that: |a(z) ˜ − az,h |2 ≤ Ch2−N |∇a(z ˜ 0 )|2 ≤ Ch−N

Z

Z

|∇g|2 dx,

Bz,h

|∇g|2 dx

2Bz,h

for all z0 ∈ Dz,h ,

which imply, exactly as in (3.40): Z Sh

|g − aπ| ˜ 2 dx ≤ Ch2

Z Sh

|∇g|2 dx,

Z

|∇a| ˜ 2 dσ (z) ≤ Ch−1

Z

S

Sh

|∇g|2 dx.

Using the standard Poincar´e inequality on surfaces, we thus get: Z Z Z  |g − a|2 dx ≤ C |g − aπ| ˜ 2 dx + h |a(z) ˜ − a|2 dσ (z) Sh Sh Z S Z  Z  2 2 ≤C h ˜ 2 dσ (z) ≤ C |∇g|2 dx. |∇g| dx + h |∇a| Sh

S

Sh

46

3 Korn’s inequality

The proof is done. Theorems 3.24 and 3.25 directly imply the following Korn-Poincar´e inequality: Corollary 3.26. Let S ⊂ RN be a smooth, closed hypersurface and assume (3.34). Then, for every v ∈ H 1 (Sh , RN ), defined on Sh in (3.33), there exists an affine map with skew-symmetric gradient w ∈ I (Sh ), satisfying: kv − wkH 1 (Sh ) ≤

C ksym∇vkL2 (Sh ) . h

The constant C above depends only on S and {Ci }3i=1 in (3.34). We close this section by a trace theorem, resulting by applying the scaled versions of the usual trace theorem to each neighbourhood Bz,h in the proof of Theorem 3.23: Lemma 3.27. Let S ⊂ RN be a smooth, closed hypersurface and assume (3.34). Then, for every g ∈ H 1 (Sh , R) there holds: kgkL2 (S) + kgkL2 (∂ Sh ) ≤ C

 1  kgkL2 (Sh ) + h1/2 k∇gkL2 (Sh ) , 1/2 h

where in the left hand side we have norms of traces of g on S and ∂ Sh . The constant C is independent of g or h  1.

3.7 Killing vector fields and Korn’s inequality on surfaces The purpose of this section is to prove the counterparts of Korn’s inequalities in Theorems 3.5 and 3.1, where the open domain Ω ⊂ RN is replaced by a N − 1 dimensional surface S ⊂ RN . Rather than carrying out proofs (for now) in the more general context of Riemannian geometry, we will apply the previous results on thin shells around S and recover Korn’s inequality on S in the vanishing limit of the shell’s thickness. In addition to the independent interest of these results, the analysis below will be essential for continuing the discussion in section 3.9. Let S be a smooth, closed hypersurface in RN . We use the notation n, Tz S and π as in section 3.6. For a vector field v ∈ H 1 (S, RN ), we denote by sym∇v the symmetric part of the (covariant) gradient of (the tangent component of) v, in: hsym∇v(z)η, τi =


1 h∂η v(z), τi + h∂τ v(z), ηi 2

for all z ∈ S, τ, η ∈ Tz S.

By ∂τ v(z) we denote the derivative of v in the tangent direction τ, i.e. if γ : (−ε, ε) → S is a C 1 curve with γ(0) = z and γ 0 (0) = τ, then ∂τ v(z) = (v ◦ γ)0 (0). By Π (z) = ∇n(z) : Tz S → Tz S we denote the shape operator (which is the negative second fundamental form) on S.

3.7 Killing vector fields and Korn’s inequality on surfaces

47

We have the following counterpart to Theorem 3.5: Theorem 3.28. [First Korn’s inequality on surfaces] Let S be a smooth, closed hypersurface in RN . There holds: Z

|∇v|2 dσ (z) ≤ C

S

Z

|v|2 + |sym∇v|2 dσ (z),

(3.41)

S

for all vector fields v ∈ H 1 (S, RN ) tangent to S, i.e. satisfying hv(z), n(z)i = 0 for a.e. z ∈ S. The constant C above depends only on S but not on v. Proof. 1. Consider the extension of v on the thin neighbourhood of Sh0 of S: v(z ˜ + tn(z)) = (Id + tΠ (z))−1 v(z) for all x ∈ Sh0 ,  h0 where Sh0 = x = z + tn(z); z ∈ S, |t| < , h0  1. 2

(3.42)

We have v˜ ∈ H 1 (Sh0 , RN ) and for every x = z + tn(z) ∈ Sh0 and τ ∈ Tz S there holds: n  o  ∂τ v(z) ˜ = ∇ (Id + tΠ (z))−1 (Id + tΠ (z))−1 τ v(z) (3.43) + (Id + tΠ (z))−1 ∇v(z)(Id + tΠ (z))−1 τ. The first component above is bounded by C|tu(x)|. Taking the scalar product of the second term with any η ∈ Tx S gives: h(Id + tΠ (z))−1 η, ∇v(z)(Id + tΠ (z))−1 τi. Consequently, we obtain:

 hsym∇v(x)τ, ˜ ηi = (Id + tΠ (z))−1 η, sym∇v(z)(Id + tΠ (z))−1 τ (3.44) + hZ(t, z), v(z)i |Z(t, z)| ≤ C. On the other hand, hn(z), v(x)i ˜ = 0, so for any τ ∈ Tz S:

 h∂τ v(x), ˜ n(z)i = − Π (z)(Id + tΠ (z))−1 τ, v(x ˜ )

 = − (Id + tΠ (z))−1 Π (z)(Id + tΠ (z))−1 v(z), τ = hτ, ∂n v(x)i. ˜ Hence it follows that:

hsym∇v(x)τ, ˜ ni = − (Id + tΠ (z))−1 Π (z)(Id + tΠ (z))−1 v(z), τi, hsym∇v(x)n, ˜ ni = 0.

(3.45)

2. We now invoke first Korn’s inequality on the open bounded smooth Sh0 , to get: Z S h0

|∇v| ˜ 2 dx ≤ C

Z Sh0

|v| ˜ 2 + |sym∇v| ˜ 2 dx,

48

3 Korn’s inequality

where C depends only on S and the chosen parameter h0 . By (3.43) and noting that: h∂τ v, ni = −hΠ τ, vi, we further obtain: Z S h0

|∇v| ˜ 2 dx ≥ c1

Z S

|∇v|2 dσ (z) − c2

Z

|v|2 dσ (z),

S

again with some uniform positive constants c1 , c2 . Further, by (3.44) and (3.45): Z Sh0

|sym∇v| ˜ 2 dx ≤ C

Z

|v|2 + |sym∇v|2 dσ (z).

S

The three above displayed inequalities imply (3.41), in view of the elementary R R ˜ 2 dx ≤ C S |v|2 dσ (z). bound Sh0 |v| In order to formulate the second Korn’s inequality on S, we necessitate the displacements whose gradients replace the constant matrices A ∈ so(N) in (3.1). Definition 3.29. We say that a (smooth) vector field w : S → RN is a Killing vector field on S, provided that: w(z) ∈ Tz S and sym∇w = 0 for all z ∈ S. The linear space (and the Lie algebra) of all Killing fields on S will be denoted by I (S). The Killing fields are infinitesimal generators of isometries on S, in the sense that for every fixed t the map S 3 z 7→ Φ(t, z) ∈ S is an isometry, where Φ is the flow of: d Φ(s, z) = w(Φ(t, z)), dt

Φ(0, z) = z.

The smoothness assumption in Definition 3.29 is not a restriction, because of the following counterpart of Lemma 3.2: Lemma 3.30. (i) Let w ∈ H 1 (S, RN ) be a tangent field satisfying sym∇w = 0 almost everywhere on S. Then w ∈ I (S). (ii) The space I (S) has finite dimension. Proof. To show smoothness of w in (i), recall the extension w˜ ∈ H 1 (Sh0 , RN ) given by the formula in (3.42). By (3.44), (3.45) we see that sym∇w˜ has the improved regularity H 1 and hence, in virtue of (3.2) we get: ∆ w˜ ∈ L2 (Sh0 , R). The result is then a consequence of the elliptic regularity and a bootstrap argument. The finite dimensionality assertion in (ii) follows from the equivalence of the L2 and the H 1 norms on I (S), which in view of (3.41) then yields: k∇wkL2 (S) ≤ CkwkL2 (S)

for all w ∈ I (S).

(3.46)

For otherwise the space (I (S), k · kH 1 ) would have a countable Hilbert (orthonormal) basis {en }∞ n=1 . The sequence {en } must then converge to 0 as n → ∞, weakly

3.7 Killing vector fields and Korn’s inequality on surfaces

49

in H 1 (S, RN ). But this implies that limn→∞ ken kL2 (S,RN ) = 0, which by the norms equivalence gives the same convergence in H 1 (S, R3 ), and a contradiction. As in the proof of Theorem 3.1, an argument by contradiction now yields: Theorem 3.31. [Korn-Poincar´e’s inequality on surfaces] Let S be a smooth, closed hypersurface in RN . For every tangent vector field v ∈ H 1 (S, RN ) there exists a Killing field w ∈ I (S) such that: kv − wkH 1 (S) ≤ Cksym∇vkL2 (S) and the constant C depends only on S. Proof. Consider the orthogonal complement I (S)⊥ of I (S) in the Hilbert space of H 1 (S, RN ) regular tangent vector fields on S. Both spaces are closed (with respect to both weak and strong convergences in H 1 (S, RN )). We will show that: kvkH 1 (S) ≤ Cksym∇vkL2 (S)

for all v ∈ I (S)⊥ ,

which clearly implies the result in the Theorem. We argue by contradiction. If the above was not true, there would exist a sequence {vn ∈ I (S)⊥ }∞ n=1 such that: kvn kH 1 (S) = 1,

ksym∇vn kL2 (S) → 0

as n → ∞.

Without loss of generality, or passing to a subsequence if necessary, {vn }n→∞ converges weakly to some v ∈ I (S)⊥ . Moreover, sym∇v = 0 by the second condition above, so Lemma 3.30 implies that v ∈ I (S). Since the spaces I (S) and I (S)⊥ are orthogonal, there must be v = 0, and hence {vn }n→∞ converges to 0 (strongly) in L2 (S, RN ). This contradicts the normalisation kvn kH 1 (S) = 1, in virtue of (3.41). A few remarks on the linear space I (S) are of interest: Remark 3.32. (i) The bound (3.46) together with an estimate of the uniform constant C (that depends on S), can be recovered directly using the following identity, valid for all Killing vector fields w ∈ I (S): ∆S

1 2
e 2 |w| = ∇w − Ric(w, w). 2

(3.47)

e = (∇w)tan is the covariant derivaHere ∆S is the Laplace-Beltrami operator on S, ∇w tive of w on S, and Ric stands for the Ricci curvature form on S. To calculate Ric(w, w) in our particular setting, note that by Gauss’ Theorema Egregium, the Riemann curvature 4-tensor on S satisfies: hR(τ, η)ξ , ϑ i = hΠ (z)τ, ϑ ihΠ (z)η, ξ i − hΠ (z)τ, ξ ihΠ (z)η, ϑ i,

50

3 Korn’s inequality

for all z ∈ S and τ, η, ξ , ϑ ∈ Tz S. The Ricci curvature 2-tensor being the appropriate trace of R, we thus obtain for all z ∈ S and η, ξ ∈ Tz S: Ric(η, ξ ) = tr (τ 7→ R(τ, η)ξ ) = tr Π (z)hΠ (z)η, ξ i − hΠ (z)ξ , Π (z)ηi


 = (tr Π (z))Π (z) − Π (z)2 η, ξ .

(3.48)

Integrating (3.47) on S and using (3.48) we arrive at: Z

e 22 = k∇wk L (S)


 (tr Π (z))Π (z) − Π (z)2 w(z), w(z) dσ (z).

(3.49)

S

To calculate the L2 norm of the full gradient ∇u on S, we use: Z n−1

e 22 = k∇wk2L2 (S) − k∇wk L (S)

∑ h∂τi w, ni2 dσ (z)

S i=1 Z n−1

∑ hw(z), Π (z)τi i2 dσ (z) =

=

S i=1

Z

|Π (z)w|2 dσ (z).

S

Hence we arrive at: Z

|∇w|2 dσ (z) =

S

Z

for all w ∈ I (S), (3.50)

tr Π (z) Π (z)w(z), w(z)i dσ (z)

S

which clearly implies (3.46). (ii) Notice that in the special case of a 2 × 2 matrix Π , when N = 3 and S is a 2d surface in R3 , the Cayley-Hamilton theorem implies: (tr Π )Π − Π 2 = (det Π )Id2 , and so (3.49) becomes: e 22 = k∇wk L (S)

Z

det Π (z)|w|2 dσ (z)

S

In this case det Π (z) coincides with the Gaussian curvature of S at z. (iii) An equivalent way of obtaining the formula (3.50), but without using the language of Riemannian geometry, is to look at the extension of w: w(z ˜ + tn(z)) = w(z)

for all x = z + tn(z) ∈ Sh0 .

Since ∂n w˜ = 0 and hw, ˜ ni = 0 on the boundary of Sh0 , by (3.2) one has: Z Sh0

|∇w| ˜ 2 dx = −2

Z Sh0

hdiv sym∇w, ˜ wi ˜ dx −

Z S h0

|div w| ˜ 2 dx.

(3.51)

Calculating hsym∇w, ˜ wi ˜ in terms of Π (z), dividing both sides of (3.51) by 2h0 and passing to the limit with h0 → 0, one recovers (3.50) directly.

3.8 Blowup of Korn’s constant in thin shells

51

We conclude this section by the following Korn-type inequality on 2d surfaces: Lemma 3.33. Let S ⊂ R3 be a smooth, closed hypersurface in R3 . Assume that A ∈ C 0,1 (S, R2×2 ) satisfies det A , 0 on S. Then there holds, for every tangent vector field v ∈ H 1 (S, R3 ), with C that depends only on S and A, but not on v:
k∇vkL2 (S) ≤ C kvkL2 (S) + k(∇v)tan − h(∇v)tan : AiAkL2 (S) . Proof. It suffices to prove the claimed bound locally, and hence below we replace S ¯ so(2)) by a single patch, with a boundary that is a Lipschitz curve. Take J ∈ C 0,1 (S, to be any skew-symmetric matrix field with nonvanishing determinant. Define v˜ = JA−1 v and note the decomposition: ∇v = AJ −1 (∇v) ˜ tan + ∇(AJ −1 )JA−1 v.

(3.52)

To estimate ∇v, ˜ we use (the local version of) Theorem 3.28: k∇vk ˜ L2 (S) ≤ C(kvkL2 (S) + ksym∇vk ˜ L2 (S) ).

(3.53)


Further, sym∇v˜ = sym (∇v) ˜ tan − h(∇v)tan : AiJ as J is skew-symmetric. Thus: ksym∇vk ˜ L2 (S) ≤ k(∇v) ˜ tan − h(∇v)tan : AiJkL2 (S) ≤ CkAJ −1 (∇v) ˜ tan − h(∇v)tan : AiAkL2 (S) ≤ Ck(∇v)tan − h(∇v)tan : AiAkL2 (S) +CkvkL2 (S) , in view of (3.52). Combining (3.52), (3.53) and the above completes the proof.

3.8 Blowup of Korn’s constant in thin shells In this section, we present an extension of the argument in Example 3.21 showing that in general, the uniform constant C = Ch in (3.1) posed on the thin film Ω = Sh with the mid-surface S, blows up quadratically: Ch ≥ hc2 as h → 0. Example 3.34. Given two smooth positive functions g1 , g2 : S → R, we now consider the family {Sh }h>0 of thin shells around S: Sh = {x = z + tn(z); z ∈ S, − hg1 (z) < t < hg2 (z)}. By nh we denote for the outward unit normal to ∂ Sh . Define the subspace of I (S): Ig1 ,g2 (S) = {w ∈ I (S); hw(z), ∇(g1 + g2 )(z)i = 0 for all z ∈ S} , consisting of those Killing fields w which satisfy: limh→0 h1 hw(z), (nh+ + nh− )i = 0, where nh+ and nh− denote, respectively, the outward unit normals to Sh at its boundary points z + hg2 (z) and z − hg1 (z).

52

3 Korn’s inequality

For w ∈ Ig1 ,g2 (S) \ {0} we now construct a family {vh ∈ H 1 (Sh , RN )}h→0 with: hvh , nh i = 0

on ∂ Sh ,

(3.54)

h for which C = Ch ≥ hc2 as h → 0 in (3.1), even though I0,∂ / Sh (S ) = {0} for all h  1. This is the case, for example, when S has no rotational symmetry, see Remark 3.11 and Theorem 3.8.

1. Namely, for all z ∈ S and all t ∈ (−hg1 (z), hg2 (z)), we set:   vh (z + tn(z)) = Id + tΠ (z) + hn(z) ⊗ ∇g2 (z) w(z)

(3.55)

and calculate directly: vh (z + tn(z)) =

 hg1 (z) + t  Id + hg2 (z)Π (z) + hn(z) ⊗ ∇g2 (z) w(z) h(g1 (z) + g2 (z))  hg2 (z) − t  + Id − hg1 (z)Π (z) − hn(z) ⊗ ∇g1 (z) w(z). h(g1 (z) + g2 (z))

The above means that each vh is a linear interpolation between the push-forward of the vector field w from S onto the external boundary ∂ + Sh and the push-forward onto the internal boundary ∂ − Sh of ∂ Sh . Indeed, derivatives of the maps in: S 3 z 7→ z ± hgi (z)n(z) ∈ ∂ ± Sh are given through: Id ± hgi (z)Π (z) ± hn(z) ⊗ ∇gi (z). In particular, we see that (3.54) holds. 2. Write now vh = w˜ + (vh − w), ˜ with:
w(x) ˜ = Id + tΠ (z) w(z), and estimate components of ∇w˜ and sym∇w. ˜ For all z ∈ S and τ ∈ Tz S, there holds: ∂n w(x) ˜ = Π (z)w(z), ∂τ w(x) ˜ =

(3.56) t∂ Π (z) w(z) + (Id + tΠ (z))∇w(z)(Id + tΠ (z))−1 τ. ∂ ((Id + tΠ (z))−1 τ)

Since hn, wi ˜ = 0 and since Π (x) commutes with (Id + tΠ (x))−1 , we get: h∂τ w, ˜ ni + h∂n w, ˜ τi = −h∂τ n, wi ˜ + h∂n w, ˜ τi

 = − Π (z)(Id + tΠ (z))−1 τ, (Id + tΠ (z))w(z) + hΠ (z)w(z), τi = 0, h∂n w, ˜ ni = 0. To estimate hsym∇w(x)τ, ˜ ηi for τ, η ∈ Tz S, note that:

(3.57)

3.9 Uniformity of Korn’s constant under tangential boundary conditions in thin shells

53

h(Id + tΠ (z))∇w(z)(Id + tΠ (z))−1 τ, ηi − h(Id + tΠ (z))−1 ∇w(z)(Id + tΠ (z))−1 τ, ηi ≤ Ct|∇w(z)|, because |(Id + tΠ (z)) − (Id + tΠ (z))−1 | ≤ Ct with C that, as usual, denotes any positive constant independent of h. Since (Id + tΠ (z))−1 τ ∈ Tz S, we obtain: |hsym∇w(x)τ, ˜ ηi| ≤ Ct(|w(z)| + |∇w(z)|).

(3.58)

Further: |∇(vh − w)(x)| ˜ ≤ Ch, and by (3.57) and (3.58): |sym∇w| ˜ ≤ Ch on Sh . 3. Consequently, it follows that: Z Sh

|sym∇vh |2 dx ≤ Ch3 .

On the other hand, inspecting the terms in ∇vh and recalling that w , 0 so that ∇w , 0 as well, we see that: Z Sh

|∇vh |2 dx ≥

1 2

Z Sh

|∇w|2 dx − c2 h3 ≥ c1 h.

The two last inequalities yield the claim.

3.9 Uniformity of Korn’s constant under tangential boundary conditions in thin shells We have shown in Example 3.34 that Korn’s constants C in (3.1) may converge to infinity as the thickness h of a thin shell Sh converges to 0. In this section, we prove that this blow-up, under tangential boundary condition is only due to the presence of Killing vector fields. In particular, if the Killing fields are treated as the kernel of the rigidity estimate, then the corresponding constants Ch on Sh are uniform in h. As in section 3.6, we consider a family {Sh }h>0 of thin shells around a smooth, closed hypersurface S ⊂ RN , given by: Sh = {x = z + tn(z); z ∈ S, − gh1 (z) < t < gh2 (z)},

(3.59)

whose boundary is determined by some positive functions {gh1 , gh2 ∈ C 1 (S, R)}h>0 . By nh we denote the outward unit normal to ∂ Sh , while n is the unit normal to S. Recall the following spaces of Killing fields on S:  I (S) = w ∈ H 1 (S, RN ); w(z) ∈ Tz S and sym∇w(z) = 0 for all z ∈ S , (3.60)  Ig1 ,g2 (S) = w ∈ I (S); hw(z), ∇(g1 + g2 )(z)i = 0 for all z ∈ S . We have the following first main result, which can be seen as the homogeneous and thickness-independent version of the Korn-Poincar´e inequality in Corollary 3.26:

54

3 Korn’s inequality

Theorem 3.35. Let S ⊂ RN be a smooth, closed hypersurface and let the boundary functions of {Sh }h>0 in (3.59) satisfy, with constants C1 ,C2 ,C3 > 0: C1 h ≤ ghi (z) ≤ C2 h,

|∇ghi (z)| ≤ C3 h

for all z ∈ S,

h  1.

(3.61)

Let α ∈ [0, 1). Then, for all v ∈ H 1 (Sh , RN ) satisfying one of the conditions: hv, nh i = 0

on ∂ + Sh = {z + gh2 (z)n(z); z ∈ S},

hv, nh i = 0

or

on ∂ − Sh = {z − gh1 (z)n(z); z ∈ S},

(3.62)

and also satisfying: Z hv(x), w(π(x))i dx ≤ αkvk 2 h kwπk 2 h L (S ) L (S ) Sh

for all w ∈ I (S), (3.63)

there holds, with C independent of v and h  1: k∇vkH 1 (S) ≤ Cksym∇vkL2 (Sh ) . As shown in the second result, replacing condition (3.61) by a more restrictive requirement (3.64) below, one can prove uniform Korn’s inequality for a larger class of vector fields, namely those forming a cone and satisfying the angle condition (3.62) with the subspace Ig1 ,g2 (S) rather than the whole I (S). We note that (3.64) implies (3.61) upon taking C1  21 min{gi (z); z ∈ S, i = 1, 2}, C2  2 maxi kgi kL∞ (S) and C3  maxi k∇gi kL∞ (S) + 1. Our second main result is: Theorem 3.36. Let S ⊂ RN be a smooth, closed hypersurface and let the boundary functions of {Sh }h>0 in (3.59) satisfy, with constants C1 ,C2 ,C3 > 0: 1 h g → gi h i

in C 1 (S, R)

as h → 0,

for i = 1, 2.

(3.64)

Let α ∈ [0, 1). Then, for all v ∈ H 1 (Sh , RN ) satisfying hv, nh i = 0 on ∂ Sh and: Z hv(x), w(π(x))i dx ≤ αkvk 2 h kwπk 2 h L (S ) L (S ) Sh

(3.65)

for all v ∈ Ig1 ,g2 (S), there holds, with C independent of v and h  1: k∇vkH 1 (S) ≤ Cksym∇vkL2 (Sh ) .

Proofs of Theorems 3.35 and 3.36 will be given in (3.10), while in this section we derive the key lemmas regarding the various components and their derivatives,

3.9 Uniformity of Korn’s constant under tangential boundary conditions in thin shells

55

of the average of a given v ∈ H 1 (Sh , RN ) in the direction of thickness of Sh : ? v(z) ¯ 

gh2 (z)

for all z ∈ S.

v(z + tn(z)) dt

(3.66)

−gh1 (z)

We will work in the context of Theorem 3.23, which under assumption (3.61) constructed an approximation A ∈ H 1 (S, so(N)) with the following properties: Z Sh

Z

Z

|∇v − Aπ|2 dx ≤ C

|∇A|2 dσ (z) ≤

S

C h3

Sh

Z Sh

|sym∇v|2 dx, (3.67)

|sym∇v|2 dx.

Fig. 3.3 The midsurface S and the lower and upper boundaries in Theorem 3.35.

Denoting by Atan (z) the restriction of A(z) to Tz (S), we have: Lemma 3.37. Let S ⊂ RN be a smooth, closed hypersurface and let the boundary functions of {Sh }h>0 in (3.59) satisfy (3.61). Given v ∈ H 1 (Sh , Rn ), let the vector field v¯ and the so(N)-valued matrix field A on S be as in (3.66), (3.67). Then: k∇v¯ − Atan kL2 (S) ≤ C h1/2 kvkH 1 (Sh ) +

1 h1/2


ksym∇vkL2 (Sh ) ,

with C that is independent of v and h  1. Proof. For every z ∈ S and τ ∈ Tz S, there holds: ? v(z) ¯ − ∂τ ≤

gh2 (z) −gh1 (z)

 ∇v(z + tn(z)), τ + t∂τ n(z) dt


C |∂τ gh1 (z)| + |∂τ gh2 (z)| h

Consequently, it follows that:

Z gh (z) 2 −gh1 (z)

|∂n v(z + tn(z))| dt ≤ C

Z gh (z) 2 −gh1 (z)

|∇u| dt.

56

3 Korn’s inequality

?

gh2 (z)

−gh1 (z)



 ∇v(z + tn(z)), τ + t∂τ n(z) − A(z)τ dt ≤C

Z gh (z) 2 −gh1 (z)

? |∇v| dt +

gh2 (z)

|∇v(z + tn(z)) − A(z)| dt,

−gh1 (z)

which by (3.67) leads to: k∇v¯ − Atan k2L2 (S) ≤ C

Z  Z gh (z) 2

h

S

−gh1 (z)

|∇v|2 dt +

1 h

?

gh2 (z)

 |∇v − Aπ|2 dt dσ (z)

−gh1 (z)


1 ≤ C hk∇vk2L2 (Sh ) + ksym∇vk2L2 (Sh ) . h The proof is done. In order to estimate the normal component of v, ¯ we will need the geometric bounds: Lemma 3.38. Let S ⊂ RN be a smooth, closed hypersurface and let the boundary functions of {Sh }h>0 in (3.59) satisfy (3.61). Recall that ∂ Sh = ∂ − Sh ∪ ∂ + Sh , with ∂ − Sh , ∂ + Sh defined in (3.62). Then: (i) |nh − nπ| ≤ Ch on ∂ + Sh and |nh + nπ| ≤ Ch on ∂ − Sh , (ii) under a stronger condition (3.64), we have: |nh +(−n+∇gh2 )π| ≤ Ch2 on ∂ + Sh and |nh + (n + ∇gh1 )π| ≤ Ch2 on ∂ − Sh . Let now v ∈ H 1 (Sh , RN ). Then, there holds: |∂n hv, nπi| ≤ |sym∇v| on Sh , and: (iii) if hv, nh i = 0 on ∂ ± Sh , then: khv, nπikL2 (∂ ± Sh ) ≤ Ch1/2 kvkW 1,2 (Sh ) , (iv) under the stronger condition (3.64) and when hv, nh i = 0 on ∂ Sh , we have: Z S

hv(z − gh (z)n(z)), ∇gh (z)i + hv(z + gh (z)n(z)), ∇gh (z)i 2 dσ (z) 1 1 2 2 ≤C h

Z Sh


|sym∇v|2 + h3 kuk2H 1 (Sh ) .

All constants C above are independent of v and h  1. Proof. The assertion (i) is obvious. To prove (ii) on ∂ + Sh , observe that the normal vector nh is parallel to nπ − ∇gh2 π + w, where |w| ≤ C|(gh2 ∇gh2 )π| ≤ Ch2 . Normalising this vector, we conclude claimed inequality. To show (iii) note first that ∂n hv, nπi = h(sym∇v)n, ni. In view of (i) and the trace estimate in Lemma 3.27, we get: khv, nπikL2 (∂ + Sh ) = khv, nπ − nh ikL2 (∂ + Sh ) ≤ Ch1/2 kvkH 1 (Sh ) . For (iv), we proceed similarly but using (ii) rather than (i) in:

3.9 Uniformity of Korn’s constant under tangential boundary conditions in thin shells

Z S

57

hv(z+gh (z)n(z)), ∇gh (z)i + hv(z − gh (z)n(z)), ∇gh (z)i 2 dσ (z) 2

≤

2

Z S

2

Z ∂ Sh

Z Z g2 (z) S

1

hv(z + gh (z)n(z)), n(z)i − hv(z − gh (z)n(z)), n(z)i 2 dσ (z)

+Ch4 =

1



−g1 (z)

1

|v|2 dσ

Z 2 |∂n hv, nπ)i(z + tn(z))|2 dt dσ (z) +Ch4

∂ Sh

Since the right hand side is bounded by Ch proof is complete.

2 Sh |sym∇v|

R

|v|2 dσ .

dx + Ch3 kuk2H 1 (Sh ) , the

The remaining key estimates towards the proof of Theorems 3.35 and 3.36 are: Lemma 3.39. In the context of Lemma 3.37 and in particular under condition (3.61) on the boundary of thin shells {Sh }h>0 , let v ∈ H 1 (Sh , R3 ). Then: (i) if hv, nh i = 0 on ∂ + Sh , then: khv, ¯ nikL2 (S) ≤ Ch1/2 kvkH 1 (Sh ) , k∇z hv, ¯ nikL2 (S) + kAnkL2 (S) ≤ C kvk ¯ L2 (S) + kvkH 1 (Sh ) + +C

1 h1/2

ksym∇vkL2 (Sh )





1/2 1 kvkH 1 (Sh ) ksym∇vkL2 (Sh ) , h

(ii) under the stronger condition (3.64), having hv, nh i = 0 on ∂ Sh implies: 1 h

Z S

|hv, ¯ ∇(gh1 + gh2 )i| dσ (z) ≤ C h1/2 kvkH 1 (Sh ) +

1 h1/2


ksym∇vkL2 (Sh ) .

Proof. 1. To prove (i), we use Lemma 3.38 (iii) and (i), so that: 2

hv(x), n(z)i ≤



|hv(z + gh2 (z)n(z)), n(z)i| +

Z gh (z) 2 −gh1 (z)

|sym∇v| dt

Z

 2 ≤ C v(z + gh2 (z)n(z)), n(z) − nh (z + gh2 (z)n(z)) +Ch

2

gh2 (z)

−gh1 (z)

Z  ≤ C h2 |v(z + gh2 (z)n(z))|2 + h

gh2 (z)

−gh1 (z)

|sym∇v|2 dt

 |sym∇v|2 dt ,

for every x = z + tn(z) ∈ Sh . Hence, by Lemma 3.27, it follows that: h

g2 (z) C |hv(z + tn(z)), n(z)i|2 dt dσ (z) h S −gh1 (z)
C 3 2 ≤ h kvkL2 (∂ Sh ) + h2 ksym∇vk2L2 (Sh ) ≤ Chk∇vk2L2 (Sh ) . h

khv, ¯ nik2L2 (S) ≤

Z Z

(3.68)

58

3 Korn’s inequality

This implies the first estimate in (i). 2. To show the second estimate in (i), note first that kAnkL2 (S) = knAtan kL2 (S) , since A ∈ so(n). We now use the Hilbert space identity: kak2 + kbk2 = ka − bk2 + 2ha, bi with a = ∇hv, ¯ ni and b = nAtan . Integration by parts yields: ha, bi

L2 (S)

Z

 = nAtan , ∇z hv, ¯ ni dσ (z)

S
¯ nikL2 (S) kAkH 1 (S) . ≤ Ckhv, ¯ nikL2 (S) kAkL2 (S) + k∇(nAtan )kL2 (S) ≤ Ckhv,

Therefore, applying the divergence theorem, (3.67) and Lemma 3.39:
1 1 ksym∇vkL2 (Sh ) + 1/2 k∇vkL2 (Sh ) h3/2 h
1 ≤ C kvkH 1 (Sh ) ksym∇vkL2 (Sh ) + kvk2H 1 (Sh ) . h

ha, bi

L2 (S)

≤ Ckhv, ¯ nik

L2 (S)

On the other hand, noting that a = n∇v¯ + hv, ¯ ni, Lemma 3.37 yields:
ka − bkL2 (S) ≤ C kvk ¯ L2 (S) + k∇v¯ − Atan kL2 (S) ≤ C kvk ¯ L2 (S) + h1/2 kvkH 1 (Sh ) +

1 h1/2


ksym∇vkL2 (Sh ) .

Combining the last two displayed inequalities concludes the proof of (i). 3. Finally, from Lemma 3.38 (iv) and from an easy bound: kuv − vπk ¯ L1 (∂ Sh ) ≤ Ch1/2 k∇vkL2 (Sh ) , we get the estimate in (ii), because: 1 h

Z S

|hv, ¯ ∇(gh1 + gh2 )i| dσ (z) 1 hu(z − gh1 (z)n(z)), ∇gh1 (z)i + hv(z + gh2 (z)n(z)), ∇gh2 (z)i dσ (z) h S +Ckv − vπk ¯ L1 (∂ Sh )
1 ≤ C 1/2 ksym∇vkL2 (S) + h3/2 kvkH 1 (Sh ) + h1/2 k∇vkL2 (Sh ) , . h

≤

Z

The proof is done.

3.10 Proofs of uniformity of Korn’s constant under tangential boundary conditions in thin...

59

3.10 Proofs of uniformity of Korn’s constant under tangential boundary conditions in thin shells Equipped with the estimate in Lemmas 3.37 and (3.39) we are now ready to prove the uniform Korn-Poincar´e inequality under the angle constraints (3.63) or (3.65). These conditions may be understood in the following way: the cosine of the angle in L2 between a given vector field v and its projection onto the linear space: W h ⊂ L2 (Sh , RN )

(3.69)

which consists of trivial extensions w ◦ π of certain Killing fields: w ∈ I (S) or w ∈ Ig1 ,g2 (S), should be smaller than α. Equivalently, one considers these vector fields v ∈ H 1 (Sh , RN ) for which with a given constant β = (1−α12 )1/2 ≥ 1 there holds: kvkL2 (Sh ) ≤ β kv − wπkL2 (Sh )

for all w ∈ I (S) (or: for all w ∈ Ig1 ,g2 (S)),

(3.70)

namely, the distance of v from the space W h controls (uniformly) the full norm kvkL2 (Sh ) . Theorems 3.35 and 3.36 then state that inside each closed cone around (W h )⊥ and of fixed angle θ < π/2 in L2 (Sh , RN ), the bound: kvkH 1 (Sh ) ≤ Cksym∇vkL2 (S) holds, with a constant C that is uniform in v and h.

Fig. 3.4 The cone of vector fields v for which uniform Korn’s inequality in Theorem 3.35 holds.

Proof of Theorems 3.35 and 3.36 1. We argue by contradiction and consider a sequence {vh ∈ H 1 (Sh , RN )}h→0 such that the assumptions of Theorem 3.35 or 3.36 hold, but: h−1/2 kvh kH 1 (Sh ) = 1

and

h−1/2 ksym∇vh kL2 (Sh ) → 0

as h → 0.

(3.71)

60

3 Korn’s inequality

For the proof of Theorem 3.35 we will assume that hvh , nh i = 0 on ∂ + Sh . The case of the tangency condition on ∂ − Sh is treated exactly the same. Notice that (3.71) immediately yields through Lemmas 3.37 and 3.39, that:   h lim khv¯h , nikL2 (S) + k∇v¯h − Atan kL2 (S) = 0, h→0   (3.72) lim k∇hv¯h , nikL2 (S) + kAh nkL2 (S) ≤ C lim kv¯h kL2 (S) , h→0

h→0

where v¯h : S → RN and Ah : S → so(N) are determined by each vh according to (3.66) and (3.67), and where the constants C above depend only on S and the families {gh1 , gh2 }h>0 in the limit of h → 1. Also, under the assumption (3.64) we get: lim

Z hv¯h , ∇(g1 + g2 )i dσ (z) = 0,

h→0 S

(3.73)

where we used the fact that {v¯h }h→0 is bounded in L1 (S), again in view of (3.71). 2. The Korn-Poincar´e inequality on hypersurfaces in Theorem 3.31, applied to the tangent components of the averaged vector fields v¯h : h v¯tan = v¯h (z) − hv¯h , nin,

yields a sequence of Killing fields wh ∈ I (S) such that: h h )kL2 (S) . − wh kH 1 (S) ≤ Cksym∇(v¯tan kv¯tan

Since Ah is so(n)-valued, for every z ∈ S and τ ∈ Tz S there holds: h h∂τ (v¯tan )(z), τi = h∂τ (v¯h (x)), τi − hv¯h , ni(z)∂τ n(z) ≤ |∂τ v¯h (z) − Ah (z)τ| +C|hv¯h , ni(z)|. Consequently, (3.72) implies that::
h h kL2 (S) + khv¯h , nikL2 (S) → 0 )kL2 (S) ≤ C k∇v¯h − Atan ksym∇(v¯tan

as h → 0,

and further: h − wh kH 1 (S) = 0. lim kv¯tan

h→0

(3.74)

Denote by P be the orthogonal projection with respect to the L2 (S) norm, of the space I (S) onto its subspace V which we take to be the whole I (S) in case of Theorem 3.35 and Ig1 ,g2 (S) in case of Theorem 3.36. Call: wh1 = Pwh ∈ V,

wh2 = wh − wh1 ∈ V ⊥ .

In both cases, condition (3.70) implies: kvh kL2 (Sh ) ≤ Ckvh − wh1 πkL2 (S) .

(3.75)

3.10 Proofs of uniformity of Korn’s constant under tangential boundary conditions in thin...

61

3. We now claim that: lim kwh2 kL2 (S) = 0.

(3.76)

h→0

In the case of Theorem 3.35, when V ⊥ = {0}, convergence in (3.76) is trivial, so we concentrate on the case of Theorem 3.36. Notice that then, (3.73) and (3.74) yield: Z S

|hwh2 , ∇(g1 + g2 )i| dσ (z) = h ≤ Ckv¯tan − wh kL1 (S) +C

Z

|hwh , ∇(g1 + g2 )i| dσ (z)

S

Z S

|hv¯h , ∇(g1 + g2 )i| dσ (z) → 0

as h → 0.

Since all norms in the finitely dimensional space V ⊥ are equivalent, we also obtain: kwh2 kL2 (S) ≤ C

Z S

|hwh2 , ∇(g1 + g2 )i| dσ (z).

Indeed, the right hand side above provides a norm on the space in question. Now, the last two displayed formulas clearly imply (3.76), as claimed. 4. By Poincar´e’s inequality on segments (−gh1 (z), gh2 (z)) and by (3.71), we get: h−1/2 kv¯h π − vh kL2 (Sh ) ≤ Ch1/2 k∇vh kL2 (Sh ) → 0

as h → 0.

(3.77)

We now note the resulting convergence of the various quantities below: h h−1/2 kv¯tan π − vh kL2 (Sh ) ≤ h−1/2 kv¯h π − vh kL2 (Sh ) +Ckhv¯h , nikL2 (S) → 0 by (3.77) and (3.72),

h−1/2 kwh π − wh1 πkL2 (Sh ) → 0

by (3.76),

h−1/2 kvh π − wh1 πkL2 (Sh ) → 0

by (3.74) and convergences above.

Consequently, it follows by (3.75): lim h−1/2 kvh kL2 (Sh ) = 0.

h→0

(3.78)

Hence we get another set of convergences: kv¯h kL2 (S) → 0, k∇hv¯h , nikL2 (S) + kAh nkL2 (S) → 0 h

kw kL2 (S) → 0

by (3.72) and the above,

(3.79)

by (3.74).

Because of the equivalence of norms on the finitely dimensional space I (S), the last formula above implies: lim h−1/2 kwh kH 1 (S) = 0.

h→0

(3.80)

62

3 Korn’s inequality

Now, we may estimate the quantity h−1 k∇vh kL2 (Sh ) by the following norms: h−1/2 k∇vh − Ah πkL2 (Sh ) , k∇hv¯h , nikL2 (S) ,

kAh nkL2 (S) ,

h − ∇v¯h kL2 (S) , kAtan

h ) − ∇wh kL2 (S) , k∇(v¯tan

k∇wh kL2 (S) ,

and use the bounds in (3.67), with (3.72), (3.79), (3.74) and (3.80) to conclude: lim h−1/2 k∇vh kL2 (Sh ) = 0.

h→0

Together with (3.78) this contradicts (3.71). Remark 3.40. One could naively expect that the space W h of trivial extensions wπ of the appropriate Killing vector fields, as discussed in (3.69), is the kernel for the uniform Korn-Poincar´e inequality in the same manner as the linear maps x 7→ Ax + b with skew gradients A ∈ so(n) constitute the kernel for the standard Korn-Poincar´e inequality, implied by Theorem 3.1. This is not exactly the case, as the uniform Korn inequality is actually true for the extensions w ◦ π. The role of the aforementioned kernel is played by the space e h of other, “smart” extensions wh given by the formula (3.55) in section 3.8. Still, W with w ◦ π replaced by wh in (3.63) or (3.65), Theorems 3.35 and 3.36 remain true. e h are asymptotically tangent as h → 0: Note that the spaces W h and W kw ◦ π − wh kL2 (Sh ) ≤ Chkw ◦ πkL2 (Sh )

for all w ∈ I (S).

Hence, if |hv, wh iL2 | ≤ αkvkL2 kwh kL2 for some α < 1, then the angle conditions: |hv, w ◦ πiL2 (Sh ) | ≤ (α +Ch)kvkL2 (Sh ) kw ◦ πkL2 (Sh ) hold with another α < 1 when h  1. Thus, the fact that we chose to work with “trivial” extensions space W h (giving a simpler condition) instead of the actual kere h , is not restrictive. nel W

3.11 Bibliographical notes Korn’s inequality was discovered in the context of the boundary value problem of linear elastostatics, see Korn [1908, 1909]. The proofs in sections 3.3 and 3.4 mostly follow Kondratiev and Oleinik [1989]. Lemma 3.13 is a particular case of Hardy’s inequality in Hardy et al. [1988], while Lemma 3.12 is taken directly from Kondratiev and Oleinik [1989]. Theorem 3.22 appeared in Lewicka and M¨uller [2016]. The proof of Theorem 3.23 consisting of the mollification of local approximations follows the construction in the nonlinear case, described in Friesecke et al. [2002]. The same approximation result has been obtained in Griso [2008] in the context of the unfolding method in linearised elasticity.

3.11 Bibliographical notes

63

Theorem 3.28 remains true also in the more general framework of Riemannian manifolds, see Chen and Jost [2002]. The finite dimensionality of the space of Killing fields is a classical result, see Kobayashi and Nomizu [1963]. The identity (3.47) can be found in Petersen [2006]. For the statement and proof of Gauss’ Theorema Egregium we refer to [Spivak, 1999, vol 3]. Results in sections 3.9 and 3.10 appeared in Lewicka and M¨uller [2011]. Korn’s inequalities remain true when the L2 (Ω , RN ) norm is replaced by any L p norm, p ∈ (1, ∞). The proof, in the spirit of the arguments presented in this chapter, is available in Reshetnyak [1994] (Theorem 3.2., chapter 3). Another proof, via constructing suitable extension operators and applying the homogeneous Korn inequality in the full space RN can be found in Nitsche [1981], while for the functionalanalytic proof via the Lions lemma and the Neˇcas estimate, we refer to Duvaut and Lions [1972]. Another classical approach is due to Payne and Weinberger [1961], 56 where Korn’s constant on a 3d ball is calculated to be 13 . For other Korn constants 3 on simply-connected bounded hypersurfaces of R we refer to Knops [2023]. For the limiting cases p = 1 and p = ∞, Korn-type inequalities fail due to counterexamples in Ornstein [1962], see also a parallel discussion in Conti et al. [2005]. Korn’s inequalities generalize to different settings, in particular: the case of nonconstant coefficients in Neff [2022], Pompe [2003], the mixed growth conditions in Conti et al. [2014], Orlicz spaces in Fuchs [2010], Breit and Diening [2012], Cianchi [2014], or SBD functions in Chambolle et al. [2016] and Friedrich [2018]. In a recent paper by Spector and Spector [2021], the authors establish a Korn inequality involving the BMO-seminorms on bounded domains and with a constant depending only on the dimension. A fractional analogue of First Korn’s inequality on bounded C 1 domains has been obtained (by means of the extension technique motivated by Nitsche [1981]) in Mengesha and Scott [2019] and also in Harutyunyan and Mikayelyan [2021]. Korn-type inequalities were proved to be valid on H¨older and John’s domains in Diening et al. [2010], Jiang and Kauranen [2017] and L´opezGarc´ıa [2018]. Piecewise Korn-type inequalities subordinate to a FEM-mesh and involving jumps across element boundaries have been investigated in Brenner [2004]. We also point out that a complete characterization of the possible linear part maps A : RN×N → RN×N to have k∇ukL p ≤ CkA [∇u]kL p follows from the foundational work of Calder´on and Zygmund [1956]. There exist several results regarding Korn-like inequality for incompatible tensor fields, see Lewintan and Neff [2021], Gmeineder and Spector [2021]. In particular, Lewintan et al. [2021] introduce the following improvement: 1 kPkL p ≤ C(ksymPkL p + ksym curl P − (trace curl P)Id3 kLr , 3 of the generalized L p -Korn inequality, valid for all P ∈ W01,p (Ω , R3×3 ). When P = ∇v, there follows First Korn’s inequality. For the complete classification of the interplay between the spatial dimension N, the integrability exponents p, r > 1, the part map A and the constant coefficient, homogeneous, linear differential operator

64

3 Korn’s inequality

of arbitrary order B appearing in Korn-Maxwell-Sobolev inequalities: kPkL p ≤ c(kA [P]kL p + kBPkLr ), we refer to Gmeineder et al. [2022a]. The characterization of the part maps A in the particular choice B = curl in the borderline case r = 1 was given in Gmeineder et al. [2022b]. We remark that the validity of Korn-type inequalities is related to the validity of other inequalities such as Babuˇska-Aziz, Friedrichs or Horgan-Payne, see Acosta and Dur´an [2017], Costabel and Dauge [2015] For older results extending Korn’s inequality, see a review by Horgan [1995] and the references therein. The relations of Korn’s constant with the measure of axisymmetry of the domain have been discussed in Desvillettes and Villani [2002]. In the context of thin shells, a new inequality, that interpolates between Korn’s first and second inequalities was introduced in Grabovsky and Harutyunyan [2014], Harutyunyan [2018]. It has subsequently been shown that Korn’s constant scales like h3/2 for cylindrical shells Ω h with midsurface having 0 Gauss curvature, see Harutyunyan [2017a], Grabovsky and Harutyunyan [2018]. If the Gaussian curvature is positive, then the optimal constant scales like h, and if the Gaussian curvature is negative, then the Korn constant scales like h4/3 , see Harutyunyan [2017b], Yao [2020]. Earlier, Korn’s inequalities in thin plates have been discussed in a series of papers by Kohn and Vogelius [1985]. Finally, we point out that thin domains are encountered in many problems in solid or fluid mechanics, such as: ocean dynamics, lubrication, meteorology, or blood circulation. These are a part of a broader study of the behaviour of various PDEs on thin N-dimensional domains. The study of the global existence and asymptotic properties of solutions to the Navier-Stokes equations in thin 3d domains began in Raugel and Sell [1993]. They proved global existence of strong solutions for large initial data and in presence of large forcing, under the boundary conditions either purely periodical or combined periodic-Dirichlet. Further generalisations to other boundary conditions followed (see the references in Iftimie et al. [2007]). In this context, Korn’s inequality arises naturally under the Navier boundary condition: (sym∇v)nh ||nh

and

hv, nh i = 0

on ∂ Sh ,

for the following reason. In order to define the corresponding Stokes operator one R R uses the symmetric bilinear form B(v, v) ¯ = hsym∇v : sym∇vi ¯ rather than h∇v : ∇vi. ¯ The energy methods give then suitable bounds for the norm ksym∇vh kL2 (Sh ) of a solution flow vh in Sh . In order to establish compactness in the limit problem as h → 0, one needs bounds for the full H 1 norm of vh . These are provided by the uniform Korn’s inequality (with constants independent of h).

Chapter 4

¨ Friesecke-James-Muller’s inequality

We now turn our attention to the nonlinear counterpart of Korn’s inequality, studied in chapter 3. Discovered in the beginning of the XXIst century by G. Friesecke, R. James and S. M¨uller, this inequality plays the pivotal role in solid mechanics, nonlinear elasticity and differential geometry, and it is based on the intuition of quantifying the well-established fact that a deformation that is locally a rigid motion, must be a global rigid motion on each connected component of its domain. The outline of this chapter is as follows. In section 4.1, we state Friesecke-JamesM¨uller’s inequality and relate it to the aforementioned Liouville’s rigidity statement. Towards its proof in section 4.4, we first derive a truncation theorem in section 4.2, and then apply it in section 4.3 to show a localized version (on balls) of the main inequality. In section 4.5 we deduce an approximation result, in which a deformation gradient on a thin shell is approximated by a field of rotations varying along the shells’ midsurface, rather than by a single rotation. This construction is essential for the dimension reduction analysis in Parts II and III of this monograph. In section 4.6 we study the Friesecke-James-M¨uller’s inequality on the plane, relying on the conformal-anticonformal decomposition of the deformation gradient. We show that the counterpart of Korn’s constant in the present nonlinear setting is the same as that of the homogeneous Korn’s inequality, namely 2, and we identify the deformations for which this optimal constant is achieved. In section 4.7 we outline the status of the rigidity statements and rigidity estimates in the conformal setting, in which the space of trace-free skew-symmetric matrices replaces so(N), while the M¨obius transforms replace the rigid motions.

4.1 Liouville’s theorem and quantitative rigidity estimate In this section we state the celebrated nonlinear counterpart of Korn’s inequality (3.1) in which the distance from the linear space so(N) is replaced by the distance from the group of proper rotations SO(N). Recall that so(N) is the tangent space at IdN to SO(N), seen as a compact manifold in RN×N . Thus, writing a given deforma© Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_4

65

66

4 Friesecke-James-M¨uller’s inequality

tion u in terms of its displacement v = u − idN , we observe that: dist2 (∇u, SO(N)) = dist2 (IdN + ∇v, SO(N)) ' dist2 (∇v, so(N)) + O(|∇v|2 ) = |sym∇v|2 + O(|∇v|2 ).

(4.1)

This formally suggests our next result, whose proof will be given in section 4.4: ¨ Theorem 4.1. [Friesecke-James-Muller’s inequality] Let Ω ⊂ RN be an open, bounded, connected, Lipschitz domain. Then, for every u ∈ H 1 (Ω , RN ) there exists a rotation matrix R ∈ SO(N) satisfying: Z

2

|∇u − R| dx ≤ C

Ω

Z

dist2 (∇u, SO(N)) dx.

(4.2)

Ω

The above constant C depends only on Ω , but not on u. Moreover, C is invariant under dilations and it can be chosen uniformly for a family of domains Ω which are bilipschitz equivalent with controlled Lipschitz constants.

Fig. 4.1 The compact manifold SO(N) and its tangent spaces.

Another heuristic connecting the above result to (3.1) arises from interpreting the right hand side of (4.2) as an energy E (u) that measures how far a given deformation u is from being an orientation-preserving isometry, that is from it satisfying: (∇u)T ∇u = IdN

and

det ∇u > 0

in Ω .

When u = idN + v and v is small, the second condition above holds automatically and we may write in accordance with (4.1): (∇u)T ∇u − IdN = 2 sym∇v + (∇v)T ∇v, so that the energy on the right hand side of (3.1) captures the lowest order term of the nonlinear energy E . We remark that this point of view will be adopted and expanded in Part III of our monograph, with IdN replaced by an arbitrary Riemannian

4.1 Liouville’s theorem and quantitative rigidity estimate

67

metric g on Ω in an attempt to model the so-called prestress of an elastic material withR the reference configuration given by Ω . In this context, a general energy E (u) ' Ω dist2 (∇u(x), SO(N)g(x)1/2 ) dx quantifies the total pointwise deviation of ∇u from g1/2 , modulo orientation-preserving rotations that do not cost any energy. Similarly to the discussion of Korn’s inequality, the inequality (4.2) can be viewed as a rigidity estimate. The corresponding rigidity statement is that the vanishing of its right hand side, i.e. having each ∇u(x) ∈ SO(N), implies the vanishing of the left hand side i.e. u being linear with gradient that is a constant rotation. As in Lemma 3.2, this last assertion can be proved directly: Lemma 4.2. [Liouville’s theorem] Let Ω ⊂ RN be an open, bounded, connected domain. If u ∈ H 1 (Ω , RN ) satisfies ∇u ∈ SO(N) a.e. in Ω , then: u(x) = Rx + b for some R ∈ SO(N), b ∈ RN . Proof. We will use the following formula, valid in the sense of distributions:
div cof ∇u = 0. (4.3)   Recall that for F ∈ RN×N , its cofactor matrix is cof F = (−1)i+ j det Fiˆ jˆ i, j=1...N , where Fiˆ jˆ ∈ R(N−1)×(N−1) stands for the minor of F obtained by removing its ith row and j-th column. In particular, cof ∇u = ∇u because ∇u ∈ SO(N). It also follows that u ∈ W 1,∞ (Ω , RN ), so we can actually use (4.3) and write:
0 = div cof ∇u = div ∇u = ∆ u. Consequently, u is harmonic and thus automatically smooth. We now compute:
0 = ∆ |∇u|2 = 2h∆ ∇u : ∇ui + 2h∇2 u : ∇2 ui = 2|∇2 u|2 . Hence, ∇u must be constant. The proof is done. The inequality (4.2) is thus the quantitative version of Lemma 4.2, in the sense that the total pointwise distance of ∇u from SO(N), measured in the L2 (Ω ) norm, yields the error in the deviation of ∇u from being a constant rotation, measured in L2 (Ω ):
distL2 (Ω ) ∇u, SO(N) 

Z

inf R∈SO(N)

|∇u − R|2 dx


1/2

Ω

(4.4)

≤ Ck dist(∇u, SO(N))kL2 (Ω ) . It is obvious that the above inequality can also be reversed:
k dist(∇u, SO(N))kL2 (Ω ) ≤ distL2 (Ω ) ∇u, SO(N) . Thus, Friesecke-James-M¨uller’s inequality states equivalence of commuting the operations of taking the distance from SO(N) and integrating.

68

4 Friesecke-James-M¨uller’s inequality

4.2 First order truncation result Moving towards a proof of Theorem 4.1, we present here a useful Lusin-type estimate, which allows us to replace a given Sobolev function by its Lipschitz “truncation”. The two functions agree on a large set, whose complement has its measure controlled (inverse proportionally) in terms of the requested Lipschitz constant. Theorem 4.3. Let Ω ⊂ RN be an open, bounded, Lipschitz domain. Then, for every g ∈ H 1 (Ω , R) and every λ > 0, there exists g¯ ∈ W 1,∞ (Ω , R) satisfying: (i) k∇gk ¯ L∞ (Ω ) ≤ Cλ , CZ ≤ |∇g| dx, (ii) {x ∈ Ω ; g(x) , g(x)} ¯ λ {|∇g|>λ } Z

(iii)

|∇g − ∇g| ¯ 2 dx ≤ C

Z

|∇g|2 dx.

{|∇g|>λ }

Ω

The constants C depend only on Ω , and can be chosen uniformly for a family of domains which are bilipschitz equivalent with controlled Lipschitz constants.

A weaker version of (ii) above follows by noting that 1
λ }

(4.5)

Towards a proof of Theorem 4.3, we recall the classical covering result: Lemma 4.4. [Vitali’s covering theorem] Let F be a given family of open balls in RN with uniformly bounded radii. Then, there exists a pairwise disjoint and countable (or finite) subfamily G ⊂ F , with: [ B∈F

B⊂

[

5B,

B∈G

where for a ball B = Br (x) we denote 5B  B5r (x). Proof. 1. Set d = sup{r; Br (x) ∈ F } and decompose F = n d d o Fn  Br (x) ∈ F ; r ∈ n , n−1 2 2

n=1 Fn ,

S∞

where:

for all n ≥ 1.

∞ We now put G = ∞ n=1 Gn , where {Gn ⊂ Fn }n=1 are each a respective subfamily of Fn as above, which we now define inductively:

S

(i) G1 is any maximal disjoint collection of balls in F1 , (ii) having G1 , . . . , Gn−1 selected, Gn is any maximal disjoint collection of balls in:

4.2 First order truncation result

69 n−1 n o [ [ B ∈ Fn ; B ∩ B0 = 0/ . k=1 B0 ∈Gk

Above, by a “maximal disjoint collection of balls” we mean a pairwise disjoint subfamily of the indicated family, which is not a proper subset of any other pairwise disjoint subfamily. The existence of such a construct is guaranteed by Zorn’s lemma. 2. We now claim that for every B ∈ F there exists B0 ∈ G with B ⊂ 5B0 . Indeed, let B = Br (x) ∈ Fn , for some n ≥ 1. By maximality of Gn , there must be: B∩

n [ [

B0 , 0. /

k=1 B0 ∈Gk

Consequently, there exists a ball B0 = Br0 (x0 ) ∈ S d and r0 > Fn and B0 ∈ nk=1 Fk , we get: r ≤ 2n−1

0 k=1 Gk and x0 ∈ B ∩ B . 1 d 2n ≥ 2 r and further:

Sn

|y − x0 | ≤ |y − x0 | + |x0 − x0 | < 2r + r0 < 5r0

Since B ∈

for all y ∈ B.

This proves the claim. We finally observe that any pairwise disjoint family of balls in RN is at most countable. This ends the proof of the lemma. We first prove Theorem 4.3 in the special case of a unit ball: Lemma 4.5. Denote Ω = B1 (0) ⊂ RN . Then, for every g ∈ H 1 (Ω , R) and every λ > 0, there exists g¯ ∈ W 1,∞ (Ω , R) satisfying conditions (i) and (ii) of Theorem 4.3, with a constant C that depends only on the dimension N. Proof. 1. For a fixed λ > 0, consider the set: ? n o K λ = x¯ ∈ Ω ; |∇g| dx ≤ 2λ for all r < 2 . Br (x)∩Ω ¯

By Vitali’s covering in Lemma 4.4, there exists a sequence of pairwise disjoint balls {Bri (xi )}∞ i=1 with centers xi ∈ Ω and radii ri < 2 such that: Ω \ Kλ ⊂

∞ [

? B5ri (xi )

i=1

|∇g| dx > 2λ for all i ≥ 1.

and Bri (xi )∩Ω

N ∞ Consequently: |Ω \K λ | ≤ ∑∞ i=1 |B5ri (xi )∩Ω | ≤ 5 ∑i=1 |Bri (xi )∩Ω |. Further, since:

  1 |∇g| dx + |∇g| dx |Bri (xi ) ∩ Ω | Bri (xi )∩Ω ∩{|∇g|>λ } Bri (xi )∩Ω ∩{|∇g|≤λ } Z 1 ≤ |∇g| dx + λ , |Bri (xi ) ∩ Ω | Bri (xi )∩Ω ∩{|∇g|>λ } Z

Z

2λ
λ } |∇g|

dx for all i ≥ 1.

70

4 Friesecke-James-M¨uller’s inequality

|Ω \ K λ | ≤

5N λ

∞

Z

|∇g| dx ≤

∑

i=1 Bri (xi )∩Ω ∩{|∇g|>λ }

5N λ

Z

|∇g| dx.

(4.6)

{|∇g|>λ }

> g dx. We 2. For each x¯ ∈ Ω and r < 2, consider the average: (g)x,r,Ω  B (x)∩Ω ¯ r ¯ claim that the following uniform Poincar´e-Wirtinger inequality holds: ? ? |g − (g)x,r,Ω | dx ≤ Cr |∇g| dx, (4.7) ¯ Br (x)∩Ω ¯

Br (x)∩Ω ¯

with C depending only on N. We begin by extending g (by the even reflection across ∂ Ω ) onto B7/4 (0) and then (by the even reflection across ∂ B7/4 (0)) onto B3 (0). The extension is still denoted by g ∈ H 1 (B3 (0), R). Since Br (x) ¯ ⊂ B3 (0), we may define > g dx and apply the classical Poincar´ e -Wirtinger inequality, to get: a  B (x) ¯ r

?

? ? |Br (x)| ¯ |Br (x)| ¯ |g − a| dx ≤ |g − a| dx ≤ Cr |∇g| dx |Br (x) ¯ ∩ Ω | Br (x) |Br (x) ¯ ∩ Ω | Br (x) ¯ ¯ Br (x)∩Ω ¯ ? Z 1 |∇g| dx ≤ Cr |∇g| dx, = Cr |Br (x) ¯ ∩ Ω | Br (x) ¯ Br (x)∩Ω ¯

where C depends only on N. The last inequality follows from the extension definition of g. In particular, the above also directly yields: ? ? |(g)x,r,Ω − a| ≤ |g − a| dx ≤ Cr |∇g| dx, ¯ Br (x)∩Ω ¯

Br (x)∩Ω ¯

so the last two displayed inequalities result in (4.7). 3. Observe that (4.7) implies, for any x, ¯ y¯ ∈ K λ and r = |x¯ − y|: ¯ ? (g)x,r,Ω − (g) ≤ |g − (g)x,r,Ω | + |g − (g)y,r,Ω | dx ¯ y,r,Ω ¯ ¯ ¯ Br (x)∩B ¯ ¯ r (y)∩Ω

≤

|Br (x) ¯ ∩Ω| |Br (x) ¯ ∩ Br (y) ¯ ∩Ω| +

?

|g − (g)x,r,Ω | dx ¯

(4.8)

Br (x)∩Ω ¯

|Br (y) ¯ ∩Ω| |Br (x) ¯ ∩ Br (y) ¯ ∩Ω|

? |g − (g)y,r,Ω | dx ≤ Crλ , ¯ Br (y∩Ω ¯

where we used the defining property of points in K λ . Further, using (4.7) again, for any x¯ ∈ Ω , r < 2 and k ≥ 0 we get that there holds: ? (g) ≤ − (g) |g − (g)x,r/2 k+1 k k ,Ω | dx x,r/2 ¯ ,Ω x,r/2 ¯ ,Ω ¯ Br/2k+1 (x)∩Ω ¯

≤

|Br/2k (x) ¯ ∩Ω| ? |Br/2k+1 (x) ¯ ∩Ω|

Br/2k (x)∩Ω ¯

|g − (g)x,r/2 k ,Ω | dx, ¯

4.2 First order truncation result

71

r leading to: (g)x,r/2 k+1 ,Ω − (g)x,r/2 k ,Ω ≤ C k ¯ ¯ 2

> Br/2k (x)∩Ω ¯

|∇g| dx. Hence, on the set:

 K˜ λ = x¯ ∈ K λ ; g(x) ¯ = lim (g)x,r,Ω , ¯ r→0

we get for all r < 2 and x¯ ∈ K˜ λ : ∞

|g(x) ¯ − (g)x,r,Ω |≤ ¯





k+1 ,Ω − (g)x,r/2 k ,Ω ∑ (g)x,r/2 ¯ ¯ k=0 ∞

r k 2 k=0

?

≤C ∑

(4.9) |∇g| dx ≤ Crλ .

Br/2k (x)∩Ω ¯

In conclusion, (4.8) and (4.9) applied to x¯ and y¯ with r = |x¯ − y|, ¯ yield: for all x, ¯ y¯ ∈ K˜ λ ,

|g(x) ¯ − g(y)| ¯ ≤ Cλ |x¯ − y| ¯

with C depending only on N. We may now invoke the Kirszbraun theorem, in order to extend the restriction g|K˜ λ to a Lipschitz function g¯ on Ω , whose Lipschitz constant equals that of g|K˜ λ displayed above. We hence proved (i). Finally: |{g , g}| ¯ ≤ |Ω \ K˜ λ | = |Ω \ K λ | because K˜ λ and K λ coincide up to a set of measure zero, in virtue of the Lebesgue differentiation theorem. This yields (ii) from (4.6). The proof is done. We can now complete the proof of the main result. The truncation function on a general Lipschitz domain Ω is defined as the linear combination, through a partition of unity, of finitely many local truncations given in Lemma 4.5. The domains of such local truncations are bilipschitz homeomorphic to a ball, and their union equals Ω . The property (ii) then follows directly, while to get (i) we need an additional estimate of the linear combination of the truncations through gradients of the partition of unity. Proof of Theorem 4.3 1. We first observe that (iii) directly follows from (i) and the weaker version of (ii) displayed in the estimate (4.5), because: Z

|∇g − ∇g| ¯ 2 dx =

Z {g,g} ¯

Ω

≤2

Z {|∇g|>λ }

|∇g − ∇g| ¯ 2 dx ≤ 2

Z

|∇g|2 + |∇g| ¯ 2 dx

{g,g} ¯

Z ≤C |∇g|2 dx +Cλ 2 {x ∈ Ω ; g(x) , g(x)} ¯

|∇g|2 dx.

{|∇g|>λ }

2. Second, we derive the claimed uniformity of C under bilipschitz diffeomorphisms with controlled Lipschitz constants. By step 1 above, it suffices to show the bounds (i) and (ii). We proceed as in the second step of proof of Theorem 3.17.

72

4 Friesecke-James-M¨uller’s inequality

Let ϕ : Ω → Ω 0 be a bilipschitz map with the Lipschitz constants of both ϕ and ϕ −1 being bounded by some L > 0. Given h ∈ H 1 (Ω 0 , RN ) and λ > 0, apply (i) and (ii) to g = h ◦ ϕ ∈ H 1 (Ω , R) and Lλ > 0 to obtain a truncated function g¯ ∈ W 1,∞ (Ω , R). We define h¯  g¯ ◦ ϕ −1 ∈ W 1,∞ (Ω 0 , R) and observe that: ¯ L∞ (Ω 0 ) ≤ Lk∇gk k∇hk ¯ L∞ (Ω ) ≤ CL2 λ , {x ∈ Ω 0 ; h(x) , h(x)} ≤ CLN {x ∈ Ω ; g(x) , g(x)} ¯ ¯ ≤

CLN Lλ

Z

|∇g| dx ≤

{|∇g|>Lλ }

CL2N λ

Z

|∇h| dx,

{|∇h|>λ }

where all constants C above depend only on Ω . 3. We now prove (i) and (ii) on a general bounded, Lipschitz domain Ω . Write S Ω = ni=1 Ωi as the finite union of open domains {Ωi }ni=1 that are bilipschitz equivalent to a ball B1 (0). Such decomposition exists, for example, by Lemma 3.15 (ii). For each restriction of a given g ∈ H 1 (Ω , R) to Ωi and for a given λ > 0, we apply Lemma 4.5 and step 2 above to obtain a family {g¯i ∈ W 1,∞ (RN , R)}ni=1 , satisfying: Lip(g¯i ) ≤ Cλ ,

C |{x ∈ Ωi ; g¯i (x) , g(x)}| ≤ λ

Z

|∇g| dx,

(4.10)

Ωi ∩{|∇g|>λ }

for all i = 1 . . . n. Observe that the domain of each g¯i may be chosen to be RN rather than Ωi , in virtue of the Kirszbraun theorem. Let {φi ∈ C ∞ (Ω , [0, 1])}ni=1 be a partition of unity subordinated to the cover {Ωi }ni=1 , so that: supp(φi ) ⊂ Ω¯ i and ∑ni=1 φi = 1 in Ω . Define: n

g¯ = ∑ φi g¯i ∈ W 1,∞ (Ω , R). i=1

Since g =

∑ni=1 φi g,

in view of (4.10) we get:

Z n C {x ∈ Ω ; g(x) ¯ , g(x)} ≤ ∑ |{x ∈ Ωi ; g¯i (x) , g(x)}| ≤ |∇g| dx, λ {|∇g|>λ } i=1

with C that depends only on Ω through the particular partition. This proves (ii). 4. To show (i), we use the first estimate in (4.10) to obtain: n n |∇g| ¯ = ∑ (φi ∇g¯i + g¯i ∇φi ) ≤ Cλ + ∑ g¯i ∇φi i=1

in Ω .

(4.11)

i=1

Since ∑ni=1 ∇φi = ∇(∑∞ i=1 φi ) ≡ 0, we further observe that: n n ∑ g¯i ∇φi = ∑ (g¯i − g¯ j )∇φi ≤ C i=1

i=1

∑

|g¯i − g¯ j |

in Ω j .

i: Ωi ∩Ω j ,0/

 Define α  min |Ωi ∩ Ω j |; Ωi ∩ Ω j , 0/ > 0 and assume first that:

(4.12)

4.3 Local rigidity estimate

73

C λ

Z

|∇g| dx ≤

{|∇g|>λ }

α . 3

(4.13)

Taking C as the uniform bounding constant in (4.10), it follows that: {x ∈ Ωi ∩ Ω j ; g¯i (x) , g(x)} + {x ∈ Ωi ∩ Ω j ; g¯ j (x) , g(x)} ≤ 2α , 3 so whenever Ωi ∩ Ω j , 0, / there exists xi j ∈ Ωi ∩ Ω j with g¯i (xi j ) = g¯ j (xi j ) = g(xi j ). Consequently, by the first bound in (4.10) we get the bound for all x ∈ Ωi ∩ Ω j , 0: / |g¯i (x) − g¯ j (x)| ≤ |g¯i (x) − g¯i (xi j )| + |g¯ j (x) − g¯ j (xi j )| ≤ Cλ · diam Ω . Recalling (4.11) and (4.12), there follows (i) under condition (4.13). On the other hand, if (4.13) does not hold then we get (i) for g¯ ≡ 0, and (ii) also holds because: Z 3C|Ω | 1 {x ∈ Ω ; g(x) ¯ , g(x)} ≤ |Ω | ≤ |∇g| dx. α λ {|∇g|>λ }

This ends the proof of Theorem 4.3.

4.3 Local rigidity estimate In this section, we prove a preliminary version of the nonlinear rigidity estimate (4.2), in which Ω is the unit ball and the error of the approximation by a rotation is estimated only on a smaller ball Bα (0) b B1 (0). In the first step, a given deformation gradient is replaced by the gradient of its truncation from Theorem 4.3. Then, we follow the general outline of the proof of Korn’s inequality in section 3.4. The approximating truncation is decomposed as the sum of the harmonic part w and the correction v that equals 0 on the boundary. A simple integration by parts yields the desired bound for v (even without shrinking the domain). To deal with w, we use Korn’s inequality together with a pointwise bound due to the harmonicity of w. Theorem 4.6. Let u ∈ H 1 (Br (x0 ), RN ) be a vector field defined on the ball Br (x0 ) ⊂ RN . Then, for any α ∈ (0, 1) there exists R ∈ SO(N) such that: Z Bαr (x0 )

|∇u − R|2 dx ≤ C

Z

dist2 (∇u, SO(N)) dx,

(4.14)

Br (x0 )

with the constant C depending only on the dimension N and the factor α. Proof. 1. By a simple scaling argument, it suffices to assume that Br (x0 ) = B1 (0). To alleviate the notation, we will write:

74

4 Friesecke-James-M¨uller’s inequality

Q0  Bα (0).

Q  B1 (0),

We now observe that it, likewise, suffices to prove the theorem under the assumption k∇ukL∞ (Q) ≤ M, for an appropriate bounding constant M > 0 that depends only on N. Indeed, √ applying Theorem 4.3 to each component of the vector field u and to λ = 2 N, we get u¯ ∈ W 1,∞ (Q, RN ) for which the following bounds hold, with the constant C associated to Q and thus depending only on N: Z

k∇uk ¯ L∞ (Q) ≤ 2NC  M,

|∇u − ∇u| ¯ 2 dx ≤ C

Z

Q

2 √ |∇u| {|∇u|>2 N}

dx.

√ Writing |∇u(x)| ≤ minR∈SO(N) |∇u(x) − R| + |R| = dist(∇u(x), SO(N)) + N, and √ noting N ≤ 12 |∇u| on the domain of integration in the second bound, this implies: Z

|∇u − ∇u| ¯ 2 dx ≤ 4C

Q

Z

dist2 (∇u, SO(N)) dx.

Q

Hence, if R is a rotation for which (4.14) holds for u, ¯ then also: Z Q0

Z

|∇u − R|2 dx ≤ 2C

dist2 (∇u, ¯ SO(N)) dx + 2

Z

Q

Z

≤C

Q0

|∇u − ∇u| ¯ 2 dx

dist2 (∇u, SO(N)) + |∇u − ∇u| ¯ 2 dx




Q

Z

≤C

dist2 (∇u, SO(N)) dx

Q

with a constant C (changing from line to line) depending only on N and α. 2. From now on, we assume that: k∇ukL∞ (Q) ≤ M.

(4.15)

We further observe another simplification. It suffices to prove the theorem for: Z

dist2 (∇u, SO(N)) dx ≤ 1,

(4.16)

Q

because in the opposite case we automatically get: Z Q

|∇u − IdN |2 dx ≤ 2

Z

≤C

Q

Z

|∇u|2 + N dx ≤ 2|B1 (0)|(M 2 + N) dist2 (∇u, SO(N)) dx.

Q

Hence from now on, we also assume (4.16). 3. Equipped with the simplifications (4.15) and (4.16), we proceed to the main part of the argument. As in the proof of Korn’s inequality in Theorem 3.18, we decompose u as the sum u = v + w, where v ∈ H01 (Q, RN ) satisfies:

4.3 Local rigidity estimate

75

∆ v = ∆ u in Q,

v = 0 on ∂ Q.

Observe the identity ∆ u = div(∇u − cof ∇u), valid since u ∈ W 1,∞ . Multiplying the resulting identity ∆ v = div(∇u − cof ∇u) by ∇v and integrating by parts leads to: Z

2

|∇v| dx =

Z

Q

 ∇v : ∇u − cof ∇u dx ≤ k∇vkL2 (Q) k∇u − cof ∇ukL2 (Q) ,

Q

which implies that

2 Q |∇v|

R

Z

dx ≤

R

Q |∇u − cof ∇u|

|∇v|2 dx ≤ C

Q

Z

2

dx. Consequently:

dist2 (∇u, SO(N)) dx.

(4.17)

Q

The bound above follows by noting that the function RN×N 3 A 7→ A − cof A ∈ RN×N is smooth, and hence Lipschitz continuous on the ball B¯ M (0) ⊂ RN×N , containing almost all values ∇u(x) by (4.15). This function also equals 0 on SO(N), so that: |∇u(x) − cof ∇u(x)| ≤ L dist(∇u(x), SO(N)), Thus, (4.17) follows with C depending on M and N, hence ultimately only on N. 4. Consider the harmonic corrector w = u − v ∈ H 1 (Q, RN ), which satisfies: ∆ w = 0 in Q,

w = u on ∂ Q.

Let Q00 = B(α+1)/2 (0) be another concentric ball, so that Q0 b Q00 b Q, and let φ ∈ C0∞ (Q, [0, 1]) be a smooth cut-off function satisfying φ|Q00 ≡ 1. Note that:


 1 ∆ |∇w|2 = ∆ ∇w : ∇w + |∇2 w|2 = |∇2 w|2 2

in Q,

by the harmonicity of each component of ∇w. Consequently: Z Q00

|∇2 w|2 dx ≤

Z

φ |∇2 w|2 dx =

Q

1 ≤ k∆ φ kL∞ (Q) 2

1 2

Z


φ ∆ |∇w|2 − N dx

Q

Z Q

|∇w|2 − N dx.

Recalling that ∇w = ∇u − ∇v, the above yields in view of (4.15) and (4.17): Z Z Z  |∇u|2 − N dx + |∇v|2 dx + 2M |∇v| dx |∇2 w|2 dx ≤ C Q00 Q Q Q Z  Z
1/2 Z
1/2  2 2 ≤C dist (∇u, SO(N)) dx + |∇v| dx + |∇v|2 dx Q Q Q Z Z
1/2  ≤C dist2 (∇u, SO(N)) dx + dist2 (∇u, SO(N)) dx ,

Z

Q

Q

where we used the same argument as in step 3 towards the pointwise bound of the form: |∇u|2 − N ≤ L dist(∇u, SO(N)). It now follows by (4.16) that:

76

4 Friesecke-James-M¨uller’s inequality

Z Q00

|∇2 w|2 dx ≤

Z

dist2 (∇u, SO(N)) dx


1/2

.

(4.18)

Q

5. Applying the mean value property to the components of the harmonic matrix fields {∇2 wi }Ni=1 , we arrive at: ? Z 2 2 2 2 |∇ w(x)| = ∇ w(y) dy ≤ C |∇2 w(y)|2 dy for all x ∈ Q0 . Q00

B(1−α)/2 (x)

By the mean value theorem and (4.18) we get, for the matrix B = ∇w(0) ∈ RN×N : |∇w(x) − B|2 ≤ k∇2 wk2L∞ (Q0 ) Z

≤C

dist2 (∇u, SO(N)) dx


1/2

(4.19)

for all x ∈ Q0 .

Q

We now claim that the same bound above also holds with B replaced by a rotation R ∈ SO(N) that is closest to B. Indeed, applying the triangle inequality, recalling ∇u − ∇w = ∇v, and invoking (4.19) with (4.17) and the assumption (4.16), we get: ? 2 dist (B, SO(N)) ≤ 3 dist2 (∇u(x), SO(N)) + |∇v(x)|2 + |∇w(x) − B|2 dx Q0

≤C

Z

dist2 (∇u, SO(N)) dx +

Z

Q

≤C

Z

dist2 (∇u, SO(N)) dx


1/2 

Q

dist2 (∇u, SO(N)) dx


1/2

.

Q

Consequently, (4.19) becomes, for some R ∈ SO(N): |∇w(x) − R|2 ≤ C

Z

dist2 (∇u, SO(N)) dx


1/2

for all x ∈ Q0 .

Q

(4.20)

6. We now use Korn’s inequality (3.1) in Theorem 3.1, to the displacement RT w− idN on the domain Q0 , to obtain A ∈ so(N) for which there holds: Z Q0

Z T sym(RT ∇w − IdN ) 2 dx R ∇w − IdN − A 2 dx ≤ C Q0

≤C

Z Q0

dist2 (∇w, SO(N)) + |∇w − R|4 dx.

In the second inequality above, we applied the linearization of the function RN×N 3 F 7→ dist2 (F, SO(N)) close to R, to the effect that:
dist(F, SO(N)) = dist(RT F, SO(N)) = dist(RT F − IdN , so(N)) + O |RT F − IdN |2
= sym(RT F − IdN ) + O |F − R|2

4.4 Proof of Friesecke-James-M¨uller’s inequality

77

Thus, using the pointwise bound (4.20) to deal with |∇w − R|4 , and (4.17) to replace R R 2 2 Q0 dist (∇w, SO(N)) dx by Q0 dist (∇u, SO(N)) dx, we arrive at: Z

∇w − R(IdN + A) 2 dx Q0 Z Z  ≤C dist2 (∇u, SO(N)) + |∇v|2 dx + dist2 (∇u, SO(N)) dx (4.21) 0 Q

≤

Z

Q

2

dist (∇u, SO(N)) dx. Q

Arguing as in step 5, we finally replace the matrix B¯ = R(IdN + A) by its closest rotation, because with a constant C that depends only on N and α, we have: ? ¯ SO(N)) ≤ 3 ¯ 2 dx dist2 (B, dist2 (∇u(x), SO(N)) + |∇v(x)|2 + |∇w(x) − B| Q0

≤C

Z

dist2 (∇u, SO(N)) dx.

Q

In view of (4.21) and (4.17), this ends the proof of the theorem.

¨ 4.4 Proof of Friesecke-James-Muller’s inequality In order to combine the local results of Theorem 4.6 towards a global statement, we will employ a version of the Whitney decomposition theorem in: Lemma 4.7. Let Ω ⊂ RN be an open, bounded domain. Then, there exists a countable collection B of open balls in Ω , satisfying: (i) each B ∈ B has the form B = B 1 dist(x,∂ Ω ) (x) for some x ∈ Ω , 4 S (ii) Ω = B∈B B, (iii) the covering of Ω by the family {2B}B∈B where we write 2B = B 1 dist(x,∂ Ω ) (x) 2 for B = B 1 dist(x,∂ Ω ) (x), has a finite covering number. More precisely, each x0 ∈ 4 Ω belongs to at most M balls in {2B}B∈B , with M depending only on N. Proof. 1. We apply Vitali’s covering in Lemma 4.4 to the family of balls:  F = B 1 dist(x,∂ Ω ) (x); x ∈ Ω 20

and obtain, in view of the boundedness of Ω , a countable, pairwise disjoint subS S family G ⊂ F satisfying B∈F ⊂ B∈G 5B. We use the notation convention 5B = B 1 dist(x,∂ Ω ) (x) for B = B 1 dist(x,∂ Ω ) (x) and set: 4

20

 B  5B; B ∈ G .

78

4 Friesecke-James-M¨uller’s inequality

Conditions (i) and (ii) are clearly satisfied. We now show that (iii) holds as well. 2. Assume that a point x0 ∈ Ω belongs to a number M of distinct balls of the form 2Bi = B 1 dist(xi ,∂ Ω ) (xi ), for some {Bi = B 1 dist(xi ,∂ Ω ) (xi ) ∈ B}M i=1 . Without loss 2 4 of generality, the radii of Bi are ordered nonincreasingly. Consequently, we have: dist(x1 , ∂ Ω ) ≥ dist(xi , ∂ Ω ) >

1 dist(x1 , ∂ Ω ) for all i = 1 . . . M, 3

(4.22)

where the second inequality follows by: dist(x1 , ∂ Ω ) ≤ dist(xi , ∂ Ω ) + |xi − x0 | + |x1 − x0 | < 32 dist(xi , ∂ Ω ) + 21 dist(x1 , ∂ Ω ). At the same time, there holds: 1 21 Bi ⊂ B1 5 5

for all i = 1 . . . M,

1 because similarly as before: |x1 − xi | + 20 dist(xi , ∂ Ω ) ≤ |x1 − x0 | + |x0 − xi | + 1 11 21 1 20 dist(xi , ∂ Ω ) < 2 dist(x1 , ∂ Ω )+ 20 dist(xi , ∂ Ω ) ≤ 20 dist(x1 , ∂ Ω ). By disjointness 1 M of { 5 Bi }i=1 and recalling (4.22), we get:

1 21 M Bi B1 21
N N 5 5 ≥∑ dist (x1 , ∂ Ω ) = 20 B1 (0) ⊂ RN i=1 B1 (0) ⊂ RN M

=∑

i=1


N 1 M dist(xi , ∂ Ω ) > N distN (x1 , ∂ Ω ). 20 60

Thus M < 63N . The proof is done. We can now achieve the goal of this chapter, by giving: Proof of Theorem 4.1. 1. We start by noting that a simple rescaling argument yields invariance of the constant C in (4.2) under dilations of Ω . We now prove the claimed inequality, together with the invariance of C under bilipschitz equivalence. Arguing as in step 1 of the proof of Theorem 4.6 and recalling Theorem 4.3, it follows that it suffices to prove the result for u ∈ W 1,∞ (Ω , RN ) satisfying k∇ukL∞ (Ω ) ≤ M, where M is some constant depending only on the domain Ω . Moreover, M may be chosen uniformly for a given family of bilipschitz equivalent domains with controlled Lipschitz constants. Again, as in the proof of Theorem 4.6, we decompose u = v + w, putting: ∆ v = ∆ u in Ω ,

v = 0 on ∂ Ω .

We apply the argument in step 3 of the proof of Theorem 4.6 to get: Z Ω

|∇v|2 dx ≤ C

Z

dist2 (∇u, SO(N)) dx.

(4.23)

Ω

The main argument of the present proof concerns the harmonic corrector w in:

4.4 Proof of Friesecke-James-M¨uller’s inequality

79

∆ w = 0 in Ω ,

w = u on ∂ Ω .

2. Let Ω = ∞ i=1 Bri (xi ) be a countable decomposition of Ω into balls with the respective radii ri = 14 dist(xi , ∂ Ω ), as described in Lemma 4.7. Applying Theorem 4.6 to the scaling factor α = 89 and to w on each ball B3ri /2 (xi ), yields: S

Z B4r /3 (xi )

|∇w − Ri |2 dx ≤ C

Z

dist2 (∇w, SO(N)) dx,

B3r /2 (xi )

i

i

for a family of rotations {Ri ∈ SO(N)}∞ i=1 . Consequently, Lemma 3.12 applied to each scalar component of the harmonic matrix field ∇w − Ri on B4ri /3 (xi ), implies: Z

|∇2 w|2 dist2 (x, ∂ B4ri /3 (xi )) dx ≤ C

B4r /3 (xi )

Z

i

dist2 (∇w, SO(N)) dx. (4.24)

B3r /2 (xi ) i

Observe now the following bound that is valid for all x ∈ Bri (xi ): dist(x, ∂ Ω ) ≤ dist(xi , ∂ Ω ) + |x − xi | ≤ 4ri + ri ≤ C dist(x, ∂ B4ri /3 (xi )), Hence, (4.24) yields: Z

|∇2 w|2 dist2 (x, ∂ Ω ) dx ≤ C

Bri (xi )

Z

dist2 (∇w, SO(N)) dx,

B3r /2 (xi ) i

with C depending only on N. Let now {φi ∈ Cc∞ (B2ri (xi ), [0, 1])}∞ i=1 be a sequence of test functions that satisfy φi ≡ 1 on B3ri /2 (xi ). Summing the above displayed bounds over all i ≥ 1 leads to: Z Ω

∞

Z

|∇2 w|2 dist2 (x, ∂ Ω ) dx ≤ C ∑

dist2 (∇w, SO(N)) dx

i=1 B3ri /2 (xi )

≤C

Z  ∞ Ω

Z  2 φ (x) dist (∇w, SO(N)) dx ≤ C dist2 (∇w, SO(N)) dx, i ∑ Ω

i=1

in view of the finite covering number property (where the covering number depends only on N) in Lemma 4.7 (iii). By Theorem 3.17 (ii), and noting the required uniformity of constants, the above yields existence of a matrix B ∈ RN×N such that: Z

2

|∇w − B| dx ≤ C

Ω

Z

dist2 (∇w, SO(N)) dx.

Ω

As in the proof of Theorem 4.6 step 5 and 6, we further observe that: Z

dist2 (B, SO(N)) dx ≤ 2

Z

Ω

dist2 (∇w, SO(N)) + |B − ∇w|2 dx

Ω

≤C

Z Ω

dist2 (∇w, SO(N)) dx,

(4.25)

80

4 Friesecke-James-M¨uller’s inequality

so the matrix B in (4.25) can be replaced by its closest rotation matrix in SO(N). Upon recalling (4.23) the proof is done.

¨ 4.5 Approximation theorem and Friesecke-James-Muller’s constant in thin shells Similarly as in section 3.6, we now proceed with approximating a deformation gradient ∇u on a thin shell by a rotation-valued field defined on the shell’s midsurface, rather than by a single rotation. This construction, based on the local application of (4.2) on cube-like domains, in view of the invariance of constants C in Theorem 4.1, will be of key importance in deriving the hierarchy of limiting shell and plate theories through the dimension reduction in Parts II and III of this monograph. Let S be a smooth, compact, connected, oriented hypersurface in RN . For two families of positive, C 1 functions {gh1 , gh2 : S → R}h>0 , we consider thin shells {Sh }h>0 with the midsurface S: Sh = {x = z + tn(z); z ∈ S, − gh1 (z) < t < gh2 (z)}.

(4.26)

We will use the same notation as in section 3.6, namely: n for the unit normal, Tz S for the tangent space to S at z ∈ S, and π for the projection onto S along n, so that: π(z + tn(z)) = z for all z ∈ S and |t|  1. We have: Theorem 4.8. Let S ⊂ RN be a smooth, compact, connected, oriented hypersurface, with its boundary ∂ S given by finitely many Lipschitz curves. Assume that {Sh }h>0 is given by (4.26) where {gh1 , gh2 ∈ C 1 (S, R)}h>0 satisfy: |∇ghi (z)| ≤ C3 h

C1 h ≤ ghi (z) ≤ C2 h,

for all z ∈ S, h  1,

(4.27)

with some positive constants C1 ,C2 ,C3 independent of h. Then, for every vector field u ∈ H 1 (Sh , RN ) there exists a matrix field R˜ ∈ H 1 (S, RN×N ), such that: Z

(i)

ZSh

|∇u − R˜ ◦ π|2 dx ≤ C

˜ 2 dσ (z) ≤ |∇R|

(ii) S

C h3

Z Sh

Z Sh

dist2 (∇u, SO(N)) dx,

dist2 (∇u, SO(N)) dx.

Further, there exists R ∈ L2 (S, SO(N)) that satisfies the same estimate as in (i). All constants C above depend on S and {Ci }3i=1 , but not on u or h  1. Proof. 1. We closely follow the outline of the proof of Theorem 3.23. Recall the definitions of the local neighbourhoods on S and Sh : Dz,h = Bh (z) ∩ S,

Bz,h = π −1 (Dz,h ) ∩ Sh

for all z ∈ S,

4.5 Approximation theorem and Friesecke-James-M¨uller’s constant in thin shells

81

where Bh (z) denotes a ball in RN . The main observation is that by Theorem 4.1, the estimate (4.2) may be applied on each domain Bz,h , with a uniform constant C that is independent of z and h, and with rotation matrices Rz,h ∈ SO(N) to yield: Z Bz,h

|∇u(x) − Rz,h |2 dx ≤ C

Z

dist2 (∇u, SO(N)) dx.

(4.28)

Bz,h

We also recall that, having chosen a cut-off function θ ∈ Cc∞ ([0, 1), [0, 2]), equal R to a nonzero constant in a neighborhood of 0, and satisfying 01 θ = 1, the induced family of mollifiers {ηz : Sh → R}z∈S given below satisfies properties in (3.36): θ (|πx − z|/h) . Sh θ (|πx − z|/h) dx

ηz (x) = R

Consequently, we define the matrix field R˜ ∈ H 1 (S, RN×N ) by: ˜ = R(z)

Z Sh

ηz (x)∇u(x) dx,

and as in (3.37), (3.39) and (3.40), show that there holds: ˜ − Rz,h |2 ≤ |R(z)

C hN

Z

dist2 (∇u, SO(N)) dx

for all z ∈ S,

(4.29)

Bz,h

together with the estimates in (i) and (ii).

Fig. 4.2 A thin shell Sh and its partition into volume elements that are bilipschitz homeomorphic n(h) to the ball Bh (0) ⊂ RN . In the proof of Theorems 4.8 and 3.23, the sets {Bzi ,h }i=1 may overlap.

2. We now want to replace the RN×N -valued field R˜ by a SO(N)-valued field R for which the bounds (i) and (ii) still hold. To this end, set ε  1 and define:  ˜ ˜ PSO(N) R(z) if dist(R(z), SO(N)) < ε R(z) = IdN otherwise,

82

4 Friesecke-James-M¨uller’s inequality

where PSO(N) is the orthogonal projection onto the compact manifold SO(N). It is ˜ − R(z)| ≤ C dist(R(z), ˜ clear that R ∈ L2 (S, SO(N)) and that |R(z) SO(N)). Hence: Z Sh

|∇u − R ◦ π|2 dx ≤ C ≤C

Z Sh

Z Sh

|∇u − R˜ ◦ π|2 + dist2 (R˜ ◦ π, SO(N)) dx

3|∇u − R˜ ◦ π|2 + dist2 (∇u, SO(N)) dx ≤ C

Z Sh

dist2 (∇u, SO(N)) dx,

as claimed. The proof is done. Remark 4.9. In the context of Theorem 4.8, the additional smallness assumption: 1 hN

Z Sh

dist2 (∇u, SO(N)) dx  1,

(4.30)

˜ implies that dist2 (R(z), SO(N)) < ε for every z ∈ S, in virtue of (4.29). Hence, under (4.30) the definition of R reduces to: ˜ R(z) = PSO(N) R(z)

for all z ∈ S,

and consequently R ∈ H 1 (S, SO(N)) with ∇R satisfying the same estimate as in (ii): 2 C R 2 S |∇R| dσ (z) ≤ h3 Sh dist (∇u, SO(N)) dx.

R

As a corollary, we readily obtain: Corollary 4.10. Let S ⊂ RN be a smooth, compact, connected, oriented hypersurface, with its boundary ∂ S given by finitely many Lipschitz curves, and assume (4.27). Then, for every u ∈ H 1 (Sh , RN ) defined on Sh in (4.26) with h  1, there exists R0 ∈ SO(N) such that: Z Sh

|∇u − R0 |2 dx ≤

C h2

Z Sh

dist2 (∇u, SO(N)) dx.

In particular, there also exists an affine map w with gradient R0 , such that: ku − wkH 1 (Sh ) ≤

C k dist(∇u, SO(N))kL2 (Sh ) . h

The constants C above depend only on S and {Ci }3i=1 in (4.27). > ˜ dσ (z). Proof. Let R˜ : S → RN×N be as in Theorem 4.8 and define: R˜ 0  S R(z) Applying additionally the Poincar´e-Wirtinger inequality on S, we obtain:

4.6 Friesecke-James-M¨uller’s inequality in the plane

Z Sh

|∇u − R˜ 0 |2 dx ≤ C ≤C C ≤ 2 h

Z Sh

Z

83

˜ 2 dx + h |∇u − Rπ|

Z S


˜ − R˜ 0 |2 dσ (z) |R(z) Z

2

Sh

Z

dist (∇u, SO(N)) dx + h

˜ 2 dσ (z) |∇R|




S 2

Sh

dist (∇u, SO(N)) dx.

Replace now R˜ 0 by R0 ∈ SO(N) such that |R˜ 0 − R0 | = dist(R˜ 0 , SO(N)). Since dist(R˜ 0 , SO(N)) ≤ dist(∇u(x), SO(N)) + |∇u(x) − R˜ 0 | for each x ∈ Sh , the same estimate above is valid for R0 as well. Finally, writing w(x) = R0 x + a where a ∈ RN is obtained by applying the same argument as in the proof of Theorem 3.25 to each component of the function u − R0 x on Sh , we obtain the second claimed bound.

¨ 4.6 Friesecke-James-Muller’s inequality in the plane In this section, we study the estimate (4.2) for Ω = R2 . We show that the nonlinear rigidity result in Theorem 4.1 is still valid, and that similarly to homogeneous Korn’s inequality, the optimal constant C in (4.2) equals to 2. The key argument in the proof of Theorem 4.11 below is special to dimension N = 2, as it relies on the conformal decomposition of matrices in R2×2 . We state the result and then continue with three auxiliary lemmas of independent interest, which are valid in any dimension. 1, Theorem 4.11. For every u ∈ Hloc (R2 , R2 ) there exists R ∈ SO(2) with:

Z

2

R2

|∇u − R| dx ≤ 2

Z R2

dist2 (∇u, SO(2)) dx.

(4.31)

Moreover, there holds: (i) the multiplicative constant 2 in the right hand side of (4.31) is optimal, 1 (R2 , R2 ) with in the sense that for every R0 ∈ SO(2) there exists u ∈ Hloc dist(∇u, SO(2)) ∈ L2 (R2 , R) \ {0} such that: Z

min

R∈SO(2) R2

|∇u − R|2 dx =

Z R2

|∇u − R0 |2 dx

Z

=2

R2

(4.32) 2

dist (∇u, SO(2)) dx,

(ii) vector fields for which the optimal constant is attained, namely: n 1 u ∈ Hloc (R2 , R2 ); dist(∇u, SO(2)) ∈ L2 (R2 , R), o u satisfies (4.32) for some R0 ∈ SO(2) ,

84

4 Friesecke-James-M¨uller’s inequality

Theorem. (continued) have the defining property that their gradients are of the form:     a b cos α − sin α ∇u = R0 R(α) + , R(α) = b −a sin α cos α

(4.33)

for some α, a, b ∈ L2 (R2 , R). Conversely, for every α ∈ L2 (R2 , R) there 1 (R2 , R2 ) with (4.32) and (4.33). exists a, b ∈ L2 (R2 , R) and u ∈ Hloc The first auxiliary result extends Liouville’s theorem according to which a bounded (from above of below) harmonic function defined on the whole RN must be constant. 2 (RN , R) satisfies ∆ w = 0, and that it can be Lemma 4.12. Assume that w ∈ Lloc written as: w = f + g for some f ∈ L2 (RN , R) and g ∈ L∞ (RN , R). Then w ≡ const.

Proof. Fix x0 , y0 ∈ RN . For any r > 0, harmonicity of w implies that: ? ? Z 1 w dx , w dx = |w(x0 ) − w(y0 )| = w dx − |Br (0)| Br (x0 )÷Br (y0 ) Br (x0 ) Br (y0 ) where ÷ stands for the symmetric difference: B1 ÷ B2 = (B1 \ B2 ) ∪ (B2 \ B1 ). Thus: |w(x0 ) − w(y0 )| ≤

1 |Br (0)|

Z

| f | dx +

Br (x0 )÷Br (y0 )

1 |Br (0)|

Z

|g| dx

Br (x0 )÷Br (y0 )

|Br (x0 ) ÷ Br (y0 )|1/2 |Br (x0 ) ÷ Br (y0 )| k f kL2 (RN ) + kgkL∞ (RN ) |Br (0)| |Br (0)|  1
|Br (x0 ) ÷ Br (y0 )|  ≤ k f kL2 (RN ) + kgkL∞ (RN ) , + |Br (0)| |Br (0)|

≤

The quantity in the first parentheses above converges to 0 as r → ∞. Consequently, we deduce w(x0 ) = w(y0 ), as claimed. 1 (RN , R) satisfy: Lemma 4.13. Let f ∈ L2 (RN , RN ) and u ∈ Hloc

∆ u = div f

in D 0 (RN ).

(4.34)

Then u can be decomposed as u = v + w, with the following properties: 2 2 v, w ∈ Lloc , ∇v ∈ L2 , ∇w ∈ Lloc , ∆ w = 0 in R2 ,

∇v = lim ∇vn n→∞

strongly in L2 (RN , RN ), for some {vn ∈ Cc∞ (RN , R)}∞ n=1 .

Proof. For each n ≥ 1, define vn ∈ H01 (Bn (0) ⊂ RN , R) as the solution to: ∆ vn = div f in Bn (0),

vn = 0 on ∂ Bn (0).

(4.35)

4.6 Friesecke-James-M¨uller’s inequality in the plane

85

Since k∇vn kL2 (Bn (0) ≤ k f kL2 (Bn (0)) ≤ k f kL2 (RN ) , then up to a subsequence one has the weak convergence ∇vn * z in L2 (RN , RN ). Also, z = ∇v for some v ∈ 2 (RN , R), and consequently ∆ v = div f in RN . This implies that ∆ w = 0 where Lloc w = u − v, proving the first line in the assertion (4.35). Since ∇vn * ∇v, Mazur’s theorem implies that ∇v is the strong L2 -limit of {∇v˜n }n→∞ , where each v˜n is comprised of some finite linear combination of {vn }∞ n=1 . Since each v˜n ∈ H01 (Brn (0), R), the last result in (4.35) then follows by density. We finally deduce an extension of the classical statement that the determinant is a null-Lagrangean to the whole space: 1 (RN , RN ) with ∇u ∈ L2 (RN , RN×N ). Then: Lemma 4.14. Let u ∈ Hloc

Z RN

det ∇u dx = 0.

Proof. Since ∆ u = div ∇u in D 0 (RN ) we may apply Lemma 4.13 to each vector field { fi  ∇ui ∈ L2 (RN , RN )}Ni=1 and write u = v + w satisfying (4.35). In particular, ∆ w = 0 and ∇w = ∇u − ∇v ∈ L2 (RN , RN×N ), so it follows from Lemma 4.12 that ∇w = 0 and further, by the last assertion in (4.35): ∇u = ∇v = lim ∇vn n→∞

strongly in L2 (RN , RN×N ) for some {vn ∈ Cc∞ (RN , RN )}∞ n=1 .

It remains to prove the claim for u ∈ Cc∞ (RN , RN ), which is a standard argument. By the expansion of the determinant, one sees that (det ∇u)IdN = (cof∇u)T ∇u, so: det ∇u =


1 1 trace (det ∇u)IdN = hcof ∇u : ∇ui. N N

Let supp u ⊂ Br (0). Integrating by parts leads to: 1 det ∇u dx = N N R

Z

1 hcof ∇u : ∇ui dx = − N Br (0)

Z

Z

hdiv cof ∇u, ui dx = 0.

Br (0)

The proof is done. We now recall the conformal–anticonformal decomposition of two-dimensional matrices. Define the following linear subspaces of R2×2 , which are mutually orthogonal and complementary, i.e. R2×2 = R2×2 ⊕ R2×2 c a :       a b a b 2×2 = ; a, b ∈ R . = ; a, b ∈ R , R R2×2 a c b −a −b a For F ∈ R2×2 , its orthogonal projections on R2×2 and R2×2 are, respectively: c a 1 F = 2 c



F11 + F22 F21 − F12

F12 − F21 F11 + F22

 ,

1 F = 2 a



F11 − F22 F12 + F21

F12 + F21 F22 − F11

 .

86

4 Friesecke-James-M¨uller’s inequality

It then follows that: F = F c + F a,

|F|2 = |F c |2 + |F a |2 ,

det F = 2(|F c |2 − |F a |2 ),

(4.36)

and since SO(2) ⊂ R2×2 c , then also: a dist(F, SO(2)) ≥ dist(F, R2×2 c ) = |F |.

(4.37)

In particular, there holds: |cof F − F| = | − 2F a | ≤ 2 dist(F, SO(2)).

(4.38)

Proof of (4.31) in Theorem 4.11 1. Without loss of generality, assume that dist(∇u, SO(2)) ∈ L2 (R2 , R). By virtue of (4.38), we deduce that f  ∇u − cof ∇u ∈ L2 (R2 , R2×2 ) so that ∆ u = div f . Recalling Lemma 4.13, we write the decomposition u = v + w, whose components satisfy conditions in (4.35). We now prove that: ∇w ≡ R0 ∈ SO(2).

(4.39)

To this end, for a small ε  1 define the projection:  PSO(2) ∇u(x) if dist(∇u(x), SO(2)) < ε g(x) = Id2 otherwise. Then: ∇w = g + h;

with g ∈ L∞ (R2 , R2×2 ) and h ∈ L2 (R2 , R2×2 ).

(4.40)

The assertion that h = ∇w − g = ∇u − g − ∇v ∈ L2 (R2 , R2×2 ) follows from the integrability of dist2 (∇u, SO(2)) as follows. We already know that ∇v ∈ L2 (R2 , R2×2 ). For h1 = ∇u − g note that |h1 (x)| = dist(∇u, SO(2)) when dist(∇u(x), SO(2)) < ε, while when dist(∇u(x), SO(2)) ≥ ε, we still have: √ |h1 (x)| = |∇u(x) − Id2 | ≤ dist(∇u(x), SO(2)) + 2 2 √ 2 2
≤ 1+ dist(∇u(x), SO(2)). ε Now, since ∇w is harmonic, it follows that ∇w is constant by Lemma 4.12. But dist(∇w, SO(2)) ≤ dist(∇u, SO(2)) + |∇v| ∈ L2 (R2 , R), implying (4.39). 2. We conclude the argument. By (4.39) there holds: ∇u = ∇v + R0 . Since R det ∇v = 0 from Lemma 4.14, the last formula in (4.36) yields: Z R2

|(∇v)c |2 dx =

Z R2

|(∇v)a |2 dx.

(4.41)

Consequently, by (4.36), (4.41), (4.37) and the fact that (R0 )a = 0, we obtain:

4.6 Friesecke-James-M¨uller’s inequality in the plane

Z R2

|∇u−R0 |2 dx = Z

=2

R2

|∇v|2 dx =

Z

R2 R2

87

|(∇v)c |2 + |(∇v)a |2 dx = 2

R2

a 2

Z

=2

Z

|(∇v + R0 ) | dx ≤ 2

Z R2

Z R2

|(∇v)a |2 dx

dist2 (∇v + R0 , SO(2)) dx

dist2 (∇u, SO(2)) dx.

This ends the proof. Proof of Theorem 4.11 (i) 1. Without loss of generality, we may assume that R0 = Id. We will look for a 1 (R2 , R2 ) such that: vector field u ∈ Hloc     a(x) b(x) cos α − sin α ∇u(x) = R(α(x)) + with R(α) = , (4.42) b(x) −a(x) sin α cos α and satisfying: ∇u − Id2 ∈ L2 (R2 , R2×2 ). c 2 R2 |(∇u) − Id2 |

R

Indeed, we have: (4.36). Hence, by (4.36) again: Z R2

|∇u − Id2 |2 dx = 2

Z R2

dx =

R

R2

|(∇u)a |2

|(∇u)a |2 dx = 2

(4.43)

dx by Lemma 4.14, (4.43) and

Z

dist2 (∇u, SO(2)) dx,

R2

because (∇u)c = R(α) ∈ SO(2). Also, R2 dist2 (∇u, SO(2)) dx = 2 R2 (a2 + b2 ) dx. On the other hand, there always exists the unique rotation R which makes the quantity in the left hand side of (4.32) finite: R

Z R2

|∇u − R|2 dx ≥

1 2

Z R2

|R − Id2 |2 dx −

R

Z R2

|∇u − Id2 |2 dx.

This proves the theorem, provided (4.42) and (4.43) hold. 2. We now show that for any α ∈ L2 (R2 , R) there exists a vector field g = (a, b) ∈ satisfying (4.42). Then (4.43) will follow automatically, since:

L2 (R2 , R2 )

Z R2

2

|R(α) − Id2 | dx = 2

Z ZR

=2

2

R2

(cos α − 1)2 + (sin α)2 dx (2 − 2 cos α) dx ≤ 2

Z R2

|α|2 dx.

The last inequality above is a consequence of noting that the function α 7→ α 2 + 2 cos α − 2 attains its minimum value 0 at α = 0, because (α 2 + 2 cos α − 2)0 = 2(α − sin α) is positive for α > 0 and negative for α < 0. 1 (R2 , R2 ) with ∇u of the form (4.33) exists Fix α ∈ L2 (R2 , R). The map u ∈ Hloc if and only if the right hand side in (4.33) is curl-free, i.e.:

88

4 Friesecke-James-M¨uller’s inequality

curl g = div f

div g = curl f ,

and

where: f = (sin α, cos α − 1) ∈ L2 (R2 , R2 ).

(4.44)

The system in (4.44) can be solved by Fourier transform, namely: g = F −1 (h),

h(x) = −

Ex Dx E x⊥ , F ( f )(x) + , F ( f )(x) , |x| |x| |x| |x|

D x⊥

(4.45)

where x⊥ = (x1 , x2 )⊥ = (−x2 , x1 ). Here F stands for the Fourier transform on L2 (R2 , C) and we identify complex variable functions with R2 -valued vector fields. From (4.45) there follows (4.44): hF (g)(x), x⊥ i = hF ( f )(x), xi,

hF (g)(x), xi = −hF ( f )(x), x⊥ i.

(4.46)

Therefore, for every f ∈ L2 (R2 , R2 ) there exists a unique g ∈ L2 (R2 , R2 ) solving (4.44). This achieves the proof of the result, and also: kgkL2 (R2 ) = khkL2 (R2 ) = kF ( f )kL2 (R2 ) = k f kL2 (R2 ) , by Plancherel’s identity and by a direct inspection of (4.45). Proof of Theorem 4.11 (ii) 1 (R2 , R2 ) satisfies: Assume that u ∈ Hloc Z R2

|∇u − R0 |2 dx = 2

Z R2

dist2 (∇u, SO(2)) dx < ∞.

Since |∇u − R|2 = |(∇u)c − R|2 + |(∇u)a |2 for all R ∈ SO(2) by (4.36), we get: dist2 (∇u, SO(2)) = dist2 ((∇u)c , SO(2)) + |(∇u)a |2 ,

(4.47)

and in particular: (∇u)a ∈ L2 (R2 ). Further, by (4.36) and Lemma 4.14, one has: Z R2

|(∇u)c − R0 |2 dx =

Z R2

|(∇u)a |2 dx.

Therefore, by (4.36) and (4.47): Z R2

|(∇u)c − R0 |2 =

1 2

Z

Z

=

R2

Z

=

R2

R2

|∇u − R0 |2 dx =

Z R2

dist2 (∇u, SO(2)) dx

dist2 ((∇u)c , SO(2)) + |(∇u)a |2 dx dist2 ((∇u)c , SO(2)) dx +

Z R2

|(∇u)c − R0 |2 dx,

which implies dist2 ((∇u)c , SO(2)) dx = 0. Hence, ∇u has the form in (4.42) and: R

R(α) − R0 = ∇u − R0 − (∇u)a ∈ L2 (R2 , R2×2 ),

4.7 Rigidity estimates in conformal setting

89

by (4.47). The proof is complete.

4.7 Rigidity estimates in conformal setting In this section we outline the status of the rigidity statements and rigidity estimates, in the so called “conformal” setting, in which the trace-free skew-symmetric matrices and the conformal matrices replace the sets so(N) and SO(N) in the respective linear and nonlinear cases already discussed. We do not provide the proofs but refer the interested reader to the literature listed in section 4.8. The following result can be seen as an improvement of First Korn’s inequality in Theorem 3.5: Theorem 4.15. Let N ≥ 3 and let Ω ⊂ RN be an open, bounded, Lipschitz domain. Then, for every v ∈ H 1 (Ω , R3 ) there holds: Z

|∇v|2 dx ≤ C

Ω

2
1 |v|2 + sym∇v − (div v) IdN dx, N Ω

Z

(4.48)

where the constant C depends only on Ω . Denote the space of conformal Killing fields on Ω :  1 I c (Ω ) = w ∈ H 1 (Ω , RN ); sym∇v − (div v) IdN = 0 N The usual argument by contradiction, as in the proofs of Theorem 3.1, Theorem 3.8 and Theorem 3.31, allows one to deduce from 4.15 the following rigidity estimate: Corollary 4.16. Let N ≥ 3 and let Ω ⊂ RN be an open, bounded, Lipschitz domain. Then, for every v ∈ H 1 (Ω , R3 ) there exists w ∈ I c (Ω ) such that: Z

|∇v − ∇w|2 dx ≤ C

Ω

sym∇v − 1 (div v)IdN 2 dx, N Ω

Z

(4.49)

where C depends only on Ω . It is possible to identify the elements of the functional kernel I c (Ω ). As in Lemma 3.2, it consists of specific smooth functions which are now quadratic polynomials rather than linear rigid motions as before:  1 I c (Ω ) = w(x) = Ax + λ x + a|x|2 − ha, xix + b; A ∈ so(N), λ ∈ R, a, b ∈ RN . 2 Clearly, I c (Ω ) is finitely dimensional. The above is valid for when N ≥ 3. On the other hand, in dimension 2 we have: Lemma 4.17. For N = 2, each w ∈ I c (Ω ) can be identified with a holomorphic function f : C → C. In particular, dim I c (Ω ) = ∞.

90

4 Friesecke-James-M¨uller’s inequality

Proof. For w = (w1 , w2 ) : Ω → R2 , one has:   1 1 ∂1 w1 − ∂2 w2 , ∂1 w2 + ∂2 w1 sym∇w − (div w)Id2 = . 2 2 ∂1 w2 + ∂2 w1 , ∂2 w2 − ∂1 w1 Thus, w ∈ I c (Ω ) if an only if: ∂1 w1 = ∂2 w2

and ∂1 w2 = −∂2 w1

in Ω .

The above are precisely the Cauchy-Riemann equations for the holomorphic functions f (x1 + ix2 ) = w1 (x1 , x2 ) + iw2 (x1 , x2 ). By Lemma 4.17 we see that Theorem 4.15 can indeed only hold for N ≥ 3. This is because (4.48) implies that the norms kvkL2 (Ω ) and k∇vkL2 (Ω ) are equivalent in I c (Ω ), so this space must be finitely dimensional as in the proof of Lemma 3.30(ii). The counterpart of (4.49) for v ∈ VΓ0 ,Γ1 (Ω ) can be formulated and proved as in Theorem 3.8, with respect to the subspace IΓc0 ,Γ1 (Ω ) = VΓ0 ,Γ1 (Ω ) ∩ I c (Ω ). Note that in the homogeneous case, i.e. under the Dirichlet boundary condition w = 0 on ∂ Ω , the maximum principle readily implies in dimension N = 2 that I∂cΩ ,0/ (Ω ) = {0}, so the infinite dimensional obstruction is no more present. Similarly to Theorem 3.6 one directly can prove: Lemma 4.18. For every open domain Ω ⊂ R2 and every v ∈ H01 (Ω , R2 ) there holds: Z

|∇v|2 dx = 2

Ω

(sym∇v) − 1 (div v) Id2 2 dx. 2 Ω

Z

Proof. It suffices to take v ∈ C0∞ (Ω , R2 ). Integrating by parts the basic formula (3.2) multiplied by v, leads to: 1 ∇v : sym∇v − (div v) Id2 i dx 2 Ω Ω Z

1 1 1 =2 sym∇v : sym∇v − (div v) Id2 i − (div v) Id2 : sym∇v − (div v) Id2 i dx 2 2 2 Ω Z 2 1 = 2 (sym∇v) − (div v) Id2 dx. 2 Ω

Z

|∇v|2 dx = 2

Z

This concludes the proof. We now pass to nonlinear conformal rigidity. Define the smooth manifold:  C+ SO(N) = αR; R ∈ SO(N), α > 0 . One can check that the tangent space to C+ SO(N) at IdN consists of those matrices that are gradients of the elements of I c (Ω ):  1 TIdN C+ SO(N) = A ∈ RN×N ; sym A − (tr A)IdN = 0 N

4.8 Bibliographical notes

91

We have the following counterpart of the Liouville’s lemma in Lemma 4.2: Theorem 4.19. [Liouville’s theorem] Let N ≥ 3 and let Ω ⊂ RN be an open, bounded and connected domain. If u ∈ 1,N W (Ω , RN ) satisfies ∇u ∈ C+ SO(N) a.e. in Ω , then u must either be affine, or an orientation preserving M¨obius transform (a composition of an affine map and an inversion with respect to a sphere), i.e.: u(x) = Qx + b

or

u(x) = QJ

x−a + b, |x − a|2

for some Q ∈ C+ SO(N), a ∈ RN \ Ω , b ∈ RN , and J = diag(1, . . . , −1). Geometrically, the above theorem relates C+ SO(N) with the special M¨obius group MN , consisting of the orientation-preserving elements of the M¨obius group that is generated by inversions with respect to spheres and reflections by hyperplanes. The quantitative version of Theorem 4.19 is: Theorem 4.20. Let N ≥ 3 and let Ω 0 b Ω ⊂ RN be two open, bounded, connected domains. Fix a compact set E ⊂ C+ SO(N) \ {0} that is has finitely many connected components and is invariant with respect to rotations, i.e. satisfies SO(N)E = E. Then, for every u ∈ H 1 (Ω , RN ) there exists w ∈ H 1 (Ω 0 , RN ) that satisfies ∇w ∈ E a.e. in Ω 0 , and such that with a constant C depending only on E, Ω , Ω 0 , there holds: Z Ω0

|∇u − ∇w|2 dx ≤ C

Z

dist2 (∇u, E) dx.

Ω

When Ω = Ω 0 is additionally Lipschitz, then the result holds if an only if E = Sn n i=1 mi SO(N) for finitely many positive constants {mi }i=1 .

4.8 Bibliographical notes Theorem 4.1 appeared in Friesecke et al. [2002]. It quantifies the classical result due to Reshetnyak [1967], see also Reshetnyak [1994], stating that if a sequence 1 N of vector fields {un }∞ n=1 converges weakly in H (Ω , R ) and the corresponding se2 ∞ 2 quence {dist (∇un , SO(N))}n=1 converges to 0 in L (Ω , R) then ∇un must converge (strongly in L2 (Ω , RN×N )) to a single rotation R ∈ SO(N). The estimate (4.2) was previously shown by John [1961, 1972] for u locally bilipschitz, with further developments along this line available in Benyamini and Lindenstrauss [2000]. The proof of Lemma 4.2 is due to Reshetnyak [1994]. The truncation results in Theorem 4.3 and Lemma 4.5 are taken from Friesecke et al. [2002], their proofs are based on [Evans and Gariepy, 1992, sections 6.6.3 and 6.6.2]. The Vitali covering theorem can be found in Evans and Gariepy [1992], while the Kirszbraun extension theorem appears in Kirszbraun [1934]; see also Azagra et al. [2021]. The proof of Lemma 4.7 follows the arguments in [Heinonen et al.,

92

4 Friesecke-James-M¨uller’s inequality

2015, Proposition 4.1.15], which are similar to the classical Whitney decomposition of open subsets of RN into the non-overlapping dyadic cubes, see [Reshetnyak, 1994, Lemma 4.3]. The approximation of deformation fields on thin shells in Theorem 4.8 can be found in Friesecke et al. [2006], with a prior proof in Friesecke et al. [2002]. All results in section 4.6 are taken from Lewicka and M¨uller [2016]. Theorem 4.15 in the general L p setting p ∈ (1, ∞) is due to Reshetnyak [1994]; see also Dain [2006] in the framework of general relativity. Other contexts, notably in Orlicz or pseudo-Euclidean spaces, are explored in Fuchs and Schirra [2009], Breit et al. [2017] and Wang [2008]. The Liouville theorem in Theorem 4.19 has been first proved in Liouville [1850] for C 3 mappings, then in Gehring [1962] for homeomorphisms in W 1,N and finally in Reshetnyak [1967], see also Reshetnyak [1994]. The regularity assumption on u has been relaxed to u ∈ W 1,p (Ω , RN ) for all p ≥ pN and some pN < N in Iwaniec [1992]. The optimal regularity, conjectured at u ∈ W 1,N/2 (Ω , RN ), has been established for even dimensions N by Iwaniec and Martin [1993]. Theorem 4.20 is due to Faraco and Zhong [2005]. Extensions of Friesecke-James-M¨uller’s inequality to multiwell problems where S the connected set SO(N) is replaced by the set of the form K = ni=1 SO(N)Ai were addressed in Shu [2000], Chaudhuri and M¨uller [2004, 2006], De Lellis and Sz´ekelyhidi [2006] and further in Chermisi and Conti [2010], Jerrard and Lorent [2013]. An interesting extension to incompatible fields, has been given in M¨uller et al. [2014]. Namely, given any P ∈ L2 (Ω , R2×2 ), there exists R ∈ SO(2) with:
kP − RkL2 ≤ C k dist2 (P, SO(2))kL2 + |Curl P|(Ω ) , provided that Curl P is a vector-valued bounded Radon measure on a bounded, Lipschitz, simply connected domain Ω ⊂ R2 . A higher dimensional analogue of this result is available in Conti and Garroni [2021], and in case of Lorentz spaces in Lauteri and Luckhaus [2017].

Part II

Dimension reduction in classical elasticity

Chapter 5

Limiting theories for elastic plates and shells: nonlinear bending

Elastic materials exhibit qualitatively different responses to different kinematic boundary conditions or body forces. A sheet of paper crumples under compressive forces, a cylinder buckles in presence of axial loads, while a clamped convex shell enjoys resistance to bending and stretching, which may collapse if a hole is pierced into it. Growing tissues such as leaves, attain non-flat elastic equilibrium configurations with non-zero stress, even in the absence of any external forces. Such observations gave rise to many interesting questions in the mathematical theory of elasticity, whose main goal is to explain these apparently different phenomena based on some common principles. Among others, the variational approach proved to be effective towards the derivation of models capturing the aforementioned distinct responses to different scaling regimes of the applied forces. The strength of this approach lies in its ability to predict the appropriate model without any a priori assumptions other than the general principles of nonlinear elasticity. The outline of this chapter is as follows. In section 5.1 we introduce two main energy functionals in the (static) variational description of nonlinearly elastic three dimensional materials: the elastic energy of deformation and the total energy which takes into account an applied force. We specify the minimal assumptions on the elastic energy density, that will be used throughout this monograph and provide a few examples. In section 5.2 we specify to thin shells, presenting the set-up for the dimension reduction analysis of the asymptotic behaviour of the families of shells in the limit of their vanishing thickness h → 0. In particular, we show how the scaling of the applied forces in order hα implies the compatible scaling hβ of the elastic energy at (approximate) minimizers of the total energy, as long as α ≥ 2 (equivalently, β ≥ 2). The key ingredient of the subsequent analysis consists in identifying the Γ -limits of the elastic energies scaled by h−β , together with the compactness properties sufficient to deduce the correspondence between minimizing sequences as h → 0, and the minimizers of the limiting energy, as explained in chapter 2. The main purpose of this chapter is to discuss the case β = 2. In section 5.3 we prove both the compactness of sequences of deformations whose elastic energy scales like h2 , and the Γ − lim inf lower bound in terms of Kirchhoff’s energy IK . This energy, defined on the set of H 2 -regular isometries of the shell’s midsurface © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_5

95

96

5 Limiting theories for elastic plates and shells: nonlinear bending

S, measures the total change in curvature (i.e. bending) induced by such isometry. Bending is quantified by the quadratic form that can be explicitly calculated from an appropriate contraction of the second derivative of the elastic energy density. In order to show that IK is also the Γ − lim sup upper bound, we necessitate the second order version of a truncation result in Sobolev spaces, that has already been used in the proof of Friesecke-James-M¨uller’s inequality in chapter 4. This result is presented in section 5.4 and proved in section 5.5 (this proof is quite involved and may be skipped at the first reading), allowing for a construction of the recovery family in section 5.6. We then combine all the previously given results pertaining to the scaling β = 2, in section 5.7, where we prove the Γ -convergence of the total energies scaled by h−2 , and further deduce convergence of their approximate minimizers. In section 5.8 we observe that under scalings β < 2, no compactness in the previously developed framework is possible. Thus, in section 5.9 we consider higher energy scalings β > 2. We prove the compactness property which limits the set of all the isometries of S to only its rigid motions as the limit of deformations whose elastic energy scales like hβ as h → 0. We then show that the displacements from these rigid motions are of order hβ /2−1 and that after rescaling they accumulate at the set of infinitesimal isometries V (S). The corresponding Γ -liminf lower bound is provided in section 5.10, and it consists of the first order (in terms of hβ /2−1 ) bending due to the change of the curvature of S displaced by an element of V (S), and the second order stretching due to the change in the metric on S produced by the limiting strain. We further identify this quantity in each of the three regimes: β ∈ (2, 4), β = 4 and β > 4. Cases β ≥ 4 will be developed in chapter 6, while the case β ∈ (2, 4) will be discussed in chapter 7 and, in a larger context, in chapter 8.

5.1 Set-up of three dimensional nonlinear elasticity Given the reference configuration Ω ⊂ R3 of an elastic body with constant temperature and density, the system of equations for balance of linear momentum for the deformation u = u(t, x) ∈ R3 reads as follows:
∂tt u − div DW (∇u) + g = 0. (5.1) Here, DW : R3×3 → R3×3 is the so-called Piola-Kirchhoff stress tensor given as the derivative of the energy density W : R3×3 → [0, ∞], the divergence of a R3×3 -valued matrix field is taken row-wise, and g : Ω → R3 is the external force. The density W is assumed to satisfy, for all F ∈ R3×3 , R ∈ SO(3), with a uniform constant c > 0: W (RF) = W (F) W (R) = 0

frame invariance normalization

W (F) ≥ c · dist2 (F, SO(3))

non-degeneracy

W is C 2 -regular in the vicinity of the group of proper rotations SO(3).

(5.2)

5.1 Set-up of three dimensional nonlinear elasticity

97

The above are the only assumptions that will be used in the mathematical analysis throughout this monograph. For physical relevance, one adds an extra condition: W|{det F≤0} ≡ +∞

and

lim W (F) = ∞

det F→0+

orientation-preservation. (5.3)

The steady state solutions to (5.1) satisfy the equilibrium equations:
div DW (∇u) = g which, expressed in their weak form, coincide with the Euler-Lagrange equations for the critical points of the total energy functional: J (u) =

Z

Z

W (∇u) dx + Ω

hg(x), u(x) − xi dx.

(5.4)

Ω

The first term above corresponds to the elastic energy of deformation u, while the second term quantifies the work done by the displacement u − id3 to move the element of mass from a typical point x ∈ Ω to the point u(x). As a first step towards understanding the dynamical problem (5.1) it is natural to study the minimizers of (5.4), in an appropriate function space. However, due to the loss of convexity of W caused by the frame invariance assumption in (5.2), these problems cannot be dealt with the direct method of the calculus of variations. One advantageous direction of research has been to restrict the attention to domains Ω which are thin in one or two directions, and hence practically reduce the theory to a 2d or 1d problem. The classical approach is to propose a formal asymptotic expansion for the solutions (in other words an Ansatz) and derive the corresponding limiting theory by considering the first terms of the 3d equations under this expansion. We have already seen in section 2.2 an example of the more rigorous variational approach through Γ -convergence, for the linear counterpart of the energy in (5.4). In the following sections we will apply the same approach in the more complex nonlinear context, leading to the derivation of a hierarchy of thin plate and shell theories. Among other features, it provides a rigorous justification of convergence of minimizers of (5.4) to minimizers of suitable lower dimensional limit energies. We now introduce a technical notation related with the energy density W satisfying (5.2). Firstly, there must be DW (R) = 0 for all R ∈ SO(3), so in view of the normalisation and frame invariance conditions, the lowest order relevant information on W in the vicinity of SO(3) is carried by its second derivative at Id3 . Definition 5.1. Assume that W : R3×3 → [0, ∞] satisfies (5.2) and that it is C 2 regular in a neighbourhood of SO(3). We define the quadratic form: Q3 (F) = D2W (Id3 )(F, F)

for all F ∈ R3×3 .

and the linear operator L3 : R3×3 → R3×3 given by:

 Q3 (F) = L3 F : F for all F ∈ R3×3 .

98

5 Limiting theories for elastic plates and shells: nonlinear bending

We observe the simple fact: Lemma 5.2. In the context of Definition 5.1, we have: (i) the quadratic form Q3 depends only on the symmetric part of its argument and it is positive definite on the space of symmetric matrices, (ii) the operator L3 depends only on the symmetric part  and it  of its argument

returns a symmetric matrix. Moreover: L3 F1 : F2 = L3 F2 : F1 for all F1 , F2 ∈ R3×3 . Proof. Fix any A ∈ so(3) and recall that etA ∈ SO(3), so DW (etA ) = 0. Fix now F ∈ R3×3 and differentiate the map R 3 t 7→ hDW (etA ) : Fi = 0 at t = 0, in order to obtain: D2W (Id3 )(F, A) = 0. Then, in particular: D2W (Id3 )(F, F) = D2W (Id3 )(F, symF) = D2W (Id3 )(symF, symF). For the second claim in (i), we use the nondegeneracy of W . For any F ∈ R3×3 sym : t2 2 D W (Id3 )(F, F) + o(t 2 ) = W (Id3 + tF) 2
≥ c dist2 (Id3 + tF, SO(3)) = c |tF|2 + o(t 2 ) , 2 . This ends the proof of (i), while (ii) follows which implies that Q 

3 (F) ≥ 2c|F| directly by writing: L3 F1 : F2 = D2W (Id3 )(F1 , F2 ).

We end this section by a discussion of two important classes of examples of W : Example 5.3. Assume, in addition to the frame invariance condition in (5.2), the isotropy of the energy density W : R3×3 → [0, ∞], namely: W (FR) = W (F)

for all F ∈ R3×3 , R ∈ SO(3).

(5.5)

According to the representation theorem in Truesdell and Noll [2004], every frame invariant and isotropic W can be written exclusively in terms of the principal invariants of the left Cauchy deformation tensor FF T , namely: W (F) = σ¯ (tr(FF T ), tr cof(FF T ), det(FF T )). In view of the formula tr cof Q = 21 (tr Q)2 − 21 tr (Q2 ) valid for any Q ∈ R3×3 , we see that there exists a scalar function σ : R3 → [0, ∞), such that: W (F) = σ (|F|2 , |FF T |2 , det F). We now calculate the derivative DW (F), wherever defined at F ∈ R3×3 . We use the convention ∂AW (F) = hDW (F) : Ai, and obtain: DW (F) = 2∂1 σ (|F|2 , |FF T |2 , det F) F + 4∂2 σ (|F|2 , |FF T |2 , det F)FF T F + ∂3 σ (|F|2 , |FF T |2 , det F) cof F,

5.2 Elasticity on shells

99

since, by a direct calculation: ∂A |F|2 = 2hF : Ai, ∂A |FF T |2 = 4hFF T F : Ai and ∂A det F = hcof F : Ai, for any F, A ∈ R3×3 . Further, the second derivatives of the same expressions, at Id3 and in the direction of any A, A1 ∈ R3×3 , are:
2 ∂A,A |F|2 , |FF T |2 , det F (Id3 ) = hA : A1 i(2, 4, 1) + hsym A : A1 i(0, 8, −2). 1 Consequently, if W (Id3 ) = 0 and W is C 2 in a neighborhood of SO(3), then: 

for all A, A1 ∈ R3×3 , D2W (Id3 )(A, A1 ) = λ (tr A)Id3 + µ sym A, A1 with the Lam´e constants λ and µ given by:

 λ = ∇2 σ (3, 3, 1) : (2, 4, 1) ⊗ (2, 4, 1) ,

 µ = ∇σ (3, 3, 1), (0, 8, −2) .

In particular, for all A ∈ R3×3 : Q3 (A) = λ |tr A|2 + µ|sym A|2 ,

L3 A = λ (tr A)Id3 + µ sym A.

(5.6)

We now additionally show that the nonnegative definiteness of Q3 is equivalent to: µ ≥ 0,

3λ + µ ≥ 0.

(5.7)

Firstly, evaluating (5.6) at a traceless A and further at A = Id3 , and using Q3 (A) ≥ 0, implies (5.7) directly. To prove the contrary, write symA as the sum of orthogonal matrices: diag(a11 , a22 , a33 ) and a traceless B. Then |sym A|2 = ∑3i=1 a2ii + |B|2 , so: 3

3

3

i=1

i=1

i=1

λ |tr A|2 + µ|sym A|2 = λ ( ∑ aii )2 + µ ∑ a2ii + µ|B|2 ≥ (λ + µ/3)( ∑ aii )2 + µ|B|2 . This proves the claim. Example 5.4. The following energy densities below satisfy (5.2) and (5.3): W1,q (F) = |(F T F)1/2 − Id3 |2 + | log det F|q for det F > 0, q for det F > 0, W2,q (F) = |(F T F)1/2 − Id3 |2 + (det F)−1 − 1 W1,q = W2,q = +∞

for det F ≤ 0,

for any exponent q ≥ 2. In either case: Q3 (F) = 8|symF|2 and L3 F = 8 symF.

5.2 Elasticity on shells In this section, we recall the notation for thin shells from sections 3.6 and 4.5 and establish the correspondence between the scaling of the applied forces in terms of

100

5 Limiting theories for elastic plates and shells: nonlinear bending

powers of shell’s thickness h, and the scaling of the infima of total energies, as h → 0. We also introduce the important contraction of the quadratic form Q3 that will play the role of the dimensionally reduced elastic energies density, obtained in the following section as the Γ -limits of the rescaled energies on thin shells. Let S be a 2-dimensional surface embedded in R3 , which is smooth, compact, connected, oriented and whose boundary ∂ S is the union of finitely many (possibly none) Lipschitz curves. By n we denote the unit normal vector to S, by Tz S the tangent space at z ∈ S, and by π the projection onto S along n, so that: π(z + tn(z)) = z

for all z ∈ S, |t|  1.

Given a vector field w ∈ H 1 (S, R3 ), its derivative in a chosen tangent direction τ ∈ Tz S at z ∈ S is ∂τ w(z) = (w ◦ γ)0 (0), where γ : (−ε, ε) → S is any C 1 curve satisfying γ(0) = z and γ 0 (0) = τ. Then, by Π (z) = ∇n : Tz S → Tz S ⊂ R3 we denote the shape operator (which is the negative second fundamental form) on S: Π (z)τ = ∂τ n(z)

for all z ∈ S, τ ∈ Tz S.

With a slight abuse of notation, we will often identify Π (z) with a R3×3 matrix, by 3×3 is a smooth symmetric matrix field on putting Π (z)n(z) = 0. Then, Π : S → Rsym S. Indeed, one can first show that the bilinear form generated by the linear operator Π (z) on Tz S is symmetric: hΠ (z)τ, ηi = hΠ (z)η, τi for any τ, η ∈ Tz S, whereas the fact that hΠ (z)τ, ni = hΠ (z)n, τi = 0 follows by the extension definition. Consider a family {Sh }h>0 of thin shells with the midsurface S:  Sh = x = z + tn(z); z ∈ S, − h/2 < t < h/2 , h  1. Given the deformation uh ∈ H 1 (Sh , R3 ), its elastic energy (per unit thickness) and the total energy in presence of the applied force f h ∈ L2 (Sh , R3 ), are given by: E h (uh ) =

1 h

Z Sh

W (∇uh ) dx,

J h (uh ) = E h (uh ) +

1 h

Z Sh

h f h , uh − id3 i dx.

We now state some useful change of variable formulas which can be checked by a straightforward calculation: Lemma 5.5. In the above context, consider the rescaling yh ∈ H 1 (Sh0 , R3 ) of uh ∈ H 1 (Sh , R3 ), and the rescaling ∇h yh of ∇uh , defined on Sh0 with thickness h0  1:
h n(z) , h0
h ∇h yh (z + tn(z)) = ∇uh z + t n(z) h0 yh (z + tn(z)) = uh z + t

for all z ∈ S, |t|
0 : f h (z + tn(z)) = hα det(Id3 + tΠ (z))−1 f (z).

(5.11)

Then the following holds, with the new exponent β ≥ 0 which equals β = α for α ≤ 2 and β = 2α − 2 for α > 2:  −Chβ ≤ inf J h (uh ) − hα m f ; uh ∈ H 1 (Sh , R3 ) ≤ 0 for all h  1.
Moreover, if lim suph→0 h1β J h (uh ) − hα m < +∞ then also E h (uh ) ≤ Chβ . All constants C above are positive and depend only on S, f and W . Proof. 1. We will prove the two claimed statements, first setting β = 2α − 2 and then β = α. The result will follow in view of the easy observation, valid for h ≤ 1:

5.2 Elasticity on shells

103

min{hα , h2α−2 } = hmax{α,2α−2} =



hα if α ≤ 2, h2α−2 if α ≥ 2.

Fig. 5.1 Dependence of the energy scaling exponent β on the applied force scaling exponent α.

Let α ≥ 0 and consider the case β = 2α − 2. Given uh ∈ H 1 (Sh , R3 ) we use Theorem 4.10 to obtain Rh ∈ SO(3) and ch ∈ R3 such that: kuh − (Rh x + ch )kH 1 (Sh ) ≤

C C k dist(∇uh , SO(3))kL2 (Sh ) ≤ 1/2 E h (uh )1/2 , (5.12) h h

where the last inequality follows from the quadratic non-degeneracy of the energy density W assumed in (5.2). Note that the linear term in J h (uh ) obeys: 1 h

Z Sh

h f h , uh − xi dx − hα m = hα = hα

Z ? S

Z ? S

h/2



h/2



  f (z), uh (z + tn(z)) − z dt dσ (z) − m f

−h/2

Z   h f (z), Rzi dσ (z) , f (z), uh (z + tn(z)) dt dσ (z) − min R∈SO(3) S

−h/2

in view of (5.10) and (5.11). Consequently, (5.12) implies: 1 h

Z

h

Sh

h

α

h f , u − xi dx − h m = h

+

Z ? h/2

S

≥ −Ch

−h/2 α−1

α

Z ? S

h/2



 f (z), uh − (Rh x + ch ) dt dσ (z)

−h/2

Z   h f (z), Rzi dσ (z) f (z), Rh z + tRh n(z) + ch ) dt dσ (z) − min R∈SO(3) S h

h

h

k f kL2 (S) ku − (R x + c )kL2 (Sh )


1/2 ≥ −Chα−1 k f kL2 (S) E h (uh )1/2 ≥ −C hβ E h (uh ) . Thus: J h (uh ) − hα m ≥ E h (uh ) −C(hβ E h (uh ))1/2 , which we rewrite as:  E h (uh ) 1/2 J h (uh ) − hα m E h (uh ) ≥ −C . hβ hβ hβ

104

5 Limiting theories for elastic plates and shells: nonlinear bending

Since the function [0, ∞) 3 s 7→ s − Cs1/2 is bounded from below and coercive, we obtain both indicated statements. > 2. Consider now the case β = α. Given uh ∈ H 1 (Sh , R3 ) we choose ch = Sh uh dx and argue as in the previous case to get: 1 h

Z Sh

h f h , uh − xi dx − hα m f Z  Z ? h/2  = hα h f (z), Rzi dσ (z) h f (z), uh − ch i dx − min S

≥ −Ch

α

R∈SO(3) S

−h/2

h

−1/2



k f kL2 (S) ku − c kL2 (Sh ) + 1 ≥ −Chα h−1/2 k∇uh kL2 (Sh ) + 1 , h

h

in virtue of (5.10), (5.11)Rand the uniform Poincar´e-type inequality in Theorem 3.25. Since by (5.2) we have: Sh |∇uh |2 dx ≤ Ch(E h (uh ) + 1), it easily follows that: k∇uh kL2 (Sh ) ≤ Ch1/2 max{E h (uh )1/2 , 1}. Consequently, we my write: n E h (uh ) 1/2 o J h (uh ) − hα m E h (uh ) ,1 . ≥ −C max hβ hβ hβ The same reasoning as before, applied to the function [0, ∞) 3 s 7→ s−C max{s1/2 , 1} completes the proof of the theorem.

5.3 Kirchhoff’s theory for shells: compactness and lower bound In this and the following sections of the present chapter, we identify the asymptotic behaviour of the approximate minimizers to the total energy functionals J h , in the limit of shells’ vanishing thickness h → 0. By Theorem 5.9, the main part of the analysis consists of identifying the Γ -limits Iβ of the energies {h−β E h }h→0 for a given scaling β > 0 induced by the scaling of the applied forces { f h }h→0 .

Fig. 5.2 The surface S deformed by the isometry y and the rotated frame as in Theorem 5.10.

5.3 Kirchhoff’s theory for shells: compactness and lower bound

105

Below we focus on the case β = 2. The first result should be seen as the realization of condition (i) in Definition 2.1 of Γ -convergence, in addition to providing the compactness properties which will allow to use Theorem 2.6 from chapter 2. The realization of (ii) in Definition 2.1 in this context will be given in the next section. Theorem 5.10. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary ∂ S given by finitely many Lipschitz curves. Assume that the family of deformations {uh ∈ H 1 (Sh , R3 )}h→0 satisfies: E h (uh ) ≤ Ch2 .

(5.13)

Then there exists {ch ∈ R3 }h→0 such that rescalings {yh ∈ H 1 (Sh0 , R3 )}h→0 in: yh (z + tn(z)) = uh z + t


h n(z) − ch h0

for all z ∈ S, |t|

2. Setting ch  Sh0 uh z + t hh0 n(z) dx, the Poincar`e inequality further yields: kyh kL2 (Sh0 ) ≤ Ck∇yh kL2 (Sh0 ) ≤ C. Hence, {yh }h→0 converges in H 1 (Sh0 , R3 ) to some y, which by (5.17) satisfies:

5.3 Kirchhoff’s theory for shells: compactness and lower bound

∂n y(z + tn(z)) = 0, ∇tan y(z + tn(z)) = R(z)(Id3 + tΠ (z))−1


|Tz S

for all z ∈ S, |t| < h0 /2.

107

(5.18)

In fact, the limiting y = y ◦ π must be a deformation of S only, and: ∇y(z) = R(z)|Tz S ,

(5.19)

consistently with (5.18), as ∇tan π = (Id3 + tΠ (z))−1 . We thus proved (i). The increased regularity of y follows by R ∈ H 1 (S, R3×3 ). To show (5.14), observe that: Z

dist2 (R˜ h (z), SO(3)) dσ (z) ≤ Ch2 ,

S

recalling (5.16) and (5.13). Consequently, R ∈ SO(3) a.e. on S, completing the proof of (ii) through (5.19). At each z ∈ S, the matrix R(z) is hence a frame of the tangent vectors and the unit normal vector N to the image surface y(S) ⊂ R3 : ∂τ y = Rτ,

∂η y = Rη,

N = Rn = ∂τ y × ∂η y,

(5.20)

valid with any orthonormal tangent frame τ, η ∈ Tz S. 3. It remains to show the lower bound assertion in (iii). First, recall the construction of the “improved” approximants, that are the rotation-valued fields {Rh ∈ L2 (S, SO(3))}h→0 in the proof of Theorem 4.8:  PSO(3) R˜ h (z) if dist(R˜ h (z), SO(3)) < ε  1, h R (z) = Id3 otherwise, for which the first bound in (5.16) still holds. In particular, {Rh }h→0 converges to R in L2 (S, R3×3 ). Consider now the rescaled strains: Zh =


1 (Rh ◦ π)T ∇h yh − Id3 ∈ L2 (Sh0 , R3×3 ). h

We observe that {Z h }h→0 is uniformly bounded, in view of: Z S h0

|Z h |2 dx ≤

C h2

Z Sh0

|∇h yh − Rh ◦ π|2 dx ≤ C.

Thus, passing to a subsequence if necessary, there exists a limit: lim Z h = Z

h→0

weakly in L2 (Sh0 , R3×3 ).

(5.21)

 Define the characteristic functions: χh = 1{x∈Sh0 ; |Z h (x)|≤h−1/2 } h→0 . Recalling the frame invariance assumption in (5.2), the change of variable in Proposition 5.5, Definition 5.1 of Q3 , and Taylor expanding W at Id3 , it follows that:

108

5 Limiting theories for elastic plates and shells: nonlinear bending


h 1 χh (x)W (Id3 + hZ h ) · det (Id3 + t Π (z))(Id3 + tΠ (z))−1 dx h0 Sh0 h0 Z
h2 = χh (x) Q3 (Z h ) + o(1)|Z h |2 · (1 + o(1)) det(Id3 + tΠ (z))−1 ) dx. 2h0 Sh0

E h (uh ) ≥

Z

Since {χh Z h }h→0 converges weakly in L2 (Sh0 , R3×3 ) to Z by (5.21), we get: lim inf h→0

Z
1 1 h h E (u ) = lim inf Q3 χh Z h · (1 + o(1)) det(Id3 + tΠ (z))−1 ) dx 2 h 2h0 h→0 Sh0 Z 1 ≥ Q3 (Z(x)) · det(Id3 + tΠ (z))−1 ) dx 2h0 Sh0 Z ? h0 /2
1 Q3 Z(z + tn(z)) dt dσ (z), = 2 S −h0 /2

in virtue of the properties of Q3 in Lemma 5.2 and the weak lower-semicontinuity
1/2 R of the norm Sh0 Q3 (·) · det(Id3 + tΠ (z))−1 ) dx on the space of L2 symmetric h matrix fields on S 0 . Recalling Definition 5.6 of Q2 (z, ·), we obtain: 1 1 lim inf 2 E h (uh ) ≥ h→0 h 2

Z ? h0 /2 S


Q2 z, Z(z + tn(z))tan dt dσ (z).

−h0 /2

(5.22)

4. It remains to identify the limiting tangential strain symZ(x)tan in the integrand of (5.22). Fix a small s > 0 and consider the difference quotients: f s,h (z + tn(z)) 


1 h y (z + (s + t)n(z) − yh (z + tn(z)) ∈ H 1 (Sh0 , R3 ). sh

By Lemma 5.5 and (5.17), { 1h ∂n yh }h→0 converges in L2 (Sh0 , R3 ) to lim f

h→0

s,h

1 (z + tn(z)) = lim h→0 h

?

t+s

∂n yh (z + rn(z)) dr = t

1 h0 (Rn) ◦ π,

so:

1 N(y(z)) h0

by (5.20). There also follows convergence of normal derivatives in L2 (Sh0 , R3 ):
1 ∇h yh (z + (s + t)n(z)) − ∇h yh (z + tn(z)) n(z) = 0. h→0 sh

lim ∂n f s,h (z + tn(z)) = lim

h→0

For the tangential derivatives, use Lemma 5.5 to get for all τ ∈ Tz S:
1 h ∇h yh (z + (s + t)n(z)) · Id3 + (t + s) Π (z) sh h0 
h − ∇h yh (z + tn(z)) · Id3 + t Π (z) (Id3 + tΠ (z))−1 τ, h0

∂τ f s,h (z + tn(z)) =

which implies the following formula involving the scaled strains:

5.4 Second order truncation result

109


h 1 ∂τ f s,h (z + tn(z)) = Rh (z) Z h (z + (s + t)n(z)) · Id3 + (t + s) Π (z) sh h0 
s h − Z h (z + tn(z)) · Id3 + t Π (z) + Π (z) (Id3 + tΠ (z))−1 τ. h0 h0 We thus obtain the weak convergence in L2 (Sh0 , R3 ): lim ∂τ f s,h (z + tn(z))   1 s = R(z) Z(z + (s + t)n(z)) − Z(z + tn(z)) + Π (z) (Id3 + tΠ (z))−1 τ. s h0

h→0

The three last displayed convergences imply that { f s,h }h→0 converges (up to a subsequence) weakly in L2 (Sh0 , R3 ) to h10 N ◦ y. Equating the derivatives, we obtain:   1 1 s ∂τ (N ◦ y)(z) = R(z) Z(z + sn(z)) − Z(z) + Π (z) τ h0 s h0 which by (5.19) leads to (see also Remark 5.11 (i)):  t  (∇y(z))T ∇(N ◦ y)(z) − Π (z) . Z(z + tn(z))tan = Z(z)tan + h0

(5.23)

5. By (5.23) we finally write: ?

h0 /2

−h0 /2


Q2 z, Z(z + tn(z))tan dt ?

h0 /2


t2 Q2 z, (∇y(z))T ∇(N ◦ y)(z) − Π (z) dt 2 −h0 /2 h0
1 for a.e. z ∈ S, ≥ Q2 z, (∇y(z))T ∇(N ◦ y)(z) − Π (z) 12

= Q2 (z, Z(z)tan ) +

which completes the proof of the theorem, in virtue of (5.22).

5.4 Second order truncation result Towards completing the asymptotic analysis of scaled energies h12 E h , in the following section 5.6 we will derive the upper bound which is a counterpart to the lower bound in (5.15). The related recovery family construction necessitates an approximation result in which a H 2 -regular function is replaced by its W 2,∞ truncation. This result is presented below, as a second order version of Theorem 4.3.

110

5 Limiting theories for elastic plates and shells: nonlinear bending

Theorem 5.12. Let Ω ⊂ RN be an open, bounded, Lipschitz domain. Then, for every g ∈ H 2 (Ω , R) and every λ > 0, there exists g¯ ∈ W 2,∞ (Ω , R) satisfying: (i) kgk ¯ W 2,∞ (Ω ) ≤ Cλ , Z ≤ C (ii) {x ∈ Ω ; g(x) , g(x)} ¯ (|g| + |∇g| + |∇2 g|)2 dx. λ 2 {|g|+|∇g|+|∇2 g|>λ } The constants C above depend only on Ω .

The proof of Theorem 5.12 will be given in the next section. We now deduce one direct consequence that will be used in the future. Corollary 5.13. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary ∂ S given by finitely many Lipschitz curves. Let y ∈ H 2 (S, R3 ) and assume that N = ∂τ y × ∂η y ∈ H 1 (S, R3 ) is the unit normal vector to the image surface y(S). Fix any small ε  1. Then, there exist approximations {yh ∈ W 2,∞ (S, R3 )}h→0 and {Nh ∈ W 1,∞ (S, R3 )}h→0 with the following properties: strongly in H 2 (S, R3 ) and Nh → N strongly in H 1 (S, R3 ),
h kyh kW 2,∞ (S) + kNh kW 1,∞ (S) ≤ ε, (5.24)  1 |S \ Ωh | → 0 where Ωh = z ∈ S; yh (z) = y(z) and Nh (z) = N(z) , h2 yh → y

for any fixed ε  1. Proof. 1. By the change of variable via a surface patch parametrization on S and the (finite) partition of unity argument, it suffices to carry the argument for S = ω ⊂ R2 . We first construct the family {Nh }h→0 . To this end, we use Theorem 4.3 to each scalar component of N and with λ = ε 2 /h, to obtain: hk∇Nh kL∞ (ω) ≤ Cε 2 , C Z 1 h ≤ {x ∈ ω; N (x) , N(x)} |∇N|2 dx → 0 as h → 0. h2 ε 4 {|∇N|>ε 2 /h} Further, noting that if Br (x) ⊂ {Nh , N} then r ≤ {Nh , N}|1/2 /(π 1/2 ), we get: kNh kL∞ (ω) ≤ kNkL∞ (ω) + k∇Nh kL∞ (ω) · sup dist(x, {Nh = N}) x∈ω

≤ 1+

Cε 2 hπ

h {N , N}|1/2 ≤ 2 1/2

as h → 0.

In particular, for a sufficiently small ε  1 we arrive at: ε hkNh kW 1,∞ (ω) ≤ Cε 2 ≤ . 2

5.4 Second order truncation result

111

It remains to prove the convergence of {Nh }h→0 to N in H 1 (ω, R3 ). Indeed: Z

kNh − Nk2L2 (ω) =

|Nh − N|2 dx

{Nh ,N} (kNh kL∞ (ω) + kNkL∞ (ω) )2 {Nh

≤

k∇Nh − ∇Nk2L2 (ω) =

Z {Nh ,N}

≤C

, N}| ≤ 9h2 → 0,

|∇Nh − ∇N|2 dx

Z {Nh ,N}

|∇N|2 dx +


Cε 4 h {N , N}| → 0 2 h

as h → 0.

2. In order construct the approximating family {yh }h→0 , we use Theorem 5.12 to the scalar components of the vector field y, with λ = ε 2 /h, to get for ε  1: ε hkyh kW 2,∞ (ω) ≤ Cε 2 ≤ , 2 C Z 1 h {y , y} ≤ 4 |y|2 + |∇y|2 + |∇2 y|2 dx → 0 h2 ε {|y|+|∇y|+|∇2 y|>ε 2 /h}

as h → 0.

Similarly as in step 1, we likewise get: kyh − yk2H 2 (ω) ≤ C

Z {yh ,y}

|yh |2 + |∇yh |2 + |∇2 yh |2 dx

Z

+C

{yh ,y}

|y|2 + |∇y|2 + |∇2 y|2 dx

Cε 4 ≤ 2 |{yh , y}| +C h

Z {yh ,y}

|y|2 + |∇y|2 + |∇2 y|2 dx → 0

as h → 0.

This implies (5.24) and completes the proof. We close this section by two preliminary lemmas of independent interest, to be used in the construction of g¯ in the proof of Theorem 5.12. Lemma 5.14. Let A be a compact, nonempty subset of some open, bounded domain Ω ⊂ RN . There exists a function d ∈ C ∞ (Ω \ A, R) such that for all x ∈ Ω \ A: 1 dist(x, A) ≤ d(x) ≤ C dist(x, A) with a constant C depending only on N, C k (ii) |∇ d(x)| ≤ Ck dist(x, A)1−k for all k ≥ 1, with constants Ck depending on k, N. (i)

Proof. 1. Observe first that the assertions of Lemma 4.7 are still valid on the open, bounded domain Ω \ A, if the distance function dist(x, ∂ (Ω \ A)) is replaced by dist(x, A). More precisely, there exists a sequence of open balls {Bri (xi )}∞ i=1 with 1 centers {xi ∈ Ω \ A}∞ and radii r = dist(x , A), such that: i i i=1 4 Ω \A ⊂

∞ [ i=1

Bri (xi )

112

5 Limiting theories for elastic plates and shells: nonlinear bending

and such that each x0 ∈ Ω \ A belongs to at most M doubled balls in {B2ri (xi )}∞ i=1 , with M depending only on N.
2. Let φ ∈ Cc∞ B3/2 (0), [0, 1] be a function satisfying φ ≡ 1 on B1 (0). Define: ∞

x − xi
for all x ∈ Ω \ A. ri

d(x) = ∑ ri φ i=1

It follows that for each x0 ∈ Ω \ A, the above sum when restricted to the open set B 1 dist(x0 ,A) (x0 ) ∩ Ω ⊂ Ω \ A, contains at most M nonzero terms. This is because, if 22 B 1 dist(x0 ,A) (x0 ) ∩ B 3 ri (xi ) , 0/ for some xi , then: 22

2

| dist(x0 , A) − dist(xi , A)| ≤ |x0 − xi |
0,

5.5 Proof of second order truncation result

113

Proof. 1. We denote V = Cc∞ B1 (0) ⊂ RN , R) and consider another linear space W N(k)

that consists of all polynomials on RN of degree at most k. We let {Pi }i=1 to be the basis of W . Define the linear map T : V → RN(k) by: N(k) Z

T (φ ) =

∑

i=1

RN

for all φ ∈ V.

φ (x)Pi (x) dx ei

We claim that T is surjective. Indeed, assume that y ∈ RN(k) is orthogonal to the image subspace Im(T ). Let φ0 ∈ V be such that φ0 > 0 on B1/2 (0) and φ0 ≡ 0 on N(k)

B1 (0) \ B3/2 (0). Then φ = φ0 ∑i=1 yi Pi ∈ V so we have: 0 = hT (φ ), yi =

N(k)

Z RN

φ (x)

∑ yi Pi (x) dx =

i=1

N(k)

Z RN

φ0 (x)


2

∑ yi Pi (x)

dx,

i=1

N(k)

which implies ∑i=1 yi Pi dx ≡ 0 on B1/2 (0). In conclusion, there must be y = 0. 2. From the surjectivity of T , it follows that there exists ψ ∈ V satisfying: Z RN

Z

ψ(x) dx = 1

and

RN

ψ(x)P(x) dx = 0

for all P ∈ W, P(0) = 0.

Consequently: RN ψ(x)P(x) dx = P(0) for all P ∈ W . Given P ∈ W , x ∈ RN and ¯ = P(x − εz): ε > 0, we now apply this identity for the new polynomial P(z) R

(ψε ∗ P)(x) =

Z RN

Z z
1 ¯ P(x − z) dz = ψ(z)P(x − εz) dz = P(0) = P(x). ψ εN ε RN

Since clearly P¯ ∈ W , the proof is done.

5.5 Proof of second order truncation result The below proof of Theorem 5.12 follows a similar strategy as described in section 4.2, in which instead of applying the Kirszbraun extension theorem one needs to define the specific extension function directly. 1. Given g ∈ H 2 (Ω , R), we may without loss of generality assume that it is a restriction to Ω of some g ∈ H 2 (RN , R), that is compactly supported in a ball BR/2 (0) containing Ω , and whose the norms kgkL2 , k∇gkL2 , k∇2 gkL2 are increased with respect to the corresponding norms on Ω by at most a factor C, depending only on Ω . In this step we follow the reasoning of step 1 in the proof of Lemma 4.5. We denote a = |∇g| + |∇g| + |∇2 g| ∈ L2 (RN , R). Fix λ > 0 and define the set:

114

5 Limiting theories for elastic plates and shells: nonlinear bending

? n N K = x¯ ∈ R ;

a2 (x) dx

λ


1/2

≤ 2λ

o for all r < 2R .

Br (x) ¯

By Vitali’s covering in Lemma 4.4, there exists a sequence of pairwise disjoint balls {Bri (xi )}∞ i=1 with radii ri < 2R such that: RN \ K λ ⊂

∞ [

? B5ri (xi )

a2 (x) dx

and


1/2

> 2λ for all i ≥ 1.

Br (xi )

i=1

Consequently: |RN \ K λ | ≤ 5N ∑∞ i=1 |Bri (xi ) |, and also:
1 a2 dx + a2 dx |Bri (xi )| Bri (xi )∩{a>λ } Bri (xi )∩{a≤λ } Z 1 a2 dx + λ 2 , ≤ |Bri (xi )| Bri (xi )∩{a>λ } Z

4λ 2 ≤

which yields: |Bri (xi )| ≤

Z

1 R a2 3λ 2 Bri (xi )∩{a>λ }

|RN \ K λ | ≤

5N 3λ 2

dx for all i ≥ 1. Hence we obtain:

Z

a2 dx.

(5.26)

{a>λ }

2. By the density of Lebesgue points, the following set K˜ λ : n o K˜ λ = x¯ ∈ K λ ; x¯ is the Lebesgue point of g, ∇g and ∇2 g , obeys the bound on its complement as in (5.26). Also, there holds: |g(x)| ¯ + |∇g(x)| ¯ + |∇2 g(x)| ¯ ≤ 2λ

for all x¯ ∈ K˜ λ ,

(5.27)

so if |RN \ K˜ λ | = 0 then we may simply put g¯ = g. The remaining part of the proof is thus carried out under the assumption that |RN \ K˜ λ | > 0. We will use the following notation for the linear Taylor polynomial of g at x¯ ∈ K˜ λ : Lx¯ (x) = g(x) ¯ + h∇g(x), ¯ x − xi ¯ We now claim that: ?
1/2 |g(x) − Lx¯ (x)|2 dx ≤ Cλ r2

for all x¯ ∈ K˜ λ and r < R.

(5.28)

Br (x) ¯

To prove (5.28), let {φε ∈ Cc∞ (Bε , [0, 1])}ε>0 be the usual family of mollifiers given by φε (x) = ε −N φ (x/ε) for some φ ∈ Cc∞ (B1 (0), [0, 1]). We will write gε = g ∗ φε . Since gε (x) ¯ → g(x) ¯ and ∇gε (x) ¯ → ∇g(x) ¯ as ε → 0 at x¯ ∈ K˜ λ , Fatou’s lemma implies: ? ? |g − Lx¯ |2 dx ≤ lim inf |gε (x) − gε (x) ¯ − h∇gε (x), ¯ x − xi| ¯ 2 dx. (5.29) Br (x) ¯

ε→0

Br (x) ¯

5.5 Proof of second order truncation result

115

On the other hand, for each ε > 0 there holds: Z 1 gε (x) − gε (x) ∇gε (x¯ + t(x − x) ¯ − ∇gε (x) ¯ dt · |x − x| ¯ ¯ − h∇gε (x), ¯ x − xi ¯ ≤ 0

Z 1 Z t 2 ∇ gε (x¯ + s(x − x) ¯ ds dt · |x − x| ¯2 ≤ 0

0

Z 1

≤

0

|∇2 gε (x¯ + s(x − x)| ¯ ds · |x − x| ¯ 2,

where we used the fundamental theorem of calculus twice and changed the order of integration for the last bound. Consequently, by another change of variable we get: ? 2 gε (x) − gε (x) ¯ − h∇gε (x), ¯ x − xi ¯ dx Br (x) ¯

?

Z 1

≤ r4 =

Br (x) ¯ 0 Z 1 4 r

|Br (x)| ¯ Z 1? = r4 0

0

|∇2 gε (x¯ + s(x − x)| ¯ 2 ds dx

1 sN

(5.30)

Z Brs (x) ¯ 2

|∇2 gε (y)|2 dy ds ? 2

|∇2 gε (y)|2 dy.

4

|∇ gε (y)| dy ds ≤ r sup s

y2 B1/2 (0) 1

dy:



x + x¯  2 ∇g(x) − ∇g(x), dy ¯ y− x+ x ¯ 2 Br/2 ( 2 ) ? = |∇g(x) − ∇g(x)| ¯ 2 ¯ 2. y21 dy = cr2 |∇g(x) − ∇g(x)| Br/2 (0)

Consequently, and recalling that r = |x − x|, ¯ the estimate (5.32) becomes: |∇g(x) − ∇g(x)| ¯ ≤ Cλ |x − x|, ¯

x¯ − x  and g(x) − g(x) ¯ + ∇g(x) + ∇g(x), ¯ ≤ Cλ |x − x| ¯ 2. 2 This proves the latter bound in (5.31), while the former is due by additionally noting:

5.5 Proof of second order truncation result

117

|g(x) − Lx¯ (x)|

¯ x¯ − x  |x − x| + |∇g(x) − ∇g(x)| ¯ ≤ g(x) − g(x) ¯ + ∇g(x) + ∇g(x), ¯ 2 2 ≤ Cλ |x − x| ¯ 2. 4. Recalling assumption |RN \ K˜ λ | > 0 from step 2, we may choose a closed set: Kˆ λ ⊂ K˜ λ

such that

|RN \ Kˆ λ | ≤ 2|RN \ K˜ λ |.

In particular, by (5.26) there holds: 2 · 5N |RN \ K˜ λ | ≤ 3λ 2

Z

a2 dx.

(5.33)

{a>λ }

We define an intermediary truncation function gˆ : RN → R by:
¯2 for all x ∈ RN g(x) ˆ = sup Lx¯ (x) − Mλ |x − x| x∈ ¯ Kˆ λ

and claim that if M > 1 is a sufficiently large constant (depending only on Ω ), then: g(x) ˆ = g(x) and

for all x ∈ Kˆ λ ,

|g(x) ˆ − Lx¯ (x)| ≤ Cλ |x − x| ¯2

for all x¯ ∈ Kˆ λ , x ∈ RN .

(5.34)

Indeed, if x ∈ Kˆ λ then by (5.31) we get g(x) ˆ ≤ g(x) provided that M ≥ C. On the other hand, by taking x¯ = x in the expression under supremum in the definition of g, ˆ it follows that g(x) ˆ ≥ g(x). This proves the first claim in (5.34). For the second claim, we fix x¯ ∈ Kˆ λ and x ∈ RN . In virtue of closedness of the set Kˆ λ , of the continuity of g, ∇g on Kˆ λ which follow from (5.31), and by recalling (5.27), the supremum in the definition of gˆ must be attained, if only M ≥ 4: g(x) ˆ = Lx¯0 (x) − Mλ |x − x¯0 |2

for some x¯0 ∈ Kˆ λ .

We now use (5.31) to bound: |Lx¯0 (x) − Lx¯ (x)| ≤ |g(x) ¯ − Lx¯0 (x)| ¯ + |h∇g(x) ¯ − ∇g(x¯0 ), x − xi| ¯

7 ≤ Cλ |x¯ − x¯0 |2 + |x¯ − x¯0 | · |x − x| ¯ ≤ Cλ 3|x − x¯0 |2 + |x − x| ¯2 . 2 Consequently, if only M ≥ 3C, we get: −Mλ |x − x| ¯ 2 ≤ g(x) ˆ − Lx¯ (x) = Lx¯0 (x) − Lx¯ (x) − Mλ |x − x¯0 |2 ≤ (3C − M)λ |x − x¯0 |2 + ≤

7C λ |x − x| ¯ 2. 2

7C λ |x − x| ¯2 2

118

5 Limiting theories for elastic plates and shells: nonlinear bending

This proves the second claim in (5.34). In particular, (5.34), (5.27) also yield: kgk ˆ L∞ ≤ Cλ .

(5.35)

5. We now define the final function g¯ ∈ W 2,∞ (RN , R), satisfying: g( ¯ x) ¯ = g( ˆ x) ¯ = g(x), ¯

∇g( ¯ x) ¯ = ∇g(x)

for all x¯ ∈ Kˆ λ ,

and kgk ¯ W 2,∞ ≤ Cλ . Namely, on Kˆ λ we set g¯ = g, ˆ while on the complement of Kˆ λ we put: Z x−y g(y) ˆ dy for all x ∈ RN \ Kˆ λ . g(x) ¯ = d(x)−N ψ d(x) RN

(5.36)

(5.37)

Above, d is the approximate distance function in Lemma 5.14 for the compact set A = Kˆ λ ∩ BR (0) and its open superset BR (0), while ψ is the convolution kernel in Lemma 5.15 corresponding to the linear polynomials (i.e. with degree k = 1). Since d, ψ are smooth, it follows that g¯ ∈ C ∞ (RN \ Kˆ λ ), while by (5.35) we get: kgk ¯ L∞ ≤ Cλ . Fix x0 ∈ RN \ Kˆ λ and let x¯ ∈ Kˆ λ be such that |x0 − x| ¯ = dist(x0 , Kˆ λ ). We denote: Rx¯ = gˆ − Lx¯ ,

where

|Rx¯ (y)| ≤ Cλ |y − x| ¯ 2 for all y ∈ RN ,

in view of (5.34). Then, Lemma 5.15 implies that for all x close to x0 : Z x−y g(x) ¯ = Lx¯ (x) + d(x)−N ψ Rx¯ (y) dy. d(x) RN Consequently, using (5.27) together with the bounds in Lemma 5.14, we obtain:  x − y  d  Rx¯ (y) dy x 7→ d(x)−N ψ d(x) RN dx |x0 Z x −y C 0 d(x0 )−N ψ˜ · |Rx¯ (y)| dy ≤ Cλ + d(x0 ) Bd(x0 ) (x0 ) d(x0 )

Z ∇g(x ¯ 0 ) = ∇g(x) ¯ +

≤ Cλ +Cλ d(x0 ) ≤ Cλ , and likewise:  x − y  d2  −N R (y) dy x → 7 d(x) ψ x ¯ 2 d(x) RN dx |x0 Z   C −N ˆ x0 − y · |Rx¯ (y)| dy ≤ Cλ . d(x ) ψ ≤ 0 d(x0 )2 Bd(x0 ) (x0 ) d(x0 )

2 Z ∇ g(x ¯ 0 ) =

(5.38)

5.6 Kirchhoff’s theory for shells: recovery family

119

In exactly the same manner, one proves that if Kˆ λ = x → x¯0 ∈ Kˆ λ , then g(x) ¯ → g(x¯0 ) and ∇g(x) ¯ → ∇g(x¯0 ). It remains to show that ∇g¯ is Lipschitz, with its Lipschitz constant bounded by Cλ . In virtue of (5.38) and (5.31), we only need to verify: |∇g(x) ¯ − ∇g(x¯0 )| ≤ Cλ |x − x¯0 |

for all x < Kˆ λ , x¯0 ∈ Kˆ λ .

(5.39)

Denoting by x¯ ∈ Kˆ λ a point such that |x − x| ¯ = dist(x, Kˆ λ ), we get: |∇g(x) ¯ − ∇g(x¯0 )| ≤ |∇g(x) ¯ − ∇g(x)| ¯ + |∇g( ¯ x) ¯ − ∇g(x¯0 )| ≤ Cλ (|x − x| ¯ + |x¯ − x¯0 |). Then (5.39) follows by observing: |x − x| ¯ + |x¯ − x¯0 | ≤ |x − x¯0 | + 2|x − x| ¯ ≤ 3|x − x¯0 |. We have thus shown that g¯ ∈ C 1,1 , with all its norms bounded by Cλ . In virtue of (5.36) and (5.33), this concludes the proof of Theorem 5.12.

5.6 Kirchhoff’s theory for shells: recovery family We stress that in Theorem 5.10 no a priori assumptions were made on the structure of deformations uh , while their limiting properties were derived solely from the energy bound (5.13). In the next result we prove that our limiting description is sharp, in the sense that it can be fully recovered through the indicated procedure. Theorem 5.16. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary ∂ S given by finitely many Lipschitz curves. Then, given any isometry y ∈ H 2 (S, R3 ) satisfying (5.14), there exists a family {uh ∈ H 1 (Sh , R3 )}h→0 such that the rescaled deformations in: yh (z + tn(z)) = uh (z + t hh0 n(z)) converge in H 1 (Sh0 , R3 ) to y ◦ π, and that we have: lim

1

h→0 h2

E h (uh ) = IK (y).

Proof. 1. Given y ∈ H 2 (S, R3 ) with (5.14), for any z ∈ S we define: N(z) = ∂τ y(z) × ∂η y(z), where τ, η ∈ Tz S are any two unit orthogonal tangent vectors. Thus N ∈ H 1 (S, R3 ) is the unit normal vector field on the image surface y(S). We further introduce the rotation-valued matrix field R ∈ H 1 (S, SO(3)) by setting: R(z)n(z) = N(z),

R(z)τ = ∂τ y(z) for all τ ∈ Tz S.

Recalling the notion of the linear map c(z, ·) from Definition 5.6 (iii), define:
d ∈ L2 (S, R3 ), d(z) = R(z)c z, (∇y)T ∇N − Π (z) for all z ∈ S. (5.40)

120

5 Limiting theories for elastic plates and shells: nonlinear bending

Let {d h ∈ W 1,∞ (S, R3 )}h→0 be a family of approximating fields, satisfying: dh → d

strongly in L2 (S, R3 ),

and

hkd h kW 1,∞ (S) → 0.

(5.41)

Note that such sequence can always be derived by reparametrizing (slowing down) a sequence of smooth approximations of d. Similarly, let {yh ∈ W 2,∞ (S, R3 )}h→0 and {Nh ∈ W 1,∞ (S, R3 )}h→0 be the approximations given in Corollary 5.13. We finally set {Rh ∈ W 1,∞ (S, R3×3 )}h→0 to be the corresponding approximants of the field R: Rh (z)n(z) = Nh (z),

Rh (z)τ = ∂τ yh (z) for all τ ∈ Tz S.

2. We may now define the family of deformations {uh ∈ W 1,∞ (Sh , R3 )}h→0 in: uh (z + tn(z)) = yh (z) + tNh (z) +

t2 h d (z) 2

h for all z ∈ S, |t| < . 2

The corresponding rescalings {yh ∈ W 1,∞ (Sh0 , R3 )}h→0 converge to y ◦ π: yh (z + tn(z)) = yh (z) + t

h h t 2 h2 N (z) + 2 d h (z), h0 h0

by (5.41) and (5.24). From Lemma 5.5 we get, for all z ∈ S, |t| < h0 /2 and τ ∈ Tz S:

−1 h h2 h ∇Nh (z) + t 2 2 ∇d h (z) Id3 + t Π (z) τ, h0 h0 2h0 h ∇h yh (z + tn(z))n(z) = Nh (z) + t d h (z). h0 ∇h yh (z + tn(z))τ = ∇yh (z) + t

In particular, by (5.41) and (5.24) we note:
dist(∇h yh , SO(3)) = dist(Rh , SO(3)) + O(h) |d h | + |∇Nh | + |∇yh | + O(h2 )|∇d h | = dist(Rh , SO(3)) + O(1)ε. To bound the first term in the right hand side above, recall that Rh = R in the set Ωh and that the Lipschitz constants of yh and Nh are bounded in (5.24). Also, if Br (z)∩S ⊂ S \Ωh then |S \Ωh | ≥ cr2 , which implies r ≤ C|S \Ωh |1/2 . Consequently: dist(Rh , SO(3)) ≤

C C dist(z, Ωh ) ≤ |S \ Ωh |1/2 , h h

which converges to 0 as h → 0, by the last condition in (5.24). We thus obtain: dist(∇h yh , SO(3)) ≤ Cε

a.e. in Sh0

for all h  1.

(5.42)

3. Polar decomposition and Taylor’s expansion can hence be used to write:

5.6 Kirchhoff’s theory for shells: recovery family

121


1 (∇h yh )T ∇h yh = W Id3 + K h + O(|K h |2 ) 2
1 1 = Q3 K h + O(|K h |2 + oε (1)|K h |2 , 2 2

W (∇h yh ) = W

q



(5.43)

where K h  (∇h yh )T ∇h yh − Id3 satisfies kK h kL∞ (S) ≤ Cε by (5.42). The symbol oε (1) denotes the quantity which converges to 0 as ε → 0 and it is the modulus of continuity of D2W at Id3 , applied to Cε with C being some universal constant. We now express h12 E (uh ) as a sum of two terms, corresponding to integration on Ω h × (− h20 , h20 ) and on (S \ Ωh ) × (− h20 , h20 ). In view of Lemma 5.5 the first term is: 1 2h2

?

Z S\Ωh

h0 /2

−h0 /2

≤



Q3



h 1 h K + O(|K h |2 + oε (1)|K h |2 det Id3 + t Π (z) dt dσ (z) 2 h0

C |S \ Ωh | → 0 h2

as h → 0,

with the last convergence resulting from (5.24). Hence we get: 1

E h (uh ) Z ? h0 /2  
1 1 1 1 h = lim K + hO(| K h |2 + oε (1)| K h |2 · Q3 h→0 2 Ωh −h0 /2 2h h h
h · det Id3 + t Π (z) dt dσ (z). h0

lim

h→0 h2

(5.44)

4. We now identify the matrix fields { 1h K h ∈ L2 (Sh0 , R3×3 )}h→0 on the set Ωh × (− h20 , h20 ), and find their convergence properties. For the tangential minors, we get:
−1 h Π (z) · h0 

−1 h h h Id2 + t Π (z) · (∇y)T ∇y + 2t (∇y)T ∇N − Id2 + 2t Π (z) h0 h0 h0
+ O(h2 ) |∇N|2 + |∇y||∇d h | + 1 + O(h4 )|∇d h |2 ,

h Ktan = Id2 + t

which by (5.41) and (5.24) implies for h  1: 
1 h 2t  Ktan = (∇y)T ∇N − Π (z) + O(1)ε |∇N| + |∇y| + 1 h h0

for z ∈ Ωh . (5.45)

Similarly, for the normal component we get:  1 h 1  h 2 hK n,ni = N + 2 dh − 1 h h h0 2t 2t = hN, d h i + O(h)|d h |2 = hN, d h i + O(1)ε|d h | h0 h0

(5.46) for z ∈ Ωh .

122

5 Limiting theories for elastic plates and shells: nonlinear bending

while for the remaining off-diagonal components there holds:

−1 h h t 2 h2 h E ∇N + 2 ∇d h Id3 + t Π (z) τ, N(z) + t d h h0 h0 h0 h0
h = t h(∇y)τ, d h i + O(h2 ) |∇N|2 + |∇y|2 + |d h |2 + |d h ||∇d h | . h0

hK h τ, ni =

D

∇y) + t

Hence, by (5.41) and (5.24) we obtain for h  1:
t 1 h hK τ, ni = h(∇y)T d h , τi + O(1)ε |∇N| + |∇y| + |d h | for z ∈ Ωh . h h0

(5.47)

We summarize (5.45), (5.46) and (5.47) by writing:  1 h 2t  (∇y)T ∇N − Π (z) + RT d h ⊗ n + e(ε, h) K = h h0

for z ∈ Ωh ,

(5.48)

where for h  1 we get, in view (5.41) and (5.40):
ke(ε, h)kL2 (Sh0 ) ≤ Cε k∇NkL2 (S) + k∇y|L2 (S) + k∇2 ykL2 (S) + 1 ≤ Cε,

(5.49)

because kd h kL2 (S) ≤ kdkL2 (S) + 1 ≤ C(k(∇y)T ∇NkL2 (S) + 1) ≤ C(k∇2 ykL2 (S) + 1). By (5.41) and (5.24), there further holds: χ

Ωh ×

 
2t (∇y)T ∇N − Π (z) + RT d h ⊗ n h  2t  (∇y)T ∇N − Π (z) + RT d ⊗ n in L2 (Sh0 , R3×3 ). → h

h h − 20 , 20

(5.50)

Finally, using a similar argument as in (5.49) for the term involving d h , we get:

χ

h h Ωh × − 20 , 20

≤ Ch2


h 1 K h 2 2 2 h L (S 0 ) h

Z


|(∇y)T ∇N|4 + |d h |4 + 1 + ε 4 |∇N|4 + |∇y|4 + |d h |4 dσ (z) S
≤ Cε 2 k∇2 yk2L2 (S) + k∇Nk2L2 (S) + k∇yk2L2 (S) + 1 ≤ Cε 2 . 5. In conclusion, (5.44) and (5.48), (5.49), (5.50), (5.51) imply that converges, modulo a term of order O(1)ε + o1 (ε), to: 1 2

1 h h E (u ) h2

Z ? h0 /2 2 t S

−h0 /2

  T T Q (∇y) ∇N − Π (z) + R d ⊗ n dt dσ (z) 3 2

h0

? Z
1  h0 /2 t 2  dt Q2 z, (∇y)T ∇N − Π (z) dσ (z) 2 2 −h0 /2 h0 S Z
1 Q2 z, (∇y)T ∇N − Π (z) dσ (z) = IK (y). = 24 S

=

(5.51)

5.7 Kirchhoff’s theory for shells: Γ -limit and convergence of minimizers

123

The above reasoning is valid for any fixed ε  1. Choosing ε → 0 and h = h(ε) → 0 we may use a diagonal argument in order to send the cumulative error quantity o1 (ε) to 0 as h → 0 for the resulting family {uh }h→0 . The proof is done.

5.7 Kirchhoff’s theory for shells: Γ -limit and convergence of minimizers In this section, we summarize the assertions of Theorems 5.10 and 5.16, using the notion of Γ -convergence from Definition 2.1. The proofs are similar to the preliminary result on Γ -convergence for the linearised elasticity in Corollary 2.13. Here, we show that the approximate minimizers to the total energies: J h (uh ) = E h (uh ) +

1 h

Z Sh

h f h , uh − id3 i dx,

involving the applied forces { f h ∈ L2 (Sh , R3 )}h→0 of order h2 (see the set-up in section 5.2), accumulate at the minimizers to the limiting total energy given by:  Z   I (y) + h f , y − idS i dσ (z) if y = y ◦ π and y ∈ H 2 (S, R3 )  K S FK (y) = (5.52) and (∇y)T ∇y = Id2 a.e. in S,    ∞ otherwise. We stress that no new analysis is developed below. It is however useful to recast our previous findings in the formalism that highlights convergence of minimizers, which is a primary motivation in several applications of calculus of variations. Corollary 5.17. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary ∂ S given by finitely many Lipschitz curves. Assume that:
−1 f h (z + tn(z)) = h2 det Id2 + tΠ (z) f (z)

h for all z ∈ S, |t| < , 2

where f ∈ L2 (S, R3 ) satisfies the orthogonality to constants condition (5.10). Given y ∈ H 1 (Sh0 , R3 ), we define uh (z + tn(z)) = y(z + t hh0 n(z)) ∈ H 1 (Sh , R3 ) and: F h (y) = J h (uh ) = Then

1 h

Z Sh

W (∇uh ) dx +

1 h

Z S

h f h , uh − id3 i dx

for all h < h0 .

1 h Γ F −→ FK as h → 0 with respect to H 1 (Sh0 , R3 ), where FK is as in (5.52). h2

Proof. 1. We start by noting that if {yh }h→0 converges in H 1 (Sh0 , R3 ) to some y, then Lemma 5.5 easily gives the following convergence of the linear term:

124

5 Limiting theories for elastic plates and shells: nonlinear bending

1 h3

Z S

h f h , uh − id3 i dx =

Z ? h0 /2

S

→

Z S

 f (z), yh (z + tn(z)) − z dt dσ (z)

−h0 /2

(5.53)

h f , y − idS i dσ (z)

as h → 0.

We now verify the two conditions in Definition 2.1. To prove (i), we take a family of rescaled deformations {yh ∈ H 1 (Sh0 , R3 )}h→0 which converges to y. Without loss of generality we may also assume that: lim inf h→0

1 h h 1 F (y ) = lim 2 F h (yh ) < ∞. h→0 h h2

We now use Theorem 5.9, which in view of h12 J h (uh ) − m ≤ C (where m is defined in there) implies that E h (uh ) ≤ Ch2 . Thus, Theorem 5.10 yields: y = y ◦ π,

y ∈ H 2 (S, R3 ),

(∇y)T ∇y = Id2 a.e. on S.

(5.54)

Further, the lower bound statement in Theorem 5.10 together with (5.53) imply: lim inf h→0

1 1 h h F (y ) = lim inf 2 E h (uh ) + h→0 h h2

Z S

h f , y − idS i dσ (z) ≥ FK (y).

2. To verify condition (ii) in Definition 2.1, consider a deformation y ∈ H 1 (Sh0 , R3 ). If y does not satisfy (5.54) so that FK (y) = ∞, then we set {yh = y}h→0 . Indeed, in this case there must be limh→0 h12 F h (yh ) = ∞ as well, for otherwise arguing as in step 1 would bring a contradiction. On the other hand, if (5.54) holds, then Theorem 5.16 yields a recovery family {yh ∈ H 1 (Sh0 , R3 )}h→0 converging to y and so that: 1 1 lim F h (yh ) = lim 2 E h (uh ) + h→0 h h→0 h2

Z S

h f , y − idS i dσ (z) = FK (y),

in view of (5.53). This completes the proof. We note in passing that existence of at least one minimizer to the limiting total energy FK can be easily deduced through a direct argument: Remark 5.18. Let {yn ∈ H 2 (S, R3 )}n→∞ be a minimizing sequence of isometries: FK (yn ) → inf FK

as n → ∞. R

By (2.2) we may without loss of generality assume that S yn dσ (z) = 0. The isomR etry condition implies that S |∇yn |2 dσ (z) ≤ C, while by Poincar´e’s inequality and recalling the positive definiteness of Q2 (z, ·) we get: C ≥ IK (yn ) −Ck f kL2 (S) · k∇yn kL2 (S) ≥ c ≥c

Z S

|∇2 yn |2 dσ (z) −C,

Z S

|(∇y)T ∇(Nn ◦ yn )|2 dσ (z) −C

5.8 Non-compactness by wrinkling

125

where Nn denotes the unit normal vector to the image surface yn (S). In conclusion, kyn kH 2 (S) ≤ C so {yn }n→∞ converges weakly in H 2 (S, R3 ), and up to a subsequence which we do not relabel, to some y ∈ H 2 (S, R3 ). Applying pointwise convergence, we also see that (∇y)T ∇y = Id2 a.e. in S. Similarly as in the proof of the last inequality displayed above, it follows that {(∇yn )T ∇(Nn ◦ yn )}n→∞ converges weakly in L2 (S, R2×2 ) to (∇y)T ∇(N ◦ y). Thus, the weak lowersemicontinuity of the norm
1/2 R L2 (S, R2×2 implies that limn→∞ FK (yn ) ≥ FK (y). sym ) 3 G 7→ S Q(z, G(z)) dσ (z) Hence, y must be a minimizer of FK . On the other hand, as previously announced, we have the convergence of minimizers result, following from Corollary 5.17: Corollary 5.19. In the context of Corollary 5.17, let {uh ∈ H 1 (Sh0 , R3 )}h→0 be a family of almost-minimizers to the energy functionals {J h }h→0 , satisfying: 1 h h J (u ) − inf J h → 0 h2

as h → 0.

(5.55)

Then, the rescalings {yh (z + tn(z)) = uh (z + t hh0 n(z)) − ch ∈ H 1 (Sh0 , R3 )}h→0 defined for some {ch ∈ R3 }h→0 converge, up to subsequence in H 1 (Sh0 , R3 ), to the limit y ∈ H 1 (Sh0 , R3 ) which is a minimizer of FK given in (5.52): FK (y) = min FK < ∞.

(5.56)

In particular, the functional FK has at least one minimizer. Conversely, every minimizing y ∈ H 1 (Sh0 , R3 ) in (5.56), is the limit in H 1 (Sh0 , R3 ) of some family of deformations {yh }h→0 whose inverse rescalings given by {uh (z + tn(z)) = yh (z + t hh0 n(z)) ∈ H 1 (Sh , R3 )}h→0 obey (5.55). There also holds: 1 inf J h → min FK h2

as h → 0.

5.8 Non-compactness by wrinkling In the proof of Theorem 5.10, the key argument was that under scaling β = 2 in: E h (uh ) ≤ Chβ ,

(5.57)

the scaled gradients {∇h yh }h→0 converge, up to a subsequence, to some R ◦ π with R ∈ H 1 (S, SO(3)), strongly in L2 (Sh0 , R3×3 ). Below, we observe that this property cannot hold with any β < 2. In the following sections we hence concentrate on the opposite scaling exponent range β > 2, and derive the corresponding compactness

126

5 Limiting theories for elastic plates and shells: nonlinear bending

properties together with Γ -limits. We will come back to the construction below in Part III of this monograph when we derive the general non-Euclidean energy scaling laws from the convex integration constructions.
2 Example 5.20. Let S = − 21 , 12 ⊂ R2 . Then {Sh }h→0 is a family of thin plates with midplate given by the unit square. We consider the elastic energies with the prototypical density W (F) = dist2 (F, SO(3)) in: E h (uh ) =

1 h

Z

dist2 (∇uh , SO(3)) dx

Sh

for all uh ∈ H 1 (Sh , R3 ).


3 Set the referential plate Sh0  S1 = − 21 , 12 . Given {θ h ∈ H 1 (R, R)}h→0 , define the scaled deformations yh ∈ H 1 (S1 , R3 ) by: yh (z1 , z2 ,t) =

Z

z1

0

cos θ h (s) ds − ht sin θ h (z1 ), z2 ,

Z z1 0

 sin θ h (s) ds + ht cos θ h (z1 ) ,

for all x = (z1 , z2 ,t) ∈ S1 . It follows that: 

cos θ h (z1 ) − ht cos θ h (z1 ) · (θ h )0 (z1 ) 0 − sin θ h (z1 )

 ∇h yh (z1 , z2 ,t) =  

0

1

0

sin θ h (z1 ) − ht sin θ h (z1 ) · (θ h )0 (z1 ) 0   = Rh (z1 ) Id3 − ht(θ h )0 (z1 )e1 ⊗ e1 ,

cos θ h (z1 )

   

with the following rotation fields {Rh ∈ H 1 (S, SO(3))}h→0 :   Rh =  



cos θ h 0 − sin θ h 0

1

 . 

0

sin θ h 0

cos θ h

Thus, if kht(θ h )0 kL∞ (R)  1, we may use polar decomposition to get: E h (uh ) = =

Z ? 1/2 S

−1/2

h2 12

Z 1/2

Z
dist2 ∇h yh , SO(3) dx =

−1/2

?

S

(θ h )0 (z1 )


2

1/2

ht(θ h )0


2

dx

−1/2

(5.58)

dz1 .

For any fixed exponents 0 < δ < γ < 1 and two constants θ1 < θ2 , we define {θ h }h→0 to be smooth periodic and such that: θh 

δ

γ

| 0, h4 − h2

 ≡ θ1 ,

θ h  hδ |

4

γ

δ

γ

+ h2 , 3h4 − h2

 ≡ θ2 ,

θ h  3hδ |

4

γ

+ h2 ,hδ

 ≡ θ1 ,

5.9 Compactness beyond Kirchhoff’s scaling

127

Then, kht(θ h )0 kL∞ (R) ≤ Ch1−γ (θ2 − θ1 ) → 0 as h → 0, and (5.58) becomes: E h (uh ) ≤

C γ−δ (θ2 − θ1 )2 h · = Ch2−δ −γ h2 h2γ

By adjusting δ , γ, we see that (5.57) may be guaranteed for any exponent β ∈ (0, 2). At the same time k∇h yh −Rh ◦πkL∞ (S1 ) → 0, however {Rh }h→0 have no subsequence that would converge strongly in L2 (S, R3×3 ) (but we do may arrange for the weak convergence, to a constant rotation by angle (θ2 − θ1 )/2 in the (z1 ,t)-plane).

5.9 Compactness beyond Kirchhoff’s scaling In the remaining sections of this chapter we start addressing the energy scalings β > 2. Towards the goal of identifying the Γ -limit Iβ of the elastic energies {h−β E h }h→0 , which will be carried out in the following chapter, we first determine the general compactness properties of families of deformations {uh ∈ H 1 (Sh , R3 )}h→0 satisfying E h (uh ) ≤ Chβ . Recall that in Theorem 5.10 the limiting deformations were shown to be H 2 regular isometries of S. We will prove in Theorem 5.23 below, that as soon as the energy bound scales higher than h2 , the set of the aforementioned isometries reduces to the set of rigid motions, and that the higher order terms in the expansion of uh , in terms of h, are determined from the linear space V (S) of infinitesimal isometries. Definition 5.21. Let S ∈ R3 be a smooth, compact, connected and oriented surface. (i) For a vector field V ∈ H 1 (S, R3 ), by sym∇V ∈ L2 (S, R2×2 ) we mean a field of bilinear forms on Tz S, given by:

 1
(sym∇V (z))τ, η = h∂τ V (z), ηi + h∂η V (z), τi 2

for all τ, η ∈ Tz S.

Equivalently, with the notation in Definition 5.6 (i): sym∇V = sym(∇V )tan . (ii) The space V (S) consists of infinitesimal isometries on S, namely: n o V (S) = V ∈ H 2 (S, R3 ); sym∇V = 0 on S . Equivalently, V ∈ H 2 (S, R3 ) belongs to V (S) if and only if there exists a matrix field A ∈ H 1 (S, so(3)) such that: ∂τ V (z) = A(z)τ for all z ∈ S and τ ∈ Tz S. A few observations about the space V (S) are now in order: Remark 5.22. (i) The skew-symmetric valued matrix field A ∈ H 1 (S, R3×3 ) above is always uniquely determined from V ∈ V (S). Consider the decomposition: V = Vtan + hV, nin,

sym∇V = sym∇Vtan + hV, niΠ .

128

5 Limiting theories for elastic plates and shells: nonlinear bending

One can then define: A(z)τ = ∂τ V (z)

for all z ∈ S, τ ∈ Tz S

A(z)n(z) = Π (z)Vtan (z) − ∇hV, ni(z) ∈ Tz S.

(5.59)

(ii) For any vector field V ∈ H 1 (S, R3 ), we may always consider the one-parameter family of deformations {φ h ∈ H 1 (S, R3 )}h>0 of S given by φ h (z) = z + hV (z). Then, the amount of stretching induced by φ h at z ∈ S, measured as the change of the first fundamental forms, has the following expansion: ∇φ h (z)T ∇φ h (z) − Id2 = 2h sym∇V (z) + h2 ∇V (z)T ∇V (z). We see that V (S) consists precisely of these H 2 -regular displacements which generate no first-order change in the lengths of curves under deformations φ h . (iii) When S = ω ⊂ R2 is open, bounded and connected, then recalling Korn’s rigidity statement in Lemma 3.2 we see that infinitesimal isometries on ω reduce to, modulo rigid motions, the out-of-plane scalar displacements v: n o V (ω) = V (z) = (αz⊥ + β , v); α ∈ R, β ∈ R2 , v ∈ H 2 (ω, R)  0 −α −∂1 v Also, for V as above, the related matrix field A =  α 0 −∂2 v . ∂1 v ∂2 v 0 

Fig. 5.3 An easy one-dimensional calculation based on the Pythagorean theorem shows that any out-of-plane displacement v is an infinitesimal isometry of first order on a flat midsurface.

The compensated regularity of V (S) will be explained in Remark 6.6. The following is the main result of this section, stating the first compactness properties in the energy regime beyond Kirchhoff’s scaling considered in sections 5.3 and 5.6.

5.9 Compactness beyond Kirchhoff’s scaling

129

Theorem 5.23. Let S ⊂ R3 be a smooth, compact, connected, oriented surface with its boundary ∂ S given by finitely many Lipschitz curves. Assume that {uh ∈ H 1 (Sh , R3 )}h→0 is a family of deformations such that: E h (uh ) ≤ Chβ

for some β > 2.

(5.60)

Then there exists rotations and translations {Qh ∈ SO(3), ch ∈ R3 }h→0 such that the normalized deformations {yh ∈ H 1 (Sh0 , R3 )}h→0 given in: yh (z + tn(z)) = (Qh )T uh (z + t

h n(z)) − ch h0

for all z ∈ S, |t|
2 which implies the satisfaction of condition (4.30). Similarly as in Theorem 4.10, we further define {Qh ∈ SO(3)}h→0 by setting: ? h h h h ˜ ˜ ˜ |Q − Q | = dist(Q , SO(3)) where Q = Rh (z) dσ (z). S

>

>

Note that dist2 (Q˜ h , SO(3)) ≤ S |Q˜ h − Rh (z)|2 dσ (z) ≤ C S |∇Rh |2 dσ (z) ≤ Chβ −2 by (5.61) and using the Poincar`e-Wirtinger’s inequality on S. Hence we get: Z Z  |Rh − Qh |2 dσ (z) ≤ C |Rh − Q˜ h |2 dσ (z) + dist2 (Q˜ h , SO(3)) ≤ Chβ −2 . S

S

 1
In view of the gradient bound (5.61) this implies that hβ /2−1 (Qh )T Rh − Id3 h→0 is bounded in H 1 (S, R3×3 ). Consequently, we obtain the following convergences, up to a subsequence that we do not relabel:

130

5 Limiting theories for elastic plates and shells: nonlinear bending

(Qh )T Rh → Id3 strongly in H 1 (S, R3×3 ),
1 (Qh )T Rh − Id3 * A ∈ H 1 (S, so(3)) β /2−1 h

weakly in H 1 (S, R3×3 ).

(5.62)


2. We will use the familiar notation ∇h yh (z + tn(z)) = (Qh )T ∇uh z + t hh0 n(z) . By Lemma 5.5 and (5.61) we obtain: Z S h0

|∇h yh − (Qh )T Rh ◦ π|2 det Id3 + t


−1 h
Π det Id3 + tΠ dx ≤ Chβ . (5.63) h0

Consequently, applying change of variable formulas in Lemma 5.5, it follows that: k∂n yh k2L2 (Sh0 ) ≤ Ch2 , 



 2

∇tan yh − (Qh )T (Rh ◦ π) Id3 + t h Π (z) Id3 + tΠ (z) −1

2 h ≤ Chβ , 0 h0 |Tz S L (S ) (5.64) where by ∇tan yh (x) = (∇yh (x))|Tz S : Tz S → R3 we denote the restriction of ∇yh (x) ∈ R3×3 to the tangent space Tz S and z = π(z). Note the similarity of the above bound to the same argument (5.17) in the context of Kirchhoff’s scaling in Theorem 5.10. Now, however, we may also derive a higher order convergence property in: ∇V h [yh ](z) = =

?

1 hβ /2−1 ?

1 hβ /2−1

h0 /2 −h0 /2

h0 /2



−h0 /2

∇tan yh (z + tn(z)) Id2 + tΠ (z)) − (Id3 )|Tz S dt

   h ∇tan yh − (Qh )T Rh (Id3 + t Π (z))(Id3 + tΠ (z))−1 · h0 |Tz S · (Id2 + tΠ (z)) dt

+

1 hβ /2−1

(Qh )T Rh (z) − Id3


|Tz S

. (5.65)

The first term in the right hand side above converges to 0 in L2 (S, R3×3 |Tz S )) by (5.64), while the second term converges to A|Tz S by (5.62). It follows that: ∇V h [yh ] → A|Tz S Ensuring that

> S

3×3 strongly in L2 (S, R|T )). zS

V h dσ (z) = 0 by putting: ch 

Z ? h0 /2 S

−h0 /2

(Qh )T uh (z + t

h n(z)) − z dt dσ (z), h0

there follows (ii) together with ∇V = A|Tz S , as claimed and consistently with (5.59). 3. To deduce (i), we observe directly from (5.64) and (5.62) that:

5.9 Compactness beyond Kirchhoff’s scaling

∂n yh → 0 ∇tan yh →



131

strongly in L2 (Sh0 , R3 ),
−1  Id3 + tΠ (z) = ∇tan π.

(5.66)

|Tz S

Writing g(z + tn) = det(Id2 + tΠ (z))−1 , we note the following bound: Z

kyh − πkL2 (Sh0 ) ≤ (yh − π) −

0 Sh

Z +

Sh

0

(yh − π) · R

g

S h0

(yh − π) · R

g

S h0

g

g

dx L2 (Sh0 )

dx .

The first term in the right hand side can be bounded by means of the weighted Poincar`e inequality by k∇(yh − π)kL2 (Sh0 ) and hence it converges to 0 in virtue of (5.66). The second term likewise converges to 0 because: Z

Sh0

Z Z (yh − π) · g dx ≤

h0

S −h0

= Ch

β /2−1

?

h0



yh − π dt dσ (z) ≤ C h

h

kV [y ]kL2 (S) ≤ Ch

−h0 β /2−1

yh − π dt L2 (S)

.

This implies convergence of yh to π in H 1 (S, R3 ). The proof is done. For the special case of plates, we have the following additional property: Remark 5.24. Let S = ω ⊂ R2 and {Sh = Ω h = ω ×(− h2 , h2 )}h By translation, it suffices to consider the case when ω z dz = 0. In view of (5.62), > (5.63) we have: | Ω 1 (∇h yh )2×2 dz − Id2 | ≤ Chβ /2−1 , so by polar decomposition: ?

(Qh0 )T

Ω1

(∇h yh )2×2 dx ∈ R2×2 sym ,

for some fixed Qh0 ∈ SO(2). We extend this planar rotation to Qh0 ∈ SO(3) and note
1 that |Qh0 − Id3 | ≤ Chβ /2−1 . Thus in particular, hβ /2−1 (Qh0 )T − Id3 converges to a matrix A1 ∈ so(3) (up to a subsequence that we do not relabel). Define y˜h (z,t)  (Qh0 )T yh (z,t) = (Qh Qh0 )T uh (z,th) − (Qh0 )T ch . There holds: ky˜h − πkH 1 (Ω 1 ) ≤ kyh − πkH 1 (Ω 1 ) + |Qh0 − Id3 |kyh kH 1 (Ω 1 ) → 0 as h → 0
1 V h [y˜h ] = (Qh0 )T V h [yh ] + β /2−1 (Qh0 )T − Id3 z → V − A1 z in H 1 (ω, R3 ). h

132

5 Limiting theories for elastic plates and shells: nonlinear bending

Since the limiting field V˜ = V − A1 z ∈ V (ω), all statements in Theorem 5.23 still hold for the modified sequence {y˜h }h→0 . On the other hand: ? ? ?
1 z dz = 0, V h [y˜h ] dz = (Qh0 )T V h [yh ] dz + β /2−1 (Qh0 )T − Id3 h ω ω ω ? ? 
1  2×2 ∇V h [y˜h ] tan dz = β /2−1 (Qh0 )T (∇h yh )2×2 dz − Id2 ∈ Rsym , h ω Ω1 as claimed in (5.67).

5.10 Lower bound beyond Kirchhoff’s scaling In the next result we deduce two further convergence properties of the family of deformations {uh ∈ H 1 (Sh , R3 )}h→0 under the sole energy scaling assumption (5.60). In particular, we will identify an asymptotic lower bound for the energies {h−β E h (uh )}h→0 , corresponding to the Γ -liminf inequality in Definition 2.1. The proof will be similar to steps 3 and 4 in the proof of Theorem 5.10 and the general form of the lower bound will be later specified for particular values of β > 2. Theorem 5.25. In the context of Theorem 5.23, there exists rotation fields {Rh ∈ H 1 (S, SO(3))}h→0 such that the following holds for the rescaled strains {Z h ∈ L2 (Sh0 , R3×3 )}h→0 , defined for all z ∈ S, |t| < h20 in: Z h (z + tn(z)) 

1  hβ /2

Rh (z)T ∇uh z + t


h n(z) − Id3 . h0

(i) {Z h }h→0 converge weakly in L2 (Sh0 , R3×3 ), up to a subsequence that we do not relabel, to some Z satisfying for all z ∈ S, |t| < h20 and τ ∈ Tz S:
t ∂τ A(z) n(z) h0 ? h0 /2 where Z0 (z) = Z(z + tn(z)) dt.

Z(z + tn(z))τ = Z0 (z)τ +

−h0 /2

The skew-symmetric matrix field A ∈ H 1 (S, so(3)) is derived from the limiting displacement field V ∈ V (S) in Theorem 5.23 (ii), through (5.59), (ii) we have the following lower bound: lim inf h→0

Z
1 h h 1 E (u ) ≥ Q2 z, sym(Z0 )tan dσ (z) I (V, Z )  0 β β 2 h S Z
1 Q2 z, sym(∇(An) − AΠ )tan dσ (z). + 24 S

5.10 Lower bound beyond Kirchhoff’s scaling

133

Proof. 1. The rotation fields {Rh } have already been defined in the proof of Theorem 5.23, by applying Theorem 4.8. Recall that we have: Zh 

1 hβ /2


(Rh ◦ π)T Qh ∇h yh − Id3 .

(5.68)

The weak convergence property of the family {Z h }h→0 follows by its boundedness in L2 (Sh0 , R3×3 ) in view of (5.63). In order to derive a formula for the limit Z, for each s, h  1 consider the vector fields on Sh0 : f s,h (z + tn(z)) 


1  h h y (z + (s + t)n(z)) − (z + (s + t)n(z)) h0 shβ /2
 h − yh (z + tn(z)) − (z + tn(z)) . h0

We claim that { f s,h }h→0 converges to f s,h (z + tn(z)) = =

1

?

1 h0 (An) ◦ π

weakly in H 1 (Sh0 , R3 ). Indeed:

t+s

h ∂n yh (z + rn(z)) − n(z) dr h0 t  ? t+s  1 h ∇ y (z + rn(z)) − Id dr n(z), 3 h β /2−1

hβ /2 h0 h

t

by Lemma 5.5. Thus the said convergence in L2 (Sh0 , R3 ) follows by (5.62), (5.63). 2. For the convergence of gradients, we first observe that: ∂n f s,h (z + tn(z)) =

  1 ∇h yh (z + (s + t)n(z)) − ∇h yh (z + tn(z)) n(z) β /2−1 sh0 h

converges to 0 in L2 (Sh0 , R3 ) again by (5.62) and (5.63). Further, for any τ ∈ Tz S: ∂τ f s,h (z + tn(z)) =

−1 h 1  = β /2 ∇h yh (z + (s + t)n(z)) Id3 + (s + t) Π (z) Id3 + tΠ (z) h0 sh

−1 h − ∇h yh (z + tn(z)) Id3 + t Π (z) Id3 + tΠ (z) h0
−1  h τ − sΠ (z) Id3 + tΠ (z) h0  
−1 1 h
= β /2 ∇h yh (z + (s + t)n(z)) − ∇h yh (z + tn(z)) Id3 + t Π Id3 + tΠ τ h0 sh  
−1 1 h ∇ y (z + (s + t)n(z)) − Id Π (z) Id + tΠ τ. + 3 3 h h0 hβ /2−1 The second term in the right hand side above converges in L2 (Sh0 , R3 ) to:

134

5 Limiting theories for elastic plates and shells: nonlinear bending


−1 1 A(z)Π (z) Id3 + tΠ (z) τ h0 by (5.62) and (5.63). Regarding the first term, it equals to:  
−1 1 h T h h
(Q ) (R (z)) Z h (z + (s + t)n(z)) − Z h (z + tn(z)) Id3 + t Π Id3 + tΠ τ s h0  
−1 1 * Z(z + (s + t)n(z)) − Z(z + tn(z)) Id3 + tΠ τ weakly in L2 (Sh0 , R3 ) s by (5.62) and the weak convergence of {Z h }h→0 to Z. Equating the weak limits of tangential derivatives above, we obtain:  1 1 1 Z(z + (s + t)n(z)) − Z(z + tn(z)) + A(z)|Tz S Π (z). ∇(An)(z) = h0 s h0 |Tz S Since ∂τ (An) = (∂τ A)n + AΠ τ, there follows (i). 3. We now prove (ii). Define the family of the indicator functions: χh  1{x∈Sh0 ; |Z h (x)|≤h−β /4 } . Recalling the frame invariance assumption in (5.2) and using the change of variable formula in Lemma 5.5 together with the Taylor expansion of W at Id3 , we get: Z
h 1 χh (x)W (Id3 + hβ /2 Z h ) · det (Id3 + t Π )(Id3 + tΠ )−1 dx h h0 S 0 h0 Z   hβ = χh (x) Q3 (Z h ) + o(1)|Z h |2 · (1 + o(1)) det(Id3 + tΠ )−1 dx. 2h0 Sh0

E h (uh ) ≥

Since {χh Z h }h→0 converges to Z, weakly in L2 (Sh0 , R3×3 ), in virtue of (i) we get: 1 1 h h E (u ) = lim inf Q3 (χh Z h ) · (1 + o(1)) det(Id3 + tΠ )−1 dx β 2h h Sh0 0 h→0 Z 1 Q3 (Z(x)) det(Id3 + tΠ (z))−1 dx ≥ 2h0 Sh0 Z ? h0 /2
1 = Q3 Z(z + tn(z)) dt dσ (z) 2 S −h0 /2 Z ? h0 /2
1 Q2 z, Z(z + tn(z))tan dt dσ (z). ≥ 2 S −h0 /2 Z

lim inf h→0

Finally, we evaluate the quadratic form on the linear in t expression of Z: ?

h0 /2

−h0 /2

?
Q2 z, Z(z + tn(z))tan dt =

h0 /2

−h0 /2

Q2 z, Z0 (z)tan +


t (∇(An) − AΠ )tan dt. h0

Since the linear in t terms integrate to zero on (− h20 , h20 ), the above equals to:

5.10 Lower bound beyond Kirchhoff’s scaling

?

135

h0 /2


t2
dt Q2 z, (∇(An) − AΠ )tan 2 −h0 /2 h0

1 = Q2 z, sym Z0 (z)tan + Q2 z, sym(∇(An) − AΠ )tan . 12


Q2 z, Z0 (z)tan +

This ends the proof of (ii) and of the theorem.

Theorem 5.26. In the context of Theorems 5.23 and 5.25, the following convergences hold as h → 0, up to a subsequence which we do not relabel: 1

1 sym∇V h [yh ] → (A2 )tan , 2 1 1 2 h h sym∇V [y ] * (A )tan + sym(Z0 )tan , h 2 1 h h sym∇V [y ] * sym(Z0 )tan . h

if β ∈ (2, 4) then:

hβ /2−1

if β = 4 then: if β > 4 then:

The first convergence above is strong, the other two are weak, in L2 (S, R2×2 ). Proof. 1. We use formula (5.65) and apply the symmetrization on both its sides. The 1 is the tangential minor of: second term in there, scaled by hβ /2−1   1  (Qh )T Rh − Id3 T  (Qh )T Rh − Id3  h T h = − sym (Q ) R − Id . 3 2 hβ −2 hβ /2−1 hβ /2−1 1

1 (Qh )T Rh − Id3 ) converges to A by (5.62) in H 1 (S, R3×3 ) and Recall that hβ /2−1 hence in L4 (S, R3×3 ) in view of the Sobolev embedding of vector fields defined on the 2-dimensional surface S. Consequently:

  1 1 h T h sym (Q ) R − Id → − AT A = A2 3 2 2 hβ −2 1

strongly in L2 (S, R3×3 )

where the last equality follows from the skew-symmetry of A. 2. The first term in (5.65), scaled by ? 1  hβ /2

h0 /2

1 h

and applied on τ ∈ Tz S equals:


∇yh (z + tn(z)) Id3 + tΠ (z) − (Qh )T Rh (z) dt τ

−h0 /2

= (Qh )T Rh (z)

?

h0 /2

 Z h (z + tn(z)) dt τ

−h0 /2

+ *

?

h0 /2

−h0 /2

1 h0 hβ /2−1

?

h0 /2


t ∇h yh (z + tn(z)) − (Qh )T Rh (z) Π (z) dt τ

−h0 /2



Z(z + tn(z)) dt τ = Z0 (z)τ

weakly in L2 (Sh0 , R3 ),

136

5 Limiting theories for elastic plates and shells: nonlinear bending

where we used the definition of the finite strain Z h in (5.68) and where the final convergence is implied by (5.62), (5.64) and the weak convergence of {Z h }h→0 to Z. The proof is done by examining the three indicated ranges of β . For the special case of plates, Theorem 5.26 and Remark 5.24 yield: Remark 5.27. Let S = ω ⊂ R2 and {Sh = Ω h = ω × (− h2 , h2 )}h 2. Definition 5.28. Let S ⊂ R3 be a smooth, compact, connected and oriented surface. The space of finite strains B(S) consists of the symmetric matrix fields in:

5.11 Bibliographical notes

137

n o B(S) = L2 − lim sym∇wh ; wh ∈ H 1 (S, R3 ) . h→0

We recall that sym∇wh ∈ L2 (S, R2×2 ) stands for a field of bilinear forms on Tz S in Definition 5.21 (i). Both the weak and the strong convergences yield the same B(S). Example 5.29. When S = ω ⊂ R2 is open, bounded and n connected, then in view o of Korn’s inequality in Theorem 3.1, we get: B(S) = sym∇w; w ∈ H 1 (ω, R2 ) . If ω is additionally simply connected, then a symmetric-valued matrix field B ∈ L2 (ω, R2×2 sym ) belongs to B(ω) if and only if curl curl B = 0; see Lemma 6.19.

5.11 Bibliographical notes For an excellent mathematical treaty on the field of elasticity, we refer to the books by Gurtin [1981] and Ciarlet [2000]. The derivation of lower dimensional models for thin structures (such as membranes, shells, or beams) has been one of the important questions since the beginning of research in elasticity, see Love [1927], Antman [1995] or Ciarlet [2000]. The presentation and results of this chapter are chiefly based on the fundamental work of Friesecke et al. [2002, 2006], extended to cover the case of shells in Friesecke et al. [2003] and Lewicka et al. [2010]. The definitions of quadratic and linear forms {Qi , Li }i=1,2 in Definitions 5.1 and 5.6 are taken from Friesecke et al. [2006]. The representation theorem in Remark 5.3 relies on Truesdell and Noll [2004]. We point out that, in general, one cannot expect the functional J in (5.4) or the total energies {J h }h→0 in Rsection 5.2, to posses a minimizer. The lowersemicontinuity of the energy E (u) = Ω W (∇u) dx, allowing for the direct method of Calculus of Variations, is tied to the quasiconvexity of the energy density, whereas the mapping F 7→ dist2 (F, SO(3)) is not even rank-one convex, as shown in [Zhang, 1997, proof of Proposition 1.6]. The Kirchhoff’s fully nonlinear bending theory, derived in sections 5.3 and 5.6, was first proposed by Kirchhoff [1850] and then obtained by Γ -convergence: in Friesecke et al. [2002] for plates and in Friesecke et al. [2003] in the general geometrical setting of thin shells. Example 5.20 is taken from Friesecke et al. [2006]. The truncation result in Theorem 5.12 can be found in Friesecke et al. [2002] and it is a special case of results by Liu [1977], Ziemer [1989] (valid in the general case of Sobolev W k,p functions). For its proof, we worked out the outline in Friesecke et al. [2006], with Lemma 5.14 and Lemma 5.15 taken from Ziemer [1989], and the crucial definition (5.37) from [Ziemer, 1989, Theorem 3.6.2]. Results in sections 5.9 and 5.10 appeared in Lewicka et al. [2010]. For plates, the term sym∇w + (∇v)⊗2 in J β to which the general stretching in Theorem 5.23 reduces, appeared in F¨oppl [1907]. The space B(S) in Definition 5.28 emerges in the context of linear elasticity and ill-inhibited surfaces, see Sanchez-Palencia [1989], Geymonat and Sanchez-Palencia [1995]. We do not discuss here the huge literature

138

5 Limiting theories for elastic plates and shells: nonlinear bending

on the derivation of lower dimensional theories from linear elasticity, see Ciarlet [2000] and references therein. The range β ∈ [0, 2) remains largely unexplored. For plates, the Γ -convergence was first established for β = 0 in Le Dret and Raoult [1995], and then for shells in Le Dret and Raoult [1996]; the limiting energy I0 which we do not discuss in this monograph, depends on the stretching and shearing produced by all H 1 regular deformation of S. See also the earlier heuristics in Fox et al. [1993]. The lower bound in the dimension reduction problem for the energy density with two wells SO(3)A1 ∪ SO(3)A2 and the energy scaling corresponding to β = 1, has been studied in Chaudhuri and M¨uller [2006]. We point out that a totally clamped plate exhibits a very rigid behavior already for β ∈ (0, 4), as shown in Conti et al. [2006]. In Conti and Maggi [2008] and in Venkaratarami [2004], energy levels of the origami patterns in paper crumpling were studied (β = 5/3). See also Cerda and Mahadevan [1998], Cerda et al. [1999], Cerda and Mahadevan [2005] for a theoretical and experimental analysis of conical singularities arising in crumpled sheets, and in M¨uller and Olbermann [2014], Olbermann [2016, 2019] for work on energy minimisation of the geometrically nonlinear plates of this singularity type (E h ∼ h2 log(1/h)). The mentioned papers do not address the dimension reduction, but rather analyze the chosen actual configuration of the sheet. Although the related time-dependent problem of nonlinear elasticity is far beyond the scope of this monograph, we briefly mention a few relevant references in this context. The question of local well-posedness for the elastodynamics system (5.1) was first addressed in Hughes et al. [1997] and then in Sabl´e-Tougeron [1988] for small Neumann data, in the framework of strong solutions. The lack of global existence was shown in Knops and Payne [1979], together with the blow-up of strong solutions in John [1984] and Gawinecki and Kacprzyk [2008]. On the other hand, the almost global existence was obtained in John [1988], Klainerman and Sideris [1996], and in Jie and Tiehu [2008] for the St. Venant-Kirchhoff model. The conditional global existence was discussed in Sideris [1996, 2000b], Agemi [2000] (under the so-called null condition) and in Sideris [2000a] (under the nonresonance assumption). For incompressible materials, the global existence literature is abundant, see for example Ebin [1993, 1996] and Sideris and Thomases [2005, 2007], Lei [2016] in the case of small data. Existence of Young measure solutions was analyzed in Demoulini [2000]. We also point out that in the context of dynamical viscoelasticity, various results on existence, asymptotics and stability have been obtained in Andrews [1980], Pego [1987], Rybka [1992] and in Friesecke and Dolzmann [1997]. In Lewicka and Mucha [2013], the local in time existence of regular solutions was shown, in presence of a general form of viscous stress tensor only assuming a Korn-type condition on its derivative. Existence and stability of viscoelastic shock profiles has been studied, among others, in Antman and Malek-Madani [1988], with the analysis extended to the numerical results in the general compressible and incompressible shear flow case in Barker et al. [2011].

Chapter 6

Limiting theories for elastic plates and shells: sublinear and linear

In this chapter, we continue the derivation of the Γ -limit of {h−β E h }h→0 , in the two energy scaling ranges corresponding to cases β = 4 and β > 4 in Theorem 5.26. Analysis for the intermediate exponent β = 4 is completed in section 6.1, in which we construct the recovery family for the von K´arm´an energy IvK , which is precisely the Γ -limit of the elastic energies of deformations on S, scaled by h−4 . This energy consists of the first order (in h) bending, due to the change of curvature of S under an infinitesimal isometry displacement in V (S), and the second order stretching computed from the combination of the first and second order displacements, encoded in the space of finite strains B(S). In section 6.2 we first combine the obtained results to show the Γ -convergence of the total energies, and then prove convergence of their approximate minimizers and the coercivity of the bending energy component in IvK . We also identify the set of the limiting rotations, allowing for the alignment of infinitesimal isometries with the direction of the applied force. Section 6.3 is devoted to the case β > 4, in which the Γ -limit of the elastic energies scaled by h−β is revealed to be the linear elasticity Ilin comprising the single bending term previously derived. We subsequently state the general Γ -convergence result in presence of external forces and analyze convergence of minimizers. In section 6.4 we outline extensions of our so-far results to the case of shells with variable thickness, given in terms of a possibly non-constant boundary function scaled by h. Section 6.5 is devoted to the analysis of the asymptotic behaviour of equilibra of the total energies (rather than convergence of minimizers that follows through the Γ -convergence) in the regime β ≥ 4. Under an extra upper bound assumption on the energy density, we prove that the said equilibra accumulate at the equilibra of their Γ -limits. For plates, when S ⊂ R2 , the corresponding limiting Euler-Lagrange equations are the celebrated von K´arm´an equations which we derive in section 6.6 together with their natural boundary conditions. We anticipate that the analysis of the so far missing regime β ∈ (2, 4), leading to the linearised Kirchhoff’s energy IlinK for plates will be done in chapter 7. The general scenario for arbitrary shells, leading to the conjectured infinite hierarchy of dimensionally reduced elastic shell theories, will be discussed in section 8. © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_6

139

140

6 Limiting theories for elastic plates and shells: sublinear and linear

6.1 Von K´arm´an’s theory for shells: recovery family We first gather the compactness and the lower bound statements achieved in Theorems 5.23, 5.25, 5.26. The corresponding upper bound in terms of the Von K´arm´an energy functional IvK in (6.2), will be proved in Theorem 6.3, which is the main objective of this section. Recalling Definition 5.28, we get: Theorem 6.1. Let S ⊂ R3 be a smooth, compact, connected, oriented surface with boundary given by finitely many Lipschitz curves. Let uh ∈ H 1 (Sh , R3 ) be a family of deformations such that: E h (uh ) ≤ Ch4 .

(6.1)

Then there exists rotations and translations {Qh ∈ SO(3), ch ∈ R3 }h→0 such that the normalized deformations {yh ∈ H 1 (Sh0 , R3 )}h→0 given in: yh (z + tn(z)) = (Qh )T uh (z + t

h n(z)) − ch h0

for all z ∈ S, |t|
4 in Theorem 6.10 for the linear elasticity. Theorem 6.3. Let S ⊂ R3 be a smooth, compact, connected, oriented surface with boundary given by finitely many Lipschitz curves. Let V ∈ V (S) and B ∈ B(S). Then, there exists a family {uh ∈ H 1 (Sh , R3 )}h→0 such that the rescaled deformations in: yh (z + tn(z)) = uh (z + t hh0 n(z)) have the following convergence properties: (i) {yh }h→0 converges in H 1 (Sh0 , R3 ) to π, (ii) the displacements {V h [yh ]}h→0 defined in Theorem 6.1 (ii) converge in H 1 (S, R3 ) to V , (iii) the scaled strains { h1 sym∇V h [yh ]}h→0 converge in L2 (S, R2×2 ) to B, (iv) there holds: 1 lim E h (uh ) = IvK (V, B). h→0 h4 Proof. 1. We write the given B ∈ B(S) as the L2 (S, R2×2 ) limit of {sym∇wh }h→0 , for some family of second order displacement fields {wh ∈ H 1 (S, R3 )}h→0 . Without loss of generality, we may assume that {wh } are smooth, and that: h1/2 kwh kW 2,∞ (S) → 0.

(6.3)

As mentioned before, the construction of {uh }h→0 will be carried out for any β ≥ 4, and we will indicate each time when we specify to the present case of β = 4. We now approximate the given V ∈ V (S). As in the proof of Corollary 5.13, we 2 fix ε  1 and apply Theorem 5.12 to λ = hβε/2−1 , together with a change of variable via a surface patch parametrisation, to get {vh ∈ W 2,∞ (S, R3 )}h→0 which satisfy: kvh −V kH 2 (S) → 0, hβ /2−1 kvh kW 2,∞ (S) ≤ ε, 1 {z ∈ S; vh (z) , V (z)} → 0. β −2 h

(6.4)

Finally, recall the notion of the linear map c(z, ·) in Definition 5.6 (ii) and consider the families of vector fields {d 0,h , d 1,h ∈ W 1,∞ (S, R3 )}h→0 satisfying:

6.1 Von K´arm´an’s theory for shells: recovery family

143

κ 2 κ κ (A )tan + A2 n + |An|2 n, 2 2 2

d 1,h → 2c z, sym (∇(An) − AΠ )tan + nT AΠ − nT ∇(An)

d 0,h → 2c z, B −




2

(6.5) 3

strongly in L (S, R ), where in the present case of β = 4 we take κ = 1. By reparametrizing (slowing down) smooth approximations, we may also ensure:
(6.6) h1/2 kd 0,h kW 1,∞ (S) + kd 1,h kW 1,∞ (S) → 0. We are ready to define the family of rescaled deformations {yh ∈ H 1 (Sh0 , R3 )}h→0 : yh (z + tn(z)) = z + hβ /2−1 vh (z) + hβ /2 wh (z)
hβ /2 h h n(z) + t Π vtan − ∇hvh , ni (z) h0 h0 β /2+1
hβ /2+1 0,h hβ /2+1 1,h h nT ∇wh (z) + t d (z) + t 2 −t d (z). h0 h0 2h20

+t

(6.7)

Above, we used the decomposition: h vh = vtan + hvh , nin.

Note that if already V ∈ W 2,∞ (S, R3 ) then one may take vh = V in which case the β /2 h − ∇hvh , ni) in (6.7) coincides with t hβ /2 An, as observed through term t hh0 (Π vtan h0 the formula (5.59) in Remark 5.22. 2. We now show the first three statements of Theorem 6.3. Convergence in (i) follows by (6.3), (6.4) and (6.6). For (ii) and (iii), notice that: h

h

V [y ] =

1 hβ /2−1

?

h0 /2

yh (z + tn(z)) − z dt = vh + hwh +

−h0 /2

1 2 1,h h d 24

1 1 h sym∇V h [yh ] = sym∇vh + sym∇wh + sym∇d 1,h . h h 24 Thus (ii) follows directly, and (iii) will be achieved once we establish that: 1 ksym∇vh kL2 (S) → 0. h

(6.8)

To this end, we use (6.4) and β ≥ 4 to note that: 1/2 1 1 ksym ∇vh kL2 (S) ≤ {vh , V } · ksym∇vh kL∞ (S) ≤ Chβ /2−2 ksym∇vh kL∞ (S) . h h Further, recall that the Lipschitz constant of each ∇vh is bounded by εh1−β /2 , and that sym∇vh = 0 on the set {vh = V }. Consequently:

144

6 Limiting theories for elastic plates and shells: sublinear and linear


|sym∇vh (z)| ≤ Ch1−β /2 dist z, {vh = V } . 1/2 Since Br (z) ∩ S ⊂ {vh , V } implies r ≤ C {vh , V } , it follows by (6.4) that
dist z, {vh = V } ≤ o(1)hβ /2−1 . Thus ksym∇vh kL∞ (S) = o(1), which yields (6.8). 3. Towards the proof of (iv), we will show that: lim sup h→0

1 h h E (u ) ≤ IvK (V, B) + oε (1), hβ

(6.9)

where oε (1) denotes any quantity which converges to 0 as ε → 0. In view of the lower bound in Theorem 6.1, this will imply (iv) for a recovery family obtained through a diagonal argument, by sending ε → 0. Clearly, the assertions (i) - (iii) will remain valid for such derived recovery family as well. We first take a closer look at the quantities ∇h yh . By Lemma 5.5 it follows that:
h (∇h yh )(z + tn(z))n(z) = n(z) + hβ /2−1 Π vtan − ∇hvh , ni t − hβ /2 nT ∇wh + hβ /2 d 0,h + hβ /2 d 1,h , h0  h (∇h yh )(z + tn(z))τ = Id3 + hβ /2−1 ∇vh + hβ /2 ∇wh + t Π (z) h0 (6.10) β /2+1
h hβ /2 h ∇ Π vtan ∇(nT ∇wh ) − ∇hvh , ni − t +t h0 h0
−1 hβ /2+1 0,h 2 hβ /2+1 1,h  h +t ∇d Id3 + t Π (z) τ, ∇d + t 2 h0 h0 2h0 for all τ ∈ Tz S. Directly by (6.3), (6.4) and (6.6) one then obtains: k∇h yh − Id3 kL∞ (Sh0 ) ≤ Cε.

(6.11)

It now follows by the polar decomposition theorem that ∇h yh is a product of a proper rotation and the well defined square root of (∇h yh )T ∇h yh . By properties of the energy density W and using Taylor’s expansion, we get: q

1 (∇h yh )T ∇h yh = W Id3 + K h + O(|K h |2 ) W (∇h yh ) = W 2
1 1 h (6.12) h 2 = Q3 K + O(|K | ) + oε (1)|K h |2 , 2 2 where K h  (∇h yh )T ∇h yh − Id3 . From (6.11), there easily follows the first uniform estimate: kK h kL∞ (Sh0 ) ≤ Cε.

(6.13)

6.1 Von K´arm´an’s theory for shells: recovery family

145

4. Using (6.10) we now calculate the limiting properties of K h in (6.12). Along the way, by Error we will cumulatively denote all the terms with the property: 1 kErrorkL2 (Sh0 ) → 0. hβ /2

(6.14)

We start with the tangential minor of K h : h
−1  Π Id2 + 2hβ /2−1 sym∇vh + 2hβ /2 sym∇wh h0
h hβ /2 h + 2t Π + 2t − ∇hvh , ni sym∇ Π vtan h0 h0 h2 + hβ −2 (∇vh )T ∇vh + t 2 2 Π 2 h0  β /2
h h
−1 + 2t sym Π ∇vh + Error Id2 + t Π − Id2 h0 h0 h
−1  2sym∇wh + hβ /2−2 (∇vh )T ∇vh = hβ /2 Id2 + t Π h0

2t 2t h
−1 h + sym∇ Π vtan Id2 + t Π + Error, − ∇hvh , ni + sym Π ∇vh h0 h0 h0

h (z + tn(z)) = Id2 + t Ktan

where we used the formulae: (Id + F)T (Id + F) = Id + 2 symF + F T F, F1−1 FF1−1 − Id = F1−1 (F − F12 )F1−1 . Notice that the quantity Error contains the term hβ /2−1 sym∇vh , resulting from the relaxation of the constraint V ∈ V (S) on the small set {vh , V }, and other product h −∇hvh , ni). The convergence of 1 ksym∇vh k terms, e.g.: hβ −1 (∇vh )T ∇(Π vtan h L2 (Sh0 ) to 0 has been already proved in (6.8). All other terms in Error can be dealt with by repeated use of (6.4), (6.3), and the H¨older and Sobolev inequalities, for example: h hβ /2−1 k(∇vh )T ∇(Π vtan − ∇hvh , ni)kL2 (S)

≤ Chβ /2−1 k∇vh kL4 (S) kvh kW 2,4 (S) 1/2

1/2

≤ Chβ /2−1 k∇vh kH 1 (S) kvh kW 2,∞ (S) kvh kH 2 (S) 1/2

≤ Chβ /2−1 kvh kW 2,∞ (S) → 0. 5. The diagonal coefficient of K h in the normal direction is calculated as:

h  K (z + tn(z))n(z), n(z)   2 h 2t = hβ /2 hβ /2−2 Π vtan − ∇hvh , ni + 2hd 0,h , ni + hd 1,h , ni + Error. h0

146

6 Limiting theories for elastic plates and shells: sublinear and linear

The remaining coefficients are, for all τ ∈ Tz S:

h  K (z + tn(z))n(z), τ D 
 h
−1 E hβ /2 h − ∇hvh , ni Id2 + t Π ∇ Π vtan τ = n, hβ /2−1 ∇vh + t h0 h0 D
hβ /2 1,h h + hβ /2−1 Π vtan − ∇hvh , ni + hβ /2 d 0,h + t d h0

hβ /2 h h Π vtan − ∇hvh , ni Π , + hβ −2 Π vtan − ∇hvh , ni ∇vh + t h0 h
−1 E τ + Error Id2 + t Π h0 Dt 

h h = hβ /2 ∇ Π vtan − ∇hvh , ni n + hβ /2−2 (∇vh )T Π vtan − ∇hvh , ni h0
t t h
−1 E h + Π Π vtan − ∇hvh , ni + d 0,h + d 1,h , Id2 + t Π τ + Error. h0 h0 h0 The estimation in Error is done as in step 4. The most troublesome convergence: 1 T h h − ∇hvh , ni)T kL2 (Sh0 ) → 0, kn ∇v + (Π vtan h can be proved as in (6.8), since the above quantity vanishes on the set {vh = V }. h − ∇hvh , ni)T k ∞ Therefore, knT ∇vh + (Π vtan L (S) converges to 0, as h → 0, and the displayed convergence follows by the last assertion in (6.4). 6. Steps 4 and 5 imply, in view of (6.14) that: 1 t K h → K1 ◦ π + K2 ◦ π + (ζ ⊗ n + n ⊗ ζ ) h0 2hβ /2

in L2 (Sh0 , R2×2 ),

(6.15)

2 h0 3 where {Ki ∈ L2 (S, R2×2 sym )}i=1,2 and ζ ∈ L (S , R ) are given by:

κ 2 (A )tan , 2
K2 = sym ∇(An) − AΠ tan ,
t
κ ζ (z + tn(z)) = c z, B − (A2 )tan + c z, sym(∇(An) − AΠ )tan , 2 h0 K1 = B −

(6.16)

with κ = 1 when β = 4, and κ = 0 when β > 4. Further, we deduce that: 1 hβ

Z S h0

|K h |4 dx → 0

as h → 0.

(6.17)

Indeed, (6.15) implies that hβ1/2 K h converges pointwise a.e. in Sh0 . Thus h1β |K h |4 converges a.e. to 0. By the boundedness of {K h } in (6.13), there holds then 1 |K h |4 ≤ C h1β |K h |2 , and the dominated convergence theorem achieves (6.17). hβ

6.2 Von K´arm´an’s theory for shells: Γ -limit and convergence of minimizers

147

Finally, we prove (6.9). By (6.17), it follows that the argument of Q3 in (6.12) scaled by hβ1/2 converges in L2 (Sh0 , R2×2 ) to the same limit as 2h1β /2 K h in (6.15). Using Lemma 5.5, convergences in (6.15), (6.16) and Definition 5.6, we obtain: lim sup h→0

Z ? h0 /2 1 h
1 h h E (u ) = lim sup W (∇h yh ) det Id2 + t Π dt dσ (z) β β h0 h S −h0 /2 h→0 h Z ? h0 /2   h
1 1 K h (z + tn) det Id + t Π dt dσ (z) Q3 ≤ lim sup 2 h→0 S −h0 /2 h0 2hβ /2 Z 1 |K h |2 dx + oε (1) lim sup β Sh0 h→0 h Z ? h0 /2   1 1 = Q3 lim β /2 K h dt dσ (z) h→0 2h 2 S −h0 /2

2

1

+ oε (1) lim β /2 K h 2 h h→0 h L (S 0 ) Z ? h0 /2   1 t ≤ Q2 z, K1 (z) + K2 (z) dt dσ (z) + oε (1) 2 S −h0 /2 h0 Z ? h0 /2
t2
1 Q2 z, K1 (z) + 2 Q2 z, K2 (z) dt dσ (z) + oε (1), = 2 S −h0 /2 h0

which implies (6.9) in view of (6.16). This ends the proof of the theorem.

6.2 Von K´arm´an’s theory for shells: Γ -limit and convergence of minimizers Similarly to section 5.7, we now combine Theorems 6.1 and 6.3, to show Γ convergence of the scaled versions of the total energies: J h (uh ) = E h (uh ) +

1 h

Z Sh

h f h , uh − id3 i dx,

and subsequently the convergence of their approximate minimizers, under the assumption that the applied forces { f h ∈ L2 (Sh , R3 )}h→0 are of order h4 (see the set-up in section 5.2). Given f ∈ L2 (S, R3 ), we have defined: Z

m f  min

h f (z), Rz − zi dσ (z),

R∈SO(3) S

Z n o M f  R ∈ SO(3); h f (z), Rz − zi dσ (z) = m f .

(6.18)

S

Note that M f consists of rotations which minimize (to m f ) the total energy J h among all rigid motions. The limiting total energy in the present context is:

148

6 Limiting theories for elastic plates and shells: sublinear and linear

 Z  ¯ i dσ if y = π and V ∈ V (S),  I (V, B) + h f , QV  vK S ¯ FvK (y,V, B, Q) = B ∈ B(S) and Q¯ ∈ M f , (6.19)    +∞ otherwise.

Corollary 6.4. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary ∂ S given by finitely many Lipschitz curves. Assume that f ∈ L2 (S, R3 ) satisfies (5.10) and define the family of external forces { f h ∈ L2 (Sh , R3 )}h→0 by:
−1 f h (z + tn(z)) = h3 det Id2 + tΠ (z) f (z)

h for all z ∈ S, |t| < . 2

For each y ∈ H 1 (Sh0 , R3 ), V ∈ H 1 (S, R3 ), B ∈ L2 (S, R2×2 ), Q¯ ∈ SO(3), define: h

¯ = F (y,V, B, Q)

 h ¯ h ) − h3 m f  J (Qu

if V = V h [y] and B = 1h sym∇V,



otherwise,

+∞

where we put uh (z + tn(z)) = y(z + t hh0 n(z)) ∈ H 1 (Sh , R3 ). Then, with respect to (strong) convergence in H 1 (Sh0 , R3 ) × H 1 (S, R3 ) × L2 (S, R2×2 ) × SO(3), we have: 1 h Γ F −→ FvK h4

as h → 0.

Proof. 1. To verify condition (i) in Definition 2.1, let {(yh ,V h , Bh , Q¯ h )}h→0 converge ¯ in H 1 (Sh0 , R3 )×H 1 (S, R3 )×L2 (S, R2×2 )×SO(3). Without loss to some (y,V, B, Q), of generality, we may assume that: lim inf h→0

1 h h h h ¯h 1 F (y ,V , B , Q ) = lim 4 F h (yh ,V h , Bh , Q¯ h ) < +∞. h→0 h h4

In particular, V h = V h [yh ], Bh = h1 sym∇V h and so B ∈ B(S). By Theorem 5.9, it follows directly that E h (uh ) = E h (Q¯ h uh ) ≤ Ch4 and Theorem 6.1 yields then existence of {Qh ∈ SO(3), ch ∈ R3 }h→0 such that, up to a (not relabeled) subsequence: in H 1 (Sh0 , R3 ), V h [(Qh )T yh − ch ] → V˜ ∈ V (S) in H 1 (S, R3 ), (Qh )T yh − ch → π

1 sym∇V h [(Qh )T yh − ch ] * B˜ h

2

(6.20) 2×2

weakly in L (S, R

).

Since V h [(Qh )T yh − ch ] = (Qh )T V h + 1h ((Qh )T − Id3 )z − 1h ch , we obtain: ∇V h [(Qh )T yh − ch ] − (Qh )T ∇V h =


1 (Qh )T − Id3 |Tz S , h

(6.21)

6.2 Von K´arm´an’s theory for shells: Γ -limit and convergence of minimizers

149

and in view of the boundedness of left hand side above in L2 (S, R3 ), we get the following convergences (again, possibly up to a subsequence): Qh → Id3 ,


1 (Qh )T − Id3 → A1 ∈ so(3) h

as h → 0.

Consequently, { h1 ch }h→0 is also bounded, thus ch → 0. Convergences in (6.20) now yield: y = π and V˜ = V + A1 z − c, for some c ∈ R3 . This implies that V ∈ V (S). 2. We now examine the affine term in 14 (J h (Q¯ h uh ) − h3 m f ): h

1 h5

1 h f h ,Q¯ h uh − id3 i dx − m f h Sh Z ? h0 /2  1 h f (z), Q¯ h yh (z + tn(z)) − zi dt dσ (z) − m f = h S −h0 /2 Z Z  1 = h f , Q¯ hV h i dσ + h f (z), Q¯ h z − zi dσ (z) − m f . h S S

Z

(6.22)

¯ i dσ , and the second The first term in the right hand side above converges to S h f , QV term is nonnegative. By the lower bound statement in Theorem 6.1, we get: R

1 ¯ i dσ , ˜ + h f , QV lim inf 4 F h (yh ,V h , Bh , Q¯ h ) ≥ IvK (V˜ , B) h→0 h S  R
and also we observe that h1 S h f (z), Q¯ h z − zi dσ (z) − m f h→0 must be bounded. In particular, that the difference in the parenthesis converges to 0 as h → 0, so there must be Q¯ ∈ M f . We now show the remaining statement: Z

˜ = IvK (V, B). IvK (V˜ , B)

(6.23)

3. First, recall that a direct calculation in (6.21) yields:

1 1 1 sym∇V h [(Qh )T yh − ch ] = sym (Qh )T ∇V h + 2 sym (Qh )T − Id3 tan . h h h The first term in the right hand side above converges to: sym(A1 ∇V ) + B, in L2 (S, R2×2 ). For the second term, we observe the following convergence:
1 1  (Qh )T − Id3 T  (Qh )T − Id3  sym (Qh )T − Id3 = − 2 h 2 h h 1 T → − A1 A1 as h → 0. 2 In conclusion, by (6.20) we get: B˜ = B + sym(A1 ∇V ) − 12 (AT1 A1 )tan . This yields:

150

6 Limiting theories for elastic plates and shells: sublinear and linear

1 B˜ + (∇V˜ )T ∇V˜ 2

T 1 1 = B + sym(A1 ∇V ) − (AT1 A1 )tan + ∇V + (A1 )|Tz S ∇V + (A1 )|Tz S (6.24) 2 2 1 T = B + (∇V ) ∇V. 2 Further, denoting by A, A˜ ∈ H 1 (S, so(3)) the matrix fields so that ∇V = A|Tz S and ∇V˜ = A˜ |Tz S , respectively, we recall that A˜ = A + A1 and hence obtain: ˜ − AΠ ˜ sym ∇(An)


tan

= sym ∇(An) + A1 Π − (A + A1 )Π
= sym ∇(An) − AΠ tan .


tan

(6.25)

The formulas (6.24) and (6.25) imply (6.23) and thus conclude the verification of condition (i) in Definition 2.1. 4. We now verify condition (ii) in Definition 2.1. Consider the quadruple: ¯ ∈ H 1 (Sh0 , R3 ) × H 1 (S, R3 ) × L2 (S, R2×2 ) × SO(3). (y,V, B, Q) If the following assertion is violated: y = π,

V ∈ V (S),

B ∈ B(S),

Q¯ ∈ M f ,

(6.26)

¯ h→0 . Indeed, in this case FvK (y,V, B, Q) ¯ = then we set {(yh ,V h , Bh , Q¯ h ) = (y,V, B, Q)} 1 h h h h h ¯ +∞, and likewise there must be limh→0 h4 F (y ,V , B , Q ) = +∞, because otherwise reasoning as in step 1 would bring a contradiction. Assume now that (6.26) holds. By Theorem 6.3 we obtain a recovery family {yh ∈ H 1 (Sh0 , R3 )}h→0 converging to y and such that: V h  V h [yh ] → V in H 1 (S, R3 ), Bh  h1 sym∇V h [yh ] → B in L2 (S, R2×2 ) and h14 E h (uh ) → IvK (V, B). It thus follows: 1 1 h h h h ¯ F y ,V , B , Q) = lim 4 E h (uh ) + 4 h→0 h h→0 h

Z

lim

S

¯ i dσ (z) = FvK (y,V, B, Q), ¯ h f , QV

in view of (6.22) applied with Q¯ h = Q¯ ∈ M f . The proof is done. Similarly as in section 5.7, we now deduce a convergence of minimizers result. It implies that it is energetically advantageous for the thin shell to perform a large rotation rather than undergo a single compression. The set M f identifies candidates for such rotations, allowing for the alignment of infinitesimal isometries with the direction of the dead load f . In general, the shell chooses both an infinitesimal isometric displacement V ∈ V (S) and its second order correcting displacement through B ∈ B(S), together with a rotation Q¯ ∈ M f in response to f .

6.2 Von K´arm´an’s theory for shells: Γ -limit and convergence of minimizers

151

Corollary 6.5. In the context of Corollary 6.4, consider a family of almostminimizers {(uh , Q¯ h ) ∈ H 1 (Sh , R3 ) × SO(3)}h→0 to the energy functionals: 1 J¯ h (uh , Q¯ h ) = J h (Q¯ h uh ) = E h (uh ) + h

Z Sh

h f h , Q¯ h uh − id3 i dx

satisfying: 1 ¯ h h ¯ h J (u , Q ) − inf J¯ h → 0 h4

as h → 0.

(6.27)

Then, the family {yh (z + tn(z)) = Qh uh (z + t hh0 n(z)) − ch ∈ H 1 (Sh0 , R3 )}h→0 defined for some {Qh ∈ SO(3)}h→0 and {ch ∈ R3 }h→0 , converges up to a subsequence in H 1 (Sh0 , R3 ), to the limit π, while we also have the following subsequential convergences: V h [yh ] 

1 h

?

h0 /2

−h0 /2

yh (z + tn(z) − z dt → V ∈ V (S) in H 1 (S, R3 ),

1 sym∇V h [yh ] * B ∈ B(S) weakly in L2 (S, R2×2 ), h Q¯ h → Q¯ ∈ M f .

(6.28)

The limiting quantities minimize the functional FvK in (6.19). Equivalently: ¯ = min I¯vK < +∞, I¯vK (V, B, Q) ¯ = IvK (V, B) + where I¯vK (V, B, Q)

Z

¯ i dσ . h f , QV

(6.29)

S

In particular, the functional I¯vK , defined on V (S) × B(S) × M f , has at least ¯ in (6.29), coincides with one minimizer. Conversely, every minimizing (V, B, Q) the limit as in (6.28), derived from some families {yh ∈ H 1 (Sh0 , R3 )}h→0 and {Q¯ h ∈ SO(3)}h→0 , for which (6.27) holds, where we define the inverse rescalings {uh (z + tn(z)) = yh (z + t hh0 n(z)) ∈ H 1 (Sh , R3 )}h→0 . There also holds: 1 inf J¯ h → min I¯vK h4

as h → 0.

The fact that the energy functional I¯vK defined in (6.29) has at least one minimizer, can be proved directly. We start by pointing out an observation on the compensated regularity effect in infinitesimal isometries: Remark 6.6. Given V ∈ H 2 (S, R3 ), there holds V ∈ V (S) if and only if: sym∇Vtan = −hV, niΠ ∈ H 2 (S, R2×2 ). Recall that we use the decomposition into tangential and normal components, writing: V = Vtan + hV, nin. Applying Korn’s inequality in Theorem 3.28, and the for-

152

6 Limiting theories for elastic plates and shells: sublinear and linear

mula (3.21) in the local coordinates, it follows that:
kVtan kH 2 (S) ≤ C kVtan kL2 (S) + khV, nikH 1 (S) .

(6.30)

We note that one can deduce that Vtan ∈ H 3 (S, R3 ) and:
kVtan kH 3 (S) ≤ C kVtan kL2 (S) + khV, nikH 2 (S) , valid for any V ∈ V (S) with C that depends only on S. We now show a coercivity result that is of independent interest. Theorem 6.7. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary ∂ S given by finitely many Lipschitz curves. Then, for every V ∈ V (S) there exist D ∈ so(3) and d ∈ R3 , so that: kV

− (Dz + d)k2H 2 (S)

≤C

Z S

(∇(An) − AΠ )tan 2 dσ .

Proof. 1. We first check that if V ∈ V (S) satisfies S |(∇(An) − AΠ )tan |2 dσ = 0, then V (z) = Dz + d with some D ∈ so(3), d ∈ R3 . Consider c ∈ H 1 (S, R3 ) such that: R

A(z)τ = c(z) × τ

for all z ∈ S, τ ∈ Tz S.

Since A represents a gradient, it follows that ∂τ c × η = ∂η c × τ for all τ, η ∈ Tz S. In particular, if τ × η = n then: h∂τ c, ni = −h∂τ c × η, τi = −h∂η c × τ, τi = 0. Since:

0 = ∂τ (An) − AΠ τ tan = ∂τ (c × n) − AΠ τ tan = (∂τ c) × n, it follows that ∇c = 0 on S, so c and hence A are constant. This yields the claim. 2. The result is proved by contradiction. Assume that for a family of infinitesimal isometries {V h ∈ V (S)}h→0 there holds as h → 0:
distH 2 (S) V h , {Dx + d; D ∈ so(3), d ∈ R3 } = 1, Z (6.31) (∇Ah )n 2 dσ → 0. and S

Since the second condition above involves only higher derivatives of V h , we may without loss of generality replace the first condition by:

h  kV h kH 2 (S) = 1 and V , Dx + d H 2 (S) = 0 for all D ∈ so(3), d ∈ R3 . (6.32) In particular, {V h }h→0 converges weakly in H 2 (S, R3 ) up to a subsequence which we do not relabel, to some V ∈ V (S). By the second statement in (6.31) and the weak

6.2 Von K´arm´an’s theory for shells: Γ -limit and convergence of minimizers

153

2 R lower semicontinuity of the L2 norm, it follows that S (∇A)n dσ = 0. Hence, in view of step 1 and the second condition in (6.32) there must be V = 0, which implies: kV h kH 1 (S) → 0

as h → 0.

(6.33)

h = V h − hV h , nin, the estimates (6.30) and (6.33) further yield: Writing Vtan h kH 2 (S) → 0. kVtan

(6.34)

On the other hand: Z S

Z 2
h (∇Ah )n 2 dσ = ∇ ΠVtan − ∇(V h n) − Ah Π dσ

=

ZS S

2 h ∇2 (V h n) + (Ah Π − Π Ah )tan − (∇Π )Vtan + (V h n)Π dσ .

Therefore:
k∇2 hV h , nikL2 (S) ≤ C k(∇Ah )nkL2 (S) + kV h kH 1 (S) , and in view of (6.33) and the assumption (6.31) we also get that: khV h , nikH 2 (S) → 0. Together with (6.34), this contradicts (6.32) and proves the result. The coercivity statement proved in Theorem 6.7 immediately yields: Corollary 6.8. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary ∂ S given by finitely many Lipschitz curves. Let f ∈ L2 (S, R3 ) satisfy (2.2). Then the functional: I¯vK : V (S) × B(S) × M f → R defined in (6.29), is bounded from below and attains its infimum. Proof. Let V ∈ V (S). By Theorem 6.7, there exist D ∈ so(3), d ∈ R3 such that for V˜ = V − (Dz + d) there holds, in view of the positive definiteness of Q2 : I¯vK (V ) ≥ ckV˜ k2H 2 (S) −

Z S

¯ i dσ = ckV˜ k2 2 − h f , QV H (S)

Z

h f , Q¯ V˜ i dσ

S

(6.35)

≥ ckV˜ k2H 2 (S) − k f kL2 (S) · kV˜ kL2 (S) . In the equality above, we used (2.2) and a consequence of the definition (6.18): Z S

¯ h f (z), QDzi dσ (z) = 0

for all Q¯ ∈ M f , D ∈ so(3).

Since the function R 3 s 7→ cs2 − Cs ∈ R is clearly bounded from below, (6.35) implies that I¯vK is likewise bounded from below.

154

6 Limiting theories for elastic plates and shells: sublinear and linear

Let now {(V h , Bh , Q¯ h )}h→0 be a minimizing sequence of I¯vK . where we may without loss of generality assume that Q¯ h → Q¯ ∈ M f . Using (6.35) and applying the positive definiteness of Q2 in the first term in IvK , there follows the (uniform in h) boundedness of the following expressions: 

2 1 ckV˜ h k2H 2 (S) − k f kL2 (S) · kV˜ h kL2 (S) + c Bh − ((Ah )2 )tan L2 (S) , 2 where V˜ h = V h − (Dh z + d h ) for some Dh ∈ so(3), d h ∈ R3 .



(6.36)

In particular, we get the following convergence, up to a not relabeled subsequence: V˜ h * V ∈ V (S)

weakly in H 2 (S, R3 ) as h → 0.

Further, the corresponding gradient fields {A˜ h = Ah − Dh }h→0 converge weakly in H 1 (S, so(3)) to A ∈ H 1 (S, so(3)). Note also that: (Ah )2 = (A˜ h )2 + (Dh )2 + (Dh Ah + Ah Dh ). Hence, the boundedness of the second term in (6.36) yields the boundedness of: Bh −



1 1 (Dh )2 + Dh Ah + Ah Dh tan = Bh − sym∇ (Dh )2 z + 2DhV h (z) , 2 2

in L2 (S, R2×2 ). We may now conclude that, up to a subsequence, the above family converges, weakly in L2 (S, R2×2 ), to some B ∈ B(S). Thus: 1 1 Bh − ((Ah )2 )tan * B − (A2 )tan 2 2

weakly in L2 (S, R2×2 ) as h → 0,

up to a subsequence that we do not relabel. By the weak lower semicontinuity of ¯ realizes the infimum of both quadratic terms in IvK we conclude that I¯vK (V, B, Q) ¯ IvK . The proof is done.

6.3 Linear elasticity for shells: recovery family, Γ -limit and convergence of minimizers In this section, we present the versions of the compactness, the lower bound, the recovery family and the convergence of minimizers statements from section 6.1, pertaining to the energy scaling exponent β > 4. We only indicate when the proofs depart from those we have already carried out. Firstly, the compactness and the lower bound results obtained in Theorems 5.23, 5.25, 5.26 are summarised in Theorem 6.9 below, while the counterpart of the upper bound result in Theorem 6.3 is given in Theorem 6.10.

6.3 Linear elasticity for shells: recovery family, Γ -limit and convergence of minimizers

155

Theorem 6.9. Let S ⊂ R3 be a smooth, compact, connected, oriented surface with boundary given by finitely many Lipschitz curves. Let uh ∈ H 1 (Sh , R3 ) be a family of deformations such that: E h (uh ) ≤ Chβ

for some β > 4.

(6.37)

Then there exists rotations and translations {Qh ∈ SO(3), ch ∈ R3 }h→0 such that the normalized deformations {yh ∈ H 1 (Sh0 , R3 )}h→0 given in: yh (z + tn(z)) = (Qh )T uh (z + t

h n(z)) − ch h0

for all z ∈ S, |t|
4. Then, given any V ∈ V (S) there exists a family {uh ∈ H 1 (Sh , R3 )}h→0 such that the rescaled deformations in: yh (z + tn(z)) = uh (z + t hh0 n(z)) have the following convergence properties: (i) {yh }h→0 converges in H 1 (Sh0 , R3 ) to π, (ii) displacements {V h [yh ]}h→0 defined in Theorem 6.9 (ii), converge in H 1 (S, R3 ) to V , (iii) there holds: 1 lim E h (uh ) = Ilin (V ). h→0 hβ

156

6 Limiting theories for elastic plates and shells: sublinear and linear

Proof. The recovery family for a given V ∈ V (S) is defined by (6.7), where we put wh = 0 and {d 0,h = 0}h→0 in (6.5). That is: yh (z + tn(z)) = z + hβ /2−1 vh (z) + t +t

h n(z) h0


hβ /2 hβ /2+1 1,h h d (z), Π vtan − ∇tan (vh n) (z) + t 2 h0 2h20

where d 1,h ∈ W 1,∞ (S, R3 ) satisfies (6.6) and the second formula in (6.5). Clearly, {yh }h→0 and {V h [yh ]}h→0 converge in H 1 (Sh0 , R3 ) to π and in H 1 (S, R3 ) to V , respectively, as in step 2 of the proof of Theorem 6.3. The convergence of the scaled energy follows as in steps 3-6, by setting B = 0 and noting that κ = 0. Remark 6.11. When S = ω ⊂ R2 , then as in Remark 6.2 (iii), the linear elasticity functional Ilin in (6.38) reduces to the sole quantification of first order bending: Ilin (v) =

1 24

Z

Q2 (∇2 v) dz,

ω

written in terms of the out-of-plane displacement v ∈ H 2 (ω, R). The counterpart of Corollary 6.4 is: Corollary 6.12. Let S ⊂ R3 be a smooth, compact, connected, oriented surface, with its boundary given by finitely many Lipschitz curves. Let β > 4. Assume that f ∈ L2 (S, R3 ) satisfies (2.2) and define the family { f h ∈ L2 (Sh , R3 )}h→0 by:
−1 f h (z + tn(z)) = hβ /2+1 det Id2 + tΠ (z) f (z)

h for all z ∈ S, |t| < . 2

Recalling the definitions of m f and M f in (6.18), for each y ∈ H 1 (Sh0 , R3 ), V ∈ H 1 (S, R3 ), Q¯ ∈ SO(3), define uh (z + tn(z)) = y(z + t hh0 n(z)) ∈ H 1 (Sh , R3 ) and put: ( h ¯ h β /2+1 m if V = V h [y], f ¯ = J (Qu ) − h F h (y,V, Q) +∞ otherwise. Then, with respect to (strong) convergence in H 1 (Sh0 , R3 ) × H 1 (S, R3 ) × SO(3): 1 h Γ F −→ Flin hβ

as h → 0,

where Flin : H 1 (Sh0 , R3 ) × H 1 (S, R3 ) × SO(3) → R¯ is given by: ( ¯ = Flin (y,V, Q)

Ilin (V ) + +∞

Z S

¯ i dσ if y = π, V ∈ V (S), Q¯ ∈ M f , h f , QV otherwise.

(6.39)

6.4 Shells with variable thickness

157

There now follows the counterpart of Corollary 6.5 on convergence of minimizers: Corollary 6.13. In the context of Corollary 6.12, consider a family of almostminimizers {(uh , Q¯ h ) ∈ H 1 (Sh , R3 ) × SO(3)}h→0 to the energy functionals: 1 J¯ h (uh , Q¯ h ) = J h (Q¯ h uh ) = E h (uh ) + h

Z Sh

h f h , Q¯ h uh − id3 i dx

satisfying, for a fixed β > 4: 1 ¯ h h ¯ h J (u , Q ) − inf J¯ h → 0 β h

as h → 0.

(6.40)

Then, the family {yh (z +tn(z)) = Qh uh (z +t hh0 n(z)) − ch ∈ H 1 (Sh0 , R3 )}h→0 defined for some {Qh ∈ SO(3), ch ∈ R3 }h→0 , converges up to a subsequence in H 1 (Sh0 , R3 ), to the limit π, while we also have the following subsequential convergences: V h [yh ] → V ∈ V (S) in H 1 (S, R3 ), Q¯ h → Q¯ ∈ M f .

(6.41)

¯ minimize the functional Flin in (6.39). Equivalently: The limiting (V, Q) ¯ = min I¯lin < +∞, I¯lin (V, Q) ¯ = Ilin (V ) + where I¯lin (V, Q)

Z

(6.42)

¯ i dσ . h f , QV

S

In particular, the functional I¯lin , defined on V (S) × M f , has at least one minimizer. ¯ in (6.42), is derived in the limit (6.41), from Conversely, every minimizing (V, Q) some {yh ∈ H 1 (Sh0 , R3 ), Q¯ h ∈ SO(3)}h→0 , for which (6.40) holds, written for the inverse rescalings {uh (z +tn(z)) = yh (z +t hh0 n(z)) ∈ H 1 (Sh , R3 )}h→0 . There holds: 1 inf J¯ h → min I¯lin hβ

as h → 0.

6.4 Shells with variable thickness In this section, we outline the extension of results in the present chapter, to shells of the more general form considered in sections 3.6, 3.8, 4.5. Let S ⊂ R3 be a smooth, compact, connected, oriented surface with boundary given by finitely many Lipschitz curves. For two positive, smooth functions g1 , g2 : S → R, define: Sh = {x = z + tn(z); z ∈ S, − hg1 (z) < t < hg2 (z)},

h  1.

For simplicity of presentation, we assume that k(g1 , g2 )kL∞ (S)  1 so that the referential shell Sh0 on which π is well defined, may be taken with h0 = 1. Then, com-

158

6 Limiting theories for elastic plates and shells: sublinear and linear

pactness properties of deformations {uh ∈ H 1 (Sh , R3 )}h→0 that satisfy E h (uh ) ≤ Ch4 , are the same as stated in Theorem 6.1 (i), (ii), (iii), when defining: V h [yh ](z) =

?

1 hβ /2−1

g2 (z)

yh (z + tn(z)) − (z + htn(z)) dt.

−g1 (z)

The lower bound in (6.2) and the upper bound in Theorem 6.3 remain valid, upon replacing the von K´arm´an functional IvK by its version relative to g1 , g2 in: g1 ,g2 (V, B) = IvK


 1 1 (g1 + g2 )Q2 z, B − (A2 )tan − sym A∇(g2 n − g1 n) dσ (z) 2 2 S Z 
 1 (g1 + g2 )3 Q2 z, ∇(An) − AΠ tan dσ (z). + 24 S 1 2

Z

When β > 4 then statements as in Theorem 6.9 (i), (ii) hold, with V h [yh ] defined above, while the bound in (iv) together with Theorem 6.10 are valid with the energy: g1 ,g2 Ilin (V ) =

1 24

Z S


 (g1 + g2 )3 Q2 z, ∇(An) − AΠ tan dσ (z).

g1 ,g2 g1 ,g2 Both IvK and Ilin are defined for V ∈ V (S) and B ∈ B(S), see Definitions 5.21 and 5.28. The matrix field A ∈ H 1 (S, so(3)) is derived from V by (5.59).

As in Remark 6.2, we may identify the stretching and bending contents: Remark 6.14. Given V ∈ V (S) and {wh ∈ (S, R3 )}h→0 with sym∇wh → B ∈ B(S) in L2 (S, R2×2 ), consider the deformations: h φ˜ h = idS + (g2 − g1 )n, 2

φ h = φ˜ h + hV + h2 wh .

Then, φ˜ h (S) can be seen as the geometric mid-surface of Sh . A direct calculation: |∂τ φ h |2 −|∂τ φ˜ h |2 = 2h2

D


 E 1 1 sym∇wh − A2 − sym A∇(g2 n−g1 n) τ, τ +O(h3 ), 2 2

valid for any τ ∈ Tz S, thus shows that the expression under Q2 in the first term of g1 ,g2 IvK describes the second order in h change of the metric on φ˜ h (S), due to V , B. g1 ,g2 On the other hand, the integrand in Ilin measures the first order in h change in h ˜ the second fundamental form of φ (S). To see this, let Π h be the shape operator on φ h (S). Then, for all τ ∈ Tz S there holds:   (∇φ h )−1 Π h (∇φ h )τ − (∇φ˜ h )−1 Π (∇φ˜ h )τ = h ∂τ (An) − AΠ τ + O(h2 ).


We also point out that the term − 21 sym A∇((g2 − g1 )n ) which is new with respect to the previous analysis, expresses the first order in h deficit for V from being an infinitesimal isometry on φ˜ h (S). Indeed, for any η = ∂τ φ˜ h (x) and τ ∈ Tz S, we have:

6.5 Convergence of equilibria

159




 h

∂η (V ◦ (φ˜ h )−1 ), η = h∂τ V, ∂τ φ˜ h i = − sym A∇((g2 − g1 )n) τ, τ . 2 The above term disappears when g1 = g2 , or equivalently when S = φ˜ h (S). Convergence of minimizers in presence of external forces may be analyzed in the following setting. Given β ≥ 4 and f ∈ L2 (S, R3 ), let { f h ∈ L2 (S, R3 )}h→0 satisfy: 1 f h → f in L2 (S, R3 ). hβ /2+1
−1 h Define extensions f h (z + tn(z)) = det Id2 + tΠ (z) f (z) ∈ L2 (Sh , R3 ), and let: Z

S

(g1 + g2 ) f h dσ (z) = 0,

1

mh  min

Z

R∈SO(3) hβ /2+2

Sh

h f h (x), Rx − xi dx

for all h  1.

The relaxation function r : SO(3) → [0, ∞] with its effective domain M , are: ¯  r(Q)

inf

Q¯ h ∈SO(3), Q¯ h →Q

n

lim inf h→0

o  1 Z h ¯ h x − xi dx − 1 mh , h f (z), Q hβ +1 Sh hβ /2−1

¯ < ∞}. M  {Q¯ ∈ SO(3); r(Q) Then, results in Corollaries 6.5 and 6.13 remain true for the total energies: 1 J¯ h (uh , Q¯ h ) = E h (uh ) + h

Z Sh

h f h , Q¯ h uh − id3 i dx − hβ /2+1 mh ,

and the following limiting functionals defined for V ∈ V (S), B ∈ B(S), Q¯ ∈ M : g1 ,g2 ¯ = I¯vK (V, B, Q) ¯ + I¯vK (V, B, Q) g1 ,g2 ¯ = I¯lin (V, Q) ¯ + I¯lin (V, Q)

Z S

Z S

¯ i dσ (z) + r(Q), ¯ (g1 + g2 )h f , QV

¯ i dσ (z) + r(Q). ¯ (g1 + g2 )h f , QV

Likewise, the Γ -convergence statements in Corollaries 6.4 and 6.12 remain true with the above modifications.

6.5 Convergence of equilibria We now return to our main setting of g1 = g2 ≡ 21 , and f ∈ L2 (S, R3 ) that satisfies the zero mean condition (2.2) and generates a family of forces { f h ∈ L2 (Sh , R3 )}h→0 : f h (z + tn(z)) = hβ /2+1 det(Id2 + tΠ (z))−1 f (z)

h for all z ∈ S, |t| < . 2

We continue to have β ≥ 4. In this section, we show that under the extra condition:

160

6 Limiting theories for elastic plates and shells: sublinear and linear

for all F ∈ R3×3 ,

|DW (F)| ≤ C(|F| + 1)

(6.43)

also the equilibria (possibly non-minimizing) of the total energies: J h (uh ) = E h (uh ) +

1 h

Z Sh

h f h , uh − id3 i dx

converge to the equilibria of the respective Γ -limits of their rescalings by h−β . The growth condition (6.43) is incompatible with the global orientation preservation condition in (5.3). The relevance of (6.43) is, however, due to: Lemma 6.15. Assume (5.2), (6.43) and β > 0. If a given family of matrix fields {Z h ∈ L2 (Sh0 , R3×3 )}h→0 converges weakly in L2 (Sh0 , R3×3 ) to some Z, then: Eh 

1 DW (Id3 + hβ /2 Z h ) * E  L3 Z hβ /2 weakly in L2 (Sh0 , R3×3 ) as h → 0.

Proof. We first note that by (5.2), condition (6.43) implies (it is equivalent to): for all F ∈ R3×3 .

|DW (Id3 + F)| ≤ C|F|

(6.44)

For any E˜ ∈ L2 (Sh0 , R3×3 ) we now consider: Z

 β /2 h β /2 h ˜ ˜ dx ≤ 1 DW (Id + h Z ) − L (h Z ), E dx hE h − L3 Z, Ei 3 3 hβ /4 S h0 Sh0 Z

 L3 (Z h − Z), E˜ dx . +

Z

S h0

The second term in the right hand side above converges to 0, by the weak convergence of {Z h }h→ 0 and the linearity of L3 . Recall now the notion of characteristic functions {χh  1{x∈Sh0 ; |Z h |≤h−β /2 } }h→0 from the proof of Theorem 5.25. We now split the first integration into the sum of: integration on the set {χh = 1} where we may use the assumption of DW being C 1 -regular close to Id3 , and on the set {χh = 0} where we use (6.44). Consequently, it follows that: Z

S h0

˜ dx ≤o(1) + o(1) sup kZ h k 2 h0 hE h − L3 Z, Ei L (S ) h

˜ 2 h0 sup kZ h k 2 h0 → 0 + kχh Ek L (S ) L (S ) h

This achieves the proof. We present the main result of this section:

as h → 0.

6.5 Convergence of equilibria

161

Theorem 6.16. Let S ⊂ R3 be a smooth, compact, connected, oriented surface with boundary given by finitely many Lipschitz curves. In addition to (5.2), assume (6.43) and let β ≥ 4. Assume that {uh ∈ H 1 (Sh , R3 )}h→0 satisfies: (a) for all h  1 and all φ h ∈ H 1 (Sh , R3 ) there holds: Z

Sh

Z  DW (∇uh ) : ∇φ h dx + h f h , φ h i dx = 0, Sh

(6.45)

(b) E h (uh ) ≤ Chβ . Then, there exists {Qh ∈ SO(3), ch ∈ R3 }h→0 such that the usual rescalings {yh (z + tn(z)) = (Qh )T uh (z + t hh0 n(z)) − ch ∈ H 1 (Sh0 , R3 )}h→0 , defined on a universal shell Sh0 with fixed thickness h0  1, have the following convergence properties, up to a subsequence that we do not relabel: ¯ in H 1 (Sh0 , R3 ), Qh → Q, ? h0 /2 1 yh (z + tn(z)) − z dt → V V h [yh ](z) = β /2−1 h −h0 /2 1 h h sym∇V [y ] * B weakly in L2 (S, R2×2 ), h yh → π

in H 1 (S, R3 ),

where V ∈ V (S) and B ∈ B(S). Moreover: (i) When β = 4, then there holds for all V˜ ∈ V (S) with A˜ ∈ L2 (S, so(3)) satisfying ∇V˜ = A˜ |Tz S as in (5.59), and all B˜ ∈ B(S):
E 1 (6.46) L2 z, B − (A2 )tan : B˜ dσ = 0, 2 S Z D E
1 ˜ − AΠ ˜ )tan dσ L2 z, (∇(An) − AΠ )tan : (∇(An) 12 S (6.47) Z D Z

E
 1 ˜ tan dσ + − L2 z, B − (A2 )tan : (AA) f , Q¯ V˜ dσ = 0. 2 S S ˜ (ii) When β > 4, then for all V ∈ V (S) there holds: Z D

1 12 +

Z D

Z S

E
˜ − AΠ ˜ )tan dσ L2 z, (∇(An) − AΠ )tan : (∇(An) (6.48)

 f , Q¯ V˜ dσ = 0.

S

Above, the operators L2 (z, ·) : R2×2 → R2×2 are as in Definition 5.6. Proof. 1. All the convergence statements follow from Theorems 5.23 and 5.26. We start proving (i) and (ii) by rewriting the equilibrium equation (6.45) in a more convenient form. Changing variables in a variation φ h ∈ H 1 (Sh , R3 ) yields:

162

6 Limiting theories for elastic plates and shells: sublinear and linear

φ h (z + tn(z)) = ψ(z + t

h0 n(z)), h

(6.49)

for the corresponding ψ ∈ H 1 (Sh0 , R3 ). Then, (6.45) becomes: β /2+2

−h

?

Z

h0 /2

ψ(z + tn(z)) dt dσ (z)

f (z) S

−h0 /2

Z ? h0 /2

=h

det Id2 + t

S

−h0 /2


h
D h h E Π DW ∇uh (z + t n(z) : ∇φ h (z + t n) dt dσ (z). h0 h0 h0

Recall the approximating rotation fields {Rh ∈ H 1 (S, SO(3))}h→0 defined in the proof of Theorem 5.23 and consider the matrix fields E h , Z h ∈ L2 (Sh0 , R3×3 ) in: 1

DW (Id3 +hβ /2 Z h ), hβ /2

Eh =

Z h (z+tn(z)) =

1  hβ /2

Rh (z)T ∇uh z+t

 h
n −Id3 . h0


In particular: hβ1/2 DW ∇uh (z + t hh0 n) = Rh (z)E h (z + tn), by the frame invariance of W . Then (6.45) becomes, after exchanging ψ to (Qh )T ψ and using Lemma 5.5: 2

−h

?

Z

h0 /2

f (z) S

Z ? h0 /2

Qh ψ(z + tn(z)) dt dσ (z)

−h0 /2

h
D h T h h E Π (Q ) R (z)E h (z + tn) : ∇φ h (z + t n) dt dσ h0 h0 S −h0 /2 Z ? h0 /2 D
h =h det(Id2 + t Π ) Qh )T (Rh ◦ π)E h |Tz S : h0 S −h0 /2 E h : (∇ψ)|Tz S (Id2 + tΠ )(Id2 + t Π )−1 dt dσ h0 Z ? h0 /2 D E h + h0 det(Id2 + t Π ) (Qh )T (Rh ◦ π)E h n, ∂n ψ(z + tn) dt dσ . h0 S −h0 /2 (6.50) =h

det Id2 + t

2. Recall that, by Theorems 5.25 and 5.26, the family {Z h }h→0 converges, up to a subsequence that we do not relabel, weakly in L2 (Sh0 , R3×3 ) to the matrix field Z whose symmetrized tangential minor satisfies: sym Z(z + tn(z))tan = B −


κ 2 t (A )tan + ∇(An) − AΠ tan , 2 h0

(6.51)

with κ = 1 for β = 4 and κ = 0 for β > 4. By Lemma 6.15 it now follows that: E h * E  L3 Z,

(Qh )T (Rh ◦ π)E h * E

weakly in L2 (Sh0 , R3×3 ),

(6.52)

up to a subsequence that we do not relabel, where in the second convergence we used the fact that (Qh )T Rh → Id3 in L2 (S, SO(3)), displayed in (5.62).

6.5 Convergence of equilibria

163

We now prove several properties of the limiting field E. Firstly, observe that: En = 0

ET = E

and

and

Etan = L2 (z, Ztan )

in Sh0 .

(6.53)

Indeed, to show E ∈ Tz S, one only needs to pass h → 0 in (6.50) and use (6.52) for: Z ? h0 /2 D

E E(z + tn)n, ∂n ψ(z + tn) dt dσ (z) = 0.

S

−h0 /2

Since any vector field φ ∈ L2 (Sh0 , R3 ) has the form φ = ∂n ψ, the above implies E(z+tn)n(z) = 0 as claimed. For the symmetry of E, recall that by frame invariance of W we have L3 H = 0 for H ∈ so(3), so hE : Hi = hL3 Z : Hi = hL3 H : Zi = 0. Further, the two established properties imply, in view of E = L3 Z and Lemma 5.7
that E = L3 Ztan + c(z, Ztan ) ⊗ n . This yields Etan = L2 (z, Ztan ). Secondly, we observe that: kskew E h kL1 (Sh0 ) ≤ Chβ /2 , 1 kskew E h kL p (Sh0 ) → 0 h

(6.54) for some p ∈ (1, 2).

The former claim is deduced by writing 0 = hDW (F) : HFi = hDW (F)F T : Hi for all F ∈ R3×3 , H ∈ so(3) (since HF is tangent to SO(3)F at F), so DW (F)F T must be a symmetric matrix. Apply this statement pointwise to F = Id3 + hβ /2 Z h : 1


 β /2 h β /2 h T skew DW (Id + h Z ) · Id + h Z 3 3 hβ /2
= skew E h + hβ /2 skew E h (Z h )T ,

0=

while recalling that kE h (Z h )T kL1 (Sh0 ) ≤ CkE h kL2 (Sh0 ) kZ h kL2 (Sh0 ) ≤ C, by (6.52). The latter claim in (6.54) is now a consequence of the above, together with the boundedness of {E h }h→0 in L2 (Sh0 ), and via the interpolation inequality:  hβ /2 θ 1 C 1 h2θ −1 , kskew E h kL p (Sh0 ) ≤ kskew E h kθL1 · kE h k1−θ ≤ hθ β /2 = C 2 L h h h h2 1 where 1p = θ + 1−θ 2 and θ ∈ (0, 1). Clearly, the above converges to 0 when θ > 2 , implying the second claim in (6.54) for all p ∈ (1, 34 ). ¯ Eˆ ∈ L2 (S, R3×3 ) given by the 0-th and 1-st moments of E: 3. Define E,

? ¯  E(z)

?

h0 /2

E(z + tn) dt, −h0 /2

ˆ  E(z)

h0 /2

tE(z + tn) dt. −h0 /2

It follows by (6.53), (6.52) and the fact that L2 (z, F) depends only on symF, that:

164

6 Limiting theories for elastic plates and shells: sublinear and linear

? E¯tan (z) =

h0 /2

−h0 /2

?

L2 (z, Ztan (z + tn)) dt = L2 z, B −

κ 2
(A )tan , 2

(6.55)

h0 /2


h0 L2 (z,tZtan (z + tn)) dt = L2 z, (∇(An) − AΠ )tan . Eˆtan (z) = 12 −h0 /2 We will now use the balance law (6.50) and the above formulas to recover the EulerLagrange equations (6.46) - (6.48) in the limit as h → 0. For (6.46), we use the variation of the form: ψ = φ ◦ π in (6.50), divide by h and pass to the limit to get: Z ? h0 /2

D h Π ) (Qh )T (Rh ◦ π)(E h )|Tz S h→0 S −h0 /2 h0 E h : (∇φ )(Id2 + t Π )−1 dt dσ h0 Z Z ? h0 /2

D E 
κ E|Tz S : (∇ψ) ◦ π dt dσ = = L2 z, B − (A2 )tan : ∇φ (z)tan dσ 2 S S −h /2 Z D 0 E
κ L2 z, B − (A2 )tan : sym∇φ (z) dσ = 2 S

0 = lim

det(Id2 + t

where we have used (6.52), (6.53) and (6.55). By density of {sym∇φ ; φ ∈ H 1 (S, R3 )} in the space B(S), the identity (6.46) follows immediately. 4. In order to verify (6.47), (6.48), let V˜ ∈ V (S) with the corresponding A˜ ∈ ˜ H 1 (S, so(3)), and apply (6.50) to a variation of the form: ψ(z + tn) = t A(z)n(z). ˜ ∈ H 1 (S, R3 ), we obtain: Upon dividing (6.50) by h and denoting η = An Z ? h0 /2

0 = lim

h→0

det(Id2 + t

S

−h0 /2

D h Π )t · (Qh )T (Rh ◦ π)(E h )Tz S : h0 : (∇η)(Id2 + t

+

h0 h

Z ? h0 /2 D S

E (Qh )T (Rh ◦ π)E h n, η dt dσ

−h0 /2

Z ? h0 /2

+ S

E h Π )−1 dt dσ h0

−h0 /2

! D 
h t trace Π + t 2 det Π (Qh )T (Rh ◦ π)E h n, η dt dσ , h0 2

where we used the identity: det(Id2 +t hh0 Π ) = 1 +t hh0 trace Π +t 2 hh2 det Π . The first 0

term in the formula displayed above, in view (6.52) ans (6.53) converges to: Z ? h0 /2

S

−h0 /2

Z

  t E|Tz S : ∇η dt dσ = Eˆtan : (∇η)tan dσ . S

In turn, the third term converges to 0, since {(Qh )T (Rh ◦ π)E h n}h→0 converge weakly in L2 (Sh0 , R3 ) to En = 0, by (6.52) (6.53). Summarizing, we get by (6.55):

6.5 Convergence of equilibria

Z ? h0 /2 D 1

lim

h→0

h

S

165

E ˜ dt dσ (Qh )T (Rh ◦ π)E h n, An

−h0 /2

=−

1 12

(6.56) Z D S

E
˜ tan dσ . L2 z, (∇(An) − AΠ )tan : (∇(An))

Next, we apply (6.50) to ψ = V˜ ◦ π and pass to the limit after dividing by h2 : −

Z

h f , Q¯ V˜ i dσ = lim

S

Z ?

= lim

h→0

S

Z

h→0 S h0 /2 D

−h0 /2

h f , QhV˜ i dσ 1 ((Qh )T Rh ◦ π − Id3 )(E h )|Tz S : h

E h adj Π ) dt dσ h0 ! Z ? h0 /2 D E h 1 h ˜ (E )|Tz S : A|Tz S (Id2 + t adj Π ) dt dσ + h0 S −h0 /2 h
 lim Ih + IIh , : (A˜ ◦ π)|Tz S (Id2 + t

(6.57)

h→0

where we used the definition of the adjoint matrix: det(Id2 + t

h h
−1 h h Π ) Id2 + t Π = adj (Id2 + t Π ) = Id2 + t adj Π . h0 h0 h0 h0

Recall that by (5.62) we further have:
1 hβ /2 1 ((Qh )T Rh −Id3 ) = 2 · β /2−1 (Qh )T Rh −Id3 * κA weakly in H 1 (S, R3×3 ). h h h Hence, the terms {Ih }h→0 in (6.57), in view of (6.53) and (6.55) converge to: lim Ih = κ

h→0

Z

  ¯ |T S : A˜ |T S dσ = κ AE¯ : A˜ dσ (AE) z z

Z

S

S

Z

Z

  ˜ tan dσ = −κ E¯ : AA˜ dσ = −κ E¯tan : (AA) S ZS D E κ 2
˜ tan dσ . = −κ L2 z, B − (A )tan : (AA) 2 S

(6.58)

5. Towards finding the limit of IIh in (6.57), consider first the contribution of the tangential minors. By (6.54) and since A˜ ∈ L p (Sh0 , so(3)) for all p ≥ 1, we get: 1 lim h→0 h

Z ? h0 /2

S

−h0 /2

 h skew Etan : A˜ tan ◦ π dt dσ = 0.

(6.59)

Hence, using (6.53) combined with the formula: A˜ tan adj Π = −(A˜ tan Π )T , which can be easily checked for A˜ tan ∈ so(2), we get:

166

6 Limiting theories for elastic plates and shells: sublinear and linear

lim

Z ? h0 /2 D 1

E h adj Π ) dt dσ h0 −h0 /2 h Z ? h0 /2 D E 1 h = lim : (A˜ ◦ π)tan adj Π dt dσ tEtan h0 h→0 S −h0 /2 Z

 1 = − lim Eˆtan : (A˜ tan Π )T dσ . h0 h→0 S

h→0 S

h Etan : (A˜ ◦ π)tan (Id2 + t

(6.60)

Further, by (6.56), (6.54), (6.59) and noting that 1h ((Qh )T Rh − Id3 ) converges to κA ˜ ∈ L4 (S, R3 ) and E¯ h n converges to 0 weakly in strongly in L4 (S, R3×3 ), while An 2 3 L (S, R ), we deduce the following convergence: lim

Z ? h0 /2 D 1

h→0 S

h

−h0 /2

1 = − lim h→0 h + lim

E (E h )T n, (A˜ T ◦ π)n dt dσ

Z ? h0 /2 D

E (Qh )T (Rh ◦ π)E h n, (A˜ ◦ π)n dt dσ

S

−h0 /2

Z ? h0 /2 D 1

h Z ? h0 /2 D 1

h→0 S

+ 2 lim

(6.61)

−h0 /2

h→0 S

1 = h0

E ((Qh )T (Rh ◦ π) − Id3 )(E h n), (A˜ ◦ π)n dσ

−h0 /2

h

E (skew E h )n, (A˜ ◦ π)n dt dσ

Z

S

 ˜ tan dσ . Eˆtan : (∇(An))

Finally, in view of {(Eˆ h )T n}h→0 converging to 0 weakly in L2 (S, R3×3 ) we get: 1 h→0 h0 lim

Z ? h0 /2 D S

−h0 /2

E ˜ T n)|T S dt dσ = 0, t((E h )T n)|Tz S , (adj Π )((A) z

(6.62)

by (6.52) and (6.53). Adding now (6.60), (6.61), (6.62) we obtain: lim IIh =

h→0

1 12

Z D S

E
˜ − AΠ ˜ )tan dσ , L2 z, (∇(An) − AΠ )tan : (∇(An)

in view of (6.55). With (6.57), (6.58), the above formula implies the two remaining identities (6.47), (6.48) and ends the proof. Remark 6.17. (i) Condition (6.45) is obtained by formally passing to the limit ε → 0 under the integral sign in the equilibrium condition for the energy J h :  1 h h J (u + εφ h ) − J h (uh ) = 0 ε→0 ε lim

for all φ h ∈ H 1 (Sh , R3 ).

(6.63)

Whether (6.63) and (6.45) are equivalent, even for local minimizers (without assuming extra regularity, e.g. their Lipschitz continuity) is an open problem. However, condition (6.43) readily implies that:

6.6 Von K´arm´an’s equations

Z

lim

ε→0 Sh

167

 1 W (∇uh + ε∇φ h ) −W (∇uh ) dx = ε

Z

Sh

 DW (∇uh ) : ∇φ h dx

because of the pointwise convergence of the integrands and of their boundedness by an L1 function independent of ε: 1 1 |DW (∇uh + εs∇φ h )| · |∇φ h | ds |W (∇uh + ε∇φ h ) −W (∇uh )| ≤ ε 0
≤ C |∇uh | + |∇φ h | + 1 |∇φ h |.

Z

Hence, in presence of (6.43), conditions (6.63) and (6.45) are equivalent, and Theorem 6.16 holds true after replacing (6.45) by its variational form (6.63). (ii) Properties (i) and (ii) in Theorem 6.16 are precisely the Euler-Lagrange equations of the Γ -limits I¯vK and I¯lin , under variations in V and B. We do not state the Euler-Lagrange equation for Q¯ because of the, in general, unknown structure of the minimizing set M f . (iii) If shells {Sh }h→0 are clamped on some lateral portion of ∂ Sh , i.e. if the deformations {uh }h→0 satisfy uh |Γ ×(−h/2,h/2) = id3 for each h  1 and some Γ ⊂ ∂ S with positive measure, then in Theorem 6.16 we get Q¯ = Id3 .

6.6 Von K´arm´an’s equations Note that (6.45) is the weak formulation of the balance law:
−div DW (∇uh ) + f h = 0 in Sh , DW (∇uh )n = 0

on ∂ Sh ,

where the operator div is understood as acting on rows of the matrix field DW (∇uh ). In case of plates when S = ω ⊂ R2 , the limiting Euler-Lagrange equations (6.46), (6.47) can be rewritten as the celebrated von K´arm´an equations, which we now derive. For a general version involving prestress, we refer to Part III of this monograph. Recall from Remarks 5.22 (iii) and 6.2 (iii) that it suffices to consider V = (0, 0, v) and B = sym∇w, with v ∈ H 2 (ω, R), w ∈ H 1 (ω, R2 ). We will also assume that the density W is isotropic as in Example 5.3, leading to the formula in Example 5.8: L2 F =

λµ (tr F)Id2 + µ sym F λ +µ

for all F ∈ R2×2 ,

with the given Lam´e constants λ , µ. We start by an important observation, used in Remark 7.2 (ii), and allowing to identify the space of symmetric gradients as the kernel of the operator curl curl, when the planar domain ω is simply connected. More precisely, let: 2 Definition 6.18. Given F = [Fi j ]i, j=1,2 ∈ H 2 (ω, R2×2 sym ) on an open domain ω ⊂ R , we define curl curl F ∈ L2 (ω, R2 ) by first applying curl to each row vector of F and

168

6 Limiting theories for elastic plates and shells: sublinear and linear

then taking curl of the resulting vector field. Namely:
curl curl F = curl ∂1 F12 − ∂2 F11 , ∂1 F22 − ∂2 F11 = ∂11 F22 − 2∂12 F12 + ∂22 F11 . Using the identification of gradient fields twice, we obtain: Lemma 6.19. Let ω ⊂ R2 be an open, bounded, simply connected domain. Assume that F ∈ L2 (ω, R2×2 sym ) and that α, β ∈ R are such that: α , 0,

α + 2β , 0.

Then the following conditions are equivalent: (i) F = α sym∇w + β (div w)Id2 for some w ∈ H 1 (ω, R2 ), β ∆ (tr F) = 0 in the sense of distributions. (ii) curlT curl F − α + 2β In particular, if F has the form F = ∇v ⊗ ∇v for some v ∈ H 2 (ω, R), then F = sym∇w for some w ∈ H 1 (ω, R2 ), if and only if det ∇2 v = 0 in ω. Proof. 1. We first prove the equivalence of (i) and (ii) for β = 0, α = 1. We compute:

2 curl curl sym∇w = curl curl ∇w + curl curl(∇w)T = curl ∇curl w = 0. On the other hand, if curl curl F = 0, then curl F = ∇φ for some φ ∈ L2 (ω, R): ∂1 F12 − ∂2 F11 = ∂1 φ ,

∂1 F22 − ∂2 F12 = ∂2 φ .

Rearranging terms, we get: ∂1 (F12 − φ ) − ∂2 F11 = 0, ∂1 F22 − ∂2 (F12 + φ ) = 0,      0 −φ 0 −φ = 0. This means that F + = ∇w for or equivalently: curl F + φ 0 φ 0 some w ∈ H 1 (ω, R2 ), yielding F = sym∇w. 2. In the general case, we will repeatedly use the following formulas: curl curl(φ Id2 ) = ∆ φ = div div (φ Id2 )

for all φ ∈ L2 (ω, R).

To prove the implication (i)⇒(ii), note that by (i): tr F = (α + 2β )div w. Thus:
curl curl F = β curl curl (div w)Id2 = β ∆ (div w) =

β ∆ (tr F). α + 2β


β To prove (ii)⇒(i) observe that by (ii): curl curl F − α+2β (tr F)Id2 = 0. Thus: F−

β (tr F)Id2 = α sym∇w, α + 2β

(6.64)

6.6 Von K´arm´an’s equations

169


2β 1 tr F = α+2β tr F. by step 1, for some w ∈ H 1 (ω, R2 ) so that: div w = α1 tr F − α+2β Together with (6.64) the above implies (i). 3. For the final claim, it suffices to check that:
curl curl ∇v ⊗ ∇v = ∂11 |∂2 v|2 − 2∂12 (∂1 v · ∂2 v) + ∂22 |∂1 v|2

= 2 |∂12 v|2 + ∂2 v · ∂112 v − 2 |∂12 v|2 + ∂2 v · ∂112 v + ∂11 v · ∂22 v + ∂1 v · ∂122 v
+ 2 |∂12 v|2 + ∂1 v · ∂122 v = 2|∂12 v|2 − 2∂11 v · ∂22 v = −2 det ∇2 v, and invoke step1. The proof is done. The main result below deduces the Euler-Lagrange equations of equilibria for plates. The corresponding boundary conditions will be derived in Remark 6.21. Theorem 6.20. Let S = ω ⊂ R2 be bounded and simply connected. Assume that W is isotropic with the Lam´e constant µ, λ satisfying: µ >0

and

3λ + µ > 0.

Then we have: (i) If β = 4 then (6.46), (6.47) are equivalent to the von K´arm´an’s system: ∆ 2Φ = −

µ(µ + 3λ ) [v, v], 2(µ + 2λ )

2λ µ + µ 2 2 ¯ 3i ∆ v = [v, Φ] − h f , Qe 12(λ + µ)

in ω.

(ii) If β > 4 then (6.48) is equivalent to the single biharmonic equation: 2λ µ + µ 2 2 ¯ 3i = 0 ∆ v + h f , Qe 12(λ + µ)

in ω.

The above equations are written in terms of the out-of-plane displacement v ∈ H 2 (ω, R) and the Airy stress potential Φ ∈ H 2 (ω, R), and using the Airy bracket notation: [v1 , v2 ]  h∇2 v1 : cof ∇2 v2 i. Proof. 1. To prove (i), we rewrite (6.46) as: Z

 M : ∇w˜ dz = 0

for all w˜ ∈ H 1 (ω, R2 )

ω

where M 



1 1 λµ div w + |∇v|2 Id2 + µ sym∇w + ∇v ⊗ ∇v . λ +µ 2 2

(6.65)

Integrating by parts and using w˜ ∈ H01 (ω, R) we get: div M = 0 in ω, where the divergence of M ∈ L2 (ω, R2×2 ) is taken row-wise. It follows that the i-th row of M (i = 1, 2) can be written as ∇⊥ ψ i for some scalar fields (ψ 1 , ψ 2 ). The symmetric cofactor matrix field cof M has thus the form ∇(ψ 2 , −ψ 1 )T , which implies that:

170

6 Limiting theories for elastic plates and shells: sublinear and linear

cof M = ∇2 Φ

for some Φ ∈ H 2 (ω, R).

The scalar field Φ is precisely the Airy stress potential. From this discussion we see that (6.46) is equivalent with: M = cof ∇2 Φ. We now use Lemma 6.19 to:  λµ 1 µ∇v ⊗ ∇v + |∇v|2 Id2 , F = cof ∇2 Φ − 2 λ +µ

(6.66)

α = µ,

β=

λµ . λ +µ

Hence (6.66) is further equivalent to (ii) in Lemma 6.19, which takes the form:
curl curl cof ∇2 Φ −


λ µ ∆ (tr cof ∇2 Φ) = curl curl ∇v ⊗ ∇v . µ + 3λ 2

Since curl curl(∇v⊗∇v) = −2 det ∇2 v, and curl curl(cof ∇2 Φ) = ∆ (tr cof ∇2 Φ) equal ∆ 2 Φ, we obtain precisely the first formula in (i), from: µ + 2λ 2 ∆ Φ = −µdet∇2 v. µ + 3λ 2. We now rewrite the second Euler-Lagrange equation (6.47) as: Z D E  1 λµ M : ∇v ⊗ ∇v˜ dz + (∆ v)Id2 + µ∇2 v : ∇2 v˜ dz 12 ω λ + µ

Z

ω

Z

+

¯ 3 iv(z) h f , Qe ˜ dz = 0

for all v˜ ∈ H 2 (ω, R).

ω

Applying the following identity valid for all symmetric two-dimensional matrices: αF + β (tr F)Id2 = (α + β )F + β cof F to F = ∇2 v and α, β as in step 1, we rewrite the above balance laws as: Z D E 2λ µ + µ 2 2 λµ ∇ v+ cof ∇2 v : ∇2 v˜ dz λ +µ λ +µ ω Z Z



1 12 +

ω

(6.67)

¯ 3 iv(z) h f , Qe ˜ dz = 0 for v˜ ∈ H 2 (ω, R).

(cof ∇2 Φ)∇v, ∇v˜ dz +

ω

Integrating by parts, using v˜ ∈ H02 (ω, R), and recalling that div cof ∇ φ = 0 for any φ ∈ H 1 (ω, R2 ), it follows that:   2λ µ + µ 2 ¯ 3 i = 0. −div (cof ∇2 Φ)∇v + div div ∇2 v + h f , Qe 12(λ + µ) We use now the following formulas:

6.6 Von K´arm´an’s equations

171




 div (cof ∇2 Φ)∇v = cof ∇2 Φ : ∇2 v ,

div div ∇2 v = ∆ 2 v,

to find the claimed equivalent form of (6.47):

 2λ µ + µ 2 2 ¯ 3 i = 0. ∆ v + h f , Qe − cof ∇2 Φ : ∇2 v + 12(λ + µ) This ends the proof of (i). Since Φ = 0 for β > 4, we also get (ii). Analysis as in the proof Theorem 6.20 may be repeated with variations w˜ and v˜ which do not vanish on the boundary ∂ ω, in order to deduce the natural (free) boundary conditions (6.69), (6.70) satisfied by Φ, v: Remark 6.21. (i) Denote by η the outward unit normal vector to ∂ ω. Integrating (6.65) by parts and taking into account the already proved div M = 0, yields: Mη = 0

on ∂ ω.

(6.68)

By (6.66) we get: (cof ∇2 Φ)η = 0 on ∂ ω, which is equivalent to: ∂τ ∇Φ = 0 for the tangent vector field τ to ∂ ω. Therefore: ∇Φ ≡ const

on ∂ ω.

Since Φ is determined up to affine functions, we may assume that Φ(x0 ) and ∇Φ(x0 ) vanish at some x0 ∈ ∂ ω. We obtain hence boundary conditions for Φ: Φ = ∂η Φ = 0

on ∂ ω.

(6.69)

(ii) To get the boundary conditions for v, we integrate by parts in (6.67), obtaining: Z





 M : ∇v ⊗ η v˜ + ∇v,Ψ ˜ ηi − div Ψ , η v˜ dσ = 0

∂ω

where Ψ 

 2λ µ + µ 2  2 λ ∇ v+ cof ∇2 v . 12(λ + µ) 2λ + µ

The first term above is null by (6.68). Writing ∇v˜ = (∂τ v)τ ˜ + (∂η v)η, ˜ we get:





 Ψ : η ⊗ η = 0, ∂τ Ψ : η ⊗ τ + divΨ , η = 0 on ∂ ω. Denoting p =

λ 2λ +µ ,

the above equations are equivalent to:



 ∇2 v : η ⊗ η + p ∇2 v : τ ⊗ τ = 0,




 (1 − p)∂τ ∇2 v : η ⊗ τ + div ∇2 v + p cof ∇2 v , η = 0.

We further obtain:
2 2 v − K∂η v = 0 v + p ∂ττ ∂ηη
2 3 v =0 v + K ∆ v + 2∂ηη (2 − p)∂τ ∂η ∂τ v + ∂ηηη

) on ∂ ω,

(6.70)

172

6 Limiting theories for elastic plates and shells: sublinear and linear

where K stands for the (scalar) curvature of ∂ ω in: ∂τ τ = Kη.

6.7 Bibliographical notes The variational approach and results presented in this chapter are based on the fundamental work by Friesecke et al. [2006], extended to shells in Lewicka et al. [2010]. The construction of recovery family in section 6.1 and the definition (6.18) in section 6.2) is taken from Lewicka et al. [2010] Remark 6.6 uses the same calculations as in Ciarlet [2000][Volume 3, page 119]. Analysis of shells with variable thickness in section 6.4 was presented in Lewicka et al. [2009]. An interesting extension to boundary conditions and stability can be found in Friesecke et al. [2006] and Lecumberry and M¨uller [2009]. The derivation of von K´arm´an’s theory and equations for incompressible elastic shells was carried out for plates in Conti and Dolzmann [2009], and for shells in Li and Chermisi [2013] Convergence of equilibria in section 6.5 follows Lewicka [2011], which is based on the analysis in case of plates by M¨uller and Pakzad [2008]. The growth condition (6.43) rules out the, physically relevant, blow-up of W (F) as det F → 0. A condition which is compatible with the blow-up is |F T DW (F)| ≤ C(W (F) + 1); under this condition one can obtain a variant of the Euler-Lagrange equation which involves the (weak) divergence of the energy-momentum tensor (rather than the stress itself) Ball [1983]. A theorem by Monneau [2003] starts from a sufficiently smooth and small solution of the von K´arm´an equations and shows that there exists a nearby solution of the three-dimensional problem. We also remark that the incompressible case has been analyzed in Lewicka and Li [2015]. The von K´arm´an equations in section 6.6 were first formulated in von K´arm´an [1910]. Their justification through formal asymptotics can be found in Ciarlet [1980], see also the extensive monograph by Ciarlet [2000].

Chapter 7

Limiting theories for elastic plates: linearised bending

Following the discussion in chapter 5, where we analyzed the Γ -limit of the scaled elastic energies {h−β E h }h→0 for β = 2, and chapter 6 that was pertaining to β ≥ 4, we now turn to the intermediate scaling exponent range β ∈ (2, 4). In section 7.1 we present one possible general lower bound, in terms of the linearised Kirchhoff energy IlinK . This energy consists of solely the first order bending term, which is due to the change of curvature of the midsurface S under an infinitesimal isometry displacement in V (S), which however necessarily has to carry no stretching of second order. The question of optimality of IlinK will occupy us throughout the rest of this chapter. For S of arbitrary geometry we are only able to construct the recovery family, needed for the upper bound result, in the restricted regime β ∈ (3, 4) and under specific regularity conditions. These restrictions are natural and will be discussed in a broader context in chapter 8. We then specify to plates i.e. to the interesting case S ⊂ R2 . In section 7.2 we identify IlinK as the biharmonic potential energy subject to the degenerate MongeAmp`ere constraint, posed on the out-of-plane displacements v that now replace the displacements in the previously studied space V (S). This theory indeed turns out to be the limiting theory for all β ∈ (2, 4) on simply connected plates, due to two key results of independent interest. The first result, proved in section 7.3, is the matching property of second order infinitesimal isometries to exact isometries: if v can be augmented by a higher order perturbation to produce no second-order stretching, then it can be “matched” by an improved, equibounded family of perturbations to produce no stretching at all, i.e. to be an (exact) isometry. This statement, valid for Lipschitz displacements, allows for the construction of the recovery sequence carried out in section 7.4. Section 7.5 is devoted to proving the second key result, which is the density of Lipschitz second order infinitesimal isometries in the space of H 2 isometries. As a consequence, we obtain the recovery family and the upper bound statement in the general case, together with the Γ -convergence and the convergence of minimizers. In the following sections we analyze the question of dimension reduction on shallow shells, that is assuming that the midsurface’s curvature scales as a power of the vanishing shell’s thickness, rather than being fixed and either large for shells, or © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_7

173

174

7 Limiting theories for elastic plates: linearised bending

equal zero for plates. In section 7.6 we set up the problem and provide a heuristic derivation of one limiting theory in this context, revealing itself as a possibly nondegenerate (in the non-zero Gaussian curvature case) Monge-Amp´ere constrained, relative bending theory. In section 7.7 we state this theory precisely and already prove the related upper bound in a restricted energy scaling range, but for an arbitrary leading profile v0 of the midsurface. The result in the full scaling range, normalised as before to (2, 4), necessitates the appropriate versions of the matching and density properties. We note that the theory reduces to IlinK when v0 = 0. The matching of second order to exact isometries on convex shallow shells is proved in section 7.8. This result states that every H¨older regular displacement v whose Hessian determinant equals the Hessian determinant of v0 , assumed to be uniformly positive, can be matched by an equibounded family of perturbation to produce no stretching with respect to v0 . Before deriving the aforementioned density property, in section 7.9 we overcome another difficulty and show that every H 2 -regular displacement with positive Hessian determinant must necessarily be locally convex. The proof is achieved by a combination of purely two-dimensional arguments, concerning maps with integrable dilatation and an estimate of the modulus of continuity of deformations in H 1 with positive Jacobian. In section 7.10 we prove the density of C 2,α maps with a prescribed constant, positive Hessian determinant in the set of such H 2 -regular maps, on the domain that is star-shaped with respect to a ball. Only in such limiting setting we are able to construct the recovery family, and hence prove optimality of the previously obtained lower bound, for the full exponent range (2, 4) and on shallow shells with constant positive referential Gaussian curvature profile. We anticipate that the analysis of shallow shells can be recast using the formalism of non-Euclidean elasticity, which will be the topic of Part III of this monograph. Also, the role of matching properties will be discussed, in a more general setting, in chapter 8. We then prove other properties of this type for diverse geometries: for shallow shells in section 7.8, for surfaces of revolution in section 8.2, for elliptic surfaces in section 9.2, and in section 10.2 for developable surfaces without flat parts. Sections 7.6-7.10 may be skipped at first reading.

7.1 Linearised Kirchhoff’s theory for shells We first restate the compactness and the lower bound results from Theorems 5.23, 5.25 and 5.26, corresponding to the general case of the mid-surface S with arbitrary geometry, and the scaling exponent range β ∈ (2, 4) which has not been fully analyzed in previous chapters. Recall that the infinitesimal isometries and the finite strain spaces V (S), B(S) were introduced in Definitions 5.21 and 5.28. We have:

7.1 Linearised Kirchhoff’s theory for shells

175

Theorem 7.1. Let S ⊂ R3 be a smooth, compact, connected, oriented surface with boundary given by finitely many Lipschitz curves. Let uh ∈ H 1 (Sh , R3 ) be a family of deformations such that: E h (uh ) ≤ Chβ

for some β ∈ (2, 4).

(7.1)

Then there exists rotations and translations {Qh ∈ SO(3), ch ∈ R3 }h→0 such that the normalized deformations {yh ∈ H 1 (Sh0 , R3 )}h→0 given in: yh (z + tn(z)) = (Qh )T uh (z + t

h n(z)) − ch h0

for all z ∈ S, |t|
0 define Ω h = ω × (− h2 , h2 ), and assume that uh ∈ H 1 (Ω h , R3 ) is a family of deformations satisfying: E h (uh ) ≤ Chβ

for some β ∈ (2, 4).

(7.12)

Then there exists rotations and translations {Qh ∈ SO(3), ch ∈ R3 }h→0 such that the normalized deformations {yh ∈ H 1 (Ω 1 , R3 )}h→0 given in: yh (z,t) = (Qh )T uh (z, ht) − ch have the following properties, up to a (not-relabeled) subsequence: (i) {yh }h→0 converges in H 1 (Ω 1 , R3 ) to π, (ii) displacements {V h [yh ]}h→0 below converge in H 1 (ω, R3 ) to some ve3 : V h [yh ](z) 

1 hβ /2−1

?

1/2

yh (z,t) − z dt,

−1/2

1 while their scaled tangential components { hβ /2−1 V h [yh ]}h→0 converge, weakly in H 1 (ω, R2 ) to some limit field w, (iii) the out-of-plane component v satisfies v ∈ H 2 (ω, R) with det ∇2 v = 0, while the in-plane component w ∈ H 1 (ω, R2 ) satisfies:

1 ∇v ⊗ ∇v + sym∇w = 0, 2

(7.13)

(iv) we have the following lower bound: lim inf h→0

1 h h 1 E (u ) ≥ IlinK (v)  β 24 h

Z


Q2 ∇2 v dz.

ω

The purpose of the next three sections is to deduce the upper bound statement of Γ -convergence as in Definition 2.1, completing the above lower bound, namely:

7.2 Linearised Kirchhoff’s theory for plates

181

Theorem 7.5. Let ω ⊂ R2 be open, bounded and simply connected, with Lipschitz boundary. For each h > 0 define Ω h = ω × (− 2h , h2 ), and assume that: β ∈ (2, 4). Then, for any v ∈ H 2 (ω, R) satisfying det ∇2 v = 0, and any w ∈ H 1 (ω, R2 ) satisfying (7.13), there exist deformations: {uh ∈ H 1 (Ω h , R3 )}h→0 such that {yh (z,t)  uh (z, ht)}h→0 have the following convergence properties: (i) {yh }h→0 converges in H 1 (Ω 1 , R3 ) to π(z,t) = z, (ii) displacements {V h [yh ]}h→0 converge in H 1 (ω, R3 ) to ve3 , n 1 o h h (iii) converge up to a subsequence in H 1 (ω, R2 ), to w, V [y ] tan h→0 hβ /2−1 1 (iv) there holds: lim β E h (uh ) = IlinK (v). h→0 h The proof will be given in section 7.5. Instead of applying the Kirchhoff-Love extension construction to the ansatz for the mid-plate deformation: uh|ω = id2 + εve3 + ε 2 w,

ε = hβ /2−1 ,

as motivated by Lemma 7.3 and Lemma 6.19, we will use its modification: uε = id2 + εve3 + ε 2 wε which is an exact isometry on ω. Existence of the appropriate family of displacements {wε }ε→0 will be referred to as the matching property on plates, and it will be proved in section 7.3 under the additional assumption that v is Lipschitz. The general case will then follow by the density of Lipschitz linearised isometries in the set of H 2 linearised isometries, and a diagonal argument. We close this section by a few general observations: Lemma 7.6. Let u ∈ H 2 (ω, R3 ) be an isometry, i.e. satisfy (∇u)T ∇u = Id2 on an open domain ω ⊂ R2 . Then there holds: (i) the image surface u(ω) has the unit normal N = ∂1 u × ∂2 u and the second fundamental form [Πi j ]i, j=1,2 where Πi j = h∂i N, ∂ j ui = −h∂i j u, Ni, (ii) for all i, j = 1, 2 there holds ∂i j u = −Πi j N, (iii) we have: det Π = 0 in ω and curl Π = 0 in ω in the sense of distributions. Proof. Assertions in (i) are obvious because of the isometry conditions. Further, observe that: 2h∂i j u, ∂i ui = ∂ j |∂i u|2 = 0 and also h∂ii u, ∂ j ui = ∂i h∂i u, ∂ j ui−h∂i j u, ∂i ui = 0, which imply:

182

7 Limiting theories for elastic plates: linearised bending

h∂i j u, ∂k ui = 0

for all i, j, k = 1, 2.

Hence each vector ∂i j u is parallel to N, proving (ii) in view of (i). For (iii), note the following identity, valid in the sense of distributions on ω: ∂22 h|∂1 u|2 − 2∂12 h∂1 , ∂2 ui + ∂11 |∂2 u|2 = 2|∂12 u|2 − 2h∂11 u, ∂22 ui. The left hand side is clearly null for an isometry u, while the right hand side equals −2 det Π by (ii). This shows that det Π = 0 in ω. Finally, using the fact that ∂i N is orthogonal to N and by (i), we get for all i = 1, 2: (curl Π )i = ∂1 Πi2 − ∂2 Πi1 = h∂i1 u, ∂2 Ni − h∂i2 u, ∂1 Ni = 0. The proof is done.

7.3 Matching infinitesimal to exact isometries on plates In this section we prove one of the two key results, allowing for the “efficient” construction of a recovery family of deformations with properties as in Theorem 7.5, for the scaling exponent β that can be arbitrarily close to the nonlinear bending exponent β = 2, already analyzed in chapter 5. Namely, we will prove that plates enjoy the Lipschitz isometry matching of 2 7→ ∞, under the simple-connectedness assumptions. This assumption is also necessary, as shown in Example 7.9. The role of matching properties will be discussed, in a more general setting, in chapter 8. In the context of plates, we have: Theorem 7.7. Let ω ⊂ R2 be an open, bounded, Lipschitz and simply connected. Given v ∈ H 2 ∩ W 1,∞ (ω, R2 ) satisfying det ∇2 v = 0, there exists an equibounded family of in-plane displacements {wε ∈ H 2 (ω, R2 )}ε→0 , such that each map below is an isometry, i.e.: (∇uε )T ∇uε = Id2 on ω: uε  id2 + εve3 + ε 2 wε ∈ H 2 (ω, R3 )

ε  1.

Moreover, {wε }ε→0 can be chosen so that:
k∇wε kL2 (ω) ≤ C k∇2 vkL2 (ω) + k∇vkL2 (ω) k∇vkL∞ (ω) , k∇2 wε kL2 (ω) ≤ Ck∇2 vkL2 (ω) k∇vkL∞ (ω) ,

(7.14)

where the constant C depends only on ω, but not on v or ε. Proof. 1. The isometry condition (∇uε )T ∇uε = Id2 is equivalent to: ∇(id2 + ε 2 wε )T ∇(id2 + ε 2 wε ) = Id2 − ε 2 ∇v ⊗ ∇v.

(7.15)

7.3 Matching infinitesimal to exact isometries on plates

183

It can be easily checked that the metric in the right hand side above has the following symmetric, positive definite matrix field as its square root, for all ε  1: Fε  Id2 − λε ∇v ⊗ ∇v ∈ H 1 ∩ L∞ (ω, R2×2 sym ), where λε =

ε2 p ∈ H 1 ∩ L∞ (ω, R), 1 + 1 − ε 2 |∇v|2

(7.16)

so that there holds (Fε )2 = Id2 − ε 2 ∇v ⊗ ∇v. Thus, (7.15) is equivalent to: ∇(id2 + ε 2 wε )(Fε )−1 ∈ SO(2)

a.e. in ω.

The above claimed existence of a rotation field which completes Fε to a gradient field, will be shown in three steps below. 2. We first prove that the following distribution has regularity L1 (ω, R), and that it is null due to the assumption det ∇2 v = 0:    curl cof (Fε )−1 curl Fε = curl

 1 Fε curl Fε = 0. det Fε

(7.17)

By the formula det(A + B) = det A + hA : cof Bi + det B valid for A, B ∈ R2×2 , we get q det Fε = 1 − λε |∇v|2 = 1 − ε 2 |∇v|2 , 0 for ε  1. Hence we note that Fε is invertible, and also: 1 1 ∈ H 1 ∩ L∞ (ω, R). = p det Fε 1 − ε 2 |∇v|2 Further, denoting (z1 , z2 )⊥ = (−z2 , z1 ) it follows that:
curl Fε = −curl λε ∇v ⊗ ∇v

= ∂2 λε ∂1 v · ∇v − ∂1 λε ∂2 v · ∇v = h∇λε , (∇v)⊥ i∇v + λε ∂1 v · ∂2 ∇v − ∂2 v · ∂1 ∇v




= h∇λε , (∇v)⊥ i∇v + λ (∇2 v)(∇v)⊥ . Consequently, we obtain:

 Fε curl Fε = curl Fε − λε curl Fε , ∇v

 = λε (∇2 v)(∇v)⊥ + (∇λε )⊥ − λε curl Fε , ∇v ∇v.

(7.18)

Observe that the second term in the right hand side above is null, since:

 (∇λε )⊥ − λε curl Fε , ∇v





 = (∇λε )⊥ , ∇v − λε ∇λε , (∇v)⊥ |∇v|2 − λε2 (∇2 v)(∇v)⊥ , ∇v .

184

7 Limiting theories for elastic plates: linearised bending

Observing that: p 1  1 − 1 − ε 2 |∇v|2 2 1 2 2 |∇v|2 λε − λε |∇v| = λε − 2 2 |∇v|2 p 1 − 1 − ε 2 |∇v|2 = λε − + ε 2 = ε 2, |∇v|2 the previous equality may be continued as:

 (∇λε )⊥ − λε curl Fε , ∇v

 = (∇λε )⊥ − λε |∇v|2 (∇λε )⊥ − λε2 (∇2 v)(∇v)⊥ , ∇v
⊥ 

1 = ∇ λε − λε2 |∇v|2 , ∇v = 0, 2 The identity in (7.18) thus becomes: Fε curl Fε = λε (∇2 v)(∇v)⊥ , so that: λε (∇2 v)(∇v)⊥ cof (Fε )−1 curl Fε = p 1 − ε 2 |∇v|2 2

⊥

2

(7.19)

2

= γε (∇ v)(∇v) ∈ L (ω, R ), with γε =

ε2 (1 +

p

1 − ε 2 |∇v|2 )

∇γε = δε (∇2 v)∇v

p ∈ H 1 ∩ L∞ (ω, R) satisfying: 1 − ε 2 |∇v|2 where δε ∈ L∞ (ω, R) for all ε  1.

3. We now compute curl of the expression in (7.19) and deduce (7.17). We have:


curl γε (∇2 v)(∇v)⊥ = ∂1 γε h∂2 ∇v, (∇v)⊥ i − ∂2 γε h∂1 ∇v, (∇v)⊥ i   = γε h∂2 ∇v, (∂1 ∇v)⊥ i − h∂1 ∇v, (∂2 ∇v)⊥ i   + ∂1 γε h∂2 ∇v, (∇v)⊥ i − ∂2 γε h∂1 ∇v, (∇v)⊥ i . For the first term in the right hand side, we use the easily verified identities: h∂2 ∇v, (∂1 ∇v)⊥ i = −h∂1 ∇v, (∂2 ∇v)⊥ i = det ∇2 v. The second term equals:   δε h∂1 ∇v, ∇vih∂2 ∇v, (∇v)⊥ i − h∂2 ∇v, ∇vih∂1 ∇v, (∇v)⊥ i = δε (det ∇2 v)|∇v|2 . Hence we get both claimed statements about the distribution in (7.17):

curl γε (∇2 v)(∇v)⊥ = 2γε + δε |∇v|2 det ∇2 v ∈ L1 (ω, R2 ). In particular, it equals 0 in view of det ∇2 v = 0. 4. By (7.17) and the regularity of the vector field in (7.19), it follows that:

7.3 Matching infinitesimal to exact isometries on plates

cof (Fε )−1 curl Fε = ∇θε

185

for some θε ∈ H 1 (ω, R).

(7.20)

R

Without loss of  generality, we may assume that ω θε dz = 0. Define the field of rocos θε − sin θε tations eiθε = ∈ H 1 ∩ L∞ (ω, R2×2 ). We will now show that eiθε Fε sin θε cos θε is a gradient field, which will imply (7.15). To this end, we compute:
curl eiθε Fε = eiθε curl Fε − eiθε (cof Fε )∇θε . Since Fε is invertible, so is cof (eiθε Fε ), and further: cof (eiθε Fε )−1 = cof ((Fε )−1 e−iθε ) = (cof (Fε )−1 )e−iθε . We may thus write:

cof (eiθε Fε )−1 curl eiθε Fε = (cof (Fε )−1 )e−iθε eiθε curl Fε − eiθε (cof Fε )∇θε = cof (Fε )−1 curl Fε − ∇θε = 0, where we used (7.20) in the last step. Consequently, eiθε Fε ∈ H 1 (ω, R2×2 ) is a gradient field, so that:
eiθε Fε = ∇ id2 + ε 2 wε for some wε ∈ H 2 (ω, R2 ). (7.21) 5. It remains to prove the uniform estimates (7.14). From (7.21), (7.16) and (7.20), (7.19), in view of |∇λε | ≤ Cε 2 |∇2 v||∇v| and γε ≤ Cε 2 we get:
ε 2 |∇2 wε | = |∇(eiθε Fε )| ≤ C |∇Fε | + |Fε ||∇θε |
≤ C |λε ||∇2 v||∇v| + |∇λε ||∇v|2 + |γε ||∇2 v||∇v| ≤ Cε 2 |∇2 v||∇v|, for ε  1 and uniformly in ω. This implies the second bound in (7.14). For the remaining bound, we observe that: ε 2 |∇wε | = |eiθε Fε − Id2 | ≤ |eiθε − Id2 | +C|∇Fε − Id2 | ≤ |θε | +Cε 2 |∇v|2 . In virtue of the Poincar´e-Wirtinger inequality and (7.20), (7.21) we conclude:
kε 2 ∇wε kL2 (ω) ≤ C kθε kL2 (ω) + ε 2 k∇vk2L4 (ω)
≤ C k∇ε θ kL2 (ω) + ε 2 k∇2 vkL2 (ω) k∇vkL∞ (ω) ≤ Cε 2 k∇2 vkL2 (ω) k∇vkL∞ (ω) . This ends the proof of (7.14) and of the theorem. We remark that an alternative proof of existence of {wε }ε→0 as in Theorem 7.7, follows through computing the Riemann curvature of the metric Id2 − ε 2 ∇v ⊗ ∇v in the right hand side of (7.15), checking that it equals 0 in ω, and invoking the general

186

7 Limiting theories for elastic plates: linearised bending

existence theorem for equidimensional isometric immersions. In the proof above we actually found wε directly, by following the steps of that general construction. It is also convenient to introduce the normalisations of {wε }ε→0 , as follows: Corollary 7.8. Let ω ⊂ R2 be an open, bounded, Lipschitz and simply connected. Given v ∈ H 2 ∩ W 1,∞ (ω, R2 ) and w ∈ H 1 (ω, R2 ) satisfying (7.13), there exists an equibounded family {wε ∈ H 2 (ω, R2 )}ε→0 , such that: wε → w

in H 1 (ω, R2 )

as ε → 0

and there holds (∇uε )T ∇uε = Id2 , where: uε  id2 + εve3 + ε 2 wε . Proof. Let {wε }ε→0 be the family of the in-plane displacements constructed in the proof of Theorem 7.7 and satisfying the uniform bound (7.14). Denote: ? ? Aε  skew ∇wε dz, A = skew ∇w dz ∈ so(2). ω

ω

We now define the modified in-plane displacements family {w˜ ε }ε→0 , for whom the assertions of Corollary will hold. Namely, let: 2

eε (A−Aε ) − Id2 z + bε , wε (z) + ε2 ? with bε ∈ R2 so that: w˜ ε − w dz = 0.

w˜ ε (z)  eε

2 (A−A

ε)

ω 2

It is straightforward that: id2 + ε 2 w˜ ε = eε (A−Aε ) (id2 + ε 2 wε ) + ε 2 bε , so conse2 quently: ∇(id2 + ε 2 w˜ ε ) = eε (A−Aε ) ∇(id2 + ε 2 wε ) satisfies (7.15) and the claimed 2 isometry condition, because eε (A−Aε ) ∈ SO(2). Since {Aε , bε }ε→0 are uniformly bounded by (7.14), there follows the equiboundedness of {w˜ ε ∈ H 2 (ω, R2 )}ε→0 . It remains to show the convergence property. Observe first that, by the isometry condition: 1 sym∇w˜ ε − sym∇w = sym∇w˜ ε + ∇v ⊗ ∇v 2 1 2 = − ε ∇w˜ ε ⊗ ∇w˜ ε → 0 2

in L2 (ω, R2×2 ),

and that the Poincar´e and Korn inequalities yield: kw˜ ε − wkH 1 (ω) ≤ Ck∇w˜ ε − ∇wkL2 (ω)

?

≤ Cksym∇w˜ ε − sym∇wkL2 (ω) +C skew

∇w˜ ε dz − A . ω

Finally, we have:

7.4 Linearised Kirchhoff’s theory for plates: recovery family for Lipschitz displacements

?

skew

187

∇w˜ ε dz − A ω

?

2
eε (A−Aε ) − Id2 − (A − Aε ) ∇wε dz + skew 2 ε ω 2 (A−A ) ε ε 2 e − Id2 ≤ C eε (A−Aε ) − Id2 + − (A − Aε ) ≤ Cε 2 . ε2

2 ≤ skew (eε (A−Aε ) − Id2 )

The last three displayed formulas result in the convergence of w˜ ε → w in H 1 (ω, R2 ), as ε → 0. The proof is done. We close the discussion by noting that assumption of ω being simply connected (which was used several times throughout the proof) is necessary in Theorem 7.7: Example 7.9. Consider the scalar field v(z) = |z| defined on the annulus: ω = B1 \ B¯ 1/2 ⊂ R2 .
Then ∇v(z) = z/|z| and ∇2 v(z) = |z|2 Id2 − z ⊗ z /|z|3 , so v ∈ H 2 (ω, R) and also det ∇2 v = 0. We compute, according to (7.16) and (7.20): ε2 z⊗z √ , 1 + 1 − ε 2 |z|2 √ √ 1 − 1 − ε 2 |z|2 Id2 − z ⊗ z ⊥ 1 − 1 − ε 2 z⊥ √ √ z = ∇θε (z) = |z|4 1 + 1 − ε2 1 + 1 − ε 2 |z|2 Fε (z) = Id2 −

for all z ∈ ω.

The fundamental theorem of line integrals thus yields, in polar coordinates: √ 1 − 1 − ε2 1
√ θε (r, α) − θε (r, 0) = α for all r ∈ , 1 , α ∈ R. 2 2 1+ 1−ε For the well definiteness of wε one then needs: √ 1 − 1 − ε2 √ 2π = θε (r, 2π) − θε (r, 0) ∈ 2πZ. θε (r, α) − θε (r, 0) = 1 + 1 − ε2 This requirement, equivalent to √ 1

1−ε 2

∈ Z, is clearly false for ε
1/h}

+Chβ

Z

(|v| + |∇v| + |∇2 v|)2 dz

(|wh | + |∇wh | + |∇2 wh |)2 dz ≤ o(h2 ).

ω

Similarly as in the proof of Theorem 4.3 (iii), it also follows that: kv¯h kH 2 (ω) + kw¯ h kH 2 (ω) ≤ C.

(7.23)

Indeed, invoking Theorem 5.12 (ii) again: kw¯ h kH 2 (ω) ≤ kwh kH 2 (ω) + kw¯ h kH 2 ({wh ,w¯ h }) ≤ C +Ch−β /2 |{wh , w¯ h }|1/2 ≤ C +C

Z

(|wh | + |∇wh | + |∇2 wh |)2 dz ≤ C,

ω

with a similar argument for {v¯h }. 2. Denote the normal (not unit) vectors and the frames on image surfaces u¯h (ω): h i Nh = ∂1 u¯h × ∂2 u¯h , Q¯ h = ∂1 u¯h , ∂2 u¯h , Nh ∈ W 1,∞ (ω, R3×3 ). We now show that: kQ¯ h − Id3 kL∞ (ω) + hk∇Nh kL∞ (ω) → 0 The key argument relies on the following result:

as h → 0.

(7.24)

7.4 Linearised Kirchhoff’s theory for plates: recovery family for Lipschitz displacements

189

Theorem. [Brezis-Wainger’s inequality] Let ω ⊂ R2 be an open, bounded and Lipschitz domain. Then, for every κ > 0 and for every f ∈ H 1 ∩W 1,∞ (ω, R) such that k f kH 1 (ω) ≤ κ, there holds:   Ck f kW 1,∞ (ω)  . k f kL∞ (ω) ≤ Cκ 1 + log1/2 1 + κ

(7.25)

The constant C above depends only on ω, but not on κ or f . We apply (7.25) to each component of f = ∇w¯ h (resp. f = ∇v¯h ) and κ as the common bounding constant of k∇w¯ h kH 1 (ω) (resp. k∇v¯h kH 1 (ω) ), to get:
k∇w¯ h kL∞ (ω) ≤ C 1 + log1/2 (1 +Ch−β /2 ) ≤ C log h−β /2 ≤ Ch−(β /2−1)/2 , (7.26)
k∇v¯h kL∞ (ω) ≤ C 1 + log1/2 (1 +Ch−1 ) ≤ C log h−1 ≤ Ch−(β /2−1)/2 . The same bounds are valid for kw¯ h kW 1,∞ (ω) and kv¯h kW 1,∞ (ω) , through (7.23). Estimates in (7.26) yield convergence of the in-plane components in (7.24):
k∇u¯h − Id3×2 kL∞ (ω) ≤ C khβ −2 ∇w¯ h kL∞ (ω) + khβ /2−1 ∇v¯h kL∞ (ω)
1 3 ≤ C h 2 (β /2−1) + h 2 (β /2−1) → 0 as h → 0. For the remaining components we similarly have, recalling (7.22): kNh − e3 kL∞ (ω) ≤ C khβ −2 ∇w¯ h kL∞ (ω) + khβ /2−1 ∇v¯h kL∞ (ω) + khβ −2 ∇w¯ h kL∞ (ω) khβ /2−1 ∇v¯h kL∞ (ω) 1

≤ Ch 2 (β /2−1) → 0 h

k∇N kL∞ (ω) ≤ C kh

β −2




as h → 0,

∇ w¯ h kL∞ (ω) + khβ /2−1 ∇2 v¯h kL∞ (ω) 2




(7.27)


· 1 + khβ −2 ∇w¯ h kL∞ (ω) + khβ /2−1 ∇v¯h kL∞ (ω)
1 3 1 as h → 0. ≤ Chβ /2−2 1 + h 2 (β /2−1) + h 2 (β /2−1) ≤ o(h) h The proof of (7.24) is done. 3. For each z ∈ ω and t ∈ (− 12 , 21 ), we now define, similarly as in (7.5): hβ /2+1 1,h d (z) 2
where d 1,h → d 1  −c ∇2 v(z) in L2 (ω, R3 )

yh (z,t) = u¯h (z) + thNh (z) + t 2

and hkd 1,h kW 1,∞ (ω) → 0 We have, by (7.22), (7.24) and (7.28):

as h → 0.

(7.28)

190

7 Limiting theories for elastic plates: linearised bending

kyh − πkH 1 (Ω 1 ) ≤ ku¯h − id2 kH 1 (ω) + hkNh kH 1 (ω) + hβ /2+1 kd 1,h kH 1 (ω) → 0 as h → 0,

1

1 kV h [yh ] − ve3 kH 1 (ω) = β /2−1 (u¯h − id2 ) + h2 d 1,h − ve3 H 1 (ω) 24 h ≤ hβ /2−1 kw¯ h kH 1 (ω) + kv¯h − vkH 1 (ω) + h2 kd 1,h kH 1 (ω) → 0 as h → 0. This proves (i) and (ii) in Theorem 7.5. For the assertion in (iii), we estimate:

1 hβ /2−1

1 V h [yh ]tan − w H 1 (ω) = kw¯ h + h3−β /2 d 1,h − wkH 1 (ω) 24 ≤ kw¯ h − wkH 1 (ω) +Ch2−β /2 khd 1,h kH 1 (ω) .

(7.29)

The last term above converges to 0 by (7.28). For the other term, we use (7.26), (7.22) and convergence of {wh }h→0 to w: kw¯ h − wk2H 1 (ω) ≤ C

Z {w˜ h ,wh }

|w¯ h |2 + |∇w¯ h |2 + |w|2 + |∇w|2 dz +Ckwh − wk2H 1 (ω)

Z ≤ Ch−(β /2−1) ω \ ωh +C ω\ωh

|w|2 + |∇w|2 dz +Ckwh − wk2H 1 (ω)

→ 0 as h → 0. Together with (7.29), this concludes the proof of (iii). 4. To obtain the convergence in (iv), we start by convergence of the error term in: i h i h t 2 hβ /2+1 ∇d 1,h ,thβ /2 d 1,h → 0 ∇h yh − Q¯ h = th∇Nh , 0 + 2

in L∞ (Ω 1 ), (7.30)

by (7.24) and the assumed convergence rate in (7.28). Using the second bound in (7.27), together with (7.22), it further follows that: k dist(∇h yh , SO(3))kL∞ (Ω 1 ) ≤ k dist(Q¯ h , SO(3))kL∞ (ω) + hk∇Nh kL∞ (ω) + hβ /2 kd 1,h kL∞ ≤ Ck∇Qh kL∞ (ω) |ω \ ωh |1/2 +Chβ /2−1 ≤ Chβ /2−2 o(h) +Chβ /2−1 ≤ Chβ /2−1 . Consequently, kW (∇h yh )kL∞ (Ω 1 ) ≤ Chβ −2 and thus: 1 hβ

Z (ω\ωh )×(− 12 , 21 )

W (∇h yh ) dx ≤

1 kW (∇h yh )kL∞ (Ω 1 ) |ω \ ωh | ≤ o(1). hβ

(7.31)

5. In this last step, we examine the limiting behaviour of the remaining term: 1 hβ

Z ωh ×(− 12 , 21 )

W (∇h yh ) dx.

7.4 Linearised Kirchhoff’s theory for plates: recovery family for Lipschitz displacements

191

Since Q¯ h ∈ SO(3) on ωh , frame invariance allows for replacing ∇h yh by: (Q¯ h )T ∇h yh = Id3 + hβ /2 (Q¯ h )T

h

i h 2 i h β /2 ¯ h T t h 1,h 1,h ∇N , 0 + h ( Q ) ∇d ,td . 2 hβ /2−1 t

At the same time, in virtue of Lemma 7.6 and (7.27), we get: " 

 h t i − (hβ /2−1 ∂i j wh , ∂i j v), Nh i, j=1,2 h T h ¯ (Q ) ∇N , 0 = t hβ /2−1 0 # " −∇2 v 0 in L2 (ω, R3×3 ). →t 0 0

0

#

0

Similarly, since |(Q¯ h )T d 1,h − d 1,h | ≤ |d 1,h − d 1 | + |(Q¯ h )T − Id3 ||d 1 | converges to 0 in L2 (ω), the convergence rate in (7.28) yields: (Q¯ h )T

h t 2h 2

i h i ∇d 1,h ,td 1,h → 0,td 1

in L2 (ω, R3×3 ).

In conclusion: 1 hβ

Z ωh ×(− 21 , 21 )

W (∇h yh ) dx

Z i  h t i h t 2h 1 1,h 1,h β /2 ¯ h T h ∇d ,td dx W Id + h ( Q ) ∇N , 0 + 3 2 hβ Ω 1 hβ /2−1 Z i h t 2h  h t i 1 h 1,h 1,h ∇N , 0 + Q3 hβ /2 (Q¯ h )T = β ∇d ,td dx + o(1) 2 2h Ω 1 h#β /2−1 "   Z 1 −t∇2 v td 1 dx as h → 0, → Q3 2 Ω1 0

≤

which implies, by invoking (5.8) and Remark 7.2 (ii): lim sup h→0

1 hβ

Z ωh ×(− 12 , 21 )

1 t 2 Q2 (∇2 v) dx 2 Ω1 Z 1 = Q2 (∇2 v) = IlinK (v). 24 ω

W (∇h yh ) dx ≤

Z

1 h h E (u ) ≤ IlinK (v) and β h h→0 thus ends the proof by virtue of the lower bound in Theorem 7.1.

The above bound, together with (7.31) yields: lim sup

192

7 Limiting theories for elastic plates: linearised bending

7.5 Density result on plates and linearised Kirchhoff’s theory: recovery family, Γ -limit and convergence of minimizers In this section, we will complete the proof of Theorem 7.5. The argument will be based on the already established construction for Lipschitz infinitesimal isometries, carried out in the previous section, and the second key property, which is the density property below. We then combine Theorems 7.1 and 7.5 to prove the corresponding Γ -convergence statement and convergence of minimizers. ¨ Theorem. [Muller-Pakzad’s approximation result] Let ω ⊂ R2 be an open, bounded and Lipschitz domain. Then, for every v ∈ H 2 (ω, R) satisfying det ∇2 v = 0, there exists an increasing, to ω, sequence {Sn ⊂ ω}n→∞ of open subdomains, and a sequence of mappings {vn ∈ H 2 ∩W 1,∞ (ω, R)}n→∞ , with: vn = v

in Sn ,

and |∇vn | ≤ n,

∇2 vn = 0

in ω.

(7.32)

In particular, det ∇2 vn = 0 in ω and vn → v in H 2 (ω, R) as h → 0. The proof of the above result is beyond the scope of our monograph. It relies on two fundamental properties of maps v: the compensated regularity and the developability. The first property states that, automatically, one has v ∈ C 1 (ω). The developability property states that each z ∈ ω is either a center of a ball on which ∇v is constant, or z belongs to a line segment that intersects ∂ ω at both ends, and on which ∇v is constant. Different line segments do not intersect in ω (see Figure 7.1). Developable surfaces without flat parts and their elastic properties in the context of dimension reduction and isometry matching, will be covered in chapter 10.

Fig. 7.1 Regions of affinity and lines of constant gradient for a developable map v defined on ω.

We now sketch the geometric construction of the approximate map vn in (7.32). First, vn is set to equal v on the open set Sn = {|∇v| < n}. Because of developability, whenever Sn , 0/ for a sufficiently large n, then ∂ Sn consists of straight line segments on which ∇v is constant, and of portions of ∂ ω. In the second step, vn is defined on each connected component of ω \ Sn as the affine extension of v restricted to the aforementioned line segment within the given component’s boundary. Finally, since v ∈ C 1 , the domains {Sn }n→∞ exhaust ω.

7.5 Density result on plates and linearised Kirchhoff’s theory: recovery family, Γ -limit...

193

We are ready to give: Proof of Theorem 7.5. Given v ∈ H 2 (ω, R) satisfying det ∇2 v = 0, let {vn }n→∞ be an approximating sequence of Lipschitz infinitesimal isometries with properties as in the M¨uller-Pakzad theorem. Also, given w ∈ H 1 (ω, R2 ) with (7.13), we may choose {wn }n→∞ so that: ? ? 1 ∇vn ⊗ ∇vn + sym∇wn = 0, skew ∇(wn − w) dz = 0, (wn − w) dz = 0. 2 ω ω Since vn → v in H 2 (ω, R), it follows sym∇wn → sym∇w in L2 (ω, R2×2 ), and so: kwn − wkH 1 (ω) ≤ Ck∇(wn − w)kL2 (ω) ≤ Cksym∇(wn − w)kL2 (ω) → 0

as n → ∞,

in virtue of the Poincar´e and the Korn inequalities. For each (vn , wn ), we now use the construction in section 7.4 and find a recovery family {yhn ∈ H 1 (Ω 1 , R3 )}h→0 which satisfies the assertions of Theorem 7.5: lim kyhn − πkH 1 (Ω 1 ) = 0,

h→0

lim

h→0

1 hβ /2−1

1 h→0 hβ lim

Z Ω1

lim kV h [yhn ] − vn e3 kH 1 (ω) = 0,

h→0

V h(n) [yhn ]tan − wn H 1 (ω) = 0,

W (∇h yh ) dx = IlinK (vn ). h(n)

Since lim IlinK (vn ) = IlinK (v), one can extract a “diagonal” sequence {yn }n→∞ , n→∞

corresponding to a strictly decreasing (to 0) sequence {h(n)}n→∞ , such that: lim kyhn (n) − πkH 1 (Ω 1 ) = 0,

n→∞

 h(n)  lim kV h yn − ve3 kH 1 (ω) = 0,

n→∞

1 h(n) V h(n) [yn ]tan − w H 1 (ω) = 0, h→0 h(n)β /2−1 Z 1 lim W (∇h yh(n) ) dx = IlinK (v). n→∞ h(n)β Ω 1

lim

By letting {h(n)}n→∞ converge to 0 sufficiently fast, and defining: 
yh  yhn for all h ∈ h(n), h(n − 1) , we ensure the validity of assertions (i)-(iv) in Theorem 7.5. Similarly as in sections 5.7 and 6.2, we now combine Theorems 7.4 and 7.5 in a single Γ -convergence statement, corresponding to the total energies of deformations of thin plates Ω h = ω × (− h2 , h2 ): J h (uh )  E h (uh ) +

1 h

Z Ωh

h f h , uh − id3 i dx.

194

7 Limiting theories for elastic plates: linearised bending

We consider a simple case of the applied forces { f h ∈ L2 (Ω h , R3 )}h→0 given by: f h (z,t) = hβ /2+1 f (z)e3 , where f ∈ L2 (ω, R) satisfies:

Z

Z

f dz = 0, ω

f (z)z dz = 0.

(7.33)

ω

Observe that, automatically, there holds: m f = 0 and M f = SO(3) in (6.18). Corollary 7.10. Let ω ⊂ R2 be an open, bounded, simply connected and Lipschitz domain. Assume that { f h ∈ L2 (Ω h , R3 )}h→0 are given by (7.33), for some β ∈ (2, 4). For each y ∈ H 1 (Ω 1 , R3 ), v ∈ H 1 (ω, R3 ), w ∈ H 1 (ω, R2 ) , Q¯ ∈ SO(3), define: ( ¯ h ) if v = vh [y], w = wh [y], J h (Qu h ¯ = F (y, v, w, Q) +∞ otherwise, denoting: uh (z,t) = y(z, ht ) ∈ H 1 (Ω h , R3 ) the normalized deformations, and: 1

?

1/2

hy, e3 i dt ∈ H 1 (ω, R), hβ /2−1 −1/2 ? 1/2 1 h ytan − id2 dt ∈ H 1 (ω, R2 ) w [y] = β −2 h −1/2 vh [y] =

their scaled displacements. Then, there holds: 1 h Γ F −→ FlinK hβ

as h → 0,

1 with respect to H 1 (Ω 1 , R3 )×H 1 (ω, R)×Hweak (ω, R2 )×SO(3) convergence, where:  Z  ¯  I (v) + Q f (z)v(z) dz if y = π and v ∈ H 2 (ω, R) 33 linK  ω ¯ = FlinK (y, v, w, Q) with 12 ∇v ⊗ ∇v + sym∇w = 0,    +∞ otherwise.

Proof. 1. To verify condition (i) in Definition 2.1, we take {(yh , vh , wh , Q¯ h )}h→0 that ¯ in H 1 (Ω 1 , R3 ) × H 1 (ω, R) × H 1 (ω, R2 ) × SO(3). converges to some (y, v, w, Q) weak Without loss of generality, we may assume that: lim inf h→0

1 1 h h h h ¯h F (y , v , w , Q ) = lim β F h (yh , vh , wh , Q¯ h ) < +∞, h→0 h hβ

so in particular vh = vh [yh ] and wh = wh [yh ]. By Theorem 5.9, it follows directly that E h (uh ) = E h (Q¯ h uh ) ≤ Chβ , whereas Theorem 7.4 yields existence of {Qh ∈ SO(3), ch ∈ R3 }h→0 such that, up to a (not-relabeled) subsequence:

7.5 Density result on plates and linearised Kirchhoff’s theory: recovery family, Γ -limit...

(Qh )T yh − ch → π

195

in H 1 (Ω 1 , R3 ), in H 1 (ω, R),

vh [(Qh )T yh − ch ] → v˜

(7.34)

weakly in H 1 (ω, R2 ), 1 where v˜ ∈ H 2 (ω, R) and ∇v˜ ⊗ ∇v˜ + sym∇w˜ = 0. 2

wh [(Qh )T yh − ch ] * w˜

Observe that:  β /2−1 h h T h   β /2−1 h h  h w [(Q ) y − ch ] w [y ] h T h − (Q ) vh [(Qh )T yh − ch ] vh [yh ] 1 1 = β /2−1 ((Qh )T − Id3 )z − β /2−1 ch . h h

(7.35)

Since the left hand side is bounded in H 1 (ω, R3 ), inspecting the right hand side implies the following convergences (again, possibly up to a subsequence): Qh → Id3 ,

1 hβ /2−1


(Qh )T − Id3 → A ∈ so(3),

1 ch → c ∈ R3 hβ /2−1

ch → 0,

as h → 0.

This already yields y = π, in view of (7.34). 2. Taking the out-of-plane component in (7.35) yields:



Qh e3 − e3  hch , e3 i vh [(Qh )T yh − ch ] = Qh e3 , hβ /2−1 wh + Qh33 vh + , z − β /2−1 . hβ /2−1 h Passing to the limit in H 1 (ω, R), the assumed convergences and (7.34) imply: v(z) ˜ = v(z) + h(A31 , A32 ), zi − hc, e3 i,

(7.36)

so v ∈ H 2 (ω, R) and det ∇2 v = 0. For the in-plane components of (7.35) we have: h T h wh [(Qh )T yh − ch ] = (Qtan ) w + vh

1 hβ /2−1

1


tan

h ctan hβ −2


1 T (Qh )tan − Id2 z − hβ −2  hT
(Q ) −Id3
T 1 T − Id (Qh )tan = − 2 2 hβ /2−1 +

Since the matrices

(Qh )T e3

hβ −2 1 2 2 (A )tan ,

(Qh )T −Id3
hβ /2−1 tan



con-

verge, as h → 0, to it follows that Atan = 0. Further, passing to the limit in the above displayed formula, weakly in H 1 (ω, R2 ), we arrive at: 1 w(z) ˜ = w(z) + v(z)(A13 , A23 ) + (A2 )tan z − d 2

where

We now compute, in virtue of (7.36) and (7.37):

h ctan → d ∈ R2 . (7.37) hβ −2

196

7 Limiting theories for elastic plates: linearised bending

1 ∇v⊗∇v + sym∇w 2 
1 = ∇v˜ ⊗ ∇v˜ − 2sym ∇v˜ ⊗ (A31 , A32 ) + (A31 , A32 ) ⊗ (A31 , A32 ) 2
1 + sym∇w˜ + sym ∇v ⊗ (A31 , A32 ) − (A2 )tan = 0, 2
where we used the fact that A = 2 skew (A13 , A23 , 0) ⊗ e3 , implying that (A2 )tan = (A13 , A23 ) ⊗ (A13 , A23 ). 3. Note that by Theorem 7.4 (iv), we get: lim

1

h→0 hβ

E h (uh ) ≥ IlinK (v) ˜ = IlinK (v).

For the remaining term within 1 hβ +1

Z Ωh



1 hβ

h

 f , Q¯ h uh − id3 dx = Z

h

= →

J h (Q¯ h uh ), we recall (7.33) and write: Z ? 1/2

1 hβ /2−1

¯h h

ω

Z

−1/2

  f (z) h(Q¯ h )T e3 , hβ /2−1 wh i + Q¯ h33 vh dz

f (z)v [Q y ] dz = Zω

 f (z)e3 , Q¯ h yh (z,t) − z dt dz

ω

¯ − IlinK (v) f (z)Q¯ 33 v(z) dz = FlinK (y, v, w, Q)

as h → 0,

ω

¯ h T e3 −e3  , z . This ends the because of vh [Q¯ h yh ] = h(Q¯ h )T e3 , hβ /2−1 wh i + Q¯ h33 vh + (Qh)β /2−1 proof of the lower bound. ¯ ∈ H 1 (Ω 1 , R3 ) × 4. We now verify condition (ii) in Definition 2.1. Let (y, v, w, Q) 1 1 2 H (ω, R) × H (ω, R ) × SO(3). If any of the following assertions is violated: y = π,

v ∈ H 2 (ω, R)

1 ∇v ⊗ ∇v + sym∇w = 0 2

(7.38)

¯ h→0 . Indeed, in this case FlinK (y, v, w, Q) ¯ = then we set {(yh , vh , wh , Q¯ h ) = (y, v, w, Q)} 1 h h h h ¯h +∞, and likewise there must be limh→0 hβ F (y , v , w , Q ) = +∞, because otherwise reasoning as in steps 1 and 2 would bring a contradiction. Assume now that (7.38) holds. By Theorem 7.5 we obtain a recovery family {yh ∈ H 1 (Ω 1 , R3 )}h→0 converging to π and such that: vh [yh ] → v in H 1 (ω, R), wh [yh ] * w weakly in H 1 (ω, R2 ) and h1β E h (uh ) → IlinK (v). Thus: 1 h h h h ¯ 1 F y , v , w , Q) = lim β E h (uh ) + Q¯ 33 h→0 hβ h→0 h ¯ = FlinK (y, v, w, Q),

Z

f (z)v(z) dz

lim

ω

in view of the same computation as in step 3. The proof is done. We finally point out that the convergence of minimizers under assumption (7.33) is now a direct consequence of Corollary 7.10. A similarly assertion has been

7.5 Density result on plates and linearised Kirchhoff’s theory: recovery family, Γ -limit...

197

discussed for other energetical regimes, considered in previous sections, namely: the linear elasticity in section 2.2, the nonlinear bending in section 5.7, the von K´arm´an’s theory in section6.2, and the linear elasticity in section 6.3. Corollary 7.11. In the context of Corollary 7.10, consider a family of almostminimizers {(uh , Q¯ h ) ∈ H 1 (Ω h , R3 ) × SO(3)}h→0 to the energy functionals: 1 J¯ h (uh , Q¯ h ) = J h (Q¯ h uh ) = E h (uh ) + h

Z Ωh

h f h , Q¯ h uh − id3 i dx,

satisfying: 1 ¯ h h ¯ h J (u , Q ) − inf J¯ h → 0 β h

as h → 0.

(7.39)

Then, the family {yh (z,t) = Qh uh (z,th) − ch }h→0 defined for some {Qh ∈ SO(3), ch ∈ R3 }h→0 , converges up to a subsequence in H 1 (Ω 1 , R3 ), to the limit π, while we also have the following subsequential convergences: V h [yh ]  1 hβ /2−1

?

1 hβ /2−1

1/2

in H 1 (ω, R3 ),

yh (z,t) − z dt → ve3 −1/2

V h [yh ]tan * w

weakly in H 1 (ω, R2 ),

(7.40)

1 where v ∈ H 2 (ω, R) and ∇v ⊗ ∇v + sym∇w = 0, 2 Q¯ h → Q¯ ∈ SO(3). The limiting quantities minimize the functional FlinK . Equivalently: ¯ = min I¯linK < +∞, I¯linK (v, Q) ¯ = IlinK (v) + Q¯ 33 where I¯linK (v, Q)

Z

f (z)v(z) dz.

(7.41)

ω

In particular, the functional I¯linK : {v ∈ H 2 (ω, R); det ∇2 v = 0} × SO(3) → R ¯ to I¯linK coinhas at least one minimizer. Conversely, every minimizing (v, Q) h 1 1 cides with limits as in (7.40), derived from some {y ∈ H (Ω , R3 )}h→0 , {Q¯ h ∈ SO(3)}h→0 , for which (7.39) holds, where we define {uh (z,t) = yh (z, ht ) ∈ H 1 (Ω h , R3 )}h→0 . There also holds: 1 inf J¯ h → min I¯linK hβ

as h → 0.

198

7 Limiting theories for elastic plates: linearised bending

7.6 Elastic shallow shells In the remaining part of this chapter, we will analyze another dimension reduction problem in the nonlinear elasticity. This problem, pertaining to shallow shells can be seen as a variation of the linearised Kirchhoff theory, intermediate to Theorem 7.1 posed on shells whose midsurface has potentially large curvature, and to Theorems 7.4, 7.5 posed on plates where the referential configuration is 0. The related analytical questions are of independent interest. They are naturally parallel to the questions tackled before: isometry matching studied in section 7.3, density and regularity of infinitesimal isometries discussed in section 7.5, and constructing the recovery family. We also anticipate that the formulation below can be recast in the setting of non-Euclidean elasticity, which will be the topic of our studies in Part III of this monograph. Below, we pose the problem, introduce the basic notation, and heuristically derive one specific limiting energy functional, which can be seen as a generalization of the linearised Kirchhoff theory IlinK on plates in Theorem 7.4. ¯ R) be a fixed out-of-plate 1. Let ω ⊂ R2 be open and bounded, and let v0 ∈ C 2 (ω, displacement. For a scaling exponent γ > 0, consider the family of surfaces: Sh = φh (ω) where φh (z) = z + hγ v0 (z)e3

for all z ∈ ω, h  1,

and the resulting family of thin shallow shells {(Sh )h }h→0 : n h h
o ˜ (Sh )h  φ˜h (z,t); z ∈ ω, t ∈ − , = φh (Ω h ). 2 2

(7.42)

Above, the Kirchhoff-Love extension φ˜h : Ω h → R3 of the mid-surface Sh parametrization φh , is defined on the thin plate Ω h , by the formula: φ˜h (x) = φh (z) + tnh (z)

h h
for all x = (z,t) ∈ Ω h = ω × − , , 2 2

(7.43)

where nh (z) denotes the unit normal to Sh at φh (z): nh =


∂1 φh × ∂2 φh 1 − hγ ∂1 v0 , −hγ ∂2 v0 , 1 . = p 2γ 2 |∂1 φh × ∂2 φh | 1 + h |∇v0 |

The thickness-averaged elastic energy of a deformation uh of (Sh )h is then: E h (uh ) =

1 h

Z (Sh )h

for all uh ∈ H 1 ((Sh )h , R3 ).

W (∇uh ) dx

Given a family of external loads { f h ∈ L2 ((Sh )h , R3 )}h→0 , the total energy reads: J h (uh ) = E h (uh ) +

1 h

Z (Sh )h

h f h , uh − id3 i dx.

7.6 Elastic shallow shells

199

In what follows, we will make the simplifying assumptions similar to (7.33): f h = hβ /2+1 f ◦ φ˜h−1 e3

for some f ∈ L2 (ω, R)

Z

Z

f dz = 0

such that ω

f (z)z dz = 0.

and

(7.44)

ω

2. By a simple change of variables, denoting u˜h = uh ◦ φ˜h ∈ H 1 (Ω h , R3 ), we get: J h (uh ) =

1 h

Z Ωh


W (∇u˜h )(∇φ˜h )−1 (det ∇φ˜h ) dx

+ hβ /2

Z Ωh

(7.45)
f (z) hu˜h (x), e3 i − t (det ∇φ˜h ) dx,

where from a direct calculation, in view of nh = (−hγ ∇v0 , 1− 12 h2γ |∇v0 |2 )+O(h3γ ): # " 2v −∇v −t∇ 0 0 + O(h3γ ) + t · O(h2γ ), ∇φ˜h = Id3 + hγ ∇v0 − 21 hγ |∇v0 |2 det ∇φ˜h = 1 − thγ ∆ v0 + o(h2γ ) = 1 + oh (1). We recall the usual assumptions on the energy density W in (5.2), and complement them by the isotropy assumption (5.5). Since by the polar decomposition of p h h T ˜ ˜ ˜ positive definite matrices ∇φh = Ra with R(x) ∈ SO(3) and a = (∇φh ) ∇φh ∈ C 0 (Ω¯ h , R), the formula in (7.45) yields: E h (uh ) =

1 h

Z Ωh


W (∇u˜h )(ah )−1 (1 + o(1)) dx,

1 with a = Id3 + h2γ (∇v0 ⊗ ∇v0 )∗ − hγ t(∇2 v0 )∗ + O(h3γ ) + t · O(h2γ ) 2

(7.46)

h

and where the uniform quantities O(h3γ ), O(h2γ ) are independent of t, and where for F ∈ R2×2 , the matrix F ∗ ∈ R3×3 is obtained by (F ∗ )tan = F and (F ∗ )i3 = (F ∗ )3i = 0. The formulation in (7.46) reduces the problem to studying deformations of the flat plate Ω h relative to the prestress tensor ah (see Part III for a broad discussion). 3. Following (7.28), we now make a simplified ansatz for the deformation that is minimizing to J h under (7.44), in the small slope regime: with: u¯h = id2 + hβ /2−1 ve3 + hβ −2 w,  β /2−1  ∂1 u¯h × ∂2 u¯h −h ∇v Nh = + o(hβ /2−1 ), = 1 |∂1 u¯h × ∂2 u¯h |

u˜h (z,t) = u¯h (z) + tNh (z)

where w : ω → R2 , v : ω → R are some in-plane and out-of-plane displacements.  Compute the gradient: ∇h yh (z,t) = ∇u˜h (z, ht) = ∇u¯h (z) + th∇Nh (z) | Nh (z) for z ∈ ω, |t| < 1, and the resulting highest order terms in the rescaled strain:

200

7 Limiting theories for elastic plates: linearised bending

(∇h yh )T ∇h yh ' Id3 + hβ −2



 ∇v ⊗ ∇v + 2sym∇w 0 − 2thβ /2 (∇2 v)∗ . (7.47) 0 |∇v|2

By frame invariance and (7.46), we write: E h (uh ) '

Z Ω1


W ah (z,th)−1,T (∇h yh )T (∇h yh )ah (z,th)−1 dx.

(7.48)

Comparing exponents in (7.46) and (7.47), we observe that for β /2 − 1 < γ the effect of the deformation uh can be neglected, while for β /2 − 1 > γ, the limiting energy is similar to that of a plate (studied in previous sections). The remaining case where the shallowness is tuned with the deformation scaling, is given by the regime: γ = β /2 − 1.

(7.49)

In what follows we will always assume (7.49). Then, the leading order terms of the argument of W in (7.48), are suggestive of the expected limiting theory:   ah (z,th)−1,T (∇h yh )T (∇h yh )ah (z,th)−1,T tan

' Id2 + hβ −2 ∇v ⊗ ∇v − ∇v0 ⊗ ∇v0 + 2sym∇w + 2thβ /2 ∇2 v0 − ∇2 v . Namely, for β > 4 we expect that h1β E h has the Γ -limit consisting of the relative lin1 R 2 2 ear elasticity functional 24 ω Q2 (∇ v−∇ v0 ) dz, while for β = 4 the corresponding Γ -limit will be the relative von K´arm´an functional of the form: 1 2

Z ω

Q2

Z

1 1 1 Q2 ∇2 v − ∇2 v0 dz. ∇v ⊗ ∇v − ∇v0 ⊗ ∇v0 + sym∇w dz + 2 2 24 ω

We will derive and analyze the aforementioned limits in Part III of our monograph, in a much more general context. In the remaining part of this chapter, we will focus on the case: β ∈ (2, 4). As deduced from the above heuristics, the expected limiting theory consists of the 1 1 1 R 2 2 relative bending 24 ω Q2 (∇ v − ∇ v0 ) dz, under the constraint 2 ∇v ⊗ ∇v − 2 ∇v0 ⊗ ∇v0 + sym∇w = 0. On a simply connected referential midplate ω this yields: det ∇2 v = det ∇2 v0 , in virtue of Lemma 6.19, reducing to the homogeneous Monge-Amp´ere constraint in Theorem 7.4 for the previously studied case v0 ≡ 0. The corresponding generalizations of the analysis in sections 7.2 - 7.5 will be our subject below.

7.7 Linearised Kirchhoff’s theory for shallow shells

201

7.7 Linearised Kirchhoff’s theory for shallow shells In this section, we make precise the dimension reduction result which we have heuristically derived in section 7.6. We first observe an energy bound and equipartition result, similar to Theorem 5.9 in the context of elasticity on shells with the midsurface having an arbitrarily large curvature: Lemma 7.12. Assume (7.44) and (7.49) with β ≥ 2. Then:  −Chβ ≤ inf J h (uh ); uh ∈ H 1 ((Sh )h , R3 ) ≤ 0 for all h  1. Moreover, if lim suph→0 h1β J h (uh ) < +∞, then also E h (uh ) ≤ Chβ , with positive constants C that depend only on ω, f and W . We omit the proof, as it is the same as in Theorem 5.9. In fact, following the arguments leading to Theorem 7.4 in section 7.2, one also gets the compactness and the corresponding lower bound estimate: Theorem 7.13. Let ω ⊂ R2 be an open, bounded, simply connected and Lipschitz domain. Assume (7.44), (7.49) and let {uh ∈ H 1 ((Sh )h , R3 )}h→0 satisfy: J h (uh ) ≤ Chβ

for some β ∈ (2, 4).

Then, there exists {Qh ∈ SO(3), ch ∈ R3 }h→0 such that for the normalized deformations below, the following properties hold, up to a subsequence: yh (x,t) = (Qh )T (uh ◦ φ˜h )(z, ht) − ch ∈ H 1 (Ω 1 , R3 ); (i) {yh }h→0 converges in H 1 (Ω 1 , R3 ) to π, > 1/2 1 yh (z,t) − z dt converge in H 1 (ω, R3 ) (ii) displacements V h [yh ] = hβ /2−1 −1/2 to some ve3 , where v ∈ H 2 (ω, R) and: det ∇2 v = det ∇2 v0 , (iii) moreover: lim inf h→0

v

(7.50)

1 v0 J h (uh ) ≥ JlinK (v), where: hβ

0 JlinK (v) =

1 24

Z ω

Z
f v dz. Q2 ∇2 v − ∇2 v0 dz +

(7.51)

ω

For the proof, instead of reproducing the tedious calculations, we anticipate that a more general statement will be deduced in the context of prestressed elasticity in chapter 14 in Part III of this monograph. We now proceed to the general upper bound result, albeit in the restricted regime of the exponent β as in section 7.4:

202

7 Limiting theories for elastic plates: linearised bending

Theorem 7.14. Let ω ⊂ R2 be an open, bounded, simply connected and Lipschitz domain. Assume (7.44), (7.49) and let: β ∈ (3, 4). Then, for every v ∈ H 2 (ω, R) satisfying (7.50), there exists a family {uh ∈ H 1 ((Sh )h , R3 )}h→0 such that for {yh (z,t)  (uh ◦ φ˜h )(z, ht))h→0 , there holds: (i) {yh }h→0 converges in H 1 (Ω 1 , R3 ) to π(z,t) = z, (ii) {V h [yh ]}h→0 as in Theorem 7.13 (ii) converge in H 1 (ω, R3 ) to ve3 , 1 v0 (iii) we have: lim β J h (uh ) = JlinK (v). h→0 h For the same result to hold in the full range β ∈ (2, 4), as in the case of v0 ≡ 0 and Theorem 7.5, a possible proof, in analogy to the arguments in section 7.5, would rely on two results. The first one is the matching property for infinitesimal to exact isometries on shallow shells, fully executed in section 7.8 below. The second one is the density of C 2,α -regular scalar fields v in the set of the H 2 fields with a prescribed, strictly positive det ∇2 v = f ∈ C 0,α (ω, R). We will prove this latter property only in the specific case when f = det ∇2 v0 is constant, leading to: Theorem 7.15. In the context of Theorem 7.14, assume additionally that ω is star-shaped with respect to an interior ball. Assume moreover that: det ∇2 v0 ≡ c0 > 0

in ω.

Then, for every β ∈ (2, 4) and for every v ∈ H 2 (ω, R) satisfying (7.50), there exists a family of deformations {uh ∈ H 1 ((Sh )h , R3 )}h→0 such that the conclusions (i), (ii) and (iii) in Theorem 7.14 hold. The proof of Theorem 7.15 is postponed to section 7.10, after the matching, regularity and density properties have been established. We now give: Proof of Theorem 7.14 1. By Lemma 6.19, the Monge-Amp´ere constraint (7.50) can be rewritten as: 1 1 sym∇w = − ∇v ⊗ ∇v + ∇v0 ⊗ ∇v0 , 2 2 for some w ∈ H 2 (ω, R2 ). The Sobolev embedding theorem in the two-dimensional domain ω implies that ∇v ∈ W 1,q (ω, R2 ) for all q < ∞, so: sym∇w ∈ W 1,p (ω, R2×2 )

for all 1 ≤ p < 2.

7.7 Linearised Kirchhoff’s theory for shallow shells

203

Fix 1 < p < 2 such that: β − 2 > 2p and that W 1,p (ω) embeds in L4 (ω). This is possible sine β < 4 and so p can be chosen as close to 2 as we wish. By Korn’s inequality and a possible modification of w by an affine mapping, we can assume: w ∈ W 2,p ∩W 1,4 (ω, R2 ). Call λ = 1/p and observe that: λ
c0 > 0. Then, for nected domain and let v0 ∈ C 2,β (ω, 2,β ¯ every v ∈ C (ω, R) such that: det ∇2 v = det ∇2 v0

(7.58)

in ω,

¯ R3 )}ε→0 such that: there exists an equibounded family {wε ∈ C 2,β (ω, ∇(id 2 + εve3 +ε 2 wh )T ∇(id 2 + εve3 + ε 2 wh ) = ∇(id 2 + εv0 e3 )T ∇(id 2 + εv0 e3 )

for all ε  1.

(7.59)

206

7 Limiting theories for elastic plates: linearised bending

Theorem 7.16 asserts that if two convex out-of-plane displacements of first order have the same determinants of Hessians, then they can be matched by a family of equibounded higher order displacements to be isometrically equivalent. In the context of shallow shells, condition (7.59) means that each deformation of the surface Sε , given by z + εv0 (z)e3 7→ z + εv(z)e3 + ε 2 wε (z) is an isometry of Sε . Note that, writing wε = wε,tan + w3ε e3 where w3ε (z) = hwε (z), e3 i ∈ R, wε,tan (z) ∈ 2 R , equation (7.59) becomes: Id3 + ε 2 (2sym∇wε,tan + ∇v ⊗ ∇v) + 2ε 3 sym(∇v ⊗ ∇w3ε ) + ε 4 ((∇wε,tan )T ∇wε,tan + ∇w3ε ⊗ ∇w3ε ) = Id3 + ε 2 ∇v0 ⊗ ∇v0 . Recall that (7.58) is equivalent to curl curl(∇v ⊗ ∇v − ∇v0 ⊗ ∇v0 ) = 0, and further ¯ R2 ). Hence the constraint to: ∇v ⊗ ∇v − ∇v0 ⊗ ∇v0 = sym∇w for some w ∈ C 2,β (ω, (7.58) is necessary and sufficient for matching the lowest order terms above. A natural way for proving the complete matching is then by implicit function theorem. Indeed, this is how we proceed, and the ellipticity assumption det ∇2 v0 > 0 is a suf2,β ¯ R) → ficient condition for the invertibility of the implicit derivative L : C0 (ω, ¯ R), L (p) = −hcof∇2 v : ∇2 pi where p is the variation in w3ε . C 0,β (ω, We first need a lemma: ¯ R) be given on an open, bounded, Lipschitz doLemma 7.17. Let v0 , v1 ∈ C 2,α (ω, main ω ⊂ R2 and define the family of metrics:
¯ R2×2 gε = [gε,i j ]i, j=1,2  Id2 + ε 2 ∇v0 ⊗ ∇v0 − ∇v1 ⊗ ∇v1 ∈ C 1,α (ω, sym ). Then, the Gaussian curvature κ(gε ) of each gε with ε  1 is given by: κ(gε ) = ε

2



det(∇2 v0 − [Γi kj ∂k v0 ]i j )
2 1 − ε 2 (giεj ∂i v0 ∂ j v0 ) det gε
2  1 − ε 2 (giεj ∂i v0 ∂ j v0 ) 2 0,α ¯ det ∇ v (ω, R), −
2 1 ∈C 2 2 1 − ε |∇v1 |

(7.60)

where we use the Einstein summation convention and where the Christoffel symbols of gε , the inverse of gε , and its determinant are:
1 kl k Γε,i j  gε ∂ j gε,il + ∂i gε, jl − ∂ j gε,i j 2 ij g−1 ε = [gε ]i, j=1,2 =

for i, j, k = 1, 2,

1 cof gε , det gε

(7.61) (7.62)

det gε = 1 − ε 4 h(∇v0 )⊥ , ∇v1 i2 + ε 2 (|∇v0 |2 − |∇v1 |2 ). Proof. Firstly, if v0 , v1 are smooth, then by [Han and Hong, 2006, Lemma 2.1.2]:

7.8 Matching isometries on shallow shells


κ Id2 − ε 2 ∇v1 ⊗ ∇v1 = −ε 2
κ gε − ε 2 ∇v0 ⊗ ∇v0 =

207

det ∇2 v1 1 − ε 2 |∇v1 |2 1


2 ,


2 · 1 − ε 2 (giεj ∂i v0 ∂ j v0 ) # " k ∂ v ] ) ε 2 det(∇2 v0 − [Γε,i j k 0 ij . · κ(gε ) −
2 1 − ε 2 (giεj ∂i v0 ∂ j v0 ) det gε

Since the two metrics above are equal, (7.60) follows directly. The formula for det gε is obtained by using: det(A+B) = det A+hcof A : Bi+det B valid for all A, B ∈ R2×2 . In the general case when v0 , v1 are only C 2,α regular, one approximates by
smooth sequences {vn0 , vn1 }n→∞ . Then, each κn  κ Id2 + ε 2 (∇vn0 ⊗ vn0 − ∇vn1 ⊗ vn1 ) is given by the formula in (7.60), and {κn }n→∞ converges in C 0,α to the right hand side in (7.60). On the other hand, {κn }n→∞ converges in distributions to κ(gε ), which follows from κ = R1212 /detg. The proof is done. We are now ready to give: Proof of Theorem 7.16. 1. By a direct calculation, (7.59) is equivalent to: ∇(id 2 + ε 2 wε,tan )T ∇(id 2 + ε 2 wε,tan ) = Id2 + ε 2 ∇v0 ⊗ ∇v0 − ε 2 (∇v + ∇ζε ) ⊗ (∇v + ∇ζε ),

(7.63)

¯ R2 ) and ζε = εhwε , e3 i ∈ C 2,α (ω, ¯ R) so that wε = (wε,tan , ζεε ) where wε,tan ∈ C 2,α (ω, is the required correction in (7.59). Denote the metric in the right hand side of (7.63): gε (ζε ) = Id2 + ε 2 ∇v0 ⊗ ∇v0 − ε 2 (∇v + ∇ζε ) ⊗ (∇v + ∇ζε ).

(7.64)

Applying Lemma 7.17 to v1 = v+ζε , we find the formula for the Gaussian curvature of gε (ζε ) and observe that it vanishes: κ(gε (ζε )) = 0

(7.65)

if and only if: Φ(ε, ζε ) = 0, where Φ(ε, ζ )  1 − ε 2 |∇v + ∇ζ |2


2


k det ∇2 v0 − [Γε,i j ∂k v0 ]i j
4 − 1 − ε 2 (giεj ∂i v0 ∂ j v0 ) d(ε, ζ ) det(∇2 v + ∇2 ζ ).

(7.66)

and d(ε, ζ ) = 1 − ε 4 h(∇v0 )⊥ , ∇(v + ζ )i2 + ε 2 (|∇v0 |2 − |∇v + ∇ζ |2 ). k and gi j are given by (7.61) and (7.62) for g = Id + ε 2 ∇v ⊗ ∇v − Above, Γε,i ε 2 0 0 ε j 2 ε (∇v + ∇ζ ) ⊗ (∇v + ∇ζ ).

2. We now consider the operator in (7.66):

208

7 Limiting theories for elastic plates: linearised bending

¯ R) → C 0,α (ω, ¯ R) Φ : (−ε0 , ε0 ) × C02,α (ω, ¯ R) of Φ(ε, ζε ) = 0 with zero boundary data. It and seek for solutions ζε ∈ C02,α (ω, is elementary to check that Φ is continuously Fr´echet differentiable at (0, 0) and: Φ(0, 0) = det ∇2 v0 − det ∇2 v = 0. ¯ R) → C 0,α (ω, ¯ R) Moreover, the partial Fr´echet derivative L = ∂ Φ/∂ ζ (0, 0) : C02,α (ω, is a linear continuous operator of the form:
1 1 L (ζ ) = lim Φ(0,tζ ) = lim det ∇2 v0 − det(∇2 v + t∇2 ζ ) t→0 t t→0 t
1 = lim − t 2 det ∇2 ζ − thcof ∇2 v : ∇2 ζ i t→0 t ¯ R). = −hcof ∇2 v : ∇2 ζ i for all ζ ∈ C02,α (ω, Clearly, L is invertible to a continuous linear operator, because of the uniform ellipticity of the matrix field ∇2 v which follows from the convexity assumption of det ∇2 v = det ∇2 v0 being strictly positive. Thus, invoking the implicit function theorem, we obtain the solution operator: ¯ R) Z : (−ε0 , ε0 ) → C02,α (ω, such that ζε = Z (ε) satisfies (7.66). Moreover, Z is differentiable at ε = 0 and:   ∂Φ Z 0 (0) = L −1 ◦ (0, 0) = 0, ∂ε which follows from the identity:   E D ∂ k ∂Φ (0, 0) = cof ∇2 v0 : ( Γε,i )∂ v 0 j k ∂ε ∂ε i, j=1,2   k
∂ ∂ + det Γε,i j ∂k v0 i, j=1,2 − d(0, 0) det ∇2 v = 0. ∂ε ∂ε Consequently: 1 khwε , e3 ikC 2,α (ω) = kζε kC 2,α (ω) → 0 ε

as ε → 0.

(7.67)

3. We have so far obtained a uniformly bounded sequence of C02,α out-of-plane displacements ζε /ε such that the Gauss curvature (7.65) of the metric gε (ζε ) in the right hand side of (7.63) is 0. By the result in Mardare [2004], it follows that for each ε  1 there exists exactly one (up to fixed rotations) orientation preserving ¯ R2 ) of gε (ζε ): isometric immersion φε ∈ C 2 (ω, ∇φεT ∇φε = gε (ζε ) and

det ∇φε > 0.

(7.68)

7.8 Matching isometries on shallow shells

209

What remains to be proven is that, φε = id 2 + ε 2 wε,tan with some equibounded ¯ R2 )}ε→0 . {wε,tan ∈ C 2,α (ω, To this end, we recall a well known calculation (see, for example, in Mardare [2004]) that (7.68) implies (is actually equivalent to):   k (7.69) ∇2 φε − Γε,i j ∂k φε i, j=1,2 = 0, k are the Christoffel symbols (7.61) of the metric g (ζ ) in (7.64). By where Γε,i ε ε j (7.68) we get k∇φε kL∞ (ω) ≤ C, and k∇2 φε kL∞ (ω) ≤ C by (7.69) , hence kφε kC 2,α (ω) ≤ k are uniformly bounded (with respect to small ε) in C 0,α so by (7.69) C. But Γε,i j there follows k∇2 φε kC 0,α (ω) ≤ C, so that:

kφε kC 2,α (ω,R ¯ 2 ) ≤ C. k k 2 Note that kΓε,i j C 0,α (ω) ≤ Cε in view of the particular structure of gε (ζε ). Hence:

k∇2 φε kC 0,α (ω) ≤ Cε 2 ,

(7.70)

by (7.69). Consequently, for some Aε ∈ R2×2 we have: k∇φε − Aε kC 1,α (ω) ≤ Cε 2 .

(7.71)

We now argue that each matrix Aε as above, can be chosen to be a rotation and hence, without loss of generality Aε = Id2 . For each z ∈ ω there holds: dist(Ah , SO(3)) ≤ |Ah − ∇φε (z)| + dist(∇φε (z), SO(3)). (7.72) p ∇φεT (z)∇φε (z) = QDQT for some Q ∈ To evaluate the last term above, write: SO(3) and D = diag(λ1 , λ2 ) with λ1 , λ2 > 0. Since det ∇φε > 0, it follows by polar decomposition theorem that: q dist(∇φε (z), SO(3)) = | ∇φεT (z)∇φε (z) − Id2 | ≤ C|D − Id2 | = C max{|λi − 1|} ≤ C max{|λi2 − 1|} ≤ C|D2 − Id2 | i T

= C|Q

i T ∇φε (z)∇φε (z)Q − Id2 | ≤ C|∇φεT ∇φε (z) − Id2 | ≤ Cε 2 .

By the above and (7.72), (7.71) we see that dist(Ah , SO(3)) ≤ Cε 2 . Hence, without loss of generality, k∇φε − Id2 kC 1,α (ω) ≤ Cε 2 and: kφε − id 2 kC 2,α (ω) ≤ Cε 2 . Consequently, φε = id 2 + ε 2 wε,tan with kwε,tan kC 2,α (ω) ≤ C. This concludes the proof of Theorem 7.16, in view of (7.63) which is equivalent to (7.59).

210

7 Limiting theories for elastic plates: linearised bending

7.9 Convexity of weakly regular displacements The second main difficulty to overcome towards the proof of Theorem 7.15 is the absence of convexity assumptions on the H 2 solutions of the Monge-Amp`ere equation(7.50). Below, we combine some key observations on deformations with integrable dilatation in dimension 2, to deduce the interior regularity of solutions: Theorem 7.18. Let v ∈ H 2 (ω, R) be defined on an open, bounded, connected domain ω ⊂ R2 , and assume that: det ∇2 v = f in ω,

where f : ω → R,

f (z) ≥ c0 > 0

a.e. in ω. (7.73)

Then v ∈ C 1 (ω, R) and, modulo a global sign change, v is locally convex. The arguments towards the proof of a strengthened version of Theorem 7.18, will rely on properties of the following types of mappings: Definition 7.19. Let ω ⊂ R2 be open, bounded and connected. We say that: (i) θ ∈ C 0 (ω, R2 ) is connectedly locally one-to-one if it is locally one-to-one outside of a closed set S ⊂ ω of measure zero, for which ω \ S is connected, (ii) θ ∈ H 1 (ω, R2 ) which satisfies det ∇θ ≥ 0 a.e. in ω, has integrable dilatation if |∇θ (z)|2 ≤ K(z) det ∇θ (z)

for all z ∈ ω,

with some function K ∈ L1 (ω, R). In the remaining part of this section, we will prove: Theorem 7.20. Let v ∈ H 2 (ω, R) be defined on an open, bounded, connected domain ω ⊂ R2 and satisfy: det ∇2 v(z) > 0

for a.e. z ∈ ω.

(7.74)

Then v ∈ C 1 (ω, R). If additionally ∇v is connectedly locally one-to-one, then modulo a global sign change, v is locally convex in ω. In particular, when ω is convex then v is either convex or concave in the whole ω. Let us first quote the following strong result: ˇ ak’s decomposition theorem] Theorem. [Iwaniec-Sver´ 1 Assume that θ ∈ H (ω, R2 ) has integrable dilation, on an open, bounded and connected domain ω ⊂ R2 . Then, there exists a homeomorphism ζ ∈ H 1 (ω 0 , ω) and a holomorphic function ϕ ∈ H 1 (ω 0 , R2 = C) such that:

7.9 Convexity of weakly regular displacements

211

θ = ϕ ◦ ζ −1 . In particular, θ is either constant or connectedly locally one-to-one, and in the latter case the singular set S = ζ ((∇ϕ)−1 {0}) is at most countable and, in fact, finite on every subset compactly contained in ω. Combining the two above Theorems, one deduces a generalization of Theorem 7.18: Corollary 7.21. In the context of Theorem 7.20, let v ∈ H 2 (ω, R) satisfy (7.74) and let ∇v have integrable dilatation. Then v ∈ C 1 (ω, R) and, modulo a global sign change, v is locally convex. Towards the proof of Theorem 7.20, we now show a key result on the modulus of continuity of 2d deformations in H 1 with positive Jacobian: Lemma 7.22. Let θ ∈ H 1 (ω, R2 ) satisfy det ∇θ > 0 in an open, bounded, connected domain ω. Then θ is continuous in ω, and for any B(z, δ ) ⊂ B(z, r¯) ⊂ ω we have: oscB(z,δ ) θ ≤

√

 r¯ −1/2 k∇θ kL2 (B(z,¯r)) . 2π ln δ

(7.75)

Proof. By a result in Vodopyanov and Goldstein [1976], the field θ is continuous ˇ ak [1988]). In fact, a key ingredient of this result is to show that θ is a (see also Sver´ monotone map, i.e. for Bρ  B(z, ρ) there holds: oscBρ θ = osc∂ Bρ θ ,

(7.76)

and hence θ has the asserted modulus of continuity by [Morrey, 1996, Theorem 4.3.4]. We sketch the last part of the proof for the convenience of the reader. By Fubini’s theorem, θ ∈ H 1 (∂ Bρ , R2 ) for almost every ρ ∈ (δ , r¯). Hence the Morrey’s theorem of embedding of H 1 into C 0 for the one-dimensional set ∂ Bρ yields: oscBδ θ ≤ oscBρ θ = osc∂ Bρ θ ≤

p

2πρ

Z

|∇θ |2

1/2

for all ρ ∈ (δ , r¯).

∂ Bρ

To conclude, one squares both sides of the above inequality, divides by ρ and integrates from δ to r¯, in order to deduce (7.75). Two observations are now in order: Corollary 7.23. Assume that {θn ∈ H 1 (ω, R2 )}n→∞ is a bounded sequence with det ∇θn > 0 for all n, on an open, bounded and connected domain ω ⊂ R2 . Then, up to a subsequence, {θn }n→∞ converges locally uniformly and also weakly in H 1 to a continuous mapping θ ∈ H 1 (ω, R2 ) satisfying det ∇θ ≥ 0 a.e. in ω. Proof. The uniform convergence follows by Ascoli-Arzel´a’s theorem in view of Lemma 7.22. Noting that det ∇θ = −h∇θ1 , ∇⊥ θ2 i, the Div-Curl Lemma implies then that the desired inequality is satisfied for the limit mapping θ .

212

7 Limiting theories for elastic plates: linearised bending

Corollary 7.24. Let {vn ∈ H 2 (ω, R)}n→∞ be a bounded sequence on an open, bounded and connected domain ω ⊂ R2 . Assume that: det ∇2 vn ≥ c0 . Then, up to a subsequence, {vn }n→∞ converges weakly in H 2 (ω, R), and locally uniformly with its gradients, to v ∈ H 2 ∩ C 1 (ω, R2 ) with det ∇2 v ≥ c0 a.e. in ω. Proof. Define an auxiliary sequence of vector fields {θn ∈ H 1 (ω, R2 )}n→∞ in: θn (z) = ∇vn (z) + (|c0 | + 1)1/2 z⊥ . Since ∇2 vn is symmetric, we further obtain: det ∇θn = det ∇2 vn + |c0 | + 1 > 0. The convergence assertion follows by Corollary 7.23. Again, the Div-Curl Lemma applied to sequence ∇vn implies the desired inequality for the limit function v. Another consequence of Lemma 7.22 is the following assertion about H 2 functions whose Hessian determinants are uniformly controlled from below: Lemma 7.25. Let ω ⊂ R2 be an open, bounded and connected domain. Assume that v ∈ H 2 (ω, R) satisfies: det ∇2 v ≥ c0 . (7.77) Then v ∈ C 1 (ω, R). Moreover, if z0 ∈ ω is a Lebesgue point for ∇2 v, i.e.: ? ω(r)  |∇2 v − A|2 dz → 0 as r → 0+ ,

(7.78)

B(z0 ,r)

holds for some A ∈ R2×2 sym , then for all ε > 0 there exists r0 > 0 such that:

v(z) − v(a) − h∇v(a), z − ai − 1 hz − a · A(z − a)i 0 ≤ εr2 , C (Dr ) 2 (7.79) εr for r < r0 , a ∈ Dr = B(z0 , r). k∇v(z) − ∇v(a) − A(z − a)kC 0 (Dr ) ≤ 2 Proof. 1. Following Kirchheim [2001] we set θ = ∇v and we write φ (z) = θ (z) + (|c0 | + 1)1/2 z⊥ . Trivially φ ∈ H 1 (ω, R2 ) and, as before: det ∇φ = det ∇2 v + |c0 | + 1 > 0. Applying Lemma 7.22 to φ shows that θ is continuous and so v ∈ C 1 (ω, R). Below, we assume without loss of generality that z0 = 0, v(0) = 0 and θ (0) = 0 (otherwise it is sufficient to translate ω and to modify v by its tangent map at 0). For r sufficiently small and for all z ∈ B2  B(0, 2) we define:

7.9 Convexity of weakly regular displacements

1 θr (z)  θ (rz), r

213

φr (z)  θr (z) + (|c0 | + 1)1/2 z⊥ ,

so that, for all z ∈ B2 there holds: ∇θr (z) = ∇θ (rz) = ∇2 v(rz),

∇φr (z) = ∇2 v(rz) + (|c0 | + 1)1/2



 0 −1 , 1 0

det ∇φr (z) = det ∇θr (z) + (|c0 | + 1) > 0. As φr ∈ H 1 (B2 , R2 ), we apply (7.75) to z ∈ B1 and δ < r¯ = 1, to get for r < r0  1: √

Z 1/2 1
−1/2  |∇φr |2 δ B(z,1) Z  √

1/2  1 −1/2 2(|m| + 1)1/2 |B1 | + |∇2 v(ry)|2 dy ≤ 2π ln δ B(z,1)  1
−1/2  1 1
−1/2 ≤ C ln (|m| + 1)1/2 + k∇2 vkB(0,2r) ≤ C ln , δ r δ

oscB(z,δ ) φr ≤

2π ln

where the constant C depends only on m and |A|. Above, we also used that B(z, 1) ⊂ B2 and that 0 is a Lebesgue point for ∇2 v. Given ε > 0 we choose δ > 0 such that: ln

1
−1/2 < ε/C. δ

Consequently, it follows that: |z − y| < δ ⇒ |φr (z) − φr (y)| < ε

for all z, y ∈ D1 and r < r0 .

2. Since θr − φr is a given linear deformation, we conclude that the family: F = {θr : D1 → R2 ; r < r0 } is equicontinuous. On the other hand: Z D1

Let θ˜r  θr −

> θ B1 r

∇θr − A 2 = πω(r) → 0

as r → 0.

and apply the Poincar´e inequality to obtain that: θ˜r → Az

in H 1 (ω, R2 )

as r → 0.

Now, equicontinuity of functions in F and the fact that θr (0) = 0 yield, by Arzel`aAscoli’s theorem, that a subsequence of {θr }r→0 (which we do not relabel) converges uniformly to a continuous function V on D1 . Since θr − θ˜ is constant, we deduce that V (z) − Az = c is constant too. But then, evaluating at 0 gives c = 0. Hence, θr uniformly converges to Az on D1 . Finally, let us fix ε > 0 and choose r0 so that:

214

7 Limiting theories for elastic plates: linearised bending

kθr (z) − AzkC 0 (D1 ) ≤

ε 4

for all r < r0 .

This implies: k∇v(z) − AzkC 0 (Dr ) ≤ εr 4 . Fixing a ∈ Dr we obtain the second estimate in (7.79). Since diam Dr = 2r, the first estimate follows as well. The final ingredients that we need are: two elementary observations and a separate result on locally supporting hyperplanes: ¯ R) on an open, bounded, connected domain ω ⊂ R2 . Lemma 7.26. Let v ∈ C 1 (ω, Fix a ∈ ω, where we suppose that: v(z) ≥ v(a) + h∇v(a), z − ai

for all z ∈ ∂ ω,

(7.80)

and that ∇v(z) , ∇v(a) for all z ∈ ω \ {a}. Then v has a supporting hyperplane at a, that is (7.80) holds for all z ∈ ω. In particular, if ω is convex then: (Cv)(a) = v(a) ¯ where Cv denotes the convexification of the function v over ω: (Cv)(a) = sup {T (a); T : ω → R is affine and T (z) ≤ v(z)

¯ . for all z ∈ ω}

Proof. Consider the tangent map T (z) = v(a) + h∇v(a), z − ai. We now claim that ¯ For otherwise, the continuous function f (z) = v(z) − T (z) T (z) ≤ v(z) for all z ∈ ω. would assume a negative minimum on ω¯ at some c ∈ ω \ {a}. Hence ∇ f (c) = 0, which is a contradiction with the second assumption as ∇v(c) = ∇T (c) = ∇v(a). Lemma 7.27. Let I ⊂ R be an open and bounded interval. Assume that φ ∈ W 2,1 (I, R) is locally convex on I \ S, where S is a set of measure 0. Then φ is convex on I. Proof. Since φ ∈ C 1 (I) and φ is locally convex on a full measure open subset of I, then φ 00 ≥ 0 a.e. in I. This immediately implies that φ 0 is increasing in I, hence φ is globally convex. Lemma. [Ball’s supporting hyperplane lemma] Let Ω ⊂ RN be open and convex, and let v ∈ C 1 (Ω , R). The necessary and sufficient condition for v to be strictly convex on Ω is: (i) ∇v is locally one-to-one, and (ii) there exists a ∈ Ω and a locally supporting hyperplane for v at a: v(z) ≥ v(a) + h∇v(a), z − ai

for all ρ > 0, x ∈ B(a, ρ).

We are now ready to prove the main result of this section: Proof of Theorem 7.20. 1. The regularity v ∈ C 1 (ω, R) is an immediate consequence of Lemma 7.22.

7.9 Convexity of weakly regular displacements

215

Recall the properties of the singular set S from Definition 7.19. Since det ∇2 v > 0 a.e. in ω, modulo a global change of sign for v we can choose z0 ∈ ω \ S a Lebesgue point of ∇2 v as in (7.78), such that the matrix A is positive definite: hξ , Aξ i ≥ λ |ξ |2

for all ξ ∈ R2 ,

(7.81)

with some λ > 0. By Lemma 7.25 for all r < r0 the estimate (7.79) holds true with ε = λ /4, and without loss of generality ∇v is also one-to-one on Dr = B(z0 , r) ⊂ ω \ S. In view of Lemma 7.26, it thus follows that the function v admits a supporting hyperplane at z0 on Dr , because (7.79) implies, for all z ∈ ∂ Dr : 1 λ v(z) − v(z0 ) − h∇v(z0 ), z − z0 i ≥ hz − z0 , A(z − z0 )i − εr2 ≥ r2 > 0. 2 4 2. The next claim is that v is locally convex in ω \ S. Since ω \ S is open and connected, it is also path-wise connected. Therefore, for a fixed z ∈ ω \ S there exists a continuous path within ω \ S connecting z and z0 , which can be covered with a finite chain of open balls {Bi ⊂ ω \ S}ni=1 , such that: z0 ∈ B1 ,

Bi ∩ Bi+1 , 0/ for all i = 1 . . . n − 1,

z ∈ Bn .

Applying Ball’s supporting hyperplane lemma consecutively to each ball Bi , we deduce that v is strictly convex on Bn , hence it is locally convex at z. To finish the proof, we fix a direction in R2 and consider the family of straight lines parallel to that direction. For almost every such line L, the 1-dimensional Lebesgue measure of L ∩ S is zero and v ∈ H 2 (L ∩ ω, R). Also, v is locally convex on (L ∩ ω) \ S in view of the previously proven claim. Thus, by Lemma 7.27 we get that v is convex on each connected component of L ∩ ω. Now, by continuity of v, this property must hold for all lines L, by approaching any given line with a selected sequence of ’good’ lines and passing to the limit in the convexity inequality. This implies that actually v is convex on any convex subset of ω, which ends the proof. For completeness, we also note another corollary of Lemma 7.25 and Lemma 7.26: Lemma 7.28. Let v ∈ H 2 (ω, R) satisfy (7.77). Assume that z0 is a Lebesgue point for ∇2 v with A in (7.78) being positive definite. Assume that ∇v is one-to-one in a neighborhood of z0 . Then v is locally convex at z0 . Proof. Let λ > 0 satisfy (7.81). By Lemma 7.25 for all r < r0 and all a ∈ Dr = B(x0 , r), the estimate (7.79) holds true with ε = λ /4. Without loss of generality, ∇v is one-to one on Dr and, by (7.79) we get: 1 λ v(z) − v(a) − h∇v(a), z − ai ≥ hz − a, A(z − a)i − εr2 ≥ r2 > 0. 2 4 for all a ∈ B(z0 , r/2) and z ∈ ∂ B(z0 , r/2). The assumptions of Lemma 7.26 are satisfied and hence v(a) = (Cv)(a) for all z ∈ B(z0 , r/2). The claim is proved.

216

7 Limiting theories for elastic plates: linearised bending

We point out that in proving Theorem 7.18 we only used the conclusion of the ˇ ak’s decomposition result, claiming that θ is locally one-to-one on a Iwaniec-Sver´ connected set of full measure. Therefore, the assumptions of Theorem 7.18 could potentially be relaxed (as in Theorem 7.20), but not to det ∇2 v > 0 a.e. Indeed, let θ = ∇u be as in Example 7.29 below. Then θ is not of integrable dilatation because: 2 |∇θ |2 , (z1 , z2 ) ≥ 2 det ∇θ z1 (1 − z22 ) and also the singular set S = {(0, z2 ); z2 ∈ (−1, 1)} coincides with the vanishing set of θ where θ is obviously not locally one-to-one. On the other hand, Theorem 7.20 can be also applied to the cases where det ∇2 v ∈ C 0 (ω, R) is positive a.e. and ω \ (det ∇2 v)−1 (0) is connected. We close this section by noting that the assumption of strict positivity in Theorem 7.18 cannot be relaxed to det ∇2 v > 0 a.e. in ω, even when v is W 2,∞ regular: Example 7.29. Let B1 be the unit disk in R2 and let v ∈ C 1 (B1 , R) be given by: ( 2 z21 ez2 /2 if z1 ≥ 0 v(z1 , z2 ) = 2 −z21 ez2 /2 otherwise. Note that v(0, z2 ) = 0 and ∇v(0, z2 ) = 0 for all z2 ∈ (−1, 1). Indeed, there holds, respectively for z1 > 0 (“plus” sign below) and z1 < 0 (“minus” sign): 2

∂1 v = ±2z1 ez2 /2 , 2

∂11 v = ±2ez2 /2 ,

2

∂2 v = ±z2 z21 ez2 /2 , 2

∂12 v = ∂21 v = ±2z1 z2 ez2 /2 ,

2

2

∂22 v = ±(z21 ez2 /2 + z21 z22 ez2 /2 ),

2

∆ v = ±ez2 /2 (2 + z21 + z21 z22 ). As a consequence, v ∈ W 2,∞ (B1 , R) is strictly convex on {(z1 , z2 ) ∈ B1 ; z1 > 0} and strictly concave on {(z1 , z2 ) ∈ B1 ; z1 < 0}. On the other hand: 2

det ∇2 v = 2z21 ez2 (1 − z22 ) ∈ C ∞ (B1 , R) is positive if z1 , 0 and z22 < 1. Also, v < C 2 (B1 , R) although it solves the MongeAmp`ere equation with smooth non-negative right hand side, a.e. in its domain.

7.10 Density result and recovery family on shallow shells In this section, we will prove Theorem 7.15. As in the case of plates and the recovery family construction discussed in section 7.5, in addition to the matching property in section 7.8, our final ingredient will be the density statement:

7.10 Density result and recovery family on shallow shells

217

Theorem 7.30. Let ω ⊂ R2 be an open, bounded domain that is star-shaped with respect to an interior ball. For a fixed constant c0 > 0, define:  Ac0  v ∈ H 2 (ω, R); det ∇2 v = c0 a.e. in ω . ¯ R) is dense in Ac0 with respect to the H 2 norm. Then Ac0 ∩ C ∞ (ω, Towards the proof, we note a consequence of Theorem 7.18 and of the monotonicity property by Vodopyanov and Goldstein used in Lemma 7.22: Theorem 7.31. Let f ∈ C k,β (ω, R) be a positive function on an open, bounded, k+2,β connected ω ⊂ R2 . Then any v ∈ H 2 (ω, R) satisfying det ∇2 v = f , is Cloc regular. Proof. 1. We first observe that v is a generalized Aleksandrov solution to (7.73). By Theorem 7.18, it is hence locally convex, so twice differentiable in the classical sense a.e. in ω, and its gradient agrees with f . By [Trudinger and Wang, 2008, Lemma 2.3], the regular part of the Monge-Amp`ere measure µu equals (det ∇2 u) dz = f dz. It suffices now to show that there is no singular part of µu , i.e. that µu is absolutely continuous with respect to the Lebesgue measure dz. By Theorem 7.18 we have ∇v ∈ C 1 (ω, R2 ) and hence: µu (U) = |∇v(U)|

for every Borel subset U ⊂ ω.

We thus need to show that ∇u satisfies Luzin’s condition (N): |∇v(U)| = 0

for all U ⊂ ω with |U| = 0.

The above claim follows directly from [Mal´y and Martio, 1995, Theorem A], in view of ∇v ∈ H 1 (ω, R2 ) and the monotonicity property (7.76) of ∇v. 2. Since f ∈ C 0,α (ω, R), then [Quang, 2009, Theorem 5.4] implies that v is locally C 2,α -regular. This statement is in fact the well-known result due to Caffarelli [1990]. Fix z0 ∈ ω. By [Trudinger and Wang, 2008, Remark 3.2] which gives an elementary proof of a result by Aleksandrov and Heinz, the displacement v as above must be strictly convex in some B(z0 , ε). By adding an affine function to v, we may without loss of generality assume that v = 0 on the boundary of the convex set: ω0 = {z ∈ ω; v(z) ≤ z(z0 ) + δ } ⊂ B(z0 , ε), for a sufficiently small δ > 0. Therefore, the statement in [Quang, 2009, Theorem 5.4] can be directly applied. Once the C 2,β regularity is established, the C k+2,β regularity follows as in [Caffarelli and Cabr´e, 1995, Proposition 9.1]. The following argument is quite elementary: Proof of Theorem 7.30. Without loss of generality we assume that ω is starshaped with respect to:

218

7 Limiting theories for elastic plates: linearised bending

B  B(0, r) ⊂ ω. Let v ∈ Ac0 and define vλ (z) 

1 v(λ z) λ2

det ∇2 uλ (x) = c0 , k∇vλ kL2 (ω) = λ −2 kvkL2 (λ ω) ,

for all 0 < λ < 1. Then: kvλ kL2 (ω) = λ −3 kvkL2 (λ ω) , k∇2 vλ kL2 (ω) = λ −1 kvkL2 (λ ω) .

As a consequence, vλ ∈ Ac0 for all λ ∈ (0, 1), and vλ → v strongly in H 2 (ω, R) as λ → 1− . So far, we only the fact that ω is starshaped with respect to the origin. Now, since ω is star-shaped with respect to B, we have λ ω¯ ⊂ ω for all 0 < λ < 1. ¯ R) ∩ Ac0 , proving the claim. Hence, by Theorem 7.31, the functions vλ ∈ C ∞ (ω, We are finally ready to deduce: Proof of Theorem 7.15. ¯ R) satisfy1. It is enough to prove the three indicated assertions for v ∈ C 2,β (ω, ing (7.50). The general case v ∈ H 2 (ω, R) follows through a diagonal argument as in the proof of Theorem 7.5, in view of the density property in Theorem 7.30. ¯ R3 )}h→0 By Theorem 7.16, there exists an equibounded family {wh ∈ C 2,β (ω, such that the deformations below are isometrically equivalent to id 2 + hβ /2−1 v0 e3 : (∇ξh )T ∇ξh = ∇(id2 + hβ /2−1 v0 e3 )T ∇(id2 + hβ /2−1 v0 e3 ), ξh (z)  z + hβ /2−1 v(z)e3 + hβ −1 wh (z),

h  1.

(7.82)

We define now the recovery family {uh ∈ C 1,β ((Sh )h , R3 )}h→0 by setting: 2

t uh ◦ φ˜h (z,t)) = u˜h (z,t) = ξh (z) + tNh (z) + hβ /2−1 d h (z), 2 where Nh is the unit normal vector to the image surface ξh (ω): Nh =


∂1 ξh × ∂2 ξh = − hβ /2−1 (∇v)∗ + e3 + O(hβ −2 ), |∂1 ξh × ∂2 ξh |

¯ R3 )}h→0 , approximating the effective while the “warping” fields {d h ∈ C 1,β (ω, ¯ R3 ), are defined so that: warping d ∈ C 0,α (ω, hβ /2−1 kd h kC 1,β (ω) → 0
d h → d  c(∇2 v0 − ∇2 v

as h → 0, in L∞ (ω, R3 ).

(7.83)

2. Because of the first condition in (7.83), the statements in Theorem 7.15 (i), (ii) easily follow. In order to establish the energy limit in (iii), we note that: Z  q 1 E h (uh ) = (∇φ˜h )−1,T (∇u¯h )T (∇u¯h )(∇φ˜h )−1 (det ∇φ˜h ) dx. (7.84) W h Ωh

7.10 Density result and recovery family on shallow shells

219

and compute all the entries of the symmetric matrix fields: K h  (∇φ˜h )−1,T (∇u¯h )T (∇u¯h )(∇φ˜h )−1 . We will compute these entries up to terms of order o(hβ /2 ). For M h  (∇u¯h )T ∇u¯h we obtain, in view of (7.82):
h = ∇(id 2 + hβ /2−1 v0 e3 )T ∇(id 2 + hβ /2−1 v0 e3 ) + 2tsym (∇ξh )T ∇Nh Mtan = Id2 + hβ /2−2 ∇v0 ⊗ ∇v0 − 2thβ /2−1 ∇2 v + o(hβ /2 ), h h h (M13 , M23 ) = thβ /2−1 dtan + o(hβ /2 ), h M33 = |Nh + thβ /2−1 d h |2 = 1 + 2thβ /2−1 hd h , e3 i + o(hβ /2 ).

By a further direct calculation, we get:
∇φ˜h tan = Id2 − x3 hα ∇2 v0 + o(hβ /2 ),

∇φ˜h e3 = nh . (∇φ˜h )31 , (∇φ˜h )32 = hβ /2−1 ∇v0 + o(hβ /2 ), and the related inverse matrix has the following structure:
(∇φ˜h )−1 tan = A + o(hβ /2 ),
T
(∇φ˜h )−1 e3 = nh , (∇φ˜h )−1 e3 tan = hβ /2−1 A∇v0 + o(hβ /2 ),
−1 where A = Id2 + hβ −2 ∇v0 ⊗ ∇v0 − thβ /2−1 ∇2 v0 .

(7.85)

3. We now compute:
h h (∇φ˜h )−1,T M h tan = Id3 + thβ /2−1 A(∇2 v0 − ∇2 v) + thβ /2−1 ntan ⊗ dtan + o(hβ /2 ) = Id2 + thβ /2−1 (∇2 v0 − ∇2 v) + o(hβ /2 ),
T
(∇φ˜h )−1,T M h e3 tan = hβ /2−1 (Id2 + hβ −2 ∇v0 ⊗ ∇v0 )A∇v0 h h + thβ /2−1 dtan + o(hβ /2 ) = hβ /2−1 ∇v0 + thβ /2−1 dtan + o(hβ /2 ),
h h (∇φ˜h )−1,T M h e3 tan = thβ /2−1 dtan + ntan + o(hβ /2 ),
(∇φ˜h )−1,T M h 33 = nh3 + 2thβ /2−1 hd h , e3 i + o(hβ /2 ),


because A Id2 + hβ −2 ∇v0 ⊗ ∇v0 − 2thβ /2−1 ∇2 v = Id2 + thβ /2−1 A(∇2 v0 − ∇2 v),
and likewise we have: hβ /2−1 Id2 +hβ −2 ∇v0 ⊗∇v0 A = hβ /2−1 Id2 +o(hβ /2 ). Hence: h h h + o(hβ /2 ) ⊗ ntan = A + thβ /2−1 (∇2 v0 − ∇2 v) + ntan Ktan   h2β −4 |∇v0 |2 ⊗2 h ⊗2 (∇v ) + o(hβ /2 ) ) + = A Id2 + thβ /2−1 (∇2 v0 − ∇2 v) + (ntan 0 1 + hβ −2 |∇v0 |2

= Id2 + 2thβ /2−1 (∇2 v0 − ∇2 v) + o(hβ /2 ),

220

7 Limiting theories for elastic plates: linearised bending

and in a similar manner: h h + o(hβ /2 ) + hnh , e3 intan (K h e3 )tan = hβ /2−1 A∇v0 + thβ /2−1 dtan h = hβ /2−1 A∇v0 + thβ /2−1 dtan −

hβ /2−1 ∇v0 + o(hβ /2 ) 1 + hβ −2 |∇v0 |2

h K33 = hβ −2 hA∇v0 , ∇v0 i + 2thβ /2−1 hd h , e3 i + hnh , e3 i2 + o(hβ /2 )

= 1 + hβ −2 hA∇v0 , ∇v0 i + 2thβ /2−1 hd h , e3 i − β /2−1

Observing that hβ /2 A∇v0 − 1+hhβ −2 |∇v

2 0|

h (K h e3 )tan = thβ /2 dtan + o(hβ /2 ),

hβ −2 |∇v0 |2 + o(hβ /2 ). 1 + hβ −2 |∇v0 |2

∇v0 = o(hβ /2 ), it follows that: h K33 = 1 + 2thβ /2 hd h , e3 i + o(hβ /2 ),

which finally yields:
K h = Id3 + 2thβ /2 (∇2 v0 − ∇2 v)∗ + sym(d h ⊗ e3 ) + o(hβ /2 ). 4. Taylor expanding W at Id3 and using (7.83) we now see that: √

W ( K h ) = W Id3 + thβ /2 (∇2 v0 − ∇2 v)∗ + sym(d h ⊗ e3 ) + o(hβ /2 )   1 = t 2 hβ −2 Q3 (∇2 v0 − ∇2 v)∗ + sym(d h ⊗ e3 ) + o(hβ ). 2 Since ∇φ˜h = 1 + O(hβ /2 ), by (7.84) and the convergence in (7.83), we get: Z √ 1 1 h h K h )(1 + O(hβ /2 )) dx E (u ) = lim W ( h→0 hβ +1 Ω h h→0 hβ Z
1 = Q2 ∇2 v0 − ∇2 v dz. 24 ω

lim

Convergence of the linear term in h1β J h follows as in the last step of the proof of Theorem 7.14. The proof of Theorem 7.15 is complete.

7.11 Bibliographical notes The linearised Kirchhoff theory for plates has been proposed and derived by means of Γ -convergence in Friesecke et al. [2006]. Theorems 7.4, 7.5 as well as the matching property in section 7.3, the construction of the recovery family in section 7.4, and usage of the density argument in section 7.5 are taken from there. The Brezis-Wainger’s inequality used in section 7.4 is due to Brezis and Wainger [1980]. The M¨uller-Pakzad approximation result in section 7.5 appeared in M¨uller and Pakzad [2005]. We remark that developability of infinitesimal isometries un-

7.11 Bibliographical notes

221

der C 2 regularity is a classical result which follows (together with a parallel result for exact isometries) from the more general discussion by Hartman and Nirenberg [1959]. An extension to C 1 isometric immersions for which the image of Gauss’s map has measure zero, can be found in [Pogorelov, 1956, Chapter 2] and [Pogorelov, 1973, Chapter 9]. A short proof under the H 2 regularity hypothesis was given by Pakzad [2004], using the ideas of Kirchheim [2001]. The fact that an H 2 infinitesimal isometry on a Lipschitz domain is C 1 , was shown in M¨uller and Pakzad ¯ [2005], together with the stronger statement that when ω is C 1,α then v ∈ C 1 (ω). However, |∇v| may indeed be unbounded for Lipschitz ω. Results on shallow shells in sections 7.6 - 7.10 are taken from Lewicka et al. [2017a]. The proof of Theorem 7.16 is somewhat similar to [Han and Hong, 2006, Theorem 4.1.1]. The proof of Theorem 7.20 is essentially a combination of arguˇ ak’s unpublished paper Sver´ ˇ ak [1991]. There, it has been shown that ments in Sver´ if v ∈ H 2 (ω, R) satisfies det ∇2 v > 0 a.e. in ω, then there exists a closed set S ⊂ ω of measure zero such that on each component of ω \ S, the function v is either locally convex or locally concave. The main step in the proof is to show that any such map ˇ ak has achieved by is locally one-to-one outside a set of measure zero, which Sver´ using consequences of a version of Lemma 7.25 and the classical degree theory. ˇ ak decomposition result in section 7.9 can be found in Iwaniec The Iwaniec-Sver´ ˇ ak [1993]. Lemma 7.22 is due to Vodopyanov and Goldstein [1976]. and Sver´ ˇ ak, 1991, Lemma 2]. The Ball’s supporting hyperplane lemma Lemma 7.26 is [Sver´ is taken from [Ball, 1980, Theorem 1].

Chapter 8

Infinite hierarchy of elastic shell models

In this short chapter, we provide a heuristic derivation of the infinite hierarchy of the dimensionally reduced thin shell models, each valid in a corresponding range of their elastic energy scaling exponent. The hierarchy contains all theories that are discussed (and rigorously derived) in this monograph. The derivation is based on a formal asymptotic expansion; it predicts the form of the two-dimensional Γ -limit in the vanishing shell’s thickness, and identifies the space of the limiting deformations as infinitesimal isometries of a given integer order n ≥ 1 determined by the elastic energy. Hence, the influence of shell’s geometry on its qualitative response to external forces, i.e. the shell’s rigidity, is reflected in a hierarchy of functional spaces of isometric deformations arising as constraints of the derived theories. This derivation is carried out in section 8.1. We observe that in certain cases, a given m-th order infinitesimal isometry can be modified by higher order corrections to yield an infinitesimal isometry of order n > m, a property to which we refer to by matching property of infinitesimal isometries. This feature is conjectured to cause the theories corresponding to orders of isometries between m and n to “collapse” all into one and the same theory. We already observed such behavior for plates in chapter 7, and we will analyze further examples in this vein: elliptic shells and developable shells with no flat parts, in the following chapters. In section 8.2 we start with a simple example of the matching property valid for surfaces of revolution, which fall into a larger class of approximately robust surfaces. We then state the recovery family construction, together with the Γ -limit and the convergence of minimizers results for rotationally invariant shells and any scaling exponent β ≥ 4 leading to the linear elasticity model introduced in chapter 6.

8.1 Heuristics on isometry matching and collapse of theories In this section, based on formal calculations, we conjecture an infinite hierarchy of elastic shell models, dimensionally reduced in the energy regimes with β > 2. © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_8

223

224

8 Infinite hierarchy of elastic shell models

1. Given a family of deformations {uh : S → R3 }h→0 of the midsurface S ⊂ R3 , consider their extensions {uh : Sh → R3 }h→0 (which we do not relabel) to deformations of thin shells around S, consistent with the Kirchhoff-Love assumption: uh (z + tn(z)) = uh (z) + tNh (z)

h for all z ∈ S, |t| < . 2

Above, Nh denotes the unit normal to the image surface uh (S). Using Lemma 5.5, calculations following the outline in the proof of Theorem 5.16 yield: E h (uh ) '

1 2

Z ? h0 /2 S

−h0 /2

Q3

1 h
K dt dσ (z), 2



where: K h (z + tn) = (∇uh )T ∇uh z + t hh0 n(z) − Id3 . Exchanging Q3 (K h ) by h ) and calculating as in (5.45) while keeping only the leading order terms: Q2 (z, Ktan
K h (z+tn(z))tan ' (∇uh )T ∇uh (z) − Id2

 h + 2t (∇uh )T ∇Nh − Π − sym (∇uh )T ∇uh − Id2 Π (z), h0 we thus get: Z ? h0 /2

1 8 

E h (uh ) ' 1 ' 8 +

Z S h2

S

−h0 /2


Q2 z, K h dt dσ (z)

 Q2 z, (∇uh )T ∇uh − Id2 dσ (z)

24

Z S

(8.1)



 Q2 z, (∇uh )T ∇Nh − Π − sym (∇uh )T ∇uh − Id2 Π dσ (z).

The two terms above correspond to the stretching and bending energies, and the factor h2 points to the fact that a shell undergoes bending more easily than stretching. Another observation is that for minimizers uh , the energy should be distributed equipartedly between stretching and bending. Thus the stretching-like contribution in the integrand of the second integral in (8.1) is negligible, and we may write: E h (uh ) '

Z S

|δ gS |2 dσ + h2

Z S

|δ ΠS |2 dσ .

(8.2)

Here | · | denotes the appropriate norm of the displayed quantity and we omit constants. By δ gS = (∇uh )T ∇uh − Id2 and δ ΠS = (∇uh )∇Nh − Π we denote, respectively, the change in the metric (first fundamental form) and in the shape operator (second fundamental form), between the surface uh (S) and the reference midsurface S. For a plate, (8.2) is known in the literature as the F¨oppl-von K´arm´an energy. 2. When E h (uh ) ' h2 , Rthen equating the order of both terms Rin (8.2) we obtain in the limit of h → 0 that S |δ gS |2 dσ ' 0 and that h12 E h (uh ) ' S |δ ΠS |2 dσ . This indeed corresponds to the Kirchhoff model, as in chapter 5, where the dimension-

8.1 Heuristics on isometry matching and collapse of theories

225

ally reduced energy IK is given by the bending term (measuring the change in the second fundamental form) under the constraint of zero stretching: δ gS = 0.

scaling exponent β β =2 Kirchhoff 2 0 depending only on S and p, such that: k(T B)tan kW 2,p (S) + khT B, nikW 1,p (S) ≤ CkBkW 1,p (S) .

9.2 Matching infinitesimal to exact isometries on elliptic surfaces

241

Indeed, extend coefficients of all the equations to the domain ωε as in the proof of Theorem 9.3, and construct the linear solution operator T associated with this larger domain, satisfying the bound (9.7) there. We now notice that the system of first order equations: sym∇v + αΠ = B for the unknowns v, α, is elliptic in the sense of Agmon, Douglis and Nirenberg and apply the local estimates.

9.2 Matching infinitesimal to exact isometries on elliptic surfaces In this section, we prove the following matching property on convex surfaces: Theorem 9.7. Assume that S ⊂ R3 is a compact, elliptic surface that is ¯ there exists smoothly homeomorphic to a disk. Given V ∈ V (S) ∩ C 2,α (S), ¯ R3 )}ε>0 , such that each map: an equibounded family {wε ∈ C 2,α (S, uε = id S + εV + ε 2 wε

ε 1

is an (exact) isometry i.e. (∇uε )T ∇uε = Id2 on S. The proof of Theorem 9.7 is based on a fixed point argument and it is inspired by the proof of openness of the set of positive curvature metrics g on the topological sphere Σ which are pull-backs of the Euclidean metric by immersions of Σ in R3 . Given an immersion r of the topological disk with pull-back metric g, we seek a new immersion with the same pull-back metric whose first order difference from r is a given infinitesimal isometry. We carry out the proof under a more restrictive assumption that for some α > 0, the surface S and its boundary ∂ S are of class C 3,α . First, we prove a H¨older-type estimate for operator T defined in section 9.1. Lemma 9.1. In the context of Theorem 9.2, one has the following uniform estimate:




T sym((∇φ )T ∇ψ) 2,α ¯ ≤ Ckφ k 2,α ¯ kψk 2,α ¯ , (9.24) C (S) C (S) C (S) ¯ R3 ). valid for all φ , ψ ∈ C 2,α (S, Proof. 1. We denote B = sym((∇φ )T ∇ψ) and call w = T (B) and γ = S B). Similarly as in Remark 9.6, by the elliptic interior estimates we obtain:
(9.25) kwkC 1,α (S) ¯ , ¯ ≤ C kBkC 1,α (S) ¯ + kwkL∞ (S) ≤ CkBkC 1,α (S) where we used Remark 9.6 for deducing: kwkL∞ ≤ kwkW 1,p ≤ CkBkW 1,p ≤ kBkC 1,α , for any p > 2. In the same manner, the Schauder estimates for elliptic systems in divergence form, together with Remark 9.4 also yield:

242

9 Limiting theories on elastic elliptic shells


kγkC 1,α (ω) ¯ ≤ C kBkC 1,α (ω) ¯ + kγkL∞ (ω) ≤ CkBkC 1,α (ω) ¯ .

(9.26)

Hence, by (9.19) and (9.25) we obtain:
kwkC 2,α (S) ¯ ≤ C kBkC 1,α (ω) ¯ + ku1 kC 1,α (ω) ¯ + ku2 kC 1,α (ω) ¯ .

(9.27)

Clearly, kBkC 1,α (ω) ¯ is bounded by the right hand side of (9.24). It remains therefore to derive suitable bounds on the correction coefficients u1 and u2 . 2. Directly from (9.20) and (9.21), in view of (9.26) we get:
≤ CkBkC 1,α (ω) k(u1 , u2 )kC 0,α (ω) ¯ . ¯ ≤ C kγkC 1,α (ω) ¯ + kBkC 1,α (ω) ¯

(9.28)

By (9.20) we check that u1 and u2 solve the following first order elliptic system: p 1 ∂1 u2 − ∂2 u1 = |g|Hγ + gi j (h j2 B1i − h j1 B2i ) 2  1  det Π ij ∂i p h u j = (∂2 c1 − ∂1 c2 ), 2 |g| where as usual we use Einstein’s summation convention. As in Remark 9.6, there hold the following interior estimates: k(u1 , u2 )kC 1,α (ω) ¯ ≤ C k(u1 , u2 )kL∞ (ω) + kγkC 0,α (ω) ¯ + kBkC 0,α (ω) ¯ + k∂2 c1 − ∂1 c2 kC 0,α (ω) ¯




(9.29)

≤ C BkC 1,α (ω) ¯ + k∂2 c1 − ∂1 c2 kC 0,α (ω) ¯ , the last bound being a consequence of (9.26) and (9.28). Also, from (9.21) we get: k∂2 c1 − ∂1 c2 kC 0,α (ω) ¯ ≤ CkBkC 1,α (ω) ¯ +Ck∂11 B22 + ∂22 B11 − 2∂12 B22 kC 0,α (ω) ¯ .

(9.30)

¯ whereas: Recall that we identify φ , ψ with φ ◦ r and φ ◦ r defined on ω,
1 h∂i r, (∇φ )T ∇ψ∂ j ri + h∂i r, (∇ψ)T ∇φ ∂ j ri 2
1 = (h∂i φ , ∂ j ψi + h∂ j φ , ∂i ψi , 2

Bi j = h∂i r, B∂ j ri =

The quantity in the last term in (9.30) equals to: ∂11 B22 + ∂22 B11 − 2∂12 B22 = ∂11 h∂2 φ , ∂2 ψi + ∂22 h∂1 φ , ∂1 ψi − ∂12 h∂1 φ , ∂2 ψi + h∂2 φ , ∂1 ψi = −h∂11 φ , ∂22 ψi + h∂22 φ , ∂11 ψi − 2h∂12 φ , ∂12 ψi. Thus, by (9.30) and (9.29) it follows that:

9.2 Matching infinitesimal to exact isometries on elliptic surfaces

k(u1 , u2 )kC 1,α (ω) ¯ ≤ C kBkC 1,α (ω) ¯ + kφ kC 2,α (ω) ¯ kψkC 2,α (ω) ¯

243




≤ Ckφ kC 2,α (S) ¯ kψkC 2,α (S) ¯ . This completes the proof in view of (9.27). Proof of Theorem 9.7. ¯ R3 ) to be an isometry, the following condiFor uε = idS + εV + ε 2 wε ∈ C 2,α (S, 2 tion must be satisfied: |∂τ uε | = 1 for every τ ∈ Tz S with |τ| = 1. Since: |∂τ uε |2 − 1 = |τ + ε∂τ V + ε 2 ∂τ wε |2 − 1 = ε 2 |∂τ V |2 + 2ε 3 h∂τ V, ∂τ wε i + ε 4 |∂τ wε |2 + 2ε 2 hτ, ∂τ wε i, the vector field uε is therefore an exact isometry if and only if: 1 sym ∇wε = − (∇V + ε∇wε )T (∇V + ε∇wε ) . 2

(9.31)

¯ R3 ) → C 2,α (S, ¯ R3 ) defined in: Consider the mapping Gε : C 2,α (S,
1 Gε (w)  − T (∇V + ε∇w)T (∇V + ε∇w) . 2 By Lemma 9.1 we note that kGε (w)kC 2,α ≤ CkV + εwk2C 2,α . Hence, putting R = ¯ the inclusion kV k2C 2,α +1, and denoting by B¯ R the closed ball of radius R in C 2,α (S), ¯ ¯ Gε (BR ) ⊂ BR follows, for all ε  1 small enough. We will show that Gε is actually a contraction on B¯ R . By the linearity of T there must be: Gε (w) − Gε (v)
1 = − T (∇V + ε∇w)T (∇V + ε∇w) − (∇V + ε∇v)T (∇V + ε∇v) 2 
1  = − T 2εsym (∇V )T (∇w − ∇v) + ε 2 (∇w)T (∇w − ∇v) + ε 2 (∇w − ∇v)T ∇v . 2 The matrix field in the argument of T above is clearly symmetric and it has the form allowing for use of Lemma 9.1. Thus, for every w, v ∈ B¯ R there holds:
2 2 kGε (w) − Gε (v)kC 2,α (S) ¯ ≤ C εkV kC 2,α + ε kwkC 2,α + ε kvkC 2,α kw − vkC 2,α
≤ Cε kV kC 2,α + 2εR kw − vkC 2,α (S) ¯ . By the Banach fixed point theorem we now conclude that for all small ε the problem (9.31) has a solution wε , such that kwε kC 2,α (S) ¯ ≤ R. The proof is done.

244

9 Limiting theories on elastic elliptic shells

9.3 Density result on elliptic surfaces In this section we prove the density result which is a necessary step for the construction of the recovery family in section 9.4. Theorem 9.8. Assume that S is a compact, elliptic surface that is smoothly homeomorphic to a disk. Then, for every V ∈ V (S) there exists a sequence ¯ R3 )}n→∞ such that: {Vn ∈ V (S) ∩ C 2,α (S, lim kVn −V kH 2 (S) = 0.

n→∞

Proof. 1. We shall only require that S is of class C 4,α up to the boundary and that ∂ S is C 3,α , for some α ∈ (0, 1). Condition V ∈ V (S) implies that: sym∇Vtan = −hV, niΠ ∈ H 2 (S, R2×2 sym ). Consequently, writing V i = (V τi ) ◦ r ∈ H 2 (ω, R) where τi = ∂i r for i = 1, 2, we notice that components of the matrix field sym∇(V 1 ,V 2 ) are of the form: ∂τi hV, τ j i + ∂τ j hV, τi i = hV, ∂τi τ j + ∂τ j τi i ∈ H 2 (ω, R). Recall now that second derivatives of any vector field w : RN → RN are linear combinations of derivatives of its symmetric gradient (see (3.21) in the proof of Korn’s inequality in section 3.4)). It follows that V 1 ,V 2 ∈ H 3 (ω, R), and so Vtan ∈ H 3 (S, R3 ). Repeating the calculations in section 9.1, we consider the scalar field as in (9.5): 1 γ = p curlVtan ∈ H 2 (S, R), |g|

(9.32)

satisfying L γ = 0. The operator p L is3,αdefined in (9.6) and its coefficients have ¯ Our first goal is to approximate γ in ¯ |g| ∈ C (ω). regularity: hi j , H ∈ C 2,α (ω), ¯ R)}n→∞ such that L γn = 0. H 2 (ω, R) by a sequence {γn ∈ C 2,α (ω, 2. Consider the space:  F  v ∈ H 2 (ω, R); L v = 0 in ω, v = 0 on ∂ ω . ¯ R). Also, F has finite By the regularity of S and ∂ S, there holds F ⊂ C 3,α (ω, dimension. One easy way of seeing it is by integrating the equation L v = 0 against v on ω. By ellipticity of the leading order coefficient matrix [hi j ]i, j=1,2 we get: k∇vkL2 (ω) ≤ CkvkL2 (ω)

for every v ∈ F .

Therefore, in F the L2 and the H 1 norms are equivalent, proving the claim. Define now the finite dimensional space of traces:

9.3 Density result on elliptic surfaces

Ftr =

n

2

p

∑

245

|g|hi j η i ∂ j v



i, j=1

|∂ ω

; v∈F

o

⊂ L2 (∂ ω, R),

where by η = (η 1 , η 2 ) we denote the outer unit normal to ∂ ω ⊂ R2 . As a consequence of the regularity of vector fields in F and the regularity of S, there holds Ftr ⊂ C 2,α (∂ ω, R). The significance of the space Ftr is the following. Given w ∈ H 2 (ω, R), the problem: Lu=Lw

u=0

in ω,

on ∂ ω

has a solution if and only if w|∂ ω ∈ (Ftr )⊥ . Indeed, by Fredholm’s alternative there must be L w ∈ F ⊥ , where the orthogonal complement is taken in L2 (ω, R), namely: Z

for all v ∈ F .

hL w, vi dz = 0

(9.33)

ω

Integrating by parts we obtain: Z

hL w, vi dz =

ω

Z

hw, L vi dz +

ω

Z ∂ω

Z



= ∂ω

2

∑



2

∑

 p |g|hi j η j ∂i v w dz

i, j=1

 p |g|hi j η i ∂ j v w dz,

i, j=1

and thus (9.33) is indeed equivalent to: w|∂ ω ∈ (Ftr )⊥ . 3. Recall that γ ∈ H 2 (ω, R) given in (9.32), satisfies L γ = 0. Hence the continuous function φ  γ|∂ ω belongs to (Ftr )⊥ . Towards approximating γ by C 2,α functions, we first approximate its trace φ . Namely, we claim that there exists a sequence {φn ∈ (Ftr )⊥ ∩ C 2,α (∂ ω, R)}n→∞ such that: n o lim inf kΦkH 2 (ω) ; Φ ∈ H 2 (ω, R), Φ|∂ ω = φn − φ = 0. (9.34) n→∞

¯ R) and: Let φ˜n = γ˜n |∂ ω , so that γ˜n ∈ C 2,α (ω, lim kγ˜n − γkH 2 (ω) = 0.

n→∞

(9.35)

Defining Ptr : L2 (∂ ω, R) → Ftr to be the orthogonal projection onto Ftr , we put: φn = (Id − Ptr )(φ˜n ). Since φ˜n → φ in L2 (∂ ω, R), then φ˜n − φn = Ptr (φ˜n ) converges in L2 (∂ ω, R), as n → ∞ to Ptr (φ ) = 0. Further, in the finitely dimensional space Ftr all norms are equivalent, so we observe that: n o lim inf kΦkH 2 (ω) ; Φ ∈ H 2 (ω, R), Φ|∂ ω = φ˜n − φn = 0. n→∞

246

9 Limiting theories on elastic elliptic shells

Together with (9.35) the above implies (9.34) and proves the claim. 4. Consider now the sequence of harmonic functions {Φn ∈ H 2 (ω, R)}n→∞ such that ∆ Φn = 0 in ω and Φn = φn − φ on ∂ ω. By (9.34) it follows that: lim kΦn kH 2 (ω) = 0.

(9.36)

n→∞

⊥ , the problem: L u = −L Φ in ω and u = 0 on ∂ ω, has a Since φn − φ ∈ (F )tr n n n solution whose regularity must be un ∈ H 2 (ω, R). Moreover, using (9.36):

kun kH 2 (ω) ≤ CkL Φn kL2 (ω) ≤ CkΦn kH 2 (ω) → 0

as n → ∞.

(9.37)

Set γn = un + γ + Φn . By (9.36) and (9.37): lim kγn − γkH 2 (ω) = 0.

n→∞

(9.38)

¯ R). Also, L γn = L γ = 0 and γn |∂ ω = φn ∈ C 2,α (∂ ω) so there must be γn ∈ C 2,α (ω, 5. Based on the approximation {γn } accomplished in the previous step, we now construct the desired approximating sequence {Vn ∈ V (S) ∩ C 2,α (ω, R3 )}n→∞ of V . According to formulas (9.19) and (9.20) where we put B = 0 to have sym∇Vn = 0, and S (B) = γn in view of L γn = 0, the gradients {∇Vn }n→∞ are given by: 1 p ∂1Vn = γn |g| g2i ∂i r + u1 n, 2 with: u1 =

1p |g| h2i ∂i γn , 2

1 p ∂2Vn = − γn |g| g1i ∂i r + u2 n, 2 u2 = −

1p |g| h1i ∂i γn . 2

¯ R) and limn→∞ k∇Vn − ∇V kH 1 (ω) = 0 by (9.38). NorConsequently, ∇Vn ∈ C 1,α (ω, malizing each displacement Vn by a constant vector so that: Z

Z

Vn dz = ω

V dz, ω

the result follows in view of Poincar´e’s inequality. Remark 9.9. Theorem 9.8 remains true with higher H¨older regularity. More precisely, if S is assumed to be C m+2,α up to the boundary with ∂ S ∈ C m+1,α , for some α ∈ (0, 1) and an integer m ≥ 2, then for every V ∈ V (S) there exists a sequence ¯ R3 )}n→∞ {Vn ∈ V (S) ∩ C m,α (S, such that: limn→∞ kVn −V kH 2 (S) = 0.

9.4 Collapse of theories beyond Kirchhoff’s scaling for elliptic shells: recovery family...

247

9.4 Collapse of theories beyond Kirchhoff’s scaling for elliptic shells: recovery family, Γ -limit and convergence of minimizers Based on Theorems 9.8 and 9.7, we may now establish the limsup part of the Γ convergence result on elliptic shells. The below construction of a recovery family is valid for any scaling exponent β > 2. recall that {Sh }h>0 is a family of shells of small thickness h around S, given through: Sh = {x = z + tn(z); z ∈ S, − h/2 < t < h/2},

0 < h < h0 .

We will assume that h < h0  1, so that π well defined on each Sh . Theorem 9.10. Assume that S ⊂ R3 is a compact, elliptic surface that is smoothly homeomorphic to a disk. Let β > 2. Then for every V ∈ V (S) there exists a family of deformations {uh ∈ H 1 (Sh , R3 )}h→0 such that: (i) the rescalings {yh (z + tn(z)) = uh (x + t hh0 n(z))}h→0 of {uh } on Sh0 converge in H 1 (Sh0 , R3 ) to π, > h0 /2 1 (ii) the displacements {V h (z) = hβ /2−1 yh (z +tn(z)) − z dσ (z)}h→0 con−h /2 0

verge in H 1 (S, R3 ) to V , Z
1 1 (iii) lim β E h (uh ) = Ilin (V ) = Q2 z, sym(∇(An) − AΠ )tan dσ (z), h→0 h 24 S where A ∈ H 1 (S, so(3)) is the augmented gradient field ∇V as in (5.59). Proof. 1. By the density result in Theorem 9.8 and the continuity of Ilin with re¯ R3 ). In spect to the strong topology of H 2 (S), we can assume V ∈ V (S) ∩ C 2,α (S, the general case the result will follow through a diagonal argument. We will write: ε = hβ /2−1 . ¯ R3 )}ε→0 , so that By Theorem 9.7 there exists an equibounded family {wε ∈ C 2,α (S, each of the following derived deformations of S: uε  idS + εV + ε 2 wε

for all ε  1,

(9.39)

is an isometry. For every z ∈ S we now denote Nε (z) the unit normal vector to the image surface uε (S) at the point uε (z). By the regularity of uε we have that ¯ R3 ), while by (9.39) we obtain the expansion: Nε ∈ C 1,α (S, Nε = n + εAn + O(ε 2 ).

(9.40)

Indeed, one can take Nε = ∂τ1 uε ×∂τ2 uε , where τ1 , τ2 ∈ Tz S are such that n = τ1 ×τ2 . Using the Jacobi identity for vector product and A ∈ so(3), we arrive at (9.40).

248

9 Limiting theories on elastic elliptic shells

Here we introduce the recovery family {uh ∈ H 1 (Sh , R3 )}h→0 as required by the statement of the theorem. Note that the definition below coincides with the construction in the proof of Theorem 5.16 corresponding to the scaling exponent β = 2: uh (z + tn(z)) = uε (z) + tNε (z) +

t2 h εd (z). 2

(9.41)

The vector fields {d h ∈ W 1,∞ (S, R3 )}h→0 are defined so that: hkd h kW 1,∞ (S) → 0, d h → 2ζ

in L∞ (S, R3 ) where ζ = c x, sym(∇(An) − AΠ )tan




(9.42)

and where c(z, ·) is as in Definition 5.6 (iii). Assertions (i) and (ii) now easily follow from the uniform bound on {wε } and the normalization (9.42). 2. To prove (iii), we use Lemma 5.5 to get: 1 1 h h E (u ) = β β h h

Z ? h0 /2



h W ∇h yh (z + tn(z)) · det Id2 + t Π (z) dt dσ (z), h0 S −h0 /2 (9.43) where ∇h yh (z + tn(z)) = ∇uh (z + t hh0 n(z)) satisfies for all z ∈ S and τ ∈ Tz S: h h εd (z), h0   t2 h h ∇h yh (x + tn)τ = ∇uε (z) + t ∇Nε (z) + 2 h2 ε∇d h (z) (Id2 + t Π (z))−1 τ. h0 h0 2h0

∇h yh (z + tn)n(z) = Nε (z) + t

From (9.39), (9.40), (9.42) we get k∇h yh − Id3 kL∞ (Sh0 ) → 0. By polar decomposition theorem (for h sufficiently small), each ∇h yh is a product of a proper rotation and the well defined square root of (∇h yh )T ∇h yh . By frame invariance of W :

1 1 W (∇h yh ) = W Id3 + K h + O(|K h |2 ) = Q3 K h + O(|K h |2 ) + oε (1)|K h |2 , 2 2 where K h = (∇h yh )T ∇h yh − Id3 . We now calculate K h , first considering its tangential minor:  h h Π )−1 Id + 2t sym((∇uε )T ∇Nε ) h0 h0  h2 h + t 2 2 (∇Nε )T ∇Nε + o(hβ /2 ) (Id + t Π )−1 − Id2 h0 h0  h h h = (Id2 + t Π )−1 2t sym((∇uε )T ∇Nε ) − 2t Π h0 h0 h0 h2 h h2  + t 2 2 (∇Nε )T ∇Nε − t 2 2 Π 2 (Id2 + t Π )−1 + o(hβ /2 ), h0 h0 h0

h Ktan = (Id2 + t

9.4 Collapse of theories beyond Kirchhoff’s scaling for elliptic shells: recovery family...

249

where we have used the fact that (∇uε )T ∇uε = Id2 , and the identity: F1−1 FF1−1 − Id = F1−1 (F − F12 )F1−1 . By (9.39) and (9.40) we also deduce: sym((∇uε )T ∇Nε ) = Π + εsym(∇(An) − AΠ ) + O(ε 2 ), (∇Nε )T ∇Nε = Π 2 + O(ε). h found above, we get: Combining these two identities with the expression of Ktan

1

 t  h h h −1 K = (Id + t Π ) 2 sym(∇(An) − AΠ ) (Id2 + t Π )−1 + o(1). 2 tan h0 h0 h0 hβ /2 As |Nε | = 1, the normal component of K h is calculated by means of as:  1

2t n, K h n = |(∇h yh )n|2 − 1 = hd h , Nε i + o(1). h0

hβ /2

The remaining coefficients of K h (z + tn(z)) are calculated by using hNε , ∇Nε i = 0: 1 hβ /2

hτ, K h ni =

t h h
−1  d , ∇uε Id2 + t Π τ + o(1) h0 h0

for all τ ∈ Tz S.

3. From the previous computations we finally deduce that: 1 t t K h → K(z)tan + sym(ζ ⊗ n) β /2 h0 h0 2h

in L∞ (Sh0 , R3×3 ),

where the vector field ζ is defined in (9.42) and Ktan ∈ L∞ (S, R2×2 sym ) is given by: Ktan = sym(∇(An) − AΠ )tan .

(9.44)

Using the dominated convergence theorem, we consequently obtain: 1 h h 1 E (u ) = β h→0 h 2

Z ? h0 /2

lim

=

Q3

S

−h0 /2

S

2 −h0 /2 h0

Z ? h0 /2 2 1 t

2

t  t K(z)tan + (ζ ⊗ n) dt dσ (z) h0 h0


Q2 z, sym(∇(An) − AΠ )tan dt dσ (z)

where the last equality is a consequence of (9.42) and (9.44). Property (iii) now follows, upon integration with respect to t in the last integral above. Recalling Theorem 6.1, it follows that a convex shell transitions directly from the linear regime described by the energy functional Ilin in section 6.3, to purely nonlinear bending described by the energy IK in chapter 5. In other words, while the von K´arm´an theory describes buckling of thin plates at β = 4, the equivalent variationally correct theory for elliptic shells is IK at β = 2. For completeness, we also state the resulting validity of Corollaries 6.12 and 6.13 in the present context.

250

9 Limiting theories on elastic elliptic shells

Corollary 9.11. Let S ⊂ R3 be a compact, elliptic surface that is smoothly homeomorphic to a disk. Let β > 2. Then, we have: (i) the exact same statement of Γ -convergence as in Corollary 8.5 is satisfied, with respect to the family of external forces { f h }h→0 in there, (ii) the exact statement of the asymptotic behaviour of approximate minimizers as in Corollary 8.6 holds.

9.5 Bibliographical notes This chapter is based on the results of Lewicka et al. [2011b]. In section 9.1 we use the notation from [Han and Hong, 2006, Section 9.2]. We remark that existence of a conformal parametrisation r of S, (although we do not require this property in our proofs) follows from [Jost, 1984, Theorem 3.1]. The explicit form of coefficients of the linear mapping D in the proof of Theorem 9.2 is given in Han and Hong [2006]. The fact that γ is a classical solution in step 1 of the proof of Theorem 9.3 results from[Gilbarg and Trudinger, 2001, Theorem 8.12]. Remark 9.6 and the bound (9.25) are based on the ellipticity condition and the classical local bounds of Agmon et al. [1964], whose applicability in the present context has been shown by Geymonat and Sanchez-Palencia [1995]. Analysis in section 9.2 is based on the proof of openness of the set of positive curvature metrics g on the topological sphere Σ which are pull-backs of the Euclidean metric by immersions of Σ in R3 , due to Nirenberg [1953]. The Schauder estimates for elliptic systems in divergence form may be found in [Gilbarg and Trudinger, 2001, Theorem 8.32]. The upgraded regularity of elements of the space F in the proof of Theorem 9.8 follows from [Gilbarg and Trudinger, 2001, Theorem 9.19], Regularity claims in step 4 of the proof of Theorem 9.8 result from [Gilbarg and Trudinger, 2001, Theorem 8.12]. We remark that an interesting result is due to Cohn-Vossen, see Spivak [1999]. They showed existence of a closed smooth surface of non-negative curvature for which C ∞ ∩ V (S) consists only of trivial fields with constant gradient, whereas C 2 ∩ V (S) contains non-trivial infinitesimal isometries. Therefore C ∞ ∩ V (S) is not dense in V (S) for this surface.

Chapter 10

Limiting theories on elastic developable shells

In this chapter, we focus on developable surfaces. This class includes smooth cylindrical shells (already included in a preliminary analysis of section 8.2) which are ubiquitous in nature and technology over a range of length scales. The common approach in studying buckling phenomenon for cylindrical tubes has been to use the von K´arm´an-Donnel equations. However as we show here, the first sublinear theory for all smooth developable shells without flat parts is the purely nonlinear bending. This finding resembles the collapse of the range of theories corresponding to the energy scaling exponent β > 2, as in the case of convex shells studied in chapter 9. The term developable is derived from the property of surfaces of vanishing Gaussian curvature, stating that through each point on such a surface passes a straight segment that lies on the surface and points in a characteristic direction. Developable surfaces are also locally identified with isometric images of domains in R2 ; this last property is heavily exploited in our analysis. The outline of this chapter is as follows. In section 10.1 we review preliminary facts about developable surfaces, and in section 10.2 we study the linearised equation of isometric immersion sym∇w = B where we prove existence of a solution operator with suitable bounds. We then use this results to prove that any C 2k−1,1 regular first order infinitesimal isometry on a developable C 2k,1 surface with a positive lower bound on the mean curvature, can be matched to an kth-order infinitesimal isometry. In section 10.3 we proceed to study the space of first order infinitesimal isometries and prove their compensated regularity and rigidity properties. Combined with a straightforward density result we are finally able to show, in section 10.4, that the limit theories for the energy scalings of the order lower than h2+2/k collapse all into the linear theory. Our method is to inductively solve the aforementioned linearised metric equation with suitably chosen right hand sides, a process during which we lose regularity: only if the surface is C ∞ we can establish the total collapse of all small slope theories to the linear elasticity.

© Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_10

251

252

10 Limiting theories on elastic developable shells

10.1 Developable surfaces Consider a surface S = r(ω) ⊂ R3 , which is given as the image of a bounded, Lipschitz domain ω ⊂ R2 through an isomteric parametrization r ∈ C 2,1 (ω, R3 ): (∇r)T ∇r = Id2

in ω.

A classical result by Hartman and Nirenberg [1959] asserts that, away from the affine regions, such S must be developable: the domain ω can be decomposed (up to a controlled remainder) into finitely many subdomains on which r is affine, and finitely many subdomains on which r admits a line of curvature parametrization (see Figure 7.1 in section 7.5). In light of this result, it is natural to restrict ourselves to the situation when ω can be covered by a single line of curvature chart. We now make the above more precise. Given T > 0, let Γ ∈ C 1,1 ([0, T ], R2 ) be an arclength parametrized curve, and let s± ∈ C 0,1 ([0, T ], R) be two positive functions. We define the unit normal vector and the scalar curvature of Γ : η  (Γ 0 )⊥ ∈ C 0,1 ([0, T ], R2 )

and

κ  hΓ 00 , ηi ∈ L∞ ((0, T ), R).

We also introduce the bounded domain Ωs± , the mapping Φ : Ωs± → R2 , and the open line segments: Ωs±  {(s,t); t ∈ (0, T ), s ∈ (−s− (t), s+ (t))} ⊂ R2 , Φ(s,t)  Γ (t) + sη(t),


[Γ (t)]  Φ(·,t) (−s− (t), s+ (t)) .

From now on we assume that: [Γ (t1 )] ∩ [Γ (t2 )] = 0/

for all t1 , t2 ∈ [0, T ].

(10.1)

Condition (10.1) will be needed to define an isometry r, which maps segments [Γ (t)] to segments in R3 . The next lemma gathers a few basic properties of the above setup: Lemma 10.1. In the above context, let κn ∈ L2 ((0, T ), R), and define:  T R = γ 0 , v, n ∈ H 1 ((0, T ), SO(3)) so that it satisfies the initial value problem for matrix fields:   0 κ κn R0 =  −κ 0 0  R, R(0) = Id3 . −κn 0 0 We set γ(t) =

(10.2)

Rt 0 0 γ (s) ds and define the mapping:

r : Φ(Ωs± ) → R3 ,

r(Φ(s,t))  γ(t) + sv(t)

for all (s,t) ∈ Ωs± .

(10.3)

10.1 Developable surfaces

253

Then, there holds: 2 (Φ(Ω ± ), R3 ) is an isometric immersion, (i) r ∈ Hloc s (ii) the matrix field (∇r) ◦ Φ ∈ C 0 (Ω¯ s± , R3×2 ) satisfies:

∇r(Φ(s,t)) = γ 0 (t) ⊗ Γ 0 (t) + v(t) ⊗ η(t) for all (s,t) ∈ Ω¯ s± . 

 and the scalar coefficients in ai j  ∂1 r × ∂2 r, ∂i j r i, j=1,2 satisfy: ∂i j r = ai j (∂1 r × ∂2 r)

(10.4)

for i, j = 1, 2,

κn (t) Γ 0 (t)Γj0 (t) ai j (Φ(s,t)) = 1 − sκ(t) i

for almost every (s,t) ∈ Ωs± ,

(10.5)

(iii) if, in addition: Z T Z s+ (t) 0

s− (t)

κn2 (t) ds dt < ∞, 1 − sκ(t)

(10.6)

then r ∈ H 2 (Φ(Ωs± )), R3 ) with: Z Φ(Ωs± )

|∇2 r|2 dx =

Z T Z s+ (t) 0

s− (t)

κn2 (t) ds dt. 1 − sκ(t)

(10.7)

Proof. 1. Solvability of the Darboux equation (10.2) to the unique rotation-valued path R follows by recalling the tangent space representation: TR SO(3) = so(3)R =  R so(3). Note also that, since ∇Φ(s,t) = η(t),Γ 0 (t) + sη 0 (t) , then there holds: " # " # 1 1 0 0 T (∇Φ) ∇Φ = = , 0 |Γ 0 + sη 0 |2 0 (sκ − 1)2 where we used that hΓ 0 , η 0 i = −hΓ 00 , ηi = −κ and Γ 00 = hΓ 00 , ηiη, so that |Γ 00 | = κ 2 . Thus, det ∇Φ(s,t) , 0 in view of (10.1), and Φ is locally invertible on Ωs± . We also observe that det ∇Φ(0,t) = det η,Γ 0 ] = −1, so consequently: det ∇Φ(s,t) = sκ(t) − 1.

(10.8)

We verify (10.4) by using equations in (10.2) and checking that:
    (∇r) ◦ Φ ∇Φ = ∇(r ◦ φ ) = v, γ 0 + sv0 = v, γ 0 (1 − sκ) ,

  and γ 0 ⊗ Γ 0 + v ⊗ η ∇Φ = γ 0 ⊗ Γ 0 + v ⊗ η η,Γ 0 + sη 0   = γ 0 ⊗ (0, 1 − sκ) + v ⊗ (1, 0) = v, γ 0 (1 − sκ) . In turn, the formula (10.4) directly implies (i), because:


(∇r)T ∇r ◦ Φ = γ 0 ⊗ Γ 0 + v ⊗ η Γ 0 ⊗ γ 0 + η ⊗ v = Γ 0 ⊗ Γ 0 + η ⊗ η = Id2 . Likewise, observe another useful consequence of (10.4):

254

10 Limiting theories on elastic developable shells




∂1 r × ∂2 r ◦ Φ

= Γ10 γ 0 + η1 v ⊗ = (|Γ10 |2 + |Γ20 |2 )n



Γ20 γ 0 + η2 v = (Γ10 η2 − Γ20 η1 )γ 0 ⊗ v = n.

(10.9)

2. We now prove (10.5). The first assertion follows from Lemma 7.6. For the second formula, we recall (10.9), (10.2), (10.8) and conclude that:  


0 


ai j ◦ Φ = n, ∇ ∂ j r ◦ Φ) (∇Φ)−1 ei = n, Γj0 γ 0 + η j v · e2 , (∇Φ)−1 ei κnΓj0



 = κnΓj0 · (∇Φ)−1,T e2 , ei = (cof ∇Φ)e2 , ei det ∇Φ κnΓj0 κ n hη ⊥ , ei i = Γ 0Γ 0 . = sκ − 1 1 − sκ j i Finally, to show (iii), we use the change of variables and (10.5), (10.8) in: Z

|∇2 r|2 dx =

Z

Φ(Ωs± )

Ωs±

Z

=


|∇2 r|2 ◦ Φ | det ∇Φ| d(s,t)

∑

Z
|∂i j r| ◦ Φ (1 − sκ) d(s,t) = 2

Ωs± i, j=1,2

Z

= Ωs±

Z

= Ωs±

∑


|ai j |2 ◦ Φ (1 − sκ) d(s,t)

Ωs± i, j=1,2

κn2 ∑ |Γi 0Γj0 |2 d(s,t) = (1 − sκ) i, j=1,2

Z Ωs±

κn2 ((Γ 0 )2 + (Γ20 )2 )2 d(s,t) (1 − sκ) 1

κn2 d(s,t). (1 − sκ)

This ends the proof of the lemma. For consistency, we make the following definition: Definition 10.2. A surface S ⊂ R3 is said to be developable (resp. developable of class C k,1 ), provided that there are Γ , η, s± , γ, v, n, κ, κn , Φ and a smooth (resp. C k,1 ) vector field r : Φ(Ωs± ) → R3 as in Lemma 10.1, such that: S = r(ω)

where we denote: ω  Φ(Ωs± ) ⊂ R2 .

Remark 10.3. (i) The curve γ is a line of curvature, and the mapping (s,t) 7→ γ(t)+sv(t) is a line of curvature parametrization of S. Also, n is the unit normal and Π = −[ai j ]i, j=1,2 is the second fundamental form of S (written in r-coordinates). Equation (10.5) implies that Gauss’s curvature det Π = det[ai j ] = 0. (ii) The moving frame R is the Darboux frame on S along γ. Thus, (10.2) indicates that the geodesic curvature of γ coincides with the curvature κ of its preimage Γ , and that its geodesic torsion vanishes. This is naturally expected as r is an isometry. In the same vein, κn is the normal curvature of γ on S. (iii) We have: κn = 0 almost everywhere on the set: n  1 1 o , t ∈ (0, T ); κ(t) ∈ − , + s (t) s (t)

10.1 Developable surfaces

255

which follows by examining the condition (10.6). Further immediate properties of the discussed quantities are gathered in: Corollary 10.4. In the above context, we have:   1 1 for almost every t ∈ (0, T ). κ(t) ∈ − − , + s (t) s (t)

(10.10)

Moreover, the Lipschitz map Φ is a homeomorphism from Ωs± onto ω. Under the extra assumption that the mean curvature of r is bounded away from zero: |trace [ai j ]| > 0

¯ in ω,

the inverse Φ −1 is Lipschitz as well, as there exists δ > 0 with:   1 1 κ(t) ∈ δ − − , + − δ for almost every t ∈ (0, T ). s (t) s (t)

(10.11)

(10.12)

In the proof of Lemma 10.5 below we use: Lemma. [Chain rule for Sobolev homeomorphisms] Let Ω1 , Ω2 ⊂ RN be two open, bounded sets and let Φ : Ω1 → Ω2 be a bilipschitz homeomorphism. Then f ∈ H 1 (Ω2 , R) if and only if f ◦ Φ ∈ H 1 (Ω1 , R). In this case:
∇( f ◦ Φ) = (∇ f ) ◦ Φ ∇Φ a.e. in Ω1 . (10.13) Lemma 10.5. Assume that S is developable of class C k,1 for some k ≥ 2 and that (10.11) holds. Then κ, κn ∈ C k−2,1 ([0, T ], R) and Φ, Φ −1 are C k−1,1 up to the boundary of their respective domains Ωs± and ω. ¯ R2×2 ). By continuProof. The hypothesis on S imply that [ai j ]i, j=1,2 ∈ C k−2,1 (ω, ity of ai j and Φ and by (10.11), we may assume without loss of generality that tr [ai j ]i=1,2 ≥ c > 0. In view of the second assertion in (10.5), this implies tr [ai j ◦ Φ]i, j=1,2 =

κn 1 − sκ

in Ωs± .

(10.14)

Since Φ is bilipschitz, we get that the right-hand side of (10.14) is Lipschitz. As κn ≥ c > 0 it follows that κ, κn are Lipschitz. Thus Γ 0 , η ∈ C 1,1 ([0, T ], R), and hence the same regularity is inherited by Φ. By (10.8) and (10.12), the Jacobian of Φ is uniformly positive on Ωs± , and so Φ −1 is likewise C 1,1 -regular. If k ≥ 3 then we return to (10.14), apply the chain rule (10.13) and argue as before to conclude that κ, κn ∈ C 1,1 ([0, T ], R). The final result follows then by iteration.

256

10 Limiting theories on elastic developable shells

10.2 Matching properties on developable surfaces The purpose of this section is to prove the 1 7→ k matching property for sufficiently regular developable surface without flat parts. As in previous chapters, this property will be further used towards defining an “efficient” recovery family for a sufficiently regular displacement. We anticipate that for smooth developable shells without flat parts, the resulting 1 7→ ∞ matching will force the dimensionally reduced energy to coincide with linear elasticity in the entire scaling range β > 2. Theorem 10.6. Assume that S is a developable surface of class C 2k,1 , for some k ≥ 2 according to Definition 10.2, and satisfying (10.11). Then, for every ¯ R3 ), there exists an equibounded infinitesimal isometry V ∈ V (S) ∩ C 2k−1,1 (S, family {wε ∈ C 1,1 (S, R3 )}ε>0 , such that the derived family of deformations: {uε = id S + εV + ε 2 wε ∈ C 1,1 (S, R3 )}ε→0 induces the change of metric on S of order ε k+1 , namely: k(∇uε )T ∇uε − Id2 kL∞ (S) = O(ε k+1 )

as ε → 0.

(10.15)

As in chapter 9, where we proved the stronger matching property 1 7→ ∞ on smooth convex surfaces, the argument below relies on analyzing the linear problem sym∇w = B. On developable surfaces, the counterpart of Theorem 9.2 is: Theorem 10.7. Assume that S is developable of class C 2,1 according to Definition 10.2 and that it satisfies (10.11). Let α ∈ (0, 1). Then, for every 0,α (S, R3 ) and B ∈ C 1,1 (S, R2×2 sym ) there exists w = wtan + hw, nin with wtan ∈ C ∞ hw, ni ∈ L (S, R) such that there holds: sym∇w = sym∇wtan + hw, niΠ = B

on S,

together with the bound: kwtan kC 0,α (S) + khw, nikL∞ (S) ≤ CkBkC 1,1 (S) .

(10.16)

2×2 ) for some k ≥ 1, then: If in addition, S ∈ C k+2,1 and B ∈ C k+1,1 (S, Rsym

kwtan kC k,1 (S) + khw, nikC k−1,1 (S) ≤ CkBkC k+1,1 (S) .

(10.17)

The constants C above depend only on S but not on B or w. Proof. 1. Define the pull-back maps on ω: w3 = hw, ni ◦ r,

w0 = (wtan ◦ r)T ∇r,

 Bi j = (B ◦ r)∂i r, ∂ j r .

10.2 Matching properties on developable surfaces

257

Using (10.5) and recalling that wtan is a tangent field on S, we get: h∂ j r, ((∇w) ◦ r)∂i ri = h∂i (w ◦ r), ∂ j ri = ∂i hw ◦ r, ∂ j ri − hw ◦ r, ∂i2j ri = ∂i w0j − w3 ai j . Hence, the problem sym∇w = B can be written in terms of the pull-back quantities: [Bi j ]i, j=1,2 = sym∇w0 − w3 [ai j ]i, j=1,2

in ω.

(10.18)

Since ω is simply connected, the above problem is equivalent to:
curl curl [Bi j ]i, j=1,2 = −curl curl w3 [ai j ]i, j=1,2

= − w3 curl curl [ai j ]i, j=1,2 − 2 ∇⊥ w3 , curl [ai j ]i, j=1,2 i 

− cof ∇2 w3 : [ai j ]i, j=1,2 . Further, h∂i2j r, ∂k ri = 0 by Lemma 10.1, so: " curl [ai j ]i, j=1,2 = −curl

2 r, ni h∂ 2 r, ni h∂11 12 2 2 r, ni h∂12 r, ni h∂22

# = 0.

Denote: θ  curl curl [Bi j ]i, j=1,2 ∈ L∞ (ω, R).

The problem (10.18) becomes thus: θ = − cof ∇2 w3 : [ai j ]i, j=1,2 i, where:

 cof ∇2 w3 : [ai j ]i, j=1,2 (Φ(s,t)) =

κn (t) ∂ 2 (w3 ◦ Φ)(s,t) 1 − sκ(t) ss

in virtue of Lemma 10.1. Consequently, (10.18) is equivalent to: ∂ss2 (w3 ◦ Φ)(s,t) = −

1 − sκ(t) (θ ◦ Φ)(s,t) κn (t)

for all (s,t) ∈ Ωs± .

(10.19)

We observe that the pair (w0 , w3 ) consisting of a solution w3 to (10.19) and the derived vector field w0 as in (10.18), is unique after choosing the boundary conditions (w3 ◦ Φ)(t, 0)) and ∂s (w3 ◦ Φ)(t, 0). Uniqueness of w0 is understood up to affine maps of the form: A(s,t)T + b with A ∈ so(2), b ∈ R2 . 2. Assume first the minimal regularity r ∈ C 2,1 (ω, R3 ), B ∈ C 1,1 . Integrating (10.19) twice in s, from (w3 ◦ Φ)(t, 0) = 0 and ∂s (w3 ◦ Φ)(t, 0) = 0, we get: khw, nikL∞ (ω) ≤ Ckθ kL∞ (ω) ≤ CkBkC 1,1 (ω) ,

(10.20)

since by (10.11) there holds |κn (t)| > c > 0 for all t ∈ [0, T ]. Solving now (10.18) for wtan with the normalisation:

258

10 Limiting theories on elastic developable shells

?

? ∇(wtan ◦ r) dz = 0

skew

wtan ◦ r dz = 0,

and

ω

ω

we obtain by means of Korn’s inequality, for any p > 1: k∇wtan kL p (S) ≤ Cksym∇(wtan ◦ r)kL p (ω) ≤ C(k[Bi j ]i, j=1,2 kL∞ (ω) + khw, nikL∞ (S) ) ≤ CkBkC 1,1 (S) , where C depends only on p and S. Combining with Poincar´e’s inequality, we get: kwtan kW 1,p (S) ≤ CkBkC 1,1 (S) . By Sobolev embedding, the above together with (10.20) yields (10.16). k+2,1 (ω, R3 ), then − 1−sκ θ ∈ C k−1,1 (ω, R) 3. When B ∈ C k+1,1 (S, R2×2 sym ) and r ∈ C κn in virtue of Lemma 10.5. Consequently, by (10.19):

khw, nikC k−1,1 (S) ≤ Ckθ kC k−1,1 (ω) ≤ CkBkC k+1,1 (S) ,

(10.21)

Recalling that ∇2 w0 can be expressed as the linear combination of partial derivatives of sym∇w0 , see (3.21), it follows from the above that: k∇2 wtan kC k−2,1 (S) ≤ CkBkC k+1,1 (S) . The estimate (10.17) results hence from (10.16), (10.21). This ends the proof. The next result is similar to Lemma 9.1 and concerns the right hand side B, in the linear system sym∇w = B, that has a product structure: Corollary 10.8. In the context of Theorem of class C k+2,1
10.7, let S be developable T k+1,1 3 for some k ≥ 1. If B = sym (∇φ ) ∇ψ where φ , ψ ∈ C (S, R ), then: kwtan kC k,1 (S) + khw, nikC k−1,1 (S) ≤ CkψkC k+1,1 (S) kφ kC k+1,1 (S) . Proof. A straightforward calculation implies that: Bi j =


1 h∂i (φ ◦ r), ∂ j (ψ ◦ r)i + h∂ j (φ ◦ r), ∂i (ψ ◦ r)i . 2

Further, in the expansion of θ , the third derivatives of ψ and φ cancel out: 2 2 2 θ = curlT curl [Bi j ]i, j=1,2 = ∂11 B22 + ∂22 B11 − 2∂12 B12 2 2 = ∂22 h∂1 (φ ◦ r), ∂1 (ψ ◦ r)i + ∂11 h∂2 (φ ◦ r), ∂2 (ψ ◦ r)i


2 h∂1 (φ ◦ r), ∂2 (ψ ◦ r)i + h∂2 (φ ◦ r), ∂1 (ψ ◦ r)i − ∂12 2 2 2 2 2 2 (ψ ◦ r)i. (φ ◦ r), ∂12 (ψ ◦ r)i + 2h∂12 (φ ◦ r), ∂11 (ψ ◦ r)i − h∂22 (φ ◦ r), ∂22 = −h∂11

As a consequence, if S is of class C k+2,1 , the solution of (10.19) satisfies:

10.2 Matching properties on developable surfaces

259

khw, nikC k−1,1 (S) ≤ Ckθ kC k−1,1 (ω) ≤ Ckφ kC k+1,1 (S) kψkC k+1,1 (S) . Reasoning as in the proof of Theorem 10.7, we obtain: kwtan kC k,1 (S) + khw, nikC k−1,1 (S) ≤ CkBkC k−1,1 (S) +Ckφ kC k+1,1 (S) kψkC k+1,1 (S) ≤ Ckφ kC k+1,1 (S) kψkC k+1,1 (S) , proving the result. We are now ready to show the main result of this section: Proof of Theorem 10.6. 1. The proof uses an inductive argument, with the following inductive step. Fix m = 2 . . . k, define m¯ = 2k − 2m + 2 and assume that: n o m−1 uε = id S + ∑ ε j w j

¯ with w j ∈ C m+1,1 (S, R3 )

ε>0

j=1

satisfies: k(∇uε )T ∇uε − Id2 kL∞ (S) = O(ε m )

as ε → 0.

¯ We will show that there is wm ∈ C m−1,1 (S, R3 ) with {u¯ε  uε +ε m wm }ε>0 satisfying:

k(∇u¯ε )T ∇u¯ε − Id2 kL∞ (S) = O(ε m+1 )

as ε → 0

m−1

kwm kC m−1,1 ¯ (S) ≤ C

¯ ¯ ∑ kws kC m+1,1 (S) kwm−s kC m+1,1 (S) .

s=1

Then, setting w1 = V ∈ C 2k−1,1 (S) and applying the above result iteratively to find {wm ∈ C 2k−2m+1,1 (S, R3 )}km=2 , we obtain the requested: wε = w2 + εw3 + · · · ε k−2 wk ∈ C 1,1 (S, R3 ). 2. We now prove the above claim. Set w0 = id S , so that u¯ε = ∑mj=0 ε j w j . Calculating the induced change of metric, we get: m

(∇u¯ε )T ∇u¯ε − Id2 =

∑ ε j A j + O(ε m+1 ),

j=1

where {A j }mj=1 measure the change of metric of j-th order, respectively: j

Aj =

∑ sym


(∇ws )T ∇w j−s .

s=0

By the assumption A1 = · · · = Am−1 = 0. To ensure that Am = 0 as well, we need:

260

10 Limiting theories on elastic developable shells

sym∇wm = −


1 m−1 sym (∇ws )T ∇wm−s . ∑ 2 s=1

Applying Theorem 10.7 and Corollary 10.8, we obtain that such wm exists, and: m−1

kwm,tan kC m,1 ¯ (S) + khwm , nikC m−1,1 ¯ (S) ≤ C

¯ ¯ ∑ kws kC m+1,1 (S) kwm−s kC m+1,1 (S) ,

s=1

¯ ¯ provided that all ws ∈ C m+1,1 (S, R3 ) and that S is of class C m+2,1 . This completes the proof of the claim and of the theorem.

10.3 Density result on developable surfaces Since the matching property in section 10.2 was valid only for sufficiently regular H¨older elements in V (S), our second key result below needs to assert the density of such displacements in V (S): Theorem 10.9. Assume that S ⊂ R3 is developable of class C k+1,1 up to the boundary, according to Definition 10.2, and that (10.11) holds. Then, for every ¯ R3 )}n→∞ such that: V ∈ V (S) there exists a sequence {Vn ∈ V (S) ∩ C k,1 (S, lim kVn −V kH 2 (S) = 0.

n→∞

Towards the proof, we first further discuss the regularity of infinitesimal isometries V ∈ V (S). Note that in view of (10.13) , we may freely determine the regularity of any mapping on S, up to C 2,1 regularity, by considering the regularity of its composition with the chart r. Thus, we write f ∈ H 2 (S) precisely if f ◦ r ∈ H 2 (ω). Lemma 10.10. Assume that S is developable of class C 2,1 . Then there holds: (i) if V ∈ V (S) then, writing V = Vtan + hV, nin, we have: Vtan ∈ C 2,1/2 (S, R3 ),

hV, ni ∈ C 1,1/2 (S, R),

(ii) a vector field V ∈ V (S) if and only if there exist a, b ∈ H 2 ((0, T ), R) such that: hV, ni(u ◦ Φ)(s,t) = a(t) + sb(t), sym ∇Vtan (u ◦ Φ)(s,t) =

a(t) + sb(t) κn (t) (Γ 0 (t) ⊗ Γ 0 (t)) 1 − sκ(t)

for a.e (s,t) ∈ Ωs± , and such that the following integrals are finite:

(10.22)

10.3 Density result on developable surfaces

261

  κ(a0 (t) + sb0 (t)) 2 d(s,t) a,b 0 J1  b (t) + < ∞, 1 − sκ(t) 1 − sκ(t) Ωs± Z

J2a,b 

  sκ 0 (t)(a0 (t) + sb0 (t)) 2 · a00 (t) + sb00 (t) − κ(t)(1 − sκ(t))b(t) + 1 − sκ(t) Ωs±

Z

·

d(s,t) < ∞. (1 − sκ(t))3 (10.23)

Proof. 1. Since r ∈ C 2,1 (ω, R3 ), from (10.5) we conclude that κn is continuous up to the boundary and that κ is continuous on the open set where κn differs from zero. Let now V ∈ V (S). By Corollary 10.4 and from (10.8) we deduce that Φ is bilipschitz on its image, on every set of the form:  δ > 0. Ωδ0  (s,t) ∈ Ωs± ; s ∈ (δ − s− (t), s+ (t) − δ ) , Indeed, Φ −1 may fail to be globally Lipschitz on the set Φ(Ωs± ) unless (10.12) holds. Denote f = V3 ◦ Φ and V3 = hV, ni ◦ r ∈ H 2 (ω, R). By a similar reasoning as in Lemma 10.5 we may conclude that Φ is a C 1,1 -regular diffeomorphism on Ωδ0 . Hence, the chain rule in (10.13) implies that f ∈ H 2 (Ωδ0 , R) with:



 ∂s f (s,t) = ∇V3 (z), η(t) , ∂t f (s,t) = (1 − sκ(t)) ∇V3 (z),Γ 0 (t) ,

 ∂ss2 f (s,t) = ∇2V3 (z)η(t), η(t) ,

 (10.24) ∂ts2 f (s,t) = (1 − sκ(t)) ∇2V3 (z)η(t),Γ 0 (t) − κ(t)h∇V3 (z),Γ 0 (t)i,

2  2 2 0 0 ∂tt f (s,t) = (1 − sκ(t)) ∇ V3 (z)Γ (t),Γ (t)

 + ∇V3 (z), κ(1 − sκ)N(t) − sκ 0Γ 0 (t) , where we write z = Φ(t, s). Moreover, by (10.19), there holds: ∂ss2 f (s,t) = 0

for all t ∈ [0, T ] with κn (t) , 0.

(10.25)

Indeed, θ = curl curl[Bi j ]i, j=1,2 = 0 in the present case where sym∇V = 0. 2. Take 0 < δ¯ < inft∈[0,T ] min{s− (t), s+ (t)}, so that (−δ¯ , δ¯ ) × (0, T ) ⊂ Ωs± . Denote by f ∗ the precise representative of f and define: a(t) = f ∗ (0,t),

1 b(t) = ¯ ( f ∗ (δ¯ ,t) − a(t)). δ

In virtue of (10.19), the definition of b does not depend on δ¯ and: f (s,t) = a(t) + sb(t)

for all (s,t) ∈ Ωs± .

(10.26)

Now, since f ∈ H 2 ((0, T ) × (−δ¯ , δ¯ ), R), for almost every pair s1 , s2 ∈ (−δ¯ , δ¯ ) the traces f (·, s1 ) and f (·, s2 ) belong to H 2 ((0, T ), R), by Fubini’s theorem. Conse-

262

10 Limiting theories on elastic developable shells

1 ( f (·, s1 )− f (·, s2 )) ∈ H 2 ((0, T ), R), and therefore a ∈ H 2 ((0, T ), R) quently, b = s1 −s 2

as well. By Sobolev embedding it follows that f ∈ C 1,1/2 (Ωs± , R). Since Φ is a C 1,1 diffeomorphism, it also follows that hV, ni ∈ C 1,1/2 (S, R) which, in turn, implies that Vtan ∈ C 2,1/2 (S, R3 ) by Remark 6.6. This proves (i). Also, the second identity in (10.22) follows from Lemma 10.1. 3. We shall now prove that, given the structure (10.26), condition hV, ni ∈ H 2 (S, R) is equivalent to a, b ∈ H 2 ((0, T ), R) satisfying (10.23). This will conclude the proof  of (ii). To this end, we use (10.24) to obtain, for all t ∈ [0, T ] such that κ(t) < s−1(t) , s+1(t) (compare Remark 10.3 (iii)):

 κ (a0 (t) + sb0 (t)), b0 (t) = ∂ts2 f (s,t) = (1 − sκ) ∇2V3 (z)η(t),Γ 0 (t) − 1 − sκ

 a00 (t) + sb00 (t) = ∂tt2 f (s,t) = (1 − sκ)2 ∇2V3 (z)Γ 0 (t),Γ 0 (t) + κ(1 − sκ)b(t) −

sκ 0 (a0 (t) + sb0 (t)). 1 − sκ

where as before we write z = Φ(s,t). Solving for ∇2V3 (z) we get:  1 a00 (t) + sb00 (t) − κ(1 − sκ)b(t) (1 − sκ)2  sκ 0 + (a0 (t) + sb0 (t)) , 1 − sκ  

2  1 κ0 ∇ V3 (z)η 0 (t),Γ 0 (t) = b0 (t) + (a0 (t) + sb0 (t) , 1 − sκ 1 − sκ

2  0 0 ∇ V3 (z)η (t), η (t) = 0.

2  ∇ V3 (z)Γ 0 (t),Γ 0 (t) =

A change of variables yields: Z ω

2

2

|∇ V3 (z)| dz =

Z

|(∇2V3 )(Φ(s,t))|2 (1 − sκ(t)) d(s,t).

Ω s±

implying that V3 ∈ H 2 (ω, R) if and only if (10.23) holds. We also note the following straightforward implication of the above calculations: Corollary 10.11. Let v ∈ H 2 (S, R) satisfy: v(r(Φ(s,t))) = a(t) + sb(t)

for a.e. (s,t) ∈ Ωs± .

Then a, b ∈ H 2 ((0, T ), R) and there exists Vtan ∈ H 2 (S, R3 ) such that hVtan , ni = 0 on S, and Vtan + vn ∈ V (S). We are now ready to give the proof of the main result of this section, that together with Theorem 10.6 will directly yield the announced upper bound result in the dimension reduction problem for elastic developable shells without flat parts:

10.4 Collapse of theories beyond Kirchhoffs scaling for developable shells: recovery family... 263

Proof of Theorem 10.9. Let a, b ∈ H 2 ((0, T ), R) be as in Lemma 10.10. Take {an , bn ∈ C ∞ ([0, T ], R)}n→∞ converging in H 2 to a, b respectively, and define: vn (s,t)  an (t) + sbn (t). By Lemma 10.10 and Corollary 10.11, there exist {Vn ∈ V (S)}n→∞ such that: hVn , ni ◦ r ◦ Φ = vn

and

khVn , ni − hV, nikH 2 (S) → 0.

Indeed, the last assertion is equivalent to Ji (a−an , b−bn ) → 0 as n → ∞, for i = 1, 2, which is established immediately after observing that 1 − sκ remains bounded away from 0 by (10.12). Note that the tangential components {(Vn )tan }n→∞ may be chosen suitably so that we also have the convergence: k(Vn )tan −Vtan kH 2 (S) → 0. ¯ R3 ), which yields that each In view of Lemma 10.5, we observe that Vn ◦ r ∈ C k,1 (ω, k,1 Vn is C -regular up to the boundary of S. The proof is done.

10.4 Collapse of theories beyond Kirchhoffs scaling for developable shells: recovery family, Γ -limit and convergence of minimizers In this section, we state the upper bound result implying the Γ -convergence and convergence of minimizing deformations of developable shells without flat parts: Theorem 10.12. Assume that S is developable of class C 2k,1 according to Definition 10.2, for some k ≥ 2, and that it satisfies (10.11). Let: 2 β > 2+ . k

(10.27)

Then, for every V ∈ V (S) there exists {uh ∈ H 1 (Sh , R3 )}h→0 such that: (i) the rescaled deformations {yh (z + tn(z)) = uh (z + t hh0 n(z))}h→0 converge in H 1 (Sh0 , R3 ) to π, > h0 /2 1 (ii) the displacements {V h (z) = hβ /2−1 yh (z +tn(z)) − z dσ (z)}h→0 con−h /2 0

verge in H 1 (S, R3 ) to V . Z
1 1 (iii) lim β E h (uh ) = Ilin (V ) = Q2 z, sym(∇(An) − AΠ )tan dσ (z). h→0 h 24 S

264

10 Limiting theories on elastic developable shells

Proof. The recovery family {uh } construction follows the same outline as in the case of convex midsurface covered in Theorem 9.10. Firstly, by the density result in ¯ R3 ). In the general Theorems 10.9 it suffices to assume that V ∈ V (S) ∩ C 2k−1,1 (S, case the claim follows through a diagonal argument. Let ε = hβ /2−1 . By Theorem 10.6, there exists an equibounded family {wε ∈ 1,1 ¯ R3 )}h→0 , such that the following holds: C (S, {uε  id S + εV + ε 2 wε }ε→0

satisfies

k(∇uε )T ∇uε kL∞ (S) = O(ε k+1 ).

Note that by (10.27) we may replace O(ε k+1 ) by o(hβ /2 ), because:
k/2 ε k+1 = h(β /2−1)(k−1)−β /2 = hβ −(2+2/k) →0 β /2 h

as h → 0.

(10.28)

¯ R3 )}ε→0 that are normal to the reDefining the unit vector fields {Nε ∈ C 0,1 (S, spective surfaces uε (S) by (9.40), and the warping fields {d h ∈ W 1,∞ (S, R3 )}h→0 by (9.42), we introduce {uh ∈ H 1 (Sh , R3 )}h→0 in: uh (z + tn(z)) = uε (z) + tNε (z) +

t2 h εd (z). 2

Even though instead of exact isometries we make use of the approximate isometries {uε }ε→0 , the k-order error in the metric is controlled in accordance with the energy scaling law in view of (10.28). The argument is now verbatim the same as in the proof of Theorem 9.10. For completeness, below we gather the full collapse of theories result on smooth developable shells, together with the usual implications for Γ -convergence in presence of forces, and the convergence of minimizers of total energies: Corollary 10.13. Assume that S is (smooth) developable according to Definition 10.2, and that it satisfies (10.11). Let β > 2. Then, we have: (i) for every V ∈ V (S) there exists a family of deformations {uh ∈ H 1 (Sh , R3 )}h→0 such that the assertions (i) - (iii) in Theorem 10.12 hold, (ii) the exact same statement of Γ -convergence as in Corollary 8.5 is satisfied, with respect to the family of external forces { f h }h→0 in there, (iii) the exact statement of the asymptotic behaviour of approximate minimizers as in Corollary 8.6 holds.

10.5 Bibliographical notes

265

10.5 Bibliographical notes An example of a recently discovered developable structure is carbon nanotubes, i.e. molecular-scale tubes of graphitic carbon with outstanding rigidity properties Harris [1999]: they are among the stiffest materials in terms of the tensile strength and elastic modulus, but they easily buckle under compressive, torsional or bending stress Jensen et al. [2007]. The von K´arm´an-Donnel equations have been studied in Mahadevan et al. [2007], Horak et al. [2006]. It seems likely that these equations could be rigorously derived and valid in another scaling limit, e.g. when the radius of the cylinder is very large as the thickness vanishes. The structure properties of developable surfaces were established for C 2 regularity in Hartman and Nirenberg [1959], for C 1 with total zero curvature in [Pogorelov, 1956, Chapter II] and [Pogorelov, 1973, Chapter IX], and for H 2 isometries in Kirchheim [2001], Pakzad [2004]. It has been shown by Pakzad [2004] that any H 2 isometry on a convex domain may be approximated in strong norm by smooth isometries, and in M¨uller and Pakzad [2005] the boundary regularity was discussed. These results were later generalized by Hornung [2011a,b]. Results of this chapter are taken from Hornung et al. [2013]. Assertions of Lemmas 10.1 and 10.4 appeared in [Hornung, 2011b, Proposition 2, Proposition 1] and [Hornung, 2011a, Proposition 10], respectively. The chain rule for Sobolev homeomorphisms can be found in [Ziemer, 1989, Theorem 2.2.2]. The importance of the solvability of sym∇w = B has been noted in SanchezPalencia [1989], see also Geymonat and Sanchez-Palencia [1995] and Choi [1997] in regards to relations of rigidity and elasticity. We also remark that a surface, e.g. a regular cylinder, may well be ill-inhibited according to the definition by SanchezPalencia [1989] and yet satisfy the adequate matching properties resulting in the aforementioned collapse. In section 10.1, the notation and the proofs of Lemmas 10.1 and 10.4 are taken from Hornung [2011b]. The notion of precise representatives of Sobolev functions is defined in [Evans and Gariepy, 1992, section 4.8]. The question of existence of a developable or non-developable surface with no flat regions which shows a different elastic behavior (i.e. validity of an intermediate theory between the linear one and the purely nonlinear bending) remains open.

Part III

Dimension reduction in prestressed elasticity

Chapter 11

Limiting theories for prestressed films: nonlinear bending

The experimental observations and numerical computations suggest a common mathematical framework to understand the origin of shape: an elastic three-dimensional body seeks to realize a configuration with a prescribed Riemannian metric. In general, the biological growth arises from changes in the fields: cell number, size, shape and motion, all of which conspire to determine the local metric, likely incompatible with the existence of an isometric immersion. The deviation from or inability to reach such immersion, due to the combination of geometric incompatibility and the requirements of elastic energy minimization, result in the observed shapes. The outline of this chapter is as follows. Excluding nonphysical deformations that change the orientation in a neighborhood of the immersion, a natural way to pose the question of the origin of shape, as explained above, is by postulating a variational principle that minimizes an elastic energy Eg which measures how far a given deformation u is from being an orientation-preserving realization of g. This is done in section 11.1. Equivalently, Eg (u) quantifies the total pointwise deviation of the deformation gradient ∇u from g1/2 , modulo orientation-preserving rotations that do not cost any energy. We observe the dichotomy: either Eg can be minimized to 0 by means of a smooth isometric immersion, or the energy infimum over all Sobolev-regular deformations is strictly positive. This leads to the question of the scaling of inf Eg on thin prestressed films, in the limit of their vanishing thickness h → 0. The set-up, parallel to that of chapter 5, is given in section 11.2. Based on a deep result in convex integration for isometric immersions, we readily observe that the scaling of order hβ , β < 2/3 is always valid. The purpose of section 11.3 is to investigate the case β = 2. We prove both the compactness and the Γ -liminf lower bound in terms of the Kirchhoff-like energy I2,g , defined on the set of H 2 -regular isometric immersions of the 2-dimensional minor of g on the film’s midplate. The energy I2,g reduces to IK studied in chapter 5, when g = Id3 . It measures the total deviation of curvature induced by an isometric immersion, from the preferred second fundamental form resulting from g; the optimality of this lower bound is proved in section 11.4 by constructing a recovery family. We also recast the obtained results in the framework of Γ -convergence and provide an example of © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_11

269

270

11 Limiting theories for prestressed films: nonlinear bending

a one-parameter family of minimal surfaces which are minimizers to I2,g , representing a continuous transformation from helicoids to a catenoid. In section 11.5, we identify the equivalent conditions for the energy scaling at minimizers to be quadratic. These require the simultaneous existence of an isometric immersion of the midplate metric of regularity H 2 , and the non-vanishing of the three Riemannian curvatures R1212 , R1213 , R1223 of g at the midplate. In the opposite case, the midplate metric and the shape operator given by the integrand of I2,g are compatible in the sense of the Gauss-Codazzi-Meinardi equations, and so there exists an automatically smooth and unique (up to rigid motions) isometric immersion y0 for whom Ig,2 (y0 ) = 0. In section 11.6 we give a coercivity result showing equivalence of I2,g with the squared H 2 distance from thus identified kernel of I2,g . In section 11.7 we compute the energy density of I2,g in case when the energy density of Eg is isotropic. In section 11.8 we explore an example of a specific prestress metric g, pertaining to the model of nematic liquid glass. More examples are discussed in section 11.9. In the final section 11.10, we refer to a few experimental techniques for imposing a target metric in a thin sheet, and point out a curious discrepancy between the experimental and the analytical findings, which is an open problem begging to be addressed in the future.

11.1 Three dimensional non-Euclidean elasticity Let Ω ⊂ R3 be an open, bounded domain, viewed as the reference configuration of an elastic, homogeneous tissue. The tissue is assumed to undergo a growth process, whose instantaneous growth is described by a smooth, symmetric, positive definite 3 matrix field A ∈ C ∞ (Ω¯ , R3×3 pos,sym ). Given a deformation u : Ω → R , we rely on the multiplicative decomposition of its gradient, similar to what is used in plasticity: ∇u = Fe A. The tensor Fe = (∇u)A−1 corresponds to the elastic part of the deformation u, accounting for the reorganization of Ω in response to the growth tensor A. The elastic energy of u is then a function of F only, and it is given by: Eg (u) =

Z

Z

W (Fe ) dx = Ω


W (∇u)A−1 dx,

(11.1)

Ω

where the energy density W : R3×3 → [0, ∞] obeys the principles of material frame invariance, normalization, non-degeneracy, and material consistency in (5.2), (5.3). An advantageous point of view is to consider a smooth Riemannian metric: g  A2 : Ω¯ → R3×3 pos,sym , whose unique symmetric, positive definite square root is given by A. Then:

11.1 Three dimensional non-Euclidean elasticity

Eg (u) =

Z

271


W (∇u)g−1/2 dx.

(11.2)

Ω

We observe that W (Fe ) = 0 if and only if Fe ∈ SO(3) in Ω , or equivalently: (∇u)T ∇u = g

and

det ∇u > 0

in Ω .

(11.3)

Thus, Eg (u) = 0 if and only if u is an orientation preserving isometric immersion of g into R3 . The minimization of the energy (11.2) is then a prescription for shape, defined in terms of the energetic cost of deviating from an isometric immersion. We point out that the above discussion assumes that it is possible to differentiate a reference configuration with respect to which all relative displacements are measured. In general, the tensor A may follow its own dynamical evolution; here, we will focus only on the relation of A to the effective elastic theory for the grown body. In the sequel, we denote |g| = det g and g−1 = [gi j ]i, j=1...3 . In all the differential geometry formulas, the Einstein summation convention will be used. We first recall the following classical result on existence of equidimensional isometric immersions: Theorem. [Fundamental theorem of Riemannian manifolds] Let g ∈ C ∞ (Ω , R3×3 pos,sym ) be a Riemannian metric on an open, bounded and simply connected domain Ω ⊂ R3 . Then, the following conditions are equivalent: (i) there exists u ∈ C ∞ (Ω , R3 ) such that (∇u)T ∇u = g in Ω , (ii) there holds Riem(g) ≡ 0 in Ω , where Riem(g) = [Riklm ]i, j,l,m=1...3 stands for the Riemann curvature tensor with components given by:

1 n p ∂kl gim + ∂im gkl − ∂km gil − ∂il gkm + gnp ΓklnΓimp − Γkm Γil , 2 (11.4)
1 where Γkln = gns ∂k gsl + ∂l gsk − ∂s gkl . 2

Riklm =

The isometric immersion u in (i) is automatically unique up to rigid motions of R3 . The main result of this section states that the infimum of Eg in absence of any forces or boundary conditions is strictly positive (pointing to existence of residual strain) for non-Euclidean metrics, i.e. metrics g for whom none of the, equivalent, conditions in the above theorem holds. More precisely, we have: Theorem 11.1. Let g ∈ C ∞ (Ω , R3×3 pos,sym ) be a Riemannian metric on an open, bounded, simply connected Ω ⊂ R3 . Then, the following are equivalent: (i) there exists u ∈ C ∞ (Ω , R3 ) such that (∇u)T ∇u = g in Ω , (ii) there  holds Riem(g) ≡ 0 in Ω , (iii) inf Eg (u); u ∈ H 1 (Ω , R3 ) = 0.

272

11 Limiting theories for prestressed films: nonlinear bending

Before the proof, we observe that any solution to (11.3) is automatically smooth, satisfying condition (i) above. The proof is a generalisation of the argument in the proof of Liouville’s theorem in Lemma 4.2 in chapter 4: Lemma 11.2. Let g ∈ C ∞ (Ω , R3×3 pos,sym ) be a Riemannian metric on an open, bounded Ω ⊂ R3 . Let u ∈ W 1,1 (Ω , R3 ) satisfy ∇u(x) ∈ SO(3)g1/2 (x) for a.e. x ∈ Ω . Then u is smooth and Riem(g) ≡ 0 in Ω . Proof. By assumption, each component {ui  hu, ei i ∈ W 1,∞ (Ω , R)}3i=1 . Moreover: p p det ∇u = |g|, cof ∇u = |g|(∇u)g−1 . Since div(cof ∇u) = 0 (we recall that the divergence of a matrix field is always taken row-wise), the Laplace-Beltrami operator of each component ui is zero: ∆g ui = 0

in Ω .

(11.5)

We recall that for a scalar field f on Ω , one defines: ∆g f = √1 ∂i ( |g|

p

|g|gi j ∂ j f ).

By (11.5), we conclude that ui ∈ C ∞ (Ω , R). The assertion Riem(g) follows by the Fundamental theorem of Riemannian manifolds, where the simple connectedness assumption is only necessary for the implication (ii)⇒(i). For the proof of Theorem 11.1, we will check the implication (iii)⇒(i), and rely on the Fundamental theorem of Riemannian manifolds. We are ready to give: Proof of Theorem 11.1. 1. Assume that for some sequence of deformations {un ∈ H 1 (Ω , R3 )}n→∞ , there holds limn→∞ Eg (un ) = 0. By the first order truncation result in Theorem 4.3, we obtain the modified, equibounded sequence of Lipschitz deformations {u¯n  uλn ∈ W 1,∞ (Ω , R3 )}n→∞ , where the truncation constant λ > 0 has been chosen so that: |F| ≤ 2 dist2 (F, SO(3)A(x))

for all x ∈ Ω and F ∈ R3×3 with |F| ≥ λ .

We recall the notation A = g1/2 . Consequently, there holds: Eg (u¯n ) ≤ CEg (un ) → 0

as n → ∞,

because: k∇u¯n − ∇un k2L2 (Ω ) ≤ C ≤C

Z {|∇un |>λ }

Z {|∇u|>λ }

|∇un |2 dx


dist2 (∇un )g−1/2 , SO(3) dx ≤ CE (un ).

Using the Poincar´e inequality, after a possible modification by a constant and after passing to a subsequence, if necessary, we get as n → ∞:

11.1 Three dimensional non-Euclidean elasticity

273

weakly in H 1 (Ω , R3 ).

u¯n * u

(11.6)

2. Consider the splitting u¯n = wn + zn , where wn satisfies, component-wise: ∆g wm n =0

m wm n = u¯n

in Ω ,

on ∂ Ω .

The correction zn = u¯n − wn ∈ H01 (Ω , R3 ) satisfies, for all φ ∈ H01 (Ω , R): Z

gi j

Ω

Z Z p p ij m ∂ φ dx = g ∂ φ dx − ∇φ (cof ∇u¯n )m−th row dx. |g|∂i zm |g|∂ u ¯ i n j n j Ω

Ω

Indeed, the last term above equals 0, since the row-wise divergence of the cofactor matrix of ∇u¯n is 0, in view of u¯n being Lipschitz continuous. We put φ = zm n to get: Z

p

2 |g||∇zm n |g dx =

Z

p

2 |g||∇g zm n |g dx

Ω

Ω

Z

=

ij ∂ j zm n g

p


|g|∂i u¯m n − (cof ∇u¯n )m j dx

Ω

≤ k∇zm n kL2 (Ω )

(11.7)

Z p 1/2 |g|(∇u¯n )g−1 − cof ∇u¯n 2 Ω

1/2 , ≤ Ck∇zm n kL2 (Ω ) E (u¯n )

where by ∇g we denote the contravariant gradient. p In order to deduce the last bound in (11.7), we consider the function f (x, F) = |g|Fg−1 − cof F, which is locally Lipschitz continuous with respect to F, uniformly in x ∈ Ω . When FA−1 ∈ SO(3) then for some R ∈ SO(3), and so: cof F = cof (RA) = (det A)RA−1 = p F = RA −1 |g|(RA)g , implying f (x, F) = 0. Hence: | f (x, ∇u¯n (x))|2 ≤ Kdist2 (∇u¯n (x)A−1 (x), SO(3)). Here, K stands for the Lipschitz constant of f on a sufficiently large ball, whose radius is equal to the common bound on {k∇u¯n kL∞ }n→∞ . Form (11.7), we now conclude that k∇zn k2L2 (Ω ) ≤ CE (u¯n ), so: zn → 0

strongly in H 1 (Ω , R3 ).

(11.8)

3. In view of (11.6), (11.8), the sequence {wn }n→∞ must be uniformly bounded in H 1 (Ω , R3 ). Hence, by the local elliptic estimates for the Laplace-Beltrami operator, each ∆g -harmonic component wm n is locally uniformly bounded in a higher norm: m kwm n kH 2 (Ω 0 ) ≤ CΩ 0 kwn kH 1 (Ω ) ≤ CΩ 0

for all Ω 0 b Ω .

1 (Ω , R3 ), and finally yields: This implies that {wn }n→∞ converge to u in Hloc

Eg (u) = 0,

274

11 Limiting theories for prestressed films: nonlinear bending

so that ∇u(x) ∈ SO(3)A(x) for a.e. x ∈ Ω . The proof is done by Lemma 11.2. We close this section by an important auxiliary observation: Remark 11.3. Given a deformation u : Ω → R3 , consider another energy functional measuring the difference between the pull-back metric on Ω and the metric g: Estr (u) 

Z

|(∇u)T ∇u − g|2 dx.

Ω

Contrary to the findings of Theorem 11.1 for the energy Eg , there always exists u ∈ W 1,∞ (Ω , R3 ) such that Estr (u) = 0. On the other hand, if the Riemann curvature tensor Riem(g) does not vanish identically, say Riem(g)(x) , 0 at some x ∈ Ω , then u must have folding structure around x; the realization u cannot be orientation preserving, or reversing, in any open neighborhood of x. These deep results are based on the techniques of convex integration.

11.2 Prestressed thin films From this section on, we turn our attention to thin films, with the notation as in chapter 7. Let ω ⊂ R2 be an open, bounded, connected, Lipschitz domain and let:  h h
Ω h  x = (z,t); z ∈ ω, t ∈ − , 2 2

for all h  1.

(11.9)

Given a Riemannian metric g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ) together with its unique sym1/2 metric, positive definite square root A = g , we pose the elastic energy (per unit thickness) of deformations uh ∈ H 1 (Ω h , R3 ) in agreement with the formula (11.2): Egh (uh )

1  h

Z Ωh


W (∇uh ) A−1 dx.

(11.10)

In the following sections, we will identify the asymptotic behaviour of the approximate minimizers of Egh , in the limit of the vanishing thickness h → 0. Similarly to Part II of this monograph, we will derive the entire hierarchy of the dimensionally reduced models, each valid in their proper scaling regime. These will be, in turn, dictated by the properties of the prestress metric g. We start by an observation of the residual energy upper bound, valid for any g: Theorem 11.4. Let ω ⊂ R2 be an open disk and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Then, there holds: inf Egh ≤ Chβ

2 for all β < . 3

11.2 Prestressed thin films

275

In fact, regularity C 2 of the metric g would suffice for the above result. Construction of the infimizing sequence of deformations will be obtained through the Kirchhoff-Love - like extension of an isometric immersion of the midplate metric G = g(·, 0)tan , whose existence is based on the following deep result: Theorem. [Delellis-Inauen-Szekelyhidi’s theorem] 2 ¯ R2×2 Let G ∈ C 2 (ω, pos,sym ) be a Riemannian metric on an open disk ω ⊂ R . Then, for 1 1,α 3 ¯ R ) such that: every α < 5 , there exists y ∈ C (ω, (∇y)T ∇y = G

in ω.

(11.11)

We remark that in the above theorem, an isometric immersion y can be found ar¯ R3 ) to any fixed “short immersion” y0 ∈ C 1 (ω, ¯ R3 ), i.e. a bitrarily close in C 0 (ω, T ¯ in the sense of quadratic forms. map that satisfies: 0 < (∇y0 ) ∇y0 < G in ω, The following will be used to define the infimizing family in Theorem 11.4: Lemma 11.5. [Commutator estimate] R Let φ ∈ Cc∞ (RN , [0, ∞)) satisfy RN φ dx = 1 and let {φε (x) = ε −N φ (x/ε)}ε→0 be the corresponding family of convolution kernels. Then, given α ∈ (0, 1], for every f , g ∈ C 0,α (RN , R) there holds: k( f g) ∗ φε − ( f ∗ φε )(g ∗ φε )kC 0 ≤ Cε 2α k f kC 0,α kgkC 0,α . Proof. Fix x ∈ RN and observe that: ( f g) ∗ φε − ( f ∗ φε )(g ∗ φε )


= ( f − f (x))(g − g(x) ∗ φε − ( f − f (x)) ∗ φε (g − g(x) ∗ φε . The first term in the right hand side above, evaluated at x, can be estimated by:
( f − f (x))(g − g(x) ∗ φε (x) ≤

Z RN

| f (y) − f (x)| · |g(y) − g(x)| · φε (x − y) dy

≤ k f kC 0,α kgkC 0,α

Z RN

|y − x|2 φε (x − y) dy ≤ ε 2α k f kC 0,α kgkC 0,α .

For the second term, we likewise have:

( f − f (x)) ∗ φε (g − g(x) ∗ φε (x) ≤

Z

Z

| f (y) − f (x)| · φε (x − y) dy ≤ |g(y) − g(x)| · φε (x − y) dy RN RN Z 2 ≤ k f kC 0,α kgkC 0,α |y − x|φε (x − y) dy ≤ ε 2α k f kC 0,α kgkC 0,α . RN

This ends the proof of the lemma. We are now ready to give:

276

11 Limiting theories for prestressed films: nonlinear bending

Proof of Theorem 11.4. ¯ R3 ) satisfy (11.11) for G = g(·, 0)tan . Define 1. Fix α ∈ (0, 15 ) and let y ∈ C 1,α (ω, 0,α 3 ¯ R ) by requesting that: the vector field b ∈ C (ω, ¯ R3×3 ) B  [∂1 y, ∂2 y, b] ∈ C 0,α (ω,

¯ satisfies: det B > 0, BT B = g(·, 0) in ω.

The last assertion above implies that: B(z)A(z0 , 0)−1 ∈ SO(3)

¯ for all z ∈ ω.

(11.12)

¯ R3 )}ε→0 by means of the Regularize now the vector fields y, b to {yε , bε ∈ C ∞ (ω, −2 family of standard convolution kernels {φε (z) = ε φ (z/ε)}ε→0 : yε = y ∗ φε ,

bε = b ∗ φε ,

Bε = B ∗ φε

and

ε = ht ,

where ε is a power of h to be chosen later (and where we extend y and b outside of ω, while keeping their norms controlled). By Lemma 11.5, we obtain:

T

Bε Bε − g(·, 0) 0 C (ω)

T

T

(11.13) ≤ Bε Bε − (B B) ∗ φε C 0 (ω) + g(·, 0) ∗ φε − g(·, 0) C 0 (ω) ≤ Cε 2α +Cε 2 ≤ Cε 2α , where the Cε2 bound results  by Taylor expanding g up to second order terms. Denoting Dε  ∂1 bε , ∂2 bε , 0 , we get the uniform bounds: kBε − BkC 0 (ω) ≤ Cε α ,

kDε kC 0 (ω) ≤ Cε α−1 .

(11.14)

2. Consider the family of deformations: {uh ∈ C ∞ (Ω¯ h , R3 )}h→0 in: uh (z,t)  yε (z) + tbε (z), so that ∇uh = Bε + tDε . In particular, k∇uh (z, ht) − B(z)kC 0 (Ω 1 ) ≤ C(ε α + hε α−1 ) and since: A(z, ht)−1 = A(z)−1 + O(h) for all (x,t) ∈ Ω 1 , by (11.12) we get:




dist (∇uh ) A−1 , SO(3) 0 h ≤ (∇uh ) A−1 (z, ht) − BA−1 (z, 0) 0 1 C (Ω )

C (Ω )

α

≤ C(ε + hε

α−1

+ h) → 0 as h → 0,

provided that hε α−1 → 0. We then use the polar decomposition of matrices and conclude that for some Rh = Rh (z, ht) ∈ SO(3) there holds:  1/2 Rh ∇uh (z, ht)A(z, ht)−1 = A−1 (∇uh )T ∇uh A−1 (z, ht)  1/2
= A(z, ht)−1 BTε Bε (z) + O(hDε ) A(z, ht)−1

11.3 Kirchhoff-like theory for prestressed films: compactness and lower bound

277

which leads to, in virtue of (11.13) and (11.14): Rh ∇uh (z, ht)A(z, ht)−1 

1/2 = A(z, 0)−1 + O(h) (g(z, 0) + O(ε 2α + hε α−1 ) A(z, 0)−1 + O(h)
1/2 = Id3 + O(h + ε 2α + hε α−1
= Id3 + O h + ε 2α + hε α−1 , Consequently, there follows the energy bound: Z


W Rh (∇uh )A−1 (z, ht) dx
2
2 ≤ C h + ε 2α + hε α−1 = C h + h2αt + h1+(α−1)t .

Egh (uh ) =

Ω1

Minimizing the right hand side above corresponds to maximizing the minimal of the three displayed exponents. We hence choose t in ε = ht so that 2αt = 1 + (α − 1)t, 1 . This leads to the estimate: namely t = α+1 4α

inf Egh ≤ Ch α+1

1 for all α < . 5

The proof is done. From the proof above, it is clear that the particular energy scaling bound in Theorem 11.4 is in a direct correspondence with the threshold regularity in the DelellisInauen-Szekelyhidi’s theorem. If one could take α < 1/3 (corresponding to the socalled “one step” in each “stage” of the Nash-Kuiper iteration scheme, on which the proof of Delellis-Inauen-Szekelyhidi’s theorem is based), then the exponent would be β < 1. Taking α < 1/2 would imply β < 4/3. In the next section, we show that existence of a H 2 isometric immersion implies that inf Egh may be decreased to Ch2 .

11.3 Kirchhoff-like theory for prestressed films: compactness and lower bound In this section, we will derive the first identification of the asymptotic behaviour of the approximate minimizers to the energies {Egh }h→0 . Similarly as in Theorem 11.1 and the proof of 11.4, there is a connection between inf Egh and the existence of isometric immersions y : ω → R3 of the midplate metric g(·, 0)tan on ω into R3 : (∇y)T ∇y = g(·, 0)tan

in ω.

(11.15)

It turns out that existence of y as in (11.15) with regularity H 2 , is equivalent to the vanishing of inf E h of order h2 . To formulate this result precisely, we extend the

278

11 Limiting theories for prestressed films: nonlinear bending

definition of the effective, dimensionally reduced energy density in Definition 5.6. In particular, one has Q2,Id3 = Q2 , L2,Id3 = L2 and cId3 = c in the following: ¯ R3×3 Definition 11.6. Given an open domain ω ⊂ R2 , the metric g ∈ C ∞ (ω, pos,sym ) and 3×3 W :R → [0, ∞] satisfying (5.2), we define the quadratic forms {Q2,g (z, ·)}z∈ω : n   o ˜ Q2,g (z, F) = min Q3 g(z, 0)−1/2 Fg(z, 0)−1/2 ; F˜ ∈ R3×3 , F˜tan = F , for all z ∈ ω, F ∈ R2×2 , where: Q3 (F) = D2W Id3 (F, F)


(11.16)

for all F ∈ R3×3 .

For each z ∈ ω, the linear map which determines the unique solution to the minimization problem above will be denoted by cg (z, ·). More precisely, we have: R2×2 3 F 7→ cg (z, F) ∈ R3  
with: Q2,g (z, F) = Q3 g(z, 0)−1/2 F ∗ + cg (z, F) ⊗ e3 g(z, 0)−1/2 .

(11.17)

Finally, the linear operator L2,g (z, ·) : R2×2 → R2×2 is given by:

 Q2,g (z, F) = L2,g (z, F) : F for all z ∈ ω, F ∈ R2×2 . As in Lemma 5.7, the forms Q2,g (z, ·) depend only on the symmetric parts of their arguments and are positive definite on the space of symmetric matrices. We will come back to the discussion of Q2,g and present a few examples in section 11.7.

Fig. 11.1 The limiting fields in the statement of Theorem 11.7: the deformation y of the midplate ω and the Cosserat vector b which induces the volume ratio det g(·, 0)1/2 .

We now state the main result of this section, generalizing the Kirchhoff’s dimensionally reduced theory in Theorem 5.10, to the setting of prestressed elasticity. Namely, we have the following compactness and the lower bound statements:

11.3 Kirchhoff-like theory for prestressed films: compactness and lower bound

279

Theorem 11.7. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz do3×3 main and let g ∈ C ∞ (Ω¯ 1 , Rpos,sym ). Assume that the family of deformations h 1 h 3 {u ∈ H (Ω , R )}h→0 satisfies: Egh (uh ) ≤ Ch2 .

(11.18)

Then, there exists {ch ∈ R3 }h→0 such that rescalings defined by {yh ∈ H 1 (Ω 1 , R3 )}h→0 in yh (z,t)  uh (z, ht) − ch , obey: (i) up to a (not relabeled subsequence), {yh }h→0 converges in H 1 (Ω 1 , R3 ) to the limit y = y(z) that depends only on the in-plane variables, (ii) there holds y ∈ H 2 (ω, R3 ) and (11.15), (iii) we have the following lower bound: 1 h h E (u ) h2 g (11.19) Z   1 1 ≥ I2,g (y)  Q2 z, (∇y(z))T ∇b(z) − ∂3 g(z, 0) dz, 24 ω 2   where b ∈ H 1 ∩L∞ (ω, R3 ) is such that ∂1 y, ∂2 y, b ∈ SO(3)g(·, 0)1/2 , and where the quadratic forms Q2 (z, ·) are given in (11.16). Equivalently: √   ∂1 y × ∂2 y det g g . (11.20) N, N = b = (∇y)(gtan )−1 13 + √ g23 |∂1 y × ∂2 y| det gtan lim inf h→0

Remark 11.8. (i) The vector b in (11.20) is the Cosserat vector, comprising the sheer directions in addition to the direction N that is normal to the surface y(ω). In the analysis of the dimensionally reduced theories of prestressed films at higher energy scalings hβ where β > 2, we will similarly encounter the higher order Cosserat vectors {bi }∞ i=1 . As above, the defining property of these vector fields will be that, together with  the gradients of higher order displacements {yi }∞ , they form frame fields ∂1 yi , ∂2 yi , bi+1 which up to rotations coincide i=1 i 1/2 with (∂3 ) g(·, 0) . (ii) When g = Id3 , the functional I2,g reduces to the Kirchhoff’s plate energy IK defined in (5.15). In the general case, I2,g measures the total difference of curvature (bending), induced by the deformation y that is an isometric immersion of the midplate metric g(·, 0)tan . The bending is quantified through the quadratic
form Q2,g applied on the difference of the shape operator Π¯ = sym (∇y)T ∇b in the ambient metric g, and the “ideal” shape operator Π = 21 ∂3 g(·, 0). Indeed, if we had (∇u)T ∇u = g for some u : Ω h → R3 and h  1, then there would be for all i, j = 1, 2: Πi j =


1 1 1 h∂i u, ∂ j3 ui + h∂ j u, ∂i3 ui = ∂3 h∂i u, ∂ j ui = (∂3 g)i j , 2 2 2

280

11 Limiting theories for prestressed films: nonlinear bending

so Π would be the shape operator of the surface that is image of ω under an isometric immersion u of g. To prove Theorem 11.7, we need an approximation result based on Theorem 4.8: Lemma 11.9. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ 1 h 3 1 3×3 ), with: C ∞ (Ω¯ 1 , R3×3 pos,sym ). Then, for every u ∈ H (Ω , R ) there is Q ∈ H (ω, R 1 h

Z Ωh

|∇u − Q ◦ π|2 dx ≤ C(h2 + Egh (u)), (11.21)


1 |∇Q| dz ≤ C 1 + 2 Egh (u) , h ω

Z

2

where the constants C depend only on ω and g, but are independent of u and h  1. Proof. 1. Define local neighbourhoods on ω and Ω h : h h
Bz,h = Dz,h × − , 2 2

Dz,h = Bh (z) ∩ ω,

for all z ∈ ω,

where Bh (z) denotes a ball in R2 . We claim that for each z ∈ ω and h  1, there exists a fixed matrix Qz,h ∈ R3×3 such that: Z Bz,h

|∇u(x) − Qz,h |2 dx ≤ C

Z


dist2 (∇u)A−1 , SO(3) dx

Bz,h



+ hk∇gk2L∞ |Bz,h |

(11.22) ,

with C depending on kgkL∞ , kg−1 kL∞ and on the domain ω. Indeed, fix z and h, and denote A0  A(z, h). We now apply Friesecke- James-M¨uller’s inequality in Theorem 1 3 4.1 to the v(x)  u(A−1 0 x) ∈ H (A0 Bz,h , R ). After changing variables, we get: Z Bz,h

2 |(∇u)A−1 0 −R| dx ≤ CA0 Bz,h

Z Bz,h

dist2 ((∇u)A−1 0 , SO(3)) dx for some R ∈ SO(3).

Since the set A0 Bz,h is a bilipschitz image of Bz,h , constants CA0 Bz,h have a uniform bound C depending on |A0 |, |A−1 0 | and Bz,h , but independent of u. By the uniformity of the Friesecke-James-M¨uller constant on bilipschitz equivalent domains and domains equivalent under dilation, this C is also independent of z and h. Further: Z Bz,h

|∇u − RA0 |2 dx ≤ |A0 |2

≤ C|A0 |

2

2 |A−1 0 |

2 ≤ C|A0 |2 |A−1 0 |

Z Bz,h

Bz,h

dist2 ((∇u)A−1 0 , SO(3)) dx

dist2 (∇u, SO(3)A0 ) dx Z
dist2 ∇u, SO(3)A(x) dx +

Z

≤ CkgkL∞ ,kg−1 kL∞

Z

Bz,h

Bz,h

Z Bz,h

|A(x) − A0 |2 dx



 dist2 (∇u)A−1 , SO(3) dx +Ck∇gk2L∞ (diam Bz,h )2 |Bz,h | .


11.3 Kirchhoff-like theory for prestressed films: compactness and lower bound

281

This proves (11.22) with Qz,h = RA0 . R 2. The final claim is achieved by defining Q(z) = Ω h ηz (x)∇u(x) dx, where the family of mollifiers {ηz : Ω h → R}z∈ω is as in the proof of Theorem 4.8. Following verbatim the proof of Theorem 3.23, where instead of Korn’s inequality we use (11.22) on each Bz,h , leads to the estimates in (11.21). We are now ready to follow the proof of Theorem 5.10 and give: Proof of Theorem 11.7. 1. By Lemma 11.9, there exists a family {Qh ∈ H 1 (ω, R3×3 )}h→0 such that: 1 2

Z Ωh

|∇uh − Qh ◦ π|2 dx ≤ Ch2 ,

Z

|∇Qh |2 dz ≤ C.

ω

In particular, {Qh }h→0 is bounded in H 1 (ω, R3×3 ), so up to a subsequence: weakly in H 1 (ω, R3×3 ).

Qh * Q We take ch = Z Ω1

> Ωh

(11.23)

uh dx and consider yh (z,t) = uh (z, ht) − ch ∈ H 1 (Ω 1 , R3 ). Since:

|∇uh (z, ht) − Q ◦ π|2 dx ≤ 2

it follows by (11.23) that:  ∂1

yh

∂2

yh

∂3 yh h

Z Ω1

|∇uh (z, ht) − Qh ◦ π|2 dx + 2

Z

|Qh − Q|2 dz,

ω

 →Q

strongly in L2 (Ω 1 , R3×3 ).

(11.24)

In particular, {∇yh }h→0 is bounded in L2 , so up to a subsequence (that we do not relabel) it converges weakly in H 1 (Ω 1 , R3 ) to some y, where we also used the Poincar´e inequality. On the other hand, (11.24) implies: ∇tan yh → Q3×2

and

∂3 yh → 0

strongly in L2 (Ω 1 ).

Consequently, convergence of {yh }h→0 is strong and y = y(z) ∈ H 2 (ω, R3 ) with: ∇y = ∇tan y = Q3×2 .

(11.25)

To conclude the proof of (i) and (ii), it suffices to check (11.15). Note that: Z ω

dist2 (Qh A(·, 0)−1 , SO(3)) dz Z Z C ≤ dist2 (∇uh A−1 , SO(3)) dx + |∇uh − Qh ◦ π|2 dx h Ωh Ωh Z  + |A−1 − (A ◦ π)−1 |2 ≤ Ch2 . Ωh

Consequently, by (11.23) we obtain:

(11.26)

282

11 Limiting theories for prestressed films: nonlinear bending

QA(·, 0)−1 ∈ SO(3)

a.e. in ω,

(11.27)

In particular, recalling (11.25) implies (11.15). We also note that, automatically: ∇y ∈ H 1 ∩ L∞ (ω, R3×2 ).

(11.28)

2. We will now prove that, assuming (11.25) and (11.15), condition (11.27) is equivalent to b = Qe3 satisfy (11.20). Firstly, using the formula a × (b × c) = ha, cib − ha, bic, we note that: |∂1 y × ∂2 y|2 = |A(·, 0)e1 × A(·, 0)e2 |2 E D = A(·, 0)e1 , (A(·, 0)e1 × A(·, 0)e2 ) × A(·, 0)e1 D  E (11.29) 

= A(·, 0)e2 , hA(·, 0)e1 , A(·, 0)e1 A(·, 0)e2 − A(·, 0)e1 , A(·, 0)e1 Ae1
= g11 g22 − g212 (·, 0) = det g(·, 0)tan . Hence, in view of (11.28), we get N ∈ H 1 ∩ L∞ (ω, R3 ) and the same holds for b. Further, writing: b = α1 ∂1 y + α2 ∂2 y + α3 N and using (11.29), we obtain: p   det Q = det ∂1 y ∂2 y α3 N = α3 |∂1 y × ∂2 y| = α3 det g(·, 0)tan . Now, (11.27) is equivalent to QT Q = g(·, 0) and det Q > 0, and further to: g13 = hb, ∂1 yi = α1 g11 + α2 g12 , g23 = hb, ∂2 yi = α1 g21 + α2 g22 , p p det g(·, 0) = det Q = α3 det g(·, 0)tan , which yields: 

   α1 g = (gtan )−1 13 (·, 0), α2 g23

√ det g α3 = √ (·, 0), det g2×2

precisely as claimed in (11.20). 3. Towards proving the lower bound in (iii), we modify the family {Qh }h→0 to {Q˜ h ∈ L2 (ω, R3×3 )}h→0 so that: Rh  Q˜ h A(·, 0)−1 ∈ SO(3)

a.e. in ω,

This is done by projecting PSO(3) onto SO(3) when possible, and setting: Q˜ h A(·, 0)−1 =



PSO(3) (Qh A(·, 0)−1 ) if Qh A(·, 0)−1 ∈ Oε (SO(3)) Id3 otherwise

with some ε  1. Then, by (11.26) we get:

11.3 Kirchhoff-like theory for prestressed films: compactness and lower bound

Z

|Q˜ h − Qh |2 dz ≤ C

Z

ω

|Q˜ h A(·, 0)−1 − Qh (·, 0)A−1 |2 dz

ω

≤C

Z

283

(11.30) dist2 (Qh A(·, 0)−1 , SO(3)) dz ≤ Ch2 .

ω

In particular, by (11.23): Q˜ h → Q

strongly in L2 (ω, R3×3 ).

(11.31)

Consider now the scaled strains {Z h ∈ L2 (Ω 1 , R3×3 )}h→0 , given by: Z h (z,t) 


1 h T h R (z) ∇u (z, ht)A(z, ht)−1 − Id3 . h

We then have, in view of (11.21) and (11.30): Z C ∇uh (z, ht) − Rh (z)A(z, ht) 2 dx |Z h |2 dx ≤ 2 h Ω1 Ω1 (11.32) Z 1 Z  1 ≤C 3 |Qh − Q˜ h |2 dz ≤ C, |∇uh − Qh |2 + |A − A ◦ π|2 dx + 2 h Ωh h ω

Z

and hence a subsequence (not relabeled) of {Z h }h→0 converges: Zh * Z

weakly in L2 (Ω 1 , R3×3 ).

(11.33)

4. To derive the formula on the strain Z, consider the difference quotients: f s,h (z,t) 


11 h y (z,t + s) − yh (z,t) ∈ L2 (Ω 1 , R3×3 ). hs

By (11.24), (11.31) and (11.33) it follows that: ? 1 s f s,h (z,t) = ∂3 yh (z,t + τ) dτ → b(z) in L2 (Ω 1 , R3 ), h 0
1 in L2 (Ω 1 , R3 ), ∂3 f s,h (z,t) = h−1 ∂3 yh (z,t + s) − h−1 ∂3 yh (z,t) → 0 s and likewise, for i = 1, 2 we get the weak convergence in L2 (Ω 1 , R3 ):
11 ∂i f s,h (z,t) = ∇uh (z, h(t + s))) − ∇uh (z, ht) ei hs  1 h  h = R (z) Z (z,t + s)A(z, h(t + s)) − Z h (z,t)A(z, ht) ei s  11 h  + R (z) A(z, h(t + s)) − A(z, ht) ei hs
1 * Q(z)A(z, 0)−1 Z(z,t + s) − Z(z,t) A(z, 0)ei + Q(z)A(z, 0)−1 ∂3 A(z, 0)ei . s Concluding, we obtain:

284

11 Limiting theories for prestressed films: nonlinear bending

f s,h * b

weakly in H 1 (Ω 1 , R3 ),

and consequently, for i = 1, 2 there holds:
1 ∂i b(z) = Q(z)A(z, 0)−1 Z(z,t + s) − Z(z,t) A(z, 0)ei s + Q(z)A(z, 0)−1 ∂3 A(z, 0)ei . By (11.27), one can replace QA(·, 0)−1 by QT,−1 A(·, 0), so that for i = 1, 2:     Z(z,t + s) − Z(z,t) A(z, 0)ei = s A(z, 0)−1 Q(z)T ∂i b(z) − ∂3 A(z, 0)ei , and in view of (11.25) we obtain:

A(z, 0)Z(z,t + s)A(z, 0) tan = A(z, 0)Z(z,t)A(z, 0) tan 
 + s (∇y(z))T ∇b(z) − A(z, 0)∂3 A(z, 0) tan .

(11.34)

5. We now compute the lower bound on the rescaled energies. Define the characteristic functions {χh = χ{x∈Ω 1 ; |Z h (z)|≤h−1/2 } }h→0 . By (11.32), (11.33), we get: χh Z h * Z

weakly in L2 (Ω 1 , R3×3 ).

Now, Taylor expanding W on the “good” sets where χh = 1, we obtain: lim inf h→0

Z   1 1 h h h E (u ) ≥ lim inf χ W Id + hZ dx 3 h h→0 h2 Ω 1 h2 g Z Z   1 1 h Q3 χh Z dx ≥ Q3 (Z) dx. = lim inf h→0 2 Ω 1 2 Ω1

(11.35)

Since the quadratic form Q3 is nonnegative definite, we finally get: 1 2

Z   1 Q3 (Z) dx ≥ Q2,g z, (A(z0 , 0)Z(z,t)A(z,0 0))tan d(z,t) 2 Ω1 Ω1 Z Z 1/2 

 1 ≥ Q2,g sym (∇y)T ∇b − sym A(·, 0)∂3 A(·, 0) tan dz t 2 dt 2 −1/2 ω Z  
1 1 Q2,g sym (∇y)T ∇b − ∂3 g(·, 0)tan dz = I2,g (y), ≥ 24 ω 2

Z

where we used (11.34). In view of (11.35), the proof is complete.

11.4 Kirchhoff-like theory for prestressed films: recovery family

285

11.4 Kirchhoff-like theory for prestressed films: recovery family In this section, we prove that the lower bound in Theorem 11.7 is optimal, in the following sense: Theorem 11.10. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz 3×3 domain and let g ∈ C ∞ (Ω¯ 1 , Rpos,sym ). Then, for every y ∈ H 2 (ω, R3 ) satisfying (11.15), there exists a family {uh ∈ H 1 (Ω h , R3 )}h→0 , such that: (i) the family {yh (z,t) = uh (z, ht)}h→0 converges in H 1 (Ω 1 , R3 ) to y ◦ π, 1 (ii) one has: lim 2 E h (uh ) = I2,g (y), where the Cosserat vector b in the h→0 h definition (11.19) of the functional I2,g is derived by (11.20). Proof. 1. Let y ∈ H 2 (Ω , R3 ) satisfy (11.15). Define the Cosserat vector field b ∈ H 1 ∩ L∞ (ω, R3 ) according to (11.20) and let:   Q  ∂1 y ∂2 y b ∈ H 1 ∩ L∞ (ω, R3×3 ). With the help of (11.17), we define the warping field {d ∈2 (ω, R3 ): 1 
1 ∇|b|2 − ∂3 g(·, 0)13,23 , d  QT,−1 c z, (∇y)T ∇b − ∂3 g(·, 0)tan − QT,−1 2 − 12 ∂3 g(·, 0)33 2 and its approximations {d h ∈ W 1,∞ (ω, R3 )}h→0 in: dh → d

in L2 (ω, R3 ),

hkd h kW 1,∞ → 0

as h → 0.

(11.36)

Further, by Lemma 5.13 we obtain the approximations {yh ∈ W 2,∞ (ω, R3 )}h→0 and {bh ∈ W 1,∞ (ω, R3 )}h→0 , with the following properties: yh → y strongly in H 2 (ω, R3 ), bh → b strongly in H 1,2 (ω, R3 ),
h kyh kW 2,∞ + kbh kW 1,∞ ≤ ε, (11.37)  1 |ω \ ωh | → 0 where ωh = z ∈ ω; yh (z) = y(z) and bh (z) = b(z) , h2 for some ε  1. We now define uh ∈ W 1,∞ (Ω h , R3 ) by: uh (z,t) = yh (z) + tbh (z) +

t2 h d (z). 2 2

The rescalings yh ∈ W 1,∞ (Ω 1 , R3 ) are yh (z,t) = yh (z) + htbh (z) + h2 t 2 d h (z), and therefore in view of (11.36) and (11.37), the assertion (i) follows directly. 2. Define the matrix fields:

286

11 Limiting theories for prestressed films: nonlinear bending

  Qh (x0 ) = ∂1 yh ∂2 yh bh ,

  Bh (x0 ) = ∂1 bh ∂2 bh d h ,

  Dh (x0 ) = ∂1 d h ∂2 d h 0 ,

Since Qh = Q in the set ωh , and since QA(·, 0)−1 ∈ SO(3) by the same argument as in step 1 of the proof of Theorem 11.7, the bound on the Lipschitz constants of yh and bh in (11.37), we obtain: dist(Qh A(z, 0)−1 , SO(3)) ≤

C C dist(z, ωh ) ≤ |ω \ ωh |1/2 ≤ oh (1). h h

Observe that: ∇uh (z,t) = Qh (z) + tBh (z) +

(11.38)

t2 h D (z), 2

so (11.38) and (11.37), imply for h  1 and ε  1:   dist ∇uh (z, ht)A(z, ht)−1 , SO(3) ≤ dist(Qh A(·, 0)−1 , SO(3)) + hkQh kL∞ (ω) + hkBh kL∞ (ω) + h2 kDh kL∞ (ω) ≤ Cε +Ch(k∇bh kL∞ (ω) + kd h kL∞ (ω) ) +Ch2 k∇d h kL∞ (ω) ≤ ε0 , where we choose ε0  1 to be such that the energy density W is bounded and C 2 regular in Oε0 (SO(3)). We will use this property on the “good sets” ωh , to Taylor expand W at the rotations QA(·, 0)−1 , where we compute:  h2 t 2 h  D (z) · ∇uh (z, ht)A(z, ht)−1 = Q(z) + htBh (z) + 2   · A(z, 0)−1 − htA(z, 0)−1 ∂3 A(z, 0)A(z, 0)−1 + O(h2 )   = Q(z)A(z, 0)−1 + ht Bh (z) − Q(z)A(z, 0)−1 ∂3 A(z, 0) A(z, 0)−1
+ O(h2 ) kQkL∞ (ω) + kBh kL∞ (ω) + kDh kL∞ (ω)   = Q(z)A(z, 0)−1 + ht Bh (z) − Q(z)A(z, 0)−1 ∂3 A(z, 0) A(z, 0)−1 + O(hε). This leads to:
1 W ∇uh (z, ht)A(z, ht)−1 h2 ⊗2 
1 + oh (ε) = D2W (QA(·, 0)−1 ) t Bh − QA(·, 0)−1 ∂3 A(·, 0) A(·, 0)−1 + O(ε) 2  
1 = Q3 t A(·, 0)−1 QT Bh − ∂3 A(·, 0) A(·, 0)−1 + O(ε) + oh (ε), 2 where we have used the frame invariance of W resulting in: D2W (R)(F, F) = D2W (Id3 )(RT F, RT F) = Q3 (RT F) valid for all R ∈ SO(3), F ∈ R3×3 . Also, by (11.37) we estimate the energy on the remaining set:

11.4 Kirchhoff-like theory for prestressed films: recovery family

1 h2


C h −1 W ∇u (z, ht)A(z, ht) dx ≤ 2 |ω \ ωh | → 0 h (ω\ωh )×(− 12 , 21 )

Z

287

as h → 0.

3. By applying a diagonal procedure in order to eliminate the error terms, the last two displayed bounds finally yield (for a “diagonal” modified family {uh }h→0 ): 1 1 h h E (u ) = lim 2 2 h→0 h h→0 h lim

Z ωh ×(− 21 , 12 )

  W ∇uh (z, ht)A(z, ht)−1 dx

Z 1/2 Z

 
1 t 2 Q3 A(·, 0)−1 QT Bh − A(·, 0)∂3 A(·, 0) A(·, 0)−1 dz dt lim 2 h→0 −1/2 ωh Z  
1 1 Q3 A(·, 0)−1 sym(QT B) − ∂3 g(·, 0) A(·, 0)−1 dz, = 24 ω 2 =

in view of the following convergence:   Bh → B  ∂1 b ∂2 b d

in L2 (ω, R3×3 ).

Now, note that by (11.36):  1 1 sym(QT B) − ∂3 g(·, 0) = sym ((∇y)T ∇b − ∂3 g(·, 0)tan )∗ 2 2    (∇y)T d + 12 ∇|b|2 − ∂3 g(·, 0)13,23 ⊗ e3 + hb, di − 21 ∂3 g(·, 0)33  
∗
1 1 = sym (∇y)T ∇b − ∂3 g(·, 0)tan + c z, (∇y)T ∇b − ∂3 g(·, 0)tan ⊗ e3 . 2 2 In particular, we obtain: 1 1 h h E (u ) = 2 h→0 h 24 lim

  1 Q2 z, (∇y)T ∇b − ∂3 g(·, 0)tan dz, 2 Ω

Z

achieving the proof. We now state three direct consequences of Theorems 11.7 and 11.10. Firstly, we get the identification of the quadratic energy scaling at minimizers, in terms of the immersability of the midplate metric into R3 , with regularity H 2 : Corollary 11.11. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz 3×3 domain and let g ∈ C ∞ (Ω¯ 1 , Rpos,sym ). Then, the following are equivalent: (i) the metric g(·, 0)tan has an isometric immersion u ∈ H 2 (ω, R3 ), i.e. (11.15) holds a.e. in ω, (ii) infH 1 (Ω h ,R3 ) E h ≤ Ch2 as h → 0, with a uniform constant C.

288

11 Limiting theories for prestressed films: nonlinear bending

Secondly, we observe that the limiting energy I2,g attains its infimum. This is in agreement with the more precise coercivity results in section 11.6. Corollary 11.12. In the context of Corollary 11.11, if any of the equivalent conditions (i) or (ii) holds, then I2,g attains its minimum among y ∈ H 2 (ω.R3 ) satisfying (11.15). Moreover, there holds: limh→0 inf Egh = min I2,g . Proof. Let {yn }∞ n=1 be a minimizing sequence of I2,g . By Theorem 11.7, there exist families {uhn ∈ H 1 (Ω h , R3 )}h→0 such that: limh→0 uhn (z, ht) = yn in H 1 (Ω 1 , R3 ) and limh→0 h12 Egh (uhn ) = I2,g (yn ), for every n ≥ 1. Taking uh  uhn(h) for a sequence

{n(h))}h→0 converging to ∞ sufficiently slowly, we obtain: Egh (uh ) ≤ Ch2 . Therefore, by Theorem 11.7 there exists a limiting deformation y ∈ H 2 (ω, R3 ) that satisfies (11.15) and (for a possibly even slower increasing sequence {n(h))}h→0 ): I2,g (y) ≤ lim inf h→0

1 h h E (u ) = lim I2,g (yn ) = inf I2,g . n→∞ h2 g

This achieves that y is a minimizer of I2,g , together with the convergence claim. The proof is done. Finally, the dimensionally reduced bounds of Theorems 11.7 and 11.7 may be restated using the language of Γ -convergence: Corollary 11.13. In the context of Corollary 11.11, the following Γ -convergence holds, with respect to the strong convergence in H 1 (Ω 1 , R3 ):  1 h Γ I2,g (y) if y ∈ H 2 (ω, R3 ) satisfies (11.15) F −→ 2 +∞ otherwise, h where F h (y)  Egh (uh ) and

uh (z,t) = y(z,t/h).

There is a one-to-one correspondence between limits of sequences of (global) approximate minimizers to the energies Egh and (global) minimizers of I2,g . We end this section by an example of a one-parameter family of minimizers to the Kirchhoff-like energy I2,g , which are minimal surfaces. The family represents a continuous transition between the ruled surfaces known as helicoids, to a catenoid.
Example 11.14. Consider the region: ω = (0, 2π) × − π2 , π2 ⊂ R2 and a prestress metric on Ω 1 = ω × ( 12 , 21 ) which depends only on the midplate variables z ∈ ω:
g(z,t)  diag (cosh z2 )2 , (cosh z2 )2 , 1 . ¯ R3 )}s∈R of deformations of ω: Given is a family {ys ∈ C ∞ (ω,   cos s sinh z2 sin z1 + sin s cosh z2 cos z1 ys (z1 , z2 )   − cos s sinh z2 cos z1 + sin s cosh z2 sin z1  for all z ∈ ω, s ∈ R. z1 cos s + z2 sin s

11.4 Kirchhoff-like theory for prestressed films: recovery family

289

Fig. 11.2 The catenoid to helicoid family of minimizers in Example 11.14. Figure thanks to Neufeld [2011].

For s ∈ π/2 + Zπ each resulting surface ys (ω) is a catenoid. For all other values of s it is a helicoid (see Figure 11.2). To compute the deformation gradient:   cos s cosh z2 sin z1 cos s sinh z2 cos z1  + sin s sinh z2 cos z1  − sin s cosh z2 sin z1    cos s sinh z sin z − cos s cosh z2 cos z1  ∇ys (z1 , z2 ) =   , (11.39) 2 1    + sin s cosh z2 cos z1 + sin s sinh z2 sin z1  cos s sin s It thus follows that the pull-back metric induced by ys , equals: (∇ys )T ∇ys = (cosh z2 )2 Id2 = g(·, 0)tan  G. We now calculate the unit normal vector N = b which is independent of s. We get that |∂1 ys × ∂2 ys | = (cosh z2 )2 and further: N(z1 , z2 ) =

T 1  ∂1 ys × ∂2 ys = cos z1 , sin z1 , −sinh z2 . |∂1 ys × ∂2 ys | cosh z2

To identify the second fundamental form Πs = (∇ys )T ∇N of ys (ω), compute:   − sin z1 cosh z2 − sinh z2 cos z1 1  cos z1 cosh z2 sinh z2 sin z1  , ∇N(z1 , z2 ) = (cosh z2 )2 0 −1 which combined with (11.39) yields:     1 sin s − cos s sin s − cos s −1/2 −1/2 . Πs = , G Πs G = − cos s − sin s (cosh z2 )2 − cos s − sin s

290

11 Limiting theories for prestressed films: nonlinear bending

Assume now that the energy density W is isotropic, so that the formula (5.9) holds. For F = sym F = G−1/2 Πs G−1/2 , we have: |F|2 = |tr F|2 − 2 det F, and hence the integrand in the Kirchhoff-like energy I2,g in (11.19), becomes: Q2,Id3 (F) =

det Πs µ 2 + 2µλ |tr F|2 − 2µ . µ +λ det G

(11.40)

The second term above is intrinsic to G, and given by its Gaussian curvature κG , det Π 2µ namely: −2µ det Gy = −2µκG = (cosh . We now claim that: z )4 2

I2,g (ys ) = min I2,g =

µ 12

Z

(cosh z2 )−4 dz

for all s.

ω

Indeed, from (11.40) it follows that I
K is minimized by any y ∈ H 2 (ω, R3 ) with (11.15) and tr G−1/2 (∇y)T (∇N)G−1/2 = 0. Both conditions hold for each ys .

11.5 Identification of Kirchhoff’s scaling regime In this section, we identify the equivalent conditions for inf Egh ' h2 in terms of curvatures of the Riemannian metric g. We begin by expressing the integrand in the residual energy I2,g in terms of the shape operator on the deformed midplate. ¯ R3×3 Lemma 11.15. Let ω ⊂ R2 be open, bounded and connected and let g ∈ C ∞ (ω, pos,sym ). 2 3 Assume that y ∈ H (ω, R ) satisfies (11.15) and define the Cosserat vector b according to (11.20). Then, for a.e. in ω, there holds:   3
1 1 1 Γ11 Γ123 (∇y)T ∇b sym − ∂3 g(·, 0)sym = p Πy + 33 (·, 0). (11.41) Γ123 Γ223 2 g g33 Above, Πy = (∇y)T ∇N ∈ H 1 (ω, R2×2 sym ) is the second fundamental form of the sur3 face y(ω) ⊂ R , and the Christoffel symbols {Γkli }i,k,l=1...3 are given in (11.4).   Proof. Denote Q = ∂1 y, ∂2 y, b . Since QT Q = g(·, 0), we obtain: h∂1 y, ∂1 bi = ∂1 h∂1 y, bi − h∂11 y, bi = ∂1 g13 − h∂11 y, bi, h∂2 y, ∂2 bi = ∂2 h∂2 y, bi − h∂22 y, bi = ∂2 g23 − h∂22 y, bi, h∂1 y, ∂2 bi + h∂2 y, ∂1 bi = ∂1 g23 + ∂2 g13 − 2h∂12 y, bi. In conclusion:
 
  (∇y)T ∇b sym = ∂i g j3 (·, 0) i, j=1,2 sym − h∂i j y, bi i, j=1,2 . On the other hand, ∂i g(·, 0) = 2 (∂i Q)T Q


sym

for i = 1, 2, results in:

(11.42)

11.5 Identification of Kirchhoff’s scaling regime

h∂i j y, ∂k yi =

291


1 ∂i gk j + ∂ j gik − ∂k gi j (·, 0), 2

(11.43)

which we rewrite as: (∇y)T ∂i j y = Γi m j



 gm1 (·, 0) gm2

for i, j = 1, 2.

Consequently, there follows the formula: 

[g13 , g23 ] (gtan )−1 (∇y)T ∂i j y = g13 , g23 , [g13 , g23 ] (gtan )−1 = gm3Γi m j −



g13 g23

   Γi 1j  Γi 2j  Γi 3j

1 3 Γ , g33 i j

where all the metric coefficients are evaluated at (·, 0). Computing the normal vector N from (11.20). and noting that det gtan / det g = g33 , we get:  p  Πi j = −h∂i j y, Ni = − g33 h∂i j y, bi − [g13 , g23 ] (gtan )−1 (∇y)T ∂i j y p p
1 g33 3 T 33 ∂3 gi j (·, 0), = g (∇y) ∇b sym,i j − p Γi j (·, 0) − 2 g33 which yields (11.41) in view of (11.42) and the compatibility (the metric compatim bility of the Levi-Civita connection) equation: ∂i g jk = gmk Γi m j + gm j Γik . The key result of this section concerns the equivalent conditions for the elastic prestressed energies at minimizers, to scale beyond the Kirchhoff-like regime: Theorem 11.16. Let ω ⊂ R2 be an open, bounded, simply connected and Lipschitz domain and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Then, the following are equivalent: 1 inf Egh = 0. h→0 h2 (ii) There exists y0 ∈ H 2 (ω, R3 ) satisfying (11.15) and such that: (i) We have: lim

1 Πy0 = − p g33



Γ113 Γ123

Γ123 Γ223

 (·, 0)

in ω,

(11.44)

where Πy0 is the second fundamental form of the surface y0 (ω). The isometric immersion y0 is automatically smooth (up to the boundary) and it is unique up to rigid motions. (iii) The following Riemann curvatures of g vanish on the midplate: R1212 (·, 0) = R1213 (·, 0) = R1223 (·, 0) = 0

in ω.

(11.45)

292

11 Limiting theories for prestressed films: nonlinear bending

Proof. 1. Equivalence of (i) and (ii) follows by Corollary 11.12 and Lemma 11.15. Regularity of y0 is a consequence, via the bootstrap, of the continuity equation: 2

∂i j y0 =

∑ γimj ∂m y0 − (Πy0 )i j N0

for i, j = 1, 2,

(11.46)

m=1

where {γimj }i, j,m=1,2 denote the Christoffel symbols of the metric g2×2 (·, 0) on ω, and N0 is the unit normal vector to the surface y0 (ω). Uniqueness of y0 is a consequence of (11.44), due to the uniqueness of an isometric immersion with prescribed second fundamental form. 2. To show the equivalence (ii) and (iii), we observe that the compatibility of the metric G  gtan (0, ·) and Πy0 given in the right hand side of (11.44) is equivalent to the satisfaction of the related Gauss-Codazzi-Mainardi equations:  2

1 1 1  2 3 m 3 m Γ1m γ12 − ∑ Γ2m γ11 ∂2 p Γ113 − ∂1 p Γ123 = p ∑ g33 g33 g33 m=1 m=1   2 2

1 1 1 3 m 3 m Γ1m γ22 − ∑ Γ2m γ12 ∂2 p Γ123 − ∂1 p Γ223 = p ∑ g33 g33 g33 m=1 m=1

(11.47)

together with the Gauss equation: Γ113 Γ223 − (Γ123 )2 = g33 κ det G.

(11.48)

Above, by γikj and κ = κ(G) we denote the Christoffel symbols and the Gaussian curvature of G, respectively. Since the summation of indices related to G extends from 1 to 2, and that related to g from 1 to 3, in what follows we temporarily drop the Einstein summation convention, to avoid confusion. We will prove that (11.47), (11.48) are equivalent to (11.45). We first relate the Christoffel symbols {γkls }s,k,l=1,2 and {Γkls }s,k,l=1...3 . Coefficients of the inverse matrices to G and g are related by the formula: Gi j = gi j −

1 3i 3 j g g g33

for all i, j = 1, 2.

We get: γkls = =

1 2 sm ∑ G (∂l Gmk + ∂k Gml − ∂m Gkl ) 2 m=1
1 2 sm 1 g3s 2 g (∂l gmk + ∂k gml − ∂m gkl ) − 33 ∑ g3m ∂l gmk + ∂k gml − ∂m gkl , ∑ 2 m=1 2 g m=1

which implies that:

11.5 Identification of Kirchhoff’s scaling regime

293

 g3s  γkls = Γkls − g3s (∂l g3k + ∂k g3l ) − 33 Γkl3 − g33 (∂l g3k + ∂k g3l ) g 3s g = Γkls − 33 Γkl3 . g

(11.49)

Also, using the metric-compatibility of the Levi-Civita connection: 3

∂i g jk = −

3

∑ gmkΓmij + ∑ gm j Γmik

m=1




,

(11.50)

m=1

we obtain, for i = 1, 2: p 1 3 m3 3 1 ∂i g33 1 = g33 ∂i ( p ) = − ∑ g Γmi . 2 g33 g33 m=1 g33

(11.51)

3. By an explicit calculation, the Codazzi-Mainardi equations (11.47) become:
1  ∂2 g33 3 ∂1 g33 3 
1 3 3 3 3 3 ∂2Γ113 − ∂1Γ123 − Γ11 − 33 Γ12 + 33 ∑ gm3 Γ2m Γ11 − Γ1m Γ12 33 2 g g g m=1  2  g32 2 3 m 3 m = ∑ Γ1m Γ12 − ∑ Γ2m Γ11 + 33 (Γ113 Γ223 − (Γ123 )2 ), g m=1 m=1  
1 ∂2 g33 3 ∂1 g33 3
1 3 3 3 3 3 ∂2Γ123 − ∂1Γ223 − Γ12 − 33 Γ22 + 33 ∑ gm3 Γ2m Γ12 − Γ1m Γ22 33 2 g g g m=1 ! 2 2 g31 3 m 3 m = ∑ Γ1m Γ22 − ∑ Γ2m Γ12 − 33 (Γ113 Γ223 − (Γ123 )2 ), g m=1 m=1 Recall the covariant-contravariant version of the curvature tensor Riem(g) in (11.4): 3

Rsijk = ∂ j Γiks − ∂k Γi sj +

3

∑ Γjms Γikm − ∑ Γkms Γi mj

m=1

m=1

3

so that Rsi jk =

∑ gsm Rmijk

for all s, i, j, k = 1 . . . 3.

m=1

Therefore, in view of (11.51) we obtain: R3121 = ∂2Γ113 − ∂1Γ123 +

3

∑ (Γ2m3 Γ11m − Γ1m3 Γ12m )

m=1

 1 1 = 33 g33 (Γ233 Γ113 − Γ133 Γ123 ) + (∂2 g33Γ113 − ∂1 g33Γ123 ) g 2 + g32 (Γ113 Γ223 − (Γ123 )2 ) −

3

∑

m=1

3 3 3 3 Γ11 − Γ1m gm3 Γ2m Γ12




 ,

294

11 Limiting theories for prestressed films: nonlinear bending

R3221 = ∂2Γ123 − ∂1Γ223 +

3

∑ (Γ2m3 Γ12m − Γ1m3 Γ22m )

m=1

 1 1 = 33 g33 (Γ233 Γ123 − Γ133 Γ223 ) + (∂2 g33Γ123 − ∂1 g33Γ223 ) g 2 − g31 (Γ113 Γ223 − (Γ123 )2 ) −

3

∑

3 3 3 3 gm3 Γ2m Γ12 − Γ1m Γ22




 .

m=1

These are equivalent to R3121 = R3221 = 0 on ω × {0} in view of (11.50). 4. We now check that the Gauss equation (11.48) is, in turn, equivalent to R1212 ≡ 0 on ω × {0}. Denoting [risjk ]s,i, j,k=1,2 and [rsi jk ]s,i, j,k=1,2 the Riemann curvatures of the two-dimensional metric G on ω, we obtain: 2 1 + g12 r212 κ det G = r1212 = g11 r212

Further, for i = 1, 2 we get by (11.49) and (11.50): 2 i i i r212 = ∂ j γ22 − ∂2 γ12 +

2

i m i m γ22 − ∑ γ2m γ12 ∑ γ1m

m=1

m=1

g3i g3i =∂1 (Γ22i − 33 Γ223 ) − ∂2 (Γ12i − 33 Γ123 ) g g 2

+

g3i

2

g3m

g3i

g3m

∑ (Γ1mi − g33 Γ1m3 )(Γ22m − g33 Γ223 ) − ∑ (Γ2mi − g33 Γ2m3 )(Γ12m − g33 Γ123 )

m=1

m=1

 1i  g3i g31 g g3i 3 3 i =R212 − 33 (∂1Γ22 − ∂2Γ12 ) + 33 − 33 2 (Γ113 Γ223 − (Γ123 )2 ) g g (g ) −

g3i 3 ∑ (Γ1m3 Γ22m − Γ2m3 Γ12m ). g33 m=1

Consequently, the Gauss equation (11.48) yields: R1212 = g11 R1212 + g12 R2212 + g13 R3212 2 1 ) + g13 R3212 ) + g12 (R2212 − r212 = κ det G + g11 (R1212 − r212 1 = g13 R3212 + 33 (Γ113 Γ223 − (Γ123 )2 ) − g13 (∂1Γ223 − ∂2Γ123 ) g   3 (1 − g13 g13 ) g33 g13 g31 3 3 3 2 ) − g − + (Γ Γ − (Γ ) 31 ∑ (Γ1m3 Γ22m − Γ2m3 Γ12m ) 11 22 12 g33 (g33 )2 m=1   3 3 m 3 m = g13 R3212 − g13 ∂1Γ223 − ∂2Γ123 + ∑ (Γ1m Γ22 − Γ2m Γ12 ) = 0. m=1

5. The simultaneous vanishing of R3121 , R3221 , R1212 is equivalent with the vanishing of R1212 , R1213 and R1223 . Indeed, assuming that R1212 = 0 in ω, there follows that R3221 = 0 if and only if R1223 = 0, and R3112 = 0 if and only if R1213 = 0. To show

11.6 Coercivity of Kirchhoff-like energy for prestressed films

295

the first claim, note that (we use the Einstein notation convention again): 0 = R1221 = g1s Rs221 ,

0 = R2221 = g2s Rs221

and

R1223 = R3221 = g3s Rs221 .

By invertibility of G it hence follows that R3221 = 0 is equivalent to: Rs221 = 0 for all s = 1 . . . 3. In view of invertibility of g, this last condition is equivalent to: R1223 = 0. The second claim follows in the same manner. The proof is done.

11.6 Coercivity of Kirchhoff-like energy for prestressed films In this section we quantify the statement in Theorem 11.16 and prove that when I2,g can be minimized to zero, then I2,g (y) measures then the distance of a given isometric immersion y from the kernel, given by all rigid motions of y0 as in (11.44):  ker I2,g = Ry0 + d; R ∈ SO(3), d ∈ R3 . Assume that the set of H 2 (ω, R3 ) isometric immersions y of g(·, 0)tan is nonempty, which in view of Corollary 11.11 is equivalent to the quadratic scaling in: inf E h ≤ Ch2 . For each such y, the continuity equation (11.46) combined with Lemma 11.15 gives the following formula, valid on ω for all i, j = 1, 2: Γi 3j m p N ∂ y + γ ∑ ij m g33 m=1  p 
1 − g33 sym(∇y)T ∇b − ∂3 g(·, 0)tan N. 2 ij 2

∂i j y =

(11.52)

recall that above {γimj }i, j,m=1,2 stand for the Christoffel symbols of the metric g(·, 0)tan on ω, while {Γi m j }i, j,m=1...3 are the Christoffel symbols of the threedimensional metric g as in (11.4). The vectors N and b are given in (11.20). Another consequence of (11.46) is:

 |∇2 y|2 = |Πy |2 + ∑ g(·, 0)tan : [γi1j , γi2j ]⊗2 on ω. i, j=1,2

By Lemma 11.15 and since |∇y|2 = trace g(·, 0)tan , this yields the bound: ?

2


y − y dz H 2 (ω,R3 ) ≤ C I2,g (y) + 1 ,

(11.53)

ω

where C is a constant independent of y. Clearly, when condition (11.45) does not
hold, so that min I2,g > 0, the right hand side C I2,g (y)+1 above may be replaced by CI2,g (y). On the other hand, in presence of (11.45), the bound (11.53) can be refined to the following coercivity result:

296

11 Limiting theories for prestressed films: nonlinear bending

Theorem 11.17. Let ω ⊂ R2 be an open, bounded, simply connected and Lipschitz domain and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume (11.45) and let y0 be the unique (up to rigid motions in R3 ) isometric immersion of g(·, 0)tan satisfying (11.44). Then, for all y ∈ H 2 (ω, R3 ) with (11.15) there holds:    (11.54) dist2H 2 (ω,R3 ) y, Ry0 + d; R ∈ SO(3), d ∈ R3 ≤ CI2,g (y), with a constant C > 0 that depends on g, ω and W but is independent of y. > > Proof. 1. Without loss of generality, we set ω y = ω y0 = 0. For any R ∈ SO(3), identity (11.52) and the fact that I2,g (Ry0 ) = 0 imply: Z Z Z ∇2 y − ∇2 (Ry0 ) 2 dz ≤ C ∇y − ∇(Ry0 ) 2 dz + |N − RN0 |2 dz ω ω ω Z 
1

sym (∇y)T ∇b − ∂3 g(·, 0)tan 2 dz 2 ω Z  Z 2
1 ∇y − ∇(Ry0 ) dz + sym (∇y)T ∇b − ∂3 g(·, 0)tan 2 dz . ≤C 2 ω ω +

Above, we denoted by N0 the unit normal vector to the smooth surface y0 (ω) and 2 R R also used that ω |N − RN0 |2 dz ≤ C ω ∇y − ∇(Ry0 ) dz. This estimate follows p from: |∂1 y×∂2 y| = |∂1 (Ry0 )×∂2 (Ry0 )| = det g(·, 0)tan . Also, the non-degeneracy of quadratic forms Q2,g (z0 , ·) in Definition 11.6 implies the uniform bound:
sym (∇y)T ∇b − 1 ∂3 g(·, 0)tan 2 dz ≤ CI2,g (y). 2 ω

Z

Taking R ∈ SO(3) as in Lemma 11.18 below, the bound (11.54) follows by (11.55). The next weak coercivity estimate was used in the proof of Theorem 11.17: Lemma 11.18. In the context of Theorem 11.17, there exists R ∈ SO(3) such that: Z

|∇y − R∇y0 |2 dz ≤ C


sym (∇y)T ∇b − 1 ∂3 g(·, 0)tan 2 dz, 2 ω

Z

ω

(11.55)

with a constant C > 0 that depends on g, ω,W but it is independent of y. Proof. 1. Consider the extensions u and u0 of y and y0 , respectively: u(z,t) = y(z) + tb(z),

u0 (z,t) = y0 (z) + tb0 (z) for all (z,t) ∈ Ω h .

Clearly, u ∈ H 1 (Ω h , R3 ) and u0 ∈ C 1 (Ω¯ h , R3 ) satisfies det ∇u0 > 0 for h  1. Write: ω=

n [ k=1

ωk ,

Ωh =

n [ k=1

Ωkh

11.6 Coercivity of Kirchhoff-like energy for prestressed films

297

as the union of n ≥ 1 open, bounded, connected domains with Lipschitz boundary, such that on each {Ωkh  ωk × (− 2h , h2 )}nk=1 , the deformation u0 |Ω h is a C 1 diffeok

morphism onto its image Ukh ⊂ R3 . 1 h 3 We first prove (11.55) when n = 1. Call u¯ = u ◦ u−1 0 ∈ H (U , R ) and apply the geometric rigidity estimate (4.2) for the existence of R ∈ SO(3) satisfying: Z Uh

|∇u¯ − R|2 dx ≤ C

Z Uh


dist2 ∇u, ¯ SO(3) dx,

(11.56)

with a constant C depending on h (and ultimately on k as well, when n > 1), but
−1 independent of u. ¯ Since ∇u(u ¯ 0 (x)) = ∇u(x) ∇u0 (x) for all x ∈ Ω h , we get: Z U

|∇u¯ − R|2 dx =

h

≥C

Z Ωh

Z

=C ≥ Ch

Ωh

Z

Z Ωh

2 (det ∇u0 ) (∇u − R∇u0 )(∇u0 )−1 dx

|∇u − R∇u0 |2 dx (11.57)   2   ∂1 y, ∂2 y, b − R ∂1 y0 , ∂2 y0 , b0 + t 2 |∇b − R∇b0 |2 dx |∇y − R∇y0 |2 dz.

ω

Similarly, the change of variables in the right hand side of (11.56) gives: Z Uh

Z
¯ SO(3) dx ≤ C dist2 ∇u,

Ωh


dist2 (∇u)(∇u0 )−1 , SO(3) dx.

(11.58)

2. Since (∇u)T ∇u(·, 0) = (∇u0 )T ∇u0 (·, 0) = g(·, 0), by polar decomposition it ˜ follows that: ∇u(z, 0)  Q(z0 ) = R(z)g(z, 0)1/2 and ∇u0 (z, 0)  Q0 (z) = R˜ 0 (z)g(z, 0)1/2 ˜ for some SO(3)-valued rotation fields R, R˜ 0 on ω. Observe further that:      ˜ ∇u(·,t) = Q + t ∂1 b, ∂2 b, 0 = Rg(·, 0)1/2 Id3 + tg(·, 0)−1 QT ∂1 b, ∂2 b, 0  
∗  T  ˜ = Rg(·, 0)1/2 Id3 + tg(·, 0)−1 (∇y)T ∇b + e3 ⊗ ∇b, 0 b , and similarly:  
∗  T  ∇u0 (·,t) = R˜ 0 g(·, 0)1/2 Id3 + tg(·, 0)−1 (∇y0 )T ∇b0 + e3 ⊗ ∇b0 , 0 b0 . Consequently, the integrand in the right hand side of (11.58) becomes:  −1 1/2 ˜ (∇u)(∇u0 ) = Rg(·, 0) Id3 + tg(·, 0)−1 S·    T −1 −1 T ∗ · Id3 + tg(·, 0) (∇y0 ) ∇b0 ) + e3 ⊗ ∇b0 , 0 b0 g(·, 0)1/2 R˜ T0 , where we denote:

(11.59)

298

11 Limiting theories for prestressed films: nonlinear bending

∗  T  T  S = (∇y)T ∇b − (∇y0 )T ∇b0 + e3 ⊗ ∇b, 0 b − ∇b0 , 0 b0 ∗  = sym (∇y)T ∇b − (∇y0 )T ∇b0 . 

The last equality follows from the easy facts that, for i, j = 1, 2, we have: 1 h∂i b, bi = h∂i b0 , b0 i = ∂i g(·, 0)33 2 h∂i y, ∂ j bi − h∂ j y, ∂i bi = h∂i y0 , ∂ j b0 i − h∂ j y0 , ∂i b0 i = ∂ j g(·, 0)i3 − ∂i g(·, 0) j3 . Thus, (11.58) and (11.59) imply: Z Uh

Z
dist2 ∇u, ¯ SO(3) dx ≤ C

Ωh

≤C

Z Ωh

(∇u)(∇u0 )−1 − R˜ R˜ T0 2 dx

tS(z,t) 2 dx


2
sym (∇y)T ∇b − sym (∇y0 )T ∇b0 dz ω Z 2
1 = C sym (∇y)T ∇b − ∂3 g(·, 0)tan dz 2 ω ≤C

Z

(11.60)

with a constant C that depends on g, ω and h, but not on y. We conclude (11.55) in view of (11.56), (11.57) and (11.60). 3. To prove (11.55) in case n > 1, let k, s : 1 . . . n be such that ωk ∩ ωs , 0. / Define: F

Z Ωkh ∩Ωsh

det ∇u0 dx

−1 Z Ωkh ∩Ωsh

(det ∇u0 )(∇u)(∇u0 )−1 dx ∈ R3×3 .

Denote by Rk , Rs ∈ SO(3) the corresponding rotations in (11.55) on ωk , ωs . For i ∈ {k, s} we have:  Z |F − Ri |2 =

Ωkh ∩Ωsh

≤C ≤

Z Ωkh ∩Ωsh

det ∇u0 dx

−1 Z Ωkh ∩Ωsh

|∇u − Ri ∇u0 |2 dx ≤ C

2
(det ∇u0 ) ∇u − Ri ∇u0 (∇u0 )−1 dx

Z Ωih

|∇u − Ri ∇u0 |2 dx

2
sym (∇y)T ∇b − 1 ∂3 g(·, 0)tan dz 2 ωi

Z

where for the sake of the last bound we applied the intermediate estimate in (11.57) to the left hand side of (11.56), as discussed in the previous step. Consequently: |Rk − Rs |2 ≤ C which leads to:

2
sym (∇y)T ∇b − 1 ∂3 g(·, 0)tan dz, 2 ω

Z

11.7 Effective energy density under isotropy condition

Z

299

Z Z  |∇y − Rs ∇y0 |2 dz ≤ 2 |∇y − Rk ∇y0 |2 dz + |Rk − Rs |2 |∇y0 |2 dz

ωk

ωk

≤C

ωk

2 sym (∇y)T ∇b − 1 ∂3 g(·, 0)tan dz 2 ω

Z




This shows that it is possible to take one and the same R = R1 on each {ωk }nk=1 , at the expense of possibly increasing the constant C by a controlled factor depending only on n. The proof of (11.55) is done. Remark 11.19. A similar reasoning as in the proof of Lemma 11.18, yields a quantitative version of the uniqueness of isometric immersion with a prescribed second fundamental form compatible to the metric by the Gauss-Codazzi-Mainardi equations. More precisely, given a smooth metric G in ω ⊂ R2 , for every two isometric immersions y1 , y2 ∈ H 2 (ω, R3 ) of G, there holds: Z

min R∈SO(3) ω

|∇y1 − R∇y2 |2 dz ≤ C

Z ω

|Πy1 − Πy2 |2 dz,

with a constant C > 0, depending on g and ω but independent of y1 and y2 .

11.7 Effective energy density under isotropy condition In this section, we identify the quadratic forms Q2,g introduced in (11.16), for the case of the isotropic energy density W . Recall Definition 5.1 of the linear operator L3 and its basic properties given in Lemma 5.2. Also, recall that F ∈ R2×2 , by F ∗ ∈ R3×3 we denote the matrix such that (F ∗ )tan = F and whose all other entries are 0. We begin with the formula for the auxiliary fields cg (z, ·): Lemma 11.20. Let ω ⊂ R2 be an open, bounded, simply connected and Lipschitz domain and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). In the context of Definition 11.6, we have:
A−1 cg (z, F) = −MA−1 L3 (A−1 F ∗ A−1 A−1 e3

for all F ∈ R2×2 sym , z ∈ ω, (11.61)

where A(z) = g(z, 0)1/2 , and where the matrix field MA : ω → R3×3 is given by:
MA ei = L3 ei ⊗ A−1 e3 A−1 e3 for all i = 1 . . . 3. Consequently, there holds:
Q2,g (F) = Q3 A−1 F ∗ A−1 E D

− MA−1 L3 A−1 F ∗ A−1 A−1 e3 , L3 (A−1 F ∗ A−1 A−1 e3 , Proof. For all F ∈ R2×2 and i : 1 . . . 3 we have:

(11.62)

300

11 Limiting theories for prestressed films: nonlinear bending

D E d Q3 (A−1 (F ∗ + c ⊗ e3 )A−1 ) = 2 L3 (A−1 (F ∗ + c ⊗ e3 )A−1 ) : A−1 ei ⊗ A−1 e3 dci D E
= 2 A−1 L3 A−1 (F ∗ + c ⊗ e3 )A−1 A−1 : ei ⊗ e3 . Therefore, at the minimizer c0 , there holds:
∇c Q3 (A−1 (F ∗ + c0 ⊗ e3 )A−1 ) = 2A−1 L3 A−1 (F ∗ + c0 ⊗ e3 )A−1 A−1 e3 = 2A−1 L3 (A−1 F ∗ A−1 + A−1 c0 ⊗ A−1 e3 )A−1 e3 = 0, which is equivalent to: −L3 (A−1 F ∗ A−1 )A−1 e3 = L3 (A−1 c0 ⊗ A−1 e3 )A−1 e3 = MA A−1 c0 , and consequently to (11.61). Then: Q2 (F2×2 ) = Q3 (A−1 F ∗ A−1 + A−1 c0 ⊗ A−1 e3 ) D E = L3 (A−1 F ∗ A−1 ) + L3 (A−1 c0 ⊗ A−1 e3 ) : A−1 F ∗ A−1 E D = L3 (A−1 F ∗ A−1 ) : A−1 F ∗ A−1 + A−1 c0 ⊗ A−1 e3

(11.63)

= Q3 (A−1 F ∗ A−1 ) D E − L3 (A−1 F ∗ A−1 ) : MA−1 L3 (A−1 F ∗ A−1 )(A−1 e3 ⊗ A−1 e3 ) , proving (11.62). We now assume that the energy density W is isotropic as in (5.5), i.e.: W (RF) = W (F)

for all F ∈ R3×3 , R ∈ SO(3).

It shown in Example 5.3, that the associated quadratic form Q3 and the bilinear operator L3 are then given in terms of the Lam´e coefficients µ ≥ 0, λ ≥ −µ/3: Q3 (F) = µ|sym F|2 + λ |tr F|2 ,

L3 (F) = µsymF + λ (trF)Id3 .

(11.64)

Then we have the following: Lemma 11.21. Assume that W is isotropic, so that (11.64) holds. In the context of Definition 11.6, we have: MA =

µ µ −1 2 |A e3 | Id3 + (λ + )(A−1 e3 ⊗ A−1 e3 ) 2 2

(11.65)

and, denoting D = A−1 F ∗ A−1 , d = A−1 e3 for F ∈ R2×2 sym , we have:  |Dd|2 hDd, di2  λµ  hDd, di 2 Q2,g (z, F) = µ |D|2 − 2 + + trD − . (11.66) |d|2 |d|4 λ +µ |d|2

11.7 Effective energy density under isotropy condition

301

Proof. By the formulas in (11.64), we obtain: MA ei = L3 (ei ⊗ d)d = (λ +

µ µ )hd, ei id + |d|2 ei 2 2

which gives (11.65). Further, it is easy to check the following general formula: (α Id3 + a ⊗ b)−1 =

1 1 Id3 − a ⊗ b. α α(α + ha, bi)

Applying it to α = µ2 |d|2 and a = (λ + µ2 )d and b = d, we get: MA−1 =

2 1 2λ + µ 1 Id3 − (d ⊗ d). 2 µ |d| µ(λ + µ) |d|4

Therefore:

 λ2 λµ hDd, di MA−1 L3 (D)d, L3 (D)d = (trD)2 + 2 (trD) λ +µ λ +µ |d|2 + 2µ

|Dd|2 (2λ + µ)µ hDd, di2 − . |d|2 λ +µ |d|4

Concluding: Q2 (·, F) =

λµ hDd, di λµ (trD)2 + µ|D|2 − 2 (trD) λ +µ λ +µ |d|2 − 2µ

|Dd|2 (2λ + µ)µ hDd, di2 + , |d|2 λ +µ |d|4

which yields (11.66). Finally, we get the main result of this section: Theorem 11.22. Assume that W is isotropic, so that (11.64) holds. In the context of Definition 11.6, denote G = g(·, 0). Then, for all F ∈ R2×2 sym there holds:   Q2,g (z, F) = Q2,Id3 (Gtan )−1/2 F(Gtan )−1/2 2 2 (11.67) λ µ = µ (Gtan )−1/2 F(Gtan )−1/2 + (Gtan )−1/2 F(Gtan )−1/2 λ +µ Proof. Recall that given F ∈ R3×3 , by Ftan ∈ R2×2 we denote the principal 2 × 2 minor of F. We will also write Fcross = (Fe3 )tan = (F13 , F23 ) ∈ R2 . Using the notation of Lemma 11.21, we identify the terms in (11.15). Call P = G−1 . Then:

302

11 Limiting theories for prestressed films: nonlinear bending

|D|2 = hPF ∗ P : F ∗ i = h(PF ∗ P)tan : Fi = hPtan FPtan : Fi , |Dd|2 = hPF ∗ Pe3 , F ∗ Pe3 i = h(PF ∗ Pe3 )tan , FPcross i = hPtan FPcross , FPcross i , hDd, di = hPF ∗ Pe3 , e3 i = hFPcross , Pcross i, |d|2 = hPe3 , e3 i = P33 , tr D = tr(PF ∗ ) = tr(Ptan F). Hence, (11.15) becomes:  hPtan FPcross , FPcross i Q2,g (·, F) = µ hPtan FPtan : Fi − 2 P33    2 λµ hFPcross , Pcross i 2 hFPcross , Pcross i . + tr(P F) − + tan (P33 )2 λ +µ P33

(11.68)

We now identify the terms in the right hand side of (11.67), using: Gtan Ptan + Gcross ⊗ Pcross = Id2 , Gtan Pcross + P33 Gcross = 0,
−1
−1 1 Pcross ⊗ Pcross . Gcross ⊗ Pcross = Ptan − Gtan = Ptan + Gtan P33 to the effect that:

(Gtan )−1/2 F(Gtan )−1/2 2 = (Gtan )−1 F(Gtan )−1 :i      1 1 Pcross ⊗ Pcross F Ptan − Pcross ⊗ Pcross : F Ptan − = P33 P33

  2

= Ptan FPtan : F − (Pcross ⊗ Pcross )FPtan : F P33  1

(Pcross ⊗ Pcross )F(Pcross ⊗ Pcross ) : F + 2 (P33 ) = hPtan FPtan : Fi − 2

hPtan FPcross , FPcross i hFPcross , Pcross i2 , + P33 (P33 )2

and also:  
tr (Gtan )−1/2 F(Gtan )−1 = tr (Gtan )−1 F   1 1 Pcross ⊗ Pcross )F = tr(Ptan F) − hFPcross , Pcross i. = tr (Ptan − P33 P33 The equality in (11.67) follows now directly by (11.68). The proof is done. We offer a couple of related observations, in the context of Theorem 11.22: ∗ + e ⊗ e , then d = e and Remark 11.23. (i) In the particular case when G = Gtan 3 3 3 Dd = De3 = 0, so (11.15) directly becomes:

11.8 Application to liquid crystal glass

303

Q2,g (·, F) = µ|D|2 +

λµ |tr D|2 , λ +µ

(ii) Call C = G−1 F ∗ and note that:
tr D = trC, |D|2 = tr C2 |Dd|2 = hC2 G−1 e3 , e3 i,

|d|2 = hG−1 e3 , e3 i,

hDd, di = hCG−1 e3 , e3 i.

Consequently, (11.15) can also be equivalently written as:  
hC2 G−1 e3 , e3 i hCG−1 e3 , e3 i2 2 Q2 (·, F) = µ tr C − 2 + hG−1 e3 , e3 i hG−1 e3 , e3 i2   2 hCG−1 e3 , e3 i λµ tr C − + , λ +µ hG−1 e3 , e3 i which is somewhat a simpler formula than the original one in (11.15).

11.8 Application to liquid crystal glass In this section, we provide an example of application of the theory developed in sections 11.3 and 11.4, pertaining to a model of nematic glass. Nematic elastomers are rubber-like, cross-linked, polymeric solids, which have both positional elasticity (due to the solid response of the polymer chains) and the orientation elasticity (due to the separately deforming director). A nematic glass is a highly cross-linked nematic elastomer such that the director is constrained to move with the elastomer matrix. In the model that we consider, the referential conformation matrix field A corresponds to a prolate ellipsoid, elongating the eigenvector n by the factor λ , while shrinking the invariant subspace n⊥ = span(v, w) by λ τ : A = λ −τ v ⊗ v + λ −τ w ⊗ w + λ n ⊗ n
= λ −τ Id3 + (λ τ+1 − 1)n ⊗ n ,

λ > 1,

|n| = 1.

The coefficient τ is experimentally verified to be in the range 21 < τ < 2. Setting τ r = λ τ+1 , and writing λ −τ = rδ with δ = − τ+1 , we obtain the metric g and its √ symmetric, positive definite square root A = g, given by: g(z,t) = G(z) = r2δ (Id3 + (r2 − 1)n ⊗ n), A(z) = rδ (Id3 + (r − 1)n ⊗ n). We start by the following observation:

(11.69)

304

11 Limiting theories for prestressed films: nonlinear bending

Theorem 11.24. Let ω ⊂ R2 be open, bounded, simply connected and Lipschitz. ∞ ¯ R3 ) satisfies: Define g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ) by (11.69), where n ∈ C (ω, n = (n1 , n2 , 0)T

and

|n| = 1, where n = (n1 , n2 ).

(11.70)

Then, the following conditions are equivalent: (i) the metric g is immersable, i.e. g = (∇u)T ∇u for some u ∈ C ∞ (Ω 1 , R3 ), (ii) the Gaussian curvature κ(Id2 + (r2 − 1)n ⊗ n) vanishes identically in ω, (iii) we have curl curl Gtan = 0 in ω, (iv) the following curvatures of g vanish: R3112 = R3221 = R1212 = 0 on ω × {0}. Proof. By the fundamental theorem of Riemannian manifolds in section 11.1, assertion (i) holds if and only if Riem(g) vanishes, which in the present case becomes:
κ  κ Id2 + (r2 − 1)n ⊗ n ≡ 0. We now calculate the Gaussian curvature κ of the 2d metric Id2 + (r2 − 1)n ⊗ n as: 1 κ = r2δ κ(Gtan ) = (r−2 − 1)curl curl (n ⊗ n). 2

(11.71)

This will achieve the lemma, because R3112 = R3221 ≡ 0 automatically, while R1212 ≡ 0 is equivalent to (ii). We write r2 − 1  γ > 0 and compute:
2 det(Id2 + γn ⊗ n) κ     1 1 1 0 q 2 e,1 f ,1 − 2 e,2 2 e,2 g,1  − det  1 e,2 − 1 g,1 e f  , = det  f,2 − 21 g,1 e f 2 2 1 1 g g f g f g ,2 ,1 2 2 where q = − 12 e,22 + f,12 − 21 g,11 and e = 1 + γn21 , f = γn1 n2 , g = 1 + γn22 . A direct calculation now gives that the right hand side above equals: (1 + γ)q + γ 3 · 0 + γ 2 ( − n22 n1,1 n2,2 − n1 n2 n2,1 n2,2 + n1 n2 n1,2 n2,2 − n1 n2 n1,1 n1,2 − n21 n1,1 n2,2 + n1 n2 n1,1 n2,1 + n22 n22,1 + n21 n21,2 ) = (1 + γ)q. This is since all the terms in the bracket multiplying γ 2 , cancel out. This can be seen by substituting (n1 , n2 ) = (cos θ , sin θ ) for an angle function θ : ω → R. Consequently, we see that κ = 0 if and only if q = 0. On the other hand: q=−


γ γ (n21 ),22 − 2(n1 n2 ),12 + (n22 ),11 = − curl curl (n ⊗ n). 2 2

γ Since det(Id2 +(r2 − 1)n ⊗ n) = 1 + γ, it finally follows that κ = − 2γ+2 curl curl (n ⊗ n), as claimed in (11.71). The proof is done.

11.8 Application to liquid crystal glass

305

We remark in passing that given any two-dimensional metric G with constant eigenvalues 0 < λ1 ≤ λ2 , namely: G  λ1 v ⊗ v + λ2 w ⊗ w = λ1 Id2 −


λ2 − λ1 w⊗w . λ1

the immersability of this metric is equivalent to curl curl (G) ≡ 0. This condition is only necessary but not sufficient, for general metrics G. Indeed, curl curl is only the leading order term in the expansion of the Gaussian curvature of a 2d metric at Id2 . Remark 11.25. In accordance with (11.69), the following metric has been put forward in Modes et al. [2010] for the description of disclination-mediated thermooptical response in nematic glass sheets: g(z,t) = αId3 + β n(z) ⊗ n(z), where α, β > 0 are constants, and n is as in (11.70) with: n1 = cos(θ + ψ),

n2 = sin(θ + ψ),

θ = arctan

x2 , x1

ψ ≡ const.

Note that θ is the polar angle and so setting the constant ψ = 0 gives the radial pattern, while ψ = π/2 gives the azimuthal pattern, and other values of ψ yield spiral patterns. It is easy to check that curl curl (n ⊗ n) ≡ 0. Therefore, if the simply connected ω does not contain 0 (since 0 is a singularity for g), then the metrics g(·, 0)tan and g are immersible by Theorem 11.24, and thus inf Egh = 0. However if 0 ∈ ω, one may not have a global immersion (implying hence a higher energy scaling; see Modes et al. [2011] for a construction with h2 scaling). We directly obtain: Theorem 11.26. Let ω ⊂ R2 be open, bounded, simply connected and Lipschitz, and assume that g ∈ C ∞ (Ω 1 , R3×3 pos,sym ) is as in (11.69) with (11.70). Then, Theorems 11.7 and 11.10 hold with the Cosserat vector b given by: b = rδ N and with the dimensionally reduced energy: Z   1 Q2,g z, rδ (∇y)T ∇N dz 24 ω Z   1 Q2,Id3 rδ (Atan )−1 (∇y)T ∇N(Atan )−1 dz. = 24 ω

I2,g (y) =

Additionally, we have: (i) Denoting α 

r−1 r ,

(Atan )−1 =

it follows that: 1  r1+δ

 1 Id2 + (r − 1)n⊥ ⊗ n⊥ = δ (Id2 − α n ⊗ n) r

and the quadratic form in the second integrand in (11.72) equals:

(11.72)

306

11 Limiting theories for prestressed films: nonlinear bending

rδ (Atan )−1 (∇y)T ∇N(Atan )−1

= r−δ (∇y)T ∇N − 2αsym (n ⊗ ∂n y)∇N + α 2 h∂n y, ∂n Ni n ⊗ n . (ii) When the elastic energy density W is isotropic as in (5.5), then: Q2,g (·, Ftan ) =

1 r

  (Id −α n⊗n)F Id −α n⊗n) Q 2 2 2,Id ( 3 4δ

for all F ∈ R2×2 sym .

We now turn to the case of the general 3-dimensional director vector n. Theorem 11.27. Let ω ⊂ R2 be open, bounded, simply connected and Lipschitz, 2 and assume that g ∈ C ∞ (Ω 1 , R3×3 pos,sym ) is as in (11.69). Let n = (n1 , n2 ) ∈ R denote the tangential component of the director vector n and let: γ

1 n23 + |n|2 r2

.

Then we have: (i) Theorems 11.7 and 11.10 hold with the Cosserat vector b in: b = (r2 − 1)n3 γ(∂n y) + r =

√

γ(rδ N)

(r2 − 1)n3 r1+δ q N ∂ y + n n23 + |n|2 r2 n23 + |n|2 r2


1 R T and with the functional I2,g (y) = 24 ω Q2,g z, (∇y) ∇b dz. (ii) When W is isotropic so that (5.5) holds, then we have: 1 



2 2 2 2 2 2 |F| − 2 (r − 1)γ |Fn| + (r − 1)γ hFn, ni r4δ 2
λµ 1  + tr F − (r2 − 1)γ hFn, ni , 4δ λ +µ r √
1 for all F ∈ R2×2 γ , the above formula is equivalent to: sym . Setting γ˜  |n|2 1 − Q2,g (·, F) = µ

Q2,g (z, F) =

1 r4δ

(

  Q2,Id3 (Id2 − γ˜ n ⊗ n)F(Id2 − γ˜ n ⊗ n)

if n3 (z)2 < 1,

Q2,Id3 (F)

if n3 (z)2 = 1.

Proof. We easily compute: det G = r2+6δ , det Gtan = r4δ (r2 − (r2 − 1)n23 )   1 (Gtan )−1 = 2δ (n23 + |n|2 r2 )−1 Id2 − (r2 − 1)n⊥ ⊗ n⊥ , r where n⊥ = (n1 , n2 )⊥ = (−n2 , n1 ). Therefore:

11.9 More examples

307

(Gtan )−1



(r2 − 1)n3 G13 n = 2 G23 n3 + |n|2 r2 

which implies the formula in (i). ˜ ⊗ n). Indeed: To prove (ii) it suffices to show that (Gtan )−1/2 = r−δ (Id2 − γn
2
˜ ⊗ n Id2 + (r2 − 1)n ⊗ n Id2 − γn

˜ 2 − 2γ)n ˜ ⊗ n Id2 + (r2 − 1)n ⊗ n = Id2 + (γ|n|

= Id2 − (r2 − 1)γn ⊗ n Id2 + (r2 − 1)n ⊗ n = Id2 , since γ˜2 |n|2 − 2γ˜ = −(r2 − 1)γ. The proof is done. Remark 11.28. The expression in Theorem 11.26 (ii) is consistent with (ii) above, 2 and γ˜ = 1 − 1/r = α. The same expression as for n3 = 0 it follows that γ = r r−1 2 is also consistent with Remark 11.23, in the following sense. Take n = e3 . Then D = r−2δ diag(1, 1, r−1 )F ∗ diag(1, 1, r−1 ) = r−2δ F ∗ . Hence:   1 λµ Q2,g (·, F) = 4δ µ|F|2 + |tr F|2 , λ +µ r while Theorem 11.27 (ii) gives the same formula directly.

11.9 More examples In this section we explore a few examples where the curvature condition of Theorem 11.16 (iii) is not satisfied. Consequently, the energy level of the minimizers to Egh drops below the h2 scaling, which as we shall see in the next chapter, automatically yields the scaling bound Ch4 . We explore various different scenarios where this phenomenon takes place. ¯ R) be a positive function and define: Example 11.29. Let λ ∈ C ∞ (ω,
g(z,t) = G(z) = diag 1, 1, λ (z) .

(11.73)

Clearly, the 2d metric Gtan = Id2 has an isometric immersion y0 = id2 with the second fundamental form Πy0 = 0. On the other hand: Γi 3j =

1 (∂i G3 j + ∂ j G3i ) = 0 2λ

for all i, j = 1, 2,

and we see that (11.44) holds, so that I2,g (y0 ) = 0. We further check that the only possibly non-zero Christoffel symbols are: 1 Γ33i = − ∂i λ , 2

Γi33 =

1 ∂i λ 2λ

for all i = 1, 2.

308

11 Limiting theories for prestressed films: nonlinear bending

In particular, it easily follows that: R3121 = R3221 = R1212 = 0, which is consistent with Theorem 11.16 (iii). At the same time g is, in general, non-immersible. To see this, recall that the Ricci curvature tensor Ric(g) = [Ri j ]i, j=1...3 is given by: l Ri j = ∂l Γi lj − ∂ j Γill + Γi lj Γlmm − ΓilmΓjm ,

for i, j = 1 . . . 3. In the present case, we compute:

1 1 (∂1 λ )2 − 2λ (∂11 λ ) , R22 = (∂2 λ )2 − 2λ (∂22 λ ) , 2 2 4λ 4λ
1 R12 = (∂1 λ )(∂2 λ ) − 2λ (∂12 λ ) , R13 = R23 = 0, 4λ 2
1 R33 = − (∂1 λ )2 − 2λ (∂11 λ ) + (∂2 λ )2 − 2λ (∂22 λ ) . 4λ R11 =

We hence see that g is immersible if and only if: ∇2 λ −

1 ∇λ ⊗ ∇λ ≡ 0 2λ

in ω.

(11.74)

Let us now consider the scaling of the 3d non-Euclidean energy Egh at the at the following sequence of smooth deformations of Ω h : uh (z,t) = z + − We have: (∇uh )g(·, 0)−1/2

√
T t2 t2 ∂1 λ , − ∂2 λ , λ t . 4 4

(11.75)

2


tan

= Id2 − t4 ∇2 λ , and:


t ∂1 λ t ∂ 2 λ
T (∇uh )g(·, 0)−1/2 e3 = − √ , − √ , 1 , 2 λ 2 λ
t ∂ λ ∂ λ
T t T √1 , √2 , 1 . (∇uh )g(·, 0)−1/2 e3 = 2 λ 2 λ Consequently:

W (∇uh )g(·, 0)−1/2 ≤ Cdist2 (∇uh )g(·, 0)−1/2 , SO(3) ≤ Ct 4 , and therefore: inf Egh ≤ Egh (uh ) ≤

C h

Z h/2

t 4 dt = Ch4 ,

(11.76)

−h/2

for any choice of λ in (11.73). In the next chapter we will show that this scaling is also optimal, provided that the condition (11.74) does not hold.

Example 11.30. Let λ : ω¯ → R be a smooth positive function and define: g(z,t)  G(z) = λ (z)Id3 .

(11.77)

11.9 More examples

309

1 (δik ∂l λ + δil ∂k λ − δkl ∂i λ ) = δik ∂l f + δil ∂k f − One checks directly that Γkli = 2λ 1 δkl ∂i f , where we denote f = 2 log λ . We then compute:

R3112 = R3221 = 0

and

1 R1212 = − λ ∆ (log λ ) = λ 2 κ(λ Id2 ) 2

on ω × {0}.

Therefore, condition (11.45) which is equivalent to min I2,g = 0 according to Theorem 11.16, holds if and only if: ∆ (log λ ) = 0,

(11.78)

or equivalently, when the 2-dimensional metric Gtan = λ Id2 is immersable in R2 . Since Γi 3j = 0 for i, j = 1, 2 then this is precisely the case when (11.44) is satisfied. We now compute the Ricci curvature of g using the conformal rescaling formula:
Ric(g) = Ric(e2 f Id3 ) = −(∇2 f − ∇ f ⊗ ∇ f )∗ − ∆ f − |∇ f |2 Id3  ∗ = − 2(∆ f )Id2 + cof(∇2 f − ∇ f ⊗ ∇ f ) − (∆ f + |∇ f |2 ) e3 ⊗ e3 . We observe that g is immersable if and only if Ric(g) = 0, i.e. when ∇ f = 0, which is equivalent to: λ ≡ const. (11.79) Clearly (11.79) implies (11.78). On the other hand, there exist non-immersible metrics g for which (11.78) holds i.e. for which the minimum of the residual energy I2,g equals 0, and it is attained by the unique (up to rigid motions) smooth isometric immersion y : ω → R2 of λ Id2 . As in Example 11.29, we now consider scaling of inf Egh , assuming (11.78). Let: uh (z,t) = y(z) + t

√

λ e3 −

∗ t2  (∇y)−1,T ∇λ . 4

(11.80)

We easily compute that ∇uh (z,t)tan = ∇y(z) + O(t 2 ) and:  t √ T
t ∇uh (z,t) e3 = − h(∇y)−1,T ∇λ , e1 i, − h(∇y)−1,T ∇λ , e2 i, λ , 2 2  t √ T
T t h √ ∂1 λ , √ ∂2 λ , λ . ∇u (z,t) e3 = 2 λ 2 λ Since (∇y)T ∇y = λ Id2 , it follows that for i = 1, 2 there holds:   (∇uh )T ∇uh = λ Id2 + O(t 2 ) 2×2   t t (∇uh )T ∇uh = − h∂i y, (∇y)−1,T ∇λ i + ∂i λ + O(t 2 ) = O(t 2 ) 2 2 3i   (∇uh )T ∇uh = λ + O(t 2 ). 33

310

11 Limiting theories for prestressed films: nonlinear bending

Therefore it follows, by polar decomposition:   1
W (∇uh )g(·, 0)−1/2 ≤ Cdist2 √ ∇uh , SO(3) λ  q  r1 (∇uh )T ∇uh , SO(3) = Cdist2 Id3 + O(t 2 ), SO(3) ≤ Ct 4 , ≤ Cdist2 λ which again yields the scaling h4 , precisely as in (11.76). We observe that a more general example of g in the same spirit as above, is given with G(z) = Gtan (z)∗ +λ (z)e3 ⊗e3 such that Gtan is immersable in R2 . Since Γi 3j = 0 for i, j = 1, 2, we see that min I2,g = 0, in virtue of Theorem 11.16. On the other hand, taking smooth y : ω → R2 such that (∇y)T ∇y
= Gtan and defining {uh }h→0 as in (11.80), it again follows that W (∇uh )g(·, 0)−1/2 ≤ Ct 4 . Consequently, the same energy scaling as in (11.76) is valid here as well. ¯ R) be such that λ3 > λ12 + λ22 . Define: Example 11.31. Let λ1 , λ2 , λ3 ∈ C ∞ (ω,   1 0 λ1   g(z,t) = G(z) =  0 1 λ2  . (11.81) λ1 λ2 λ3 We directly compute the Christoffel symbols involved in (11.44): Γ113 =

∂1 λ1 , λ3 − (λ12 + λ22 )

Γ123 =

1 2 (∂1 λ2 + ∂2 λ1 ) , λ3 − (λ12 + λ22 )

Γ223 =

∂2 λ2 . λ3 − (λ12 + λ22 )

Hence, if |∂1 λ1 | + |∂2 λ2 | . 0 in ω, it follows by Theorem 11.16 (ii), that the isometric immersion y0 = id2 of Gtan = Id2 is certainly not the immersion for which I2,g (y0 ) = 0. Of course, this does not preclude the possibility that there exists another immersion y : ω → R3 of Gtan (now necessarily non-immersable), for which I2,g (y) = 0. As we shall see below, both scenarios are possible. (i) Consider a subcase of (11.81), where λ1 = 0 and λ3 = λ22 + 1, so that:   1 0 0   g(z,t) = G(z) =  0 1 λ2  . 0 λ2 λ22 + 1 Consequently: Γ113 = 0,Γ123 = 21 ∂1 λ2 ,Γ223 = ∂2 λ2 . We further compute: 1 Γ133 = λ2 ∂1 λ2 , 2

1 Γ122 = − λ2 ∂1 λ2 , 2 1 3 3 3 2 3 3 3 1 3 2 R112 = ∂1Γ12 + (Γ12Γ12 + Γ13Γ12 ) − (Γ12Γ11 + Γ22Γ11 ) = ∂11 λ2 . 2

Γ111 = Γ112 = 0,

11.9 More examples

311

We see that when ∂11 λ2 . 0 in ω, then min I2,g > 0, in view of Theorem 11.16. In particular, there is no isometric immersion of Gtan satisfying (11.44). (ii) In a further subcase of (11.81), take λ1 = 0, λ2 = −z2 and λ3 = z22 + 1, and:   0 0 1   g(z,t) = G(z2 ) =  0 1 −z2  . 2 0 −z2 z2 + 1 As before, we get: Γ113 = Γ123 = 0 and Γ223 = −1. Let y(z) = (z1 , sin z2 , cos z2 ) be an isometric immersion of Gtan into a cylinder. This immersion has the second fundamental form Πy given by: " # # " 1 Γ113 Γ123 0 0 T . =−√ Πy = (∇y) ∇N = 3 3 0 1 G33 Γ12 Γ22 Hence, by Theorem 11.16 it follows that I2,g (y) = 0. In particular, R3112 = R3221 = R1212 ≡ 0 in ω. The metric g is, however, non-immersible: S = G11 R11 + G22 R22 + G33 R33 + 2G23 R23 = −2 + 2z22 . 0, by the above direct calculation of its scalar Ricci curvature.

Example 11.32. In this example we will have Gtan non-immersable in R2 . Let ω¯ ⊂ {z ∈ R2 ; x1 > x1 > 0} and define:   1 + z21 z1 z2 b1 + z1 b3   g(z,t) = G(z) =  z1 z2 1 + z22 b2 + z2 b3  , (11.82) b1 + z1 b3 b2 + z2 b3 |b|2 T  1 1 1 where b = (b1 , b2 , b3 ) = − z31 , z32 , (z21 − z22 ) . 3 3 2 We see that:


1 y(z) = z1 , z2 , (z21 + z22 ) 2

is an isometric immersion of Gtan in R3 . Therefore: κ(Gtan ) =


−2 ∂11 y3 ∂22 y3 − (∂12 y3 )2 = 1 + z21 + z22 , 0. 2 2 (1 + |∇y3 | )


By Theorem 11.16, we have: min I2,g = 0 if and only if sym ∇btan +∇b3 ⊗z) = 0. Given a scalar field b3 , there exists btan so that this condition holds, if and only if: 0 = curl curl(∇b3 ⊗ z) = −∆ b3 .

312

11 Limiting theories for prestressed films: nonlinear bending

We see that indeed b3 in (11.82) is harmonic, and that btan = (b1 , b2 ) satisfy: sym∇btan = diag(−z21 , z22 ), which implies I2,g (y) = 0, with G = QT Q and Q = [∂1 y, ∂2 y, b] such that det Q > 0 ¯ One can check that g is nonimmersable in R3 , by calculating: in ω. S=

∑

Gi j Ri j =

i, j=1...3

12 , 0. 2z21 + 2z22 + 3

R the scalar Ricci curvature S of g. Above, we have used Maple .

11.10 Connection to experiments From an experimental perspective, there is now ample evidence of how relative growth leads to variations in shape in such contexts as leaves, swelling sheets of gels, the vertebrate gut, or mammalian brains, see in Kim et al. [2012], Gladman et al. [2016], Kempaiah and Nie [2014], Wei et al. [2014], Dias et al. [2011]. The experimental observations and numerical computations are largely aligned with the common mathematical framework of non-Euclidean elasticity, as a tool to understand the origin of shape: an elastic three-dimensional body Ω seeks to realize a configuration with a prescribed Riemannian metric g, by means of an isometric immersion. The deviation from or inability to reach such a state, is due to a combination of geometric incompatibility and the requirements of elastic energy minimization, resulting in the observed shapes.

Fig. 11.3 Imposing nontrivial target metrics in sheets of NIPA gels. The figures shown are: (a) radially symmetric discs cast by injecting the solution into the gap between two flat glass plates through a central hole [Klein et al., 2007, Reprinted with permission from AAAS], (b) nonaxisymmetric swelling patterns constructed by half-tone gel litography in [Kim et al., 2012, Reprinted with permission from AAAS].

There exist various techniques for construction of self-actuating elastic sheets with prescribed target metrics, resulting in spontaneous formation of complex shapes (see Figure 11.3). These structures arise as both large-scale buckling and

tr t0 # r . K ' Ktar For varying thickness, two qualitatively different behavconst iors are observed (Fig. 1): All disks of Ktar > 0 attain a 11.10 Connection to experiments domelike configuration. On the other hand, the shape of the 313 disks of K forms. < 0 In strongly depends on the thickness. Thick small-scale wrinkling tar Klein et al. [2007], gels were produced by mixdisks (Fig. 1,BIS topcross-linker middle panel) attain a singleof saddle ing NIPA monomers with in water. The addition catalysts iniconfiguration. As t decreases, this configuration is re0 tiates polymerization of a cross-linked elastic hydrogel, which undergoes a sharp, BðrÞ ' 4Hð placed by multinode wavy configurations. Unlike previreversible, volume reduction transition at a threshold temperature, above which its studied sheets [1,2,8,9],experiments all observed configurations equilibriumously volume decreases. Calibration provided the relation between the monomer concentration the mode. shrinkage ratio of thepatterns activatedare, gel. contain only a singleand wave Multiscale Anotherthus, method been reported in et al. [2012], where photopatterning ofaverage of nothas a requirement inKim hyperbolic sheets. polymer films,Alikewise, yielded of temperature-responsive gel sheets. Using halftone configuration n nodes can be written as zðr; !Þ ¼ gel lithography with two photomasks, cross-linked dots embeddedfuncin a lightly An ðrÞ!ðn!Þ, where ! ishighly an unspecified normalized cross-linked matrix access to nearly tion andprovided is characterized by thecontinuous number ofpatterns nodes of n swelling. and the This method was then used to fabricate surfaces with constant Gaussian curvature (spheramplitude profile A ðrÞ. Plotting n verses the thickness t0 ical caps, saddles, and cones) or nzero mean curvature (Ennepers surfaces), as well shows a series shape transformations, a refineas more complex and nearlyof closed shapes, see Figure 11.3namely, (b). ment of the wavelength with decreasing thickness, where

Fig. 11.4 AsFIG. a function of theonline). thickness of a swollen of sheet, one can achieve either elliptical or 1 (color Variation configurations with thick&2shape converges to the limit implied hyperbolic geometries. In the limit vanishing thickness, the ness. Disks of of K ¼ 0:0011 mm (left) and K tar tar ¼ by the Γ -convergence the elliptic case (toinitial a spherical dish), but shows &0:0011result mmin&2Corollary (middle11.13 andfor right) of various thicknesses an increasing preponderance to wrinkling on finer scales in the hyperbolic case. The multi-lobed (t0 ¼wrinkling 0:75, 0.6, 0.25, 0.19, and 0.125 mm,[Klein top ettoal.,bottom). swelling-induced begs the question of the limiting behavior 2011, Reprinted For K > 0 the disks keep the same basic shape, a hemisphere, tar with permission, copyright (2021) by the American Physical Society].

with minor variations along the edge. The disks of Ktar < 0 undergo a set of bifurcations, in which the number of nodes n In relation to the compactness and thickness. Γ -convergence statements in Theorems increases with decreasing Surface amplitude of hyper- 11.7, 11.10, we bolic point disks out on(right) the related experimental findings in Klein et of al. a[2011]nodes, n ¼ 2 shows that each configuration consists that constructed a thickness-parametrised family of axially symmetric AðrÞ by n p single wavy mode (the color bar, in mm, is common to hydrogel all disks (see Figure 11.4). The explicit control on the radial concentration c(r) of thenAn ðrÞ ¼ j figures). temperature-responsive polymer (N-isopropylacrylamide) resulted in the ability to control the (locally isotropic) shrinkage factors of distances η(r) = η(c(r)) and led to the target metric G = dr2 + κ −1 sin(ρκ 1/2 )2 dθ 2 on the midplate ω =118303-2 B(0, R),

314

11 Limiting theories for prestressed films: nonlinear bending

written in polar coordinates (r, θ ) and in terms of the prescribed constant Gaussian curvature κ ≡ ±0.0011. While decreasing the thickness h, all disks with κ > 0 kept the same basic dome-like shape, with minor variations along the edge (see left column figures in Figure 11.4. The energy related to Eg for g(z,t) = G(z)∗ was observed to stabilize as h → 0, approaching a constant multiple of h2 and exhibiting equipartition between bending and stretching; Hence, discs with positive curvature minimize their energy via the scenario in Corollary 11.13, by settling near the isometric immersion that is of the lowest bending content. On the other hand, for κ < 0, the disks were observed to undergo a set of bifurcations in which the number of nodes (within a single wave configuration) increased and was roughly proportional to h−1/2 . Measuring the bending content in this case led to Egh ' h which seems to be linked to a stretching-driven process: the sharp increase of the bending content is compensated by simultaneous decrease in the stretching content. Hence, hyperbolic disks minimize their energy via a set of bifurcations, despite the existence of the smooth immersions y.

11.11 Bibliographical notes For an early monograph about prestressed elasticity, we refer to Ies¸an [1988]. The multiplicative decomposition ∇u = Fe A in section 11.1 has been proposed in Rodriguez et al. [1994] in the context of the growth formalism. All results of section 11.1 are valid in any spacial dimension N, although we restricted our attention to the physically relevant case N = 3. Proof of the fundamental theorem of Riemannian manifolds can be found in [Spivak, 1999, Vol. II, Chapter 4]. For a more general result on the regularity of isometries between equi-dimensional Riemannian manifolds, than Lemma 11.2, see Calabi and Hartman [1970]. Theorem 11.1 appeared in Lewicka and Pakzad [2011]. An alternative proof, using Young measures instead of harmonic decomposition, and generalizing the result to non-Euclidean target geometries is due to Kupferman et al. [2019]. Theorem 11.4 is from Lewicka and Mahadevan [2022]. Delellis-Inauen-Szekelyhidi’s theorem in section 11.2 is taken from De Lellis et al. [2018] and the commutator estimate in Lemma 11.5 is from [Conti et al., 2010, Lemma 1]. The statement of existence of a Lipschitz isometric immersion in Remark 11.3, can be strengthened to the statement that the set of such immersions is actually dense in the set of “short immersions”. Namely, for every u0 ∈ C 1 (Ω¯ , RN ) such that (∇u0 (x))T ∇u0 (x) and g(x) − ∇u0 (x)T ∇u0 (x) are positive definite at each x ∈ Ω ⊂ RN , there exists a sequence {un ∈ W 1,∞ (Ω , RN )}n→∞ approximating u0 in C 0 and such that Eg (un ) = 0. This statement is an example of the h-principle in differential geometry and it follows through the method of convex integration, see Gromov [1986]. For an intuitive example in dimension N = 1, set g ≡ 1 on Ω = (−1, 1) ⊂ R1 and observe that any u0 : (−1, 1) → R with Lipschitz constant less than 1 can be uniformly approximated by piecewise linear functions {un }n→∞ such that |u0n | = 1.

11.11 Bibliographical notes

315

The question of existence of local isometric immersions of a given two-dimensional Riemannian manifold into R3 is a longstanding problem in differential geometry, its main feature consisting of finding the optimal regularity. By a classical result in Kuiper [1955], a C 1 isometric embedding can be obtained by means of convex integration. This statement was improved in Borisov [1965] to C 1,α regularity for all α < 1/7 and analytic metrics G, and in Conti et al. [2010] to C 2 metrics. In fact, this last reference covered the case of N-dimensional metrics and the density of their immersions into RN+1 , within the set of short immersions, for the regularity C 1,α and 1 all α < 1+N+N 2 . The result in De Lellis et al. [2018] is specific to the 2-dimensional case, by taking advantage of the theory of conformal maps. The mentioned results at optimal (so far) exponents, both for general N and N = 2 cases, were extended to compact manifolds by Cao and Sz´ekelyhidi [2022]. We observe that all these regularities are far from H 2 , where information about the second derivatives is also available. On the other hand, a smooth isometric immersion exists for some special cases, e.g. for smooth metrics with uniformly positive or negative Gaussian curvatures κ on bounded domains in R2 , see [Han and Hong, 2006, Theorems 9.0.1 and 10.0.2]. Counterexamples to such theories are largely unexplored. By Iaia [1992], there exists an analytic metric with nonnegative curvature on the two-dimensional sphere, with no local C 3 isometric embedding. However such metric always admits a C 1,1 embedding by Guan and Li [1994], Hong and Zuily [1995]; for a related example see also Pogorelov [1971]. Of interest is the result in De Lellis and Inauen [2020], stating that under the regularity C 1,α with α > 1/2, the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any α < 1/2 this property fails. Theorems 11.7 and 11.10 were first proved in the special case of g(z,t) = G(z) = G(z)tan + e3 ⊗ e3 in Lewicka and Pakzad [2011] and then in Bhattacharya et al. [2016] for the t-dependent case. A generalization to the abstract setting of Riemannian manifolds appeared in Kupferman and Solomon [2014]. Lemma 11.15 is from Lewicka and Luci´c [2020], while Theorem 11.16 is from Bhattacharya et al. [2016]. The coercivity results in section 11.6 were proved in Lewicka and Luci´c [2020]. The calculation of the effective energy density from section 11.7 appeared in Lewicka and Pakzad [2011] and Bhattacharya et al. [2016]. The model of nematic glass described in section 11.8 is described in Warner and Terentjev [2003], Modes et al. [2011]. Results about the liquid glass elastomers and other examples listed in sections 11.8 and 11.9 are from Bhattacharya et al. [2016]. We also mention that the generalization of the “prior” membrane theory for pre-stressed bodies (case β = 0, originally derived for the classical nonlinear elasticity in Le Dret and Raoult [1996]), appeared in Kupferman and Maor [2014]. The fundamental open problem, which arises from the mismatch of the energy scaling bounds in Theorems 11.4 and 11.7 is to analyze the limiting behaviour of minimizing deformations in the intermediate scaling regime inf Egh ' Chβ for β ∈ [2/3, 2). Some outstanding questions are: Is this behaviour necessarily guided by an isometric immersion of some prescribed regularity? What are the Γ -limits of scaled energies h1β Egh as h → 0? We now mention the related results in this context.

316

11 Limiting theories for prestressed films: nonlinear bending

In Jin and Sternberg [2001], Ben Belgacem et al. [2000, 2002], energies leading to the buckling- or compression- driven blistering in a thin film breaking away from its substrate, were discussed (β = 1). Work by Bella and Kohn [2014] displays dependence of the energy minimization on boundary conditions and classes of admissible deformations, while Cerda et al. [2004], Bella and Kohn [2017] study coarsening of folds in hanging drapes, where the energy identifies the number of generations of coarsening. In Tobasco [2021], wrinkling patterns are obtained corresponding to a thin shell placed on a liquid bath (and the range of β between 0 and 1), depending on the strengths of the elastic and substrate forces. Kohn and O’Brien [2018b,a] analyse wrinkling in the center of a stretched and twisted ribbon (β = 4/3). These papers do not address the dimension reduction, but rather analyze the chosen actual configuration of the prestressed sheet. Closely related is also the literature on shape selection in non-Euclidean plates, exhibiting hierarchical buckling patterns in zero-strain plates (β = 2), where the complex morphology is due to the non-smooth energy minimization, see Gemmer and Venkataramani [2011, 2013], Gemmer et al. [2016]. In Audoly and Boudaoud [2004] various geometrically nonlinear thin plate theories have been used to analyze the self-similar structures with metric asymptotically flat at infinity, for a disk with edge-localized growth in Efrati et al. [2009], the shape of a long leaf in Liang and Mahadevan [2009], or torn plastic sheets in Sharon et al. [2007]. In Dervaux and Ben Amar [2008], Dervaux et al. [2009] a variant of the F¨oppl-von K´arm´an equilibrium equations has been formally derived from finite incompressible elasticity, via the multiplicative decomposition of deformation gradient Rodriguez et al. [1994] similar to ours. See also models related to wrinkling of paper in areas with high ink or paint coverage or grass blades and studies of movement of micro-organisms in Dervaux et al. [2009], Ben Amar et al. [2012], Arroyo and De Simone [2014].

Chapter 12

Limiting theories for prestressed films: von K´arm´an-like theory

Following the analysis in section 11, we now turn to the derivation of the dimensionally reduced energy of prestressed films, in the next scaling exponent range, beyond the already covered case of β = 2 in {h−β Egh }h→0 . This case is naturally equivalent to having the energy I2,g in chapter 11 achieve its zero minimum, and it turns out that there automatically holds inf Egh ≤ Ch4 , while the corresponding Γ -limit is then the von K´arm´an-like energy, generalized from its Euclidean form developed in chapter 6, to the present case involving the prestress. This energy quantisation, contrary to the Euclidean case in which any β ∈ (2, 4) was viable and led to the linearised bending in chapter 7, is proved in section 12.1. There, we also provide the approximation lemmas based on the Friesecke-JamesM¨uller’s inequality from chapter 4, stating the closeness of deformation gradients with the energy level Ch4 , to a (non-symmetric) square root of the prestress of order one, in the thickness variable expansion. This expansion is given in terms of the unique isometric immersion y0 of the midplate metric for whom Ig,2 (y0 ) = 0, and of its induced Cosserat vectors up to order two. Higher order expansions of this type and higher order Cosserat vectors will play the crucial role in the forthcoming analysis of the infinite hierarchy of prestressed films’ energies, in chapter 13. In section 12.2 we prove both the compactness and the Γ -liminf lower bound in terms of the von K´arm´an-like energy I4,g , defined on the spaces of H 2 -regular infinitesimal isometries and the limiting strains on the midplate deformed by y0 . This energy reduces to IvK studied in chapter 6, when g = Id3 . It consists of three terms: the stretching, the bending, and a completely new term comprising the only potentially nonzero Riemannian curvatures R1313 , R1323 , R2323 of g at the midplate, since the vanishing the other three curvatures is guaranteed whenever β > 2. The optimality of the obtained lower bound is deduced in section 12.3 by constructing a suitable recovery family, where we also deduce the Γ -convergence result and convergence of the minima. In section 12.4 we identify the equivalent conditions for having inf Egh ' Ch4 . These, naturally, require the vanishing of R1212 , R1213 , R1223 and the non-vanishing of any of the remaining curvatures, at any midplate point. In the case when the full Riemann tensor is null on the midplate, the kernel of I4,g is thus nonempty and consists of a single displacement-strain couple, up to rigid mo© Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_12

317

318

12 Limiting theories for prestressed films: von K´arm´an-like theory

tions. We further prove a coercivity result in which the squared H 2 -distance from the aforementioned kernel is revealed to coincide with the bending term in I4,g . Despite the kernel’s finite dimensionality, the full coercivity estimate involving the stretching term is false, even in case of the classical von K´arm´an energy on plates. In section 12.5 we illustrate the findings of this chapter for two model prestress metrics, that are independent of the thickness variable. Section 12.6 contains another example, where the metric is conformal and, in turn, depends only on the thickness variable. We show that the scaling exponent β of the corresponding non-Euclidean energy at minimizers, equals twice the order to which the conformal factor vanishes at the midplate. In particular, all even integers β = 2n are viable. This analysis will be continued in the next chapter, where we also show that no other scaling exponents are possible and find the Γ -limits of {h−2n Egh }h→0 for the remaining cases n ≥ 3.

12.1 Energy quantisation and approximation lemmas 3×3 Given a Riemannian metric g ∈ C ∞ (Ω 1 , Rpos,sym ) on a family of thin films {Ω h =
h h 2 ω × − 2 , 2 }h→0 with the midplate ω ⊂ R that is an open, bounded, simply connected and Lipschitz domain, in this and the next sections we assume that:

lim

1

h→0 h2

inf Egh = 0.

(12.1)

Recall that by Theorem 11.16 the condition (12.1) is equivalent to the existence of a smooth (and unique up to rigid motions) vector field y0 : ω¯ → R3 satisfying: (∇y0 )T ∇y0 = g(·, 0)tan ,


1 sym (∇y0 )T ∇b1 = ∂3 g(·, 0)tan 2

on ω.

(12.2)

¯ R3 ) is as in (11.20) uniquely determined by: The primary Cosserat field b1 ∈ C ∞ (ω, B0  [∂1 y0 , ∂2 y0 , b1 ]

satisfies: BT0 B0 = g(·, 0) with det B0 > 0 on ω. (12.3)

In particular, the above implies: B0 A(·, 0)−1 ∈ SO(3), where we recall that A = g1/2 . ¯ R3 ) through: We now introduce the new vector field b2 ∈ C ∞ (ω, B1  [∂1 b1 , ∂2 b1 , b2 ]

satisfies:


1 sym BT0 B1 = ∂3 g(·, 0) 2

on ω,

(12.4)

as justified by (12.2) and in agreement with the construction of the recovery family for the Kirchhoff limiting energies in Theorem 11.10. Explicitly, we have:    1 (∇b1 )T b1 b2 = B0T,−1 ∂3 g(·, 0)e3 − ∂3 g(·, 0)33 e3 − . 0 2 For convenience, we also note the following easy expansion:

12.1 Energy quantisation and approximation lemmas

319

A(·,t)−1 =   (12.5)
t2 A−1 Id3 − t∂3 A + (2(∂3 A)A−1 (∂3 A)A−1 − ∂33 A) A−1 (·, 0) + O(t 3 ). 2 With the aid of b2 we now construct the sequence of deformations with low energy: Theorem 12.1. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Then (12.1) implies: inf Egh ≤ Ch4

for all h  1,

where the constant C depends only on ω, g and W , but is independent of h. Proof. Define the deformations {uh ∈ C ∞ (Ω h , R3 )}h→0 from (12.2), (12.4): uh (z,t)  y0 (z) + tb1 (z) +

t2 b2 (z). 2

(12.6)


Using (12.5) up to first order terms: A(·,t)−1 = A−1 −tA−1 (∂3 A)A−1 (·, 0)+O(t 2 ), it follows that:
(∇uh )A−1 = B0 A(·, 0)−1 Id3 + tSh + O(t 2 ) , (12.7)
where: Sh = A(·, 0)−1 BT0 B0 − (A−1 ∂3 A)(·, 0) A(·, 0)−1 . Recall that B0 A(·, 0)−1 ∈ SO(3), so that the frame invariance of W implies:


W (∇uh A−1 )(z,t) = W Id3 + tSh (z) + O(h2 ) = W Id3 − tsymSh (z) + O(h2 )
= W Id3 + O(h2 ) = O(h4 ), where we also used the fact that Sh ∈ so(3), in view of (12.2). This implies that Egh (uh ) = O(h4 ) as well, proving the claim. By a combination of the change of variable argument in the proof of Lemma 11.18 and the low energy deformation construction in the result above, we now derive an approximation result that generalizes (11.22) to the present higher order case, and replaces Friesecke-James-M¨uller’s rigidity estimate in the analysis to follow. Lemma 12.2. In the context of Theorem 12.1, for any open, connected, Lipschitz domain V ⊂ ω, we denote: h h
Vh = V × − , , 2 2

Egh (u,V h ) =

1 h

Z Vh


W (∇u)A−1 dx.

If y0 in (12.2) is injective on V , then for every u ∈ H 1 (V h , R3 ) there holds:

320

12 Limiting theories for prestressed films: von K´arm´an-like theory

1 h

Z Vh


2
∇u − R¯ B0 + tB1 dx ≤ C Egh (u,V h ) + h3 |V h | ,

(12.8)

with some R¯ ∈ SO(3). The constant C above is uniform for all subdomains V h ⊂ Ω h which are bi-Lipschitz equivalent with controlled Lipschitz constants. Proof. Denote the vector fields in (12.6) by {Y h ∈ C ∞ (Ω h , R3 )}h→0 and observe: B0 + tB1 = ∇Y h (·,t) + O(h2 ). For sufficiently small h > 0, each Y|Vh h is a smooth diffeomorphism onto its image U h = Y h (V h ) ⊂ R3 , satisfying uniformly: det ∇Y h > c > 0. We now consider the composition u¯ = u ◦ (Y h )−1 ∈ H 1 (U h , R3 ). By the rigidity estimate (4.2) we get: Z Uh

¯ 2 dx ≤ C |∇u¯ − R|

Z Uh


¯ SO(3) dx, dist2 ∇u,

(12.9)

for some rotation R¯ ∈ SO(3). Noting that: (∇u) ¯ ◦Y h = (∇u)(∇Y h )−1 on the set V h , the change of variable formula yields for the left hand side in (12.9): Z Uh

Z

2 (det ∇Y h ) (∇u)(∇Y h )−1 − R¯ dx Vh Z   2 ≥c ∇u − R¯ B0 + tB1 + O(h2 ) dx Vh Z Z
2 ≥c O(h4 ) dx. ∇u − R¯ B0 + tB1 dx − c

¯ 2 dx = |∇u¯ − R|

Vh

Vh

Similarly, the right hand side in (12.9) can be estimated by: Z

¯ SO(3) dx = (det ∇Y h ) dist2 (∇u)(∇Y h )−1 , SO(3) dx dist2 ∇u, Vh Uh Z   ≤C dist2 (∇u)A−1 , SO(3)(∇Y h )A−1 dx.

Z

Vh


Recall that from (12.7) we have: (∇Y h )A−1 ∈ SO(3) Id3 + tSh + O(h2 ) . This last
set is a subset of the set SO(3) Id3 + O(h2 ) , since Sh ∈ so(3). Consequently: Z Uh

Z
dist2 ∇u, ¯ SO(3) dx ≤ C

≤C

ZV

h

Vh

  dist2 (∇u)A−1 , SO(3)(Id3 + O(h2 )) dx
dist2 (∇u)A−1 , SO(3) + O(h4 ) dx

The estimate (12.8) follows now in view of (12.9) and by the lower bound on W . The local approximation in Lemma 12.2 can be lifted to the global approximation result as in Lemma 11.9, yielding a higher order counterpart of Theorem 4.8:

12.1 Energy quantisation and approximation lemmas

321

Corollary 12.3. In the context of Theorem 12.1, let the family of deformations {uh ∈ H 1 (Ω h , R3 )}h→0 satisfy (12.1). Then, there exists rotation-valued maps {Rh ∈ H 1 (ω, SO(3))}h→0 , so that: 1 h

Z Ωh

h

∇u − (Rh ◦ π) B0 + tB1 2 dx ≤ C h4 + Egh (uh ) , (12.10)


1 |∇R | dz ≤ C h + 2 Egh (uh ) . h ω

Z

h 2

2

Proof. 1. We apply Lemma 12.2 on the sets as in Theorems 4.8 and 3.23: Dz,h = Bh (z) ∩ ω,

h h
Bz,h = Dz,h × − , 2 2

for all z ∈ ω, h  1,

thus obtaining fixed rotations R¯ z,h ∈ SO(3) with the property that: 1 h

Z Bz,h

h

∇u − R¯ z,h B0 + tB1 2 dx ≤ C h3 |Bz,h | + Egh (uh , Bz,h ) ,

and the constant C independent of z, h, uh . Define now R˜ h ∈ H 1 (ω, R3×3 ) by setting: R˜ h (z) 

Z Ωh

ηz (x)∇uh (x) B0 + tB1


−1

dx

where the mollifiers {ηz : Ω h → R}z∈ω are as in the proof of Theorem 4.8. Hence: 1 h

Z Bz,h


|∇uh − (R˜ h ◦ π) B0 + tB1 |2 dx

C C |∇uh − R¯ z,h (B0 + tB1 )|2 dx + |R˜ h ◦ π − R¯ z,h |2 dx h Bz,h h Bz,h Z   |R˜ h (z0 ) − R¯ z,h |2 dz0 . ≤ C Egh (uh , Bz,h ) + h3 |Bz,h | +C

≤

Z

Z

(12.11)

Dz,h

Denoting 2Bz,h = Dz,2h × (− 2h , h2 ), we take z0 ∈ Dz,h and estimate: Z 2
−1 |R˜ h (z0 ) − R¯ z,h |2 = ηz0 (x)∇uh (x) B0 + tB1 dx − R¯ z,h Ωh Z
−1 2
= dx ηz0 (x) ∇uh (x) − R¯ z,h (B0 + tB1 ) B0 + tB1 Ωh Z Z    h ∇u − R¯ z0 ,h (B0 + tB1 ) 2 dx ≤C |ηz0 |2 dx Bz0 ,h

≤

C h3

Z Bz0 ,h

Bz0 ,h

h ∇u − R¯ z0 ,h (B0 + tB1 ) 2 dx

 C ≤ 2 Egh (uh , 2Bz,h ) + h3 |2Bz,h | , h

(12.12)

322

12 Limiting theories for prestressed films: von K´arm´an-like theory

In particular, one can replace R˜ h by Rh  PSO(3) R˜ h ∈ H 1 (ω, SO(3)). Indeed, this projection is well defined in view of (12.12), since: dist2 (R˜ h (z), SO(3)) ≤ |R˜ h (z) − R¯ z,h |2 ≤


C h h Eg (u ) + h4 → 0 2 h

2. As in the proof of (12.12) and using that |∇R˜ h (z0 )|2 = =

Z Ωh

Bz0 ,h

≤C ≤

h4

Ω h ∇z ηz (x) dx = 0, it follows on Dz,h :


−1

dx

2


2 (∇z ηz0 ) ∇uh (B0 + tB1 )−1 − R¯ z0 ,h dx

Z

2

Ωh

C

R

(∇z ηz0 )(∇uh ) B0 + tB1

Z

as h → 0.

|∇z ηz | dx

 Z Bz0 ,h

 h ∇u − R¯ z0 ,h (B0 + tB1 ) 2 dx

 Egh (uh , 2Bz,h ) + h3 |2Bz,h | .

Hence, from (12.12) we obtain: 1 h

Z Bz,h

  |R˜ h ◦ π − R¯ z,h |2 dx ≤ C Egh (uh , 2Bz,h ) + h3 |2Bz,h | ,

and therefore by (12.11) we further see that: 1 h

Z Bz,h

  |∇uh − (R˜ h ◦ π)(B0 + tB1 )|2 dx ≤ C Egh (uh , 2Bz,h ) + h3 |2Bz,h | .

The above yields as well: 1 h

Z Bz,h

|∇uh − (Rh ◦ π)(B0 + tB1 )|2 dx

Z Z h C ∇u − (R˜ h ◦ π)(B0 + tB1 ) 2 dx +C |R˜ h (z0 ) − Rh (z0 )|2 dz0 (12.13) h Bz,h Dz,h   ≤ C Egh (uh , 2Bz,h ) + h3 |2Bz,h | .

≤

n(h)

3. Covering Ω h by a family {Bzi ,h }i=1 , such that the intersection number of {2Bzi ,h } is independent of h and summing (12.13) over the covering yields the first bound in (12.10). In a similar fashion we obtain: Z

|∇R˜ h (z0 )|2 dz0 ≤

Dz,h

 C h h 3 E (u , 2B ) + h |B | . z,h z,2h g h2

and then by the same covering argument: Z ω

|∇R˜ h |2 dz ≤


C h h Eg (u ) + h4 . 2 h

12.2 Von K´arm´an-like theory for prestressed films: compactness and lower bound

This ends the proof, because

R ω

|∇Rh |2 dz ≤ C

R ω

323

|∇R˜ h |2 dz.

12.2 Von K´arm´an-like theory for prestressed films: compactness and lower bound In this section, we prove that the deformations {uh }h→0 with energy of order h4 converge, together with their properly defined displacements of order one and two relative relative to the isometric immersion y0 in (12.2). We also show the Γ -liminf bound of the energies h14 Egh , in terms of the functional I4,g that generalizes the von K´arm´an energy IvK derived in Theorem 6.1 in the context of shells. Definition 12.4. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ) satisfy any of the equivalent conditions in Theorem 11.16. We introduce the following spaces and the energy defined on them, relative to y0 : (i) The space of first order infinitesimal isometries: n o
V (y0 ) = V ∈ H 2 (ω, R3 ); sym (∇y0 )T ∇V = 0 in ω . Equivalently, V ∈ H 2 (ω,R3 ) belongs to V (y0 ) if and only if there exists d ∈ H 1 (ω, R3 ) such that sym BT0 [∂1V, ∂2V, d] ≡ 0. The vector field d is uniquely associated with V by: (∇y0 )T d = −(∇V )T b1

and

hd, b1 i = 0.

(12.14)

(ii) The space of finite strains: n o
B(y0 ) = L2 − lim sym (∇y0 )T ∇wh ; wh ∈ H 1 (ω, R3 ) . h→0

(iii) The von K´arm´an energy for prestressed films: I4,g (V, S)  Z
1 1 1 1  Q2,g z, S + (∇V )T ∇V + (∇b1 )T ∇b1 − ∂33 g(·, 0) dz 2 ω 2 24 48 Z (12.15)
1 T T + Q2,g z, (∇y0 ) ∇d + (∇b1 ) ∇V dz 24 ω Z
1 1 + Q2,g z, (∇y0 )T ∇b2 + (∇b1 )T ∇b1 − ∂33 g(·, 0)tan dz, 1440 ω 2 where d is related to V by means of (12.14), and where b2 is given in (12.4). Remark 12.5. (i) The notions of spaces V (y0 ) and B(y0 ) can be derived from Definitions 5.21 and 5.28 for the deformed midplate y0 (ω), in the sense that:

324

12 Limiting theories for prestressed films: von K´arm´an-like theory

 V (y0 ) = V ◦ y0 ; V ∈ V (y0 (ω)) ,  B(y0 ) = (∇y0 )T B(∇y0 ); B ∈ B(ω(y0 )) . T From Example 5.29, we have B(id2 ) = {S ∈ L2 (ω, R2×2 sym ); curl curl S = 0}. ¯ then When the Gauss curvature κ((∇y0 )T ∇y0 ) = κ g(·, 0)2×2 ) > 0 on ω, B(y0 ) = L2 (ω, R2×2 ), by Theorem 9.2. sym (ii) We anticipate that in Lemma 12.11, the entries of the metric-related argument under Q2,g in the last term in I4,g , will be identified as R1313 , R1323 , R2323 (·, 0). Recall that in virtue of Theorem 11.16, these are precisely the only potentially non-vanishing components of the Riemann curvature tensor Riem(g) on the midplate ω × {0}. We may thus write, informally:

I4,g (V, S) =

1 2

Z

Q2,g (z, stretching of order h2 ) dz

ω

1 Q2,g (z, bending of order h) dz 24 ω Z 1 + Q2,g (z, Riem(g)(z, 0)) dz, 1440 ω Z

+

where the “stretching” and “bending” generalize the tensors in IvK in (6.2). (iii) In the “flat” case of g ≡ Id3 , the referential solution to (12.2) is: y0 = id2 , b1 = e3 and further, by Remark 5.22 and Example 5.29:  V (id2 ) = V (ω) = V (z) = (αz⊥ + β , v); α ∈ R, β ∈ R2 , v ∈ H 2 (ω, R) ,  B(id2 ) = B(ω) = S = sym∇w; w ∈ H 1 (ω, R2 ) . Given V ∈ V (y0 ), we have p = (−∇v, 0) and hence: I4,Id3 (V, S) =

1 2


1 Q2,Id3 z, sym∇w + (α 2 Id2 + ∇v ⊗ ∇v) dz 2 ω Z
1 2 Q2,Id3 z, ∇ v dz. + 24 ω Z

Absorbing the stretching α 2 Id2 into sym∇w, the above energy can be expressed in the familiar form that we observed in Remark 6.2 in section 6: I4 (v, w) =

1 2


1 Q2 sym∇w + ∇v ⊗ ∇v dz 2 ω Z
1 2 Q2 ∇ v dz, + 24 ω Z

(12.16)

written a function of the out-of-plane scalar displacement v and the in-plane vector displacement w. The following is the main result of this section:

12.2 Von K´arm´an-like theory for prestressed films: compactness and lower bound

325

Theorem 12.6. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and h 1 h 3 let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume that {u ∈ H (Ω , R )}h→0 satisfies: Egh (uh ) ≤ Ch4 . Then, there exists {ch ∈ R3 , R¯ h ∈ SO(3)}h→0 such that the renormalizations: yh (z,t)  (R¯ h )T uh (z, ht) − ch ∈ H 1 (Ω 1 , R3 )

(12.17)

have the convergence properties below, up to a subsequence that we do not relabel, and relative to the smooth fields y0 , b1 as (12.2), (12.4): (i) yh → y0 ◦ π in H 1 (Ω 1 , R3 ) and h1 ∂3 yh → b1 ◦ π in L2 (Ω 1 , R3 ), (ii) the scaled average displacements: 1 V [y ](z)  h h

h

?

1 2

− 21


yh − (y0 + htb1 ) dt

(12.18)

converge in H 1 (ω, R3 ) to some V ∈ V (y0 ),
(iii) the scaled strains: { h1 sym (∇y0 )T ∇V h [yh ] }h→0 converge weakly in L2 (ω, R2×2 ) to some limit S ∈ B(y0 ), (iv) we have the following lower bound, in terms of the energy in (12.15): lim inf h→0

1 h h E (u ) ≥ I4,g (V, S). h4 g

(12.19)

Proof. 1. To prove the claimed convergence properties for (12.17), we first set: ?  h ¯ R = PSO(3) (∇uh )B−1 0 dx . Ωh

The projection above is well defined, as for every z ∈ ω we have, in view of (12.10): ? ? 2
h dist2 (∇uh )B−1 (∇uh )B−1 dx, SO(3) ≤ 0 0 dx − R (z) Ωh Ωh ? ? h 2 2 h −1 h (∇u )B0 − R dx +C R (z) − ≤C Rh dz h ω ?Ω ? 2
−1 2 h h ≤C ∇u − R (B0 + tB1 ) B0 dx +C Rh (z) − Rh dz h ω ?Ω ? ≤C |∇uh − Rh (B0 + tB1 )|2 dx +C Rh (z) − Rh dz h Ω ω ? h 2 4 h ≤ Ch +C R (z) − R dz . ω

Taking the average on ω, and applying the Poincar´e-Wirtinger inequality we get:

326

12 Limiting theories for prestressed films: von K´arm´an-like theory

? dist

2

(∇u Ωh

h

)B−1 0

Z
4 dx, SO(3) ≤ Ch +C |∇Rh |2 dz ≤ Ch2 , ω

by the second bound in (12.10). In particular, we observe that: ? ¯ h 2 ≤ Ch2 . (∇uh )B−1 0 dx − R

(12.20)

Ωh

Further, the above yields: ? ? |Rh − R¯ h |2 dx = |Rh − R¯ h |2 dx ω Ωh  ?  ? ? ? h 2 h 2 h h −1 ≤C R − R dz + R dz − (∇u )B0 dx dx Ωh ω Ωh ? ?ω h 2 R¯ − dx + (∇uh )B−1 0 h h Ω Ω ? ? ≤C |∇Rh |2 dz +C |∇uh − Rh (B0 + tB1 )|2 dx +Ch2 ≤ Ch2 .

(12.21)

Ωh

ω

Let R¯ ∈ SO(3) be an accumulation point of the rotations {R¯ h }h→0 . We note the easy consequence of the above (up to a not relabeled subsequence): in H 1 (ω, SO(3)),

Rh → R¯

R¯ h → R¯

as h → 0.

(12.22)

Let now ch ∈ R3 be such that ω V h [yh ] dz = 0 where each V h [yh ] is defined as in (12.18). Denote by ∇h yh the matrix whose columns are given by ∂1 yh , ∂2 yh and ∂3 yh /h, so that: ∇h yh (z,t) = (R¯ h )T ∇uh (z, ht). By (12.10) and (12.21), there holds: ? Z |∇h yh − B0 |2 dx ≤ C |∇uh − R¯ h B0 |2 dx Ω1 Ωh ? ? ?
h h 2 h 2 ≤C |∇u − R (B0 + tB1 )| dx + |tR B1 | dx + |Rh − R¯ h |2 dz ≤ Ch2 . R

Ωh

Ωh

ω

Therefore, {∇h yh }h→0 converges in L2 (Ω 1 , R3×3 ) to B0 . Passing to a subsequence, if necessary and using Poincar´e’s inequality, we get that, in fact, {yh }h→0 converges weakly in H 1 (Ω 1 , R3 ), implying (i) in view of the strong convergence of {∇yh }h→0 . 2. For every z ∈ ω we now decompose the tangential displacement gradients in: ∇V h (z) = = 

1 h

?

1 h

?

1/2

−1/2

∇h yh − B0 dt


3×2

1/2

∇h yh − (R¯ h )t Rh (B0 + htB1 ) dt

−1/2 h I1 + I2h .

The first term above converges to 0. Indeed:


3×2

+


1 ¯h T h (R ) R − Id3 )B0 3×2 h

12.2 Von K´arm´an-like theory for prestressed films: compactness and lower bound

327

?

h T h C (R¯ ) ∇u (z, ht) − (R¯ h )t Rh (B0 (z) + htB1 ) 2 dx h2 Ω 1 ? h C ∇u (z,t) − Rh (B0 + tB1 ) 2 dx ≤ Ch2 . ≤ 2 h Ωh

kI1h k2L2 (ω) ≤

(12.23)

Towards analyzing the second term I2h , denote: Sh 


1 ¯h t h (R ) R − Id3 . h

By (12.21) and (12.10), it follows that: kSh k2L2 (ω) ≤

C h2

Z

|Rh − R¯ h |2 dz ≤ C

and

ω

k∇Sh k2L2 (ω) ≤

C h2

Z

|∇Rh |2 dz ≤ C.

ω

Passing to a subsequence, we can thus assume the convergence: Sh * S

weakly in H 1 (ω, R3×3 ),

which implies that I2h → (SB0 )3×2 in L2 (ω, R3×2 ). Consequently, by (12.23): ∇V h → (SB0 )3×2

in L2 (ω, R3×2 ).

(12.24)

As before, we conclude that {V h }h→0 converges in H 1 (ω, R3 ) and that the limit V belongs to H 2 (ω, R3 ), since ∇V = (SB0 )3×2 ∈ H 1 (ω, R3×2 ). We now prove that V ∈ V (y0 ). Indeed, by the definition of Sh , we observe that: h ksym Sh kL2 (ω) ≤ k − (Sh )T Sh kL2 (ω) ≤ ChkSh k2L4 (ω) ≤ ChkSh k2H 1 (ω) ≤ Ch. 2 Consequently, S is a skew symmetric field. But (∇y0 )T ∇V = (BT0 SB0 )tan ∈ so(2) as well, as claimed. This ends the proof of (ii). For future use, we also observe that, by the definition of the vector d ∈ H 1 (ω, R3 ) in (12.14), there holds:   ∂1V, ∂2V, d = SB0 . (12.25) 3. We now establish convergence in (iii). In view of step 2, we write:

1 1 1 sym BT0 ∇V h tan = sym Bt0 I1h )tan + sym BT0 Sh B0 tan  J1h + J2h . h h h

(12.26)

We first deal with the family {J2h }h→0 . Since Sh → S in L4 (ω, R3×3 ), we get: 1 1 1 1 sym Sh = − (Sh )T Sh → − ST S = S2 h 2 2 2

in L2 (ω, R3×3 ).

Therefore: J2h → −


1 1 T T B0 S SB0 tan = − (∇V )T ∇V 2 2

in L2 (ω, R2×2 ).

(12.27)

328

12 Limiting theories for prestressed films: von K´arm´an-like theory

We now prove convergence of {J1h }h→0 . Recall that by (12.26): ? 1/2  
1 Z h (·,t) dt J1h = sym BT0 I1h tan = sym BT0 (R¯ h )T h tan −1/2
1 where: Z h (z,t) = 2 ∇uh (z, ht) − Rh (z)(B0 (z) + thB1 (z)) on Ω 1 . h

(12.28)

In view of (12.10), the family {Z h ∈ L2 (Ω 1 , R3×3 )}h→0 is equibounded. Therefore, up to a subsequence. that we do not relabel, there holds: Zh * Z

weakly in L2 (Ω 1 , R3×3 ).

(12.29)

Consequently, the following convergence yields (iii) by (12.26), (12.27), (12.22): ?  J1h * J1  sym BT0 R¯ T

1/2

Z(z,t) dt −1/2

 tan

weakly in L2 (ω, R2×2 ).

(12.30)

4. We now investigate the structure of the weak limit S of h1 sym(BT0 ∇V h )tan in terms of the fields V and Z. We have just seen that: 1 S = J1 − (∇V )T ∇V, 2

(12.31)

where J1 is defined in (12.30). As a tool, for every s  1 consider { f s,h }h→0 : f s,h (z,t) 



1 h s y (z,t + s) − yh (z,t) − hs b1 + h(t + )b2 . 2 h s 2

We will show that f s,h * d weakly in H 1 (Ω 1 , R3 ) as h → 0, for any s. Write: ? τ 1 ∂3 yh (z,t + τ) − h(b1 + h(t + τ)b2 ) dτ f s,h (z,t) = 2 h 0 and observe that:

1 1 ∂3 yh − h(b1 + htb2 ) = (R¯ h )T ∇uh (z, ht) − (B0 + htB1 ) e3 2 h h
1 ¯h T h = (R ) ∇u (z, ht) − Rh (B0 + htB1 ) e3 + Sh (B0 + htB1 )e3 h = h(R¯ h )T Z h (z,t)e3 + Sh (B0 + htB1 )e3 . The first term in the right-hand side above converges to 0 in L2 (Ω 1 , R3 ) because {Z h }h→0 is bounded in L2 (Ω 1 , R3×3 ). The second term converges to SB0 e3 = Sb1 = d in L2 (Ω 1 , R3 ). Therefore, f s,h → d in L2 (Ω 1 , R3 ) as h → 0. We now deal with the derivatives of the studied sequence. Firstly:

12.2 Von K´arm´an-like theory for prestressed films: compactness and lower bound

329


1 1 ∂3 yh (z,t + s) − h(b1 + h(t + s)b2 ) ∂3 f s,h (z,t) = s h2
 1 − 2 ∂3 yh (z,t) − h(b1 + htb2 ) h converges to 0 in L2 (Ω 1 , R3 ). For i = 1, 2, the in-plane derivatives read as: 1  ¯h T h (R ) ∂i u (z, h(t + s)) h2 s  s
− (R¯ h )T ∂i uh (z, ht) − hs ∂i b1 + h(t + ) ∂i b2 2  1  ¯h T h = (R ) Z (z,t + s) − (R¯h )T Z h (z,t) ei s  1  + 2 (R¯ h )T Rh (B0 + h(t + s)B1 ) − (R¯ h )T Rh (B0 + htB1 ) ei h s
s
1 − B1 ei + h t + ∂i b2 . h 2
The last two terms above can be written as: Sh B1 ei − t + 2s ∂i b2 , hence: ∂i f s,h (z,t) =


1 ¯ T ∂i f s,h (z,t) * (R) Z(z,t + s) − Z(z,t) ei s s
+ SB1 ei − t + ∂i b2 2

weakly in L2 (Ω 1 , R3 ).

Consequently, f s,h * d weakly in H 1 (Ω 1 , R3 ) and, for i = 1, 2:

¯ T Z(z,t + s) − Z(z,t) ei + sSB1 ei − s t + s ∂i b2 , s∂i d = (R) 2

(12.32)

which proves that Z(z, ·)ei has polynomial form and that:


t2 R¯ T Z(z,t) 3×2 = R¯ T Z(z, 0) 3×2 + t ∇d − (SB1 )3×2 + ∇b2 . 2

(12.33)

By (12.29), it now follows that:

1 J1 = sym BT0 R¯ T Z(·, 0) tan + sym BT0 ∇b2 tan . 24 With (12.31), we finally arrive at the following identity that links S with V and Z: S = sym BT0 R¯ T Z(·, 0)


tan

+


1 1 sym BT0 ∇b2 tan − (∇V )T ∇V. 24 2

(12.34)

5. We now prove (iv). Recall that the definition of {Z h }h→0 implies that: ∇uh (·, ht) = Rh (B0 + htB1 ) + h2 Z h (z,t). Consequently, as in (12.7), recalling (12.5) and Q0 A(·, 0)−1 ∈ SO(3), we get:

330

12 Limiting theories for prestressed films: von K´arm´an-like theory



W (∇uh A−1 )(·, ht) = W (B0 A(·, 0)−1 )T (Rh )T (∇uh A−1 )(·, ht)
= W Id3 + htK1 + h2 (t 2 K2 + K3h (·,t)) + O(h3 )|Z h | , ¯ R3×3 ), {K3h ∈ L2 (Ω 1 , R3×3 )}h→0 and where the following matrix fields: K2 ∈ C ∞ (ω, ∞ ¯ so(3)) which is skew-symmetric valued in view of (12.4), are given by: K1 ∈ C (ω,
K1 = A−1 BT0 B1 − A(∂3 A) A−1 (·, 0),
1 K2 = A−1 A(∂3 A)A−1 (∂3 A) − A(∂33 A) − BT0 B1 A−1 (∂3 A) A−1 (·, 0), 2 K3h (·,t) = A(·, 0)−1 BT0 (Rh )T Z h (·,t)A(·, 0)−1 . Also, by (12.29) and (12.22) we get: K3h * K3  A(·, 0)−1 BT0 R¯ T ZA(·, 0)−1

weakly in L2 (Ω 1 , R3×3 ).

(12.35)

As in the proof of Theorem 11.7, we now identify the “good” sets {|Z h | ≤ h−1/2 } ⊂ Ω 1 and employ the above to write there the following Taylor’s expansion:
W (∇uh A−1 )(z, ht)   
= W e−htK1 Id3 + htK1 + h2 t 2 K2 + K3h (·,t) + O(h5/2 )  
(12.36) t2 = W Id3 + h2 t 2 K2 − (K1 )2 + K3h + O(h5/2 ) 2 2

h4  t = Q3 sym t 2 K2 − (K1 )2 + K3h + o(h4 ) 1 + |K3h |2 . 2 2 Above, we repeatedly used the frame invariance of W and the exponential formula: h2t 2 (K1 )2 + O(h3 ). 2 Since the weak convergence in (12.29) implies convergence |Z h | ≥ h−1/2 → 0 as h → 0, with the help of (12.36) and (12.35) we finally arrive at: e−htK1 = Id3 − htK1 +

lim inf h→0

Z
1 t2 1 h h 2 (u ) ≥ lim inf E (K1 )2 + K3h dx t K − Q 2 3 g 4 −1/2 h h→0 2 {|Z |≤h h 2 } Z   2 1 t ≥ Q3 t 2 K2 − (K1 )2 + K3 dx 2 Ω1 2 Z    
t2 1 dz. Q2,g z, A(z, 0) t 2 K2 − (K1 )2 + K3 A(z, 0) ≥ 2 Ω1 2 tan

(12.37)

6. We see that the argument in the integrand of the right hand side of (12.37) is a quadratic polynomial in the variable t, so that: lim inf h→0

1 1 h h E (u ) ≥ h4 g 2

Z Ω1


Q2,g z, I(z) + tIII(z) + t 2 II(z) dx,

(12.38)

12.2 Von K´arm´an-like theory for prestressed films: compactness and lower bound

331

where by (12.33), (12.34) and (12.35) there holds:
1 1 I = S + (∇V )T ∇V − sym (∇y0 )T ∇b2 , 2 24  
1 T
1 II = sym A(·, 0) K2 + K1 K1 A(·, 0) + sym (∇y0 )T ∇b2 , 2 2 tan
III = sym (∇y0 )T ∇d + (∇b1 )T ∇V .

(12.39)


In computing III, we used sym(BT0 SB1 )tan = −sym(BT1 SB0 )tan = −sym (∇b1 )T ∇V , which is implied by (12.25). To identify II, we observe that:  
1 sym A(·, 0) K2 + K1T K1 A(·, 0) 2 tan 1
1 = sym BT1 B1 − 2sym BT1 B0 A−1 ∂3 A + (∂3 A)2 2 2  1 (·, 0) + A(∂3 A)A−1 (∂3 A) − A(∂33 A) 2 tan   1 = sym BT1 B1 − (∂3 A)2 − A(∂33 A) (·, 0) 2 tan 1 1 = (∇b1 )T ∇b1 − ∂33 g(·, 0)tan , 2 4
because ∂33 g = 2(∂3 A)2 + 2sym A(∂33 A) and also by (12.4) we have: 2sym(BT0 B1 ) = ∂3 g(·, 0) = ((∂3 A)A + A(∂3 A))(·, 0). Thus, the formula for II in (12.39) becomes:
1 1 1 II = (∇b1 )T ∇b1 + sym (∇y0 )T ∇b2 − ∂33 g(·, 0). 2 2 4

(12.40)

We now rewrite the right hand side of (12.38). Recall the linear mappings L2 (z, ·) in Definition 11.6, and keep in mind the obvious fact that all the odd powers of t integrate to 0 on the symmetric interval (− 21 , 12 ). We then obtain: Z Ω1


Q2,g z, I + tIII + t 2 II dx Z

=

Q2,g (z, I) dz + (

ω

Z 1/2

+( −1/2

t 4 dt)

Z

Z 1/2 −1/2

t 2 dt)

Z

Q2,g (z, III) dz

ω

Z 1/2

Q2,g (z, II) dz + 2(

ω

t 2 dt)

−1/2

1 1 = Q2,g (z, I) dz + Q2,g (z, III) dz + 12 ω 80 ω Z

 2 L2,g (z, I) : II dz, + 12 ω Z

Z

Z

 L2,g (z, I) : II dz

ω

Z ω

Q2,g (z, II) dz

332

12 Limiting theories for prestressed films: von K´arm´an-like theory

and consequently: Z Ω1


Q2,g z, I + tIII + t 2 II dx Z Z Z  1 1 1  Q2,g (z, III) dz + Q2,g (z, II) dz. = Q2,g z, I + II dz + 12 12 ω 180 ω ω

By (12.38) together with the formulas in (12.39), (12.40), we obtain (12.19) where the lower bound is given by the functional defined in (12.15). The proof is done.

12.3 Von K´arm´an-like theory for prestressed films: recovery family In this section we complete the derivation of the dimensionally reduced limit of the non-Euclidean energies {Egh }h→0 by proving the optimality of the lower bound in Theorem 12.6. We then conclude the Γ -convergence and convergence of minima, as done in section 11.4 in the prior Kirchhoff-like scaling regime. Theorem 12.7. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz do∞ ¯ R3 ) solves (12.2), main, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume that y0 ∈ C (ω, together with the associated Cosserat vector b1 given in (12.3). Then, for every V ∈ V (y0 ) and S ∈ B(y0 ), there exists a family {uh ∈ H 1 (Ω h , R3 )}h→0 such that the rescaled family {yh (z,t)  uh (z, ht)}h→0 satisfies: (i) yh → y0 in H 1 (Ω 1 , R3 ) and h1 ∂3 yh → b1 in L2 (Ω 1 , R3 ), (ii) V h [yh ] defined in (12.18) converge in H 1 (ω, R3 ) to V ,
(iii) 1h sym (∇y0 )T ∇V h [yh ] → S in L2 (ω, R2×2 ), (iv) we have the following convergence to the limit energy in (12.15): 1 lim Egh (uh ) = h→0 h4

I4,g (V, S).

(12.41)

Proof. 1. Given y0 , b1 , b2 and the admissible V and S, we fix ε > 0 and first define the recovery family {uh ∈ W 1,∞ (Ω h , R3 )}h→0 which yields (12.41) with an error of order Cε. The ultimate argument will be obtained via a diagonal procedure. We set: uh (·,t) = y0 + hvh + h2 wh + tb1 +

t3 t2 t2 b2 + k + ht ph + h rh + h2tqh . 2 6 2

¯ R3 ) are as in (12.2), (12.4). We introduce The Cosserat vector fields b1 , b2 ∈ C ∞ (ω, ¯ R3 )}h→0 is such that: other terms in the above expansion. The family {wh ∈ C ∞ (ω,

12.3 Von K´arm´an-like theory for prestressed films: recovery family

 1
 →S (∇y0 )T ∇ wh + b2 24 sym h1/2 kwh kW 2,∞ (ω,R3 ) → 0,

333

strongly in L2 (ω, R2×2 ),

(12.42)

as h → 0,

Existence of such a sequence is guaranteed by the fact that S ∈ B)y0 ), where we “slow down” the approximations {wh }h→0 to guarantee the blow-up rate of order less that h−1/2 . Further, the truncated sequence {vh ∈ W 2,∞ (ω, R3 )}h→0 is chosen according to Theorem 5.12 in a way that: strongly in H 2 (ω, R3 ) as h → 0, 1 hkvh kW 2,∞ (ω,R3 ) ≤ ε and lim 2 {z ∈ ω; vh (z) , V (z)} = 0. h→0 h

vh → V

(12.43)

¯ R3 ) and {ph , qh ∈ W 1,∞ (ω, R3 )}h→0 , are defined by: The vector fields k ∈ C ∞ (ω,   −(∇vh )T b1 T h B0 d = , 0   (∇vh )T d h    1 (∇wh )T b1 h T h T h T h , B0 q = c ·, (∇y0 ) ∇w + (∇v ) ∇v − 1 h 2 − 0 2 2 |d | (12.44)   1 T T T B0 k = c ·, (∇y0 ) ∇b2 + (∇b1 ) ∇b1 − (∂33 g(·, 0)tan     2 1 (∇b1 )T b2 (∇b2 )T b1 − + + ∂33 g(·, 0)e3 − ∂33 g(·, 0)33 e3 . 1 2 0 2 2 |b2 | Finally, we choose rh ∈ W 1,∞ (ω, R3 )}h→0 , to satisfy:     (∇vh )T b2 ∆ rh  rh − c ·, (∇y0 )T ∇d h + (∇vh )T ∇b1 + →0 hd h , b2 i

in L2 (ω, R3 ),

h1/2 krh kW 1,∞ (ω,R3 ) → 0. 2. Assertions (i)-(iii) follow easily by direct inspection. We will now concentrate on proving (iv). Observe that for all x = (z,t) ∈ Ω 1 there holds:         t2  ∇b2 , k ∇uh (·, ht) = B0 + h ∇vh , d h + tB1 + h2 ∇wh , qh + t ∇d h , rh + 2   h h 3 + O(h ) 1 + |∇q | + |∇r | . Consequently, by (12.5) it follows that:
(∇uh )A−1 (z, ht)   = B0 A(·, 0)−1 Id3 + hA(·, 0)−1 J1h A(·, 0)−1 + h2 A(·, 0)−1 J2h A(·, 0)−1 + J3h , where:

334

12 Limiting theories for prestressed films: von K´arm´an-like theory

   ∇vh , d h + tB1 − A∂3 A)(·, 0),   t2     J2h = BT0 ∇wh , qh + t ∇d h , rh + ∇b2 , k 2

1 −1 h − J1 A ∂3 A (·, 0) − A∂33 A (·, 0) 2

J1h = BT0

and where J1h , J2h , J3h satisfy the uniform bounds (independent of ε):
|J1h | ≤ C 1 + |∇vh | ,
|J2h | ≤ C 1 + |∇wh | + |∇vh |2 + |∇2 vh | + |∇rh | ,
|J3h | ≤ Ch3 1 + |∇wh | + |∇2 wh | + |∇vh |2 + |∇2 vh | + |∇vh | · |∇2 vh | + |∇rh | + o(h2 ).
¯ 0)−1 | is as small as In particular, dist (∇uh )A−1 , SO(3) ≤ |(∇uh )A−1 − B0 A(·, needed, uniformly in z ∈ ω for h  1. Thus, the argument (∇uh )A−1 of the frame invariant density W in E h (uh ) may be replaced by its polar decomposition factor:
T
1/2 (∇uh )A−1 (∇uh )A−1 1/2 
1 = Id3 + 2h2 A(·, 0)−1 sym J2h + (J1h )T g(·, 0)−1 J1h A(·, 0)−1 + Error 2
1 h T h 2 −1 = Id3 + h A(·, 0) sym J2 + (J1 ) g(·, 0)−1 J1h A(·, 0)−1 + Error, 2



where Error stands for a quantity obeying the following bound:

Error = O(h)|sym J1h | + O(h3 ) 1 + |∇vh | 1 + |∇wh | + |∇vh |2 + |∇2 vh | + |∇rh | + O(h3 )|∇2 wh | + o(h2 ). In conclusion, Taylor’s expansion of W at Id3 gives: 1 h4

Z


 (∇uh )A−1 (z, ht) dx Ω1 Z q 
T
1 W = 4 (∇uh )A−1 (∇uh )A−1 (z, ht) dx h Ω1 Z  
1 1 1 ≤ Q3 A(·, 0)−1 sym J2h + (J1h )T g(·, 0)−1 J1h A(·, 0)−1 + 2 Error dx 2 Ω 2 h Z Z O(1) |Error|3 dx. |J2h |3 + |J1h |6 dx + 4 + O(h2 ) h Ω1 Ω1 W



The residual terms above are estimated similarly to the proof of Theorem 6.3 in chapter 6, using (12.42), (12.43), (12.44). We thus have: h2

Z Ω!

|J2h |3 + |J1h |6 dx ≤ h2

Z Ω1

1 + |∇wh |3 + |∇vh |6 + |∇2 vh |3 + |∇rh |3 dx ≤ oh (1),

12.3 Von K´arm´an-like theory for prestressed films: recovery family

where we have used the bounds: h2 R 2

h

ω

|∇2 vh |3 dz ≤ εh

1 h4

Z Ω1

ω

Z

+ O(h2 )

|∇vh |6 dz ≤ Ch2 k∇vh k6H 1 (ω) = oh (1) and

1 h2

Z


sym (∇y0 )T ∇vh 2 dz

ω



1 + |∇vh |2 1 + |∇wh |2 + |∇vh |4 + |∇2 vh |2 + |∇rh |2 dz

+ O(h ) Zω

ω

|∇2 vh |2 dz = oh (1). Further:

R

|Error|2 dx ≤ 2

R

335

|∇2 wh |2 dz + oh (1)

ω

= oh (1) + O(h2 )

Z

|∇vh | · |∇2 vh |2 dz ≤ Cε,

ω

because the last condition in (12.43) implies: 1 h2

Z


sym (∇y0 )T ∇vh 2 dz

ω

C ≤ 2 k∇2 vh kL∞ (ω) h ≤

Z Cε 2

h4

{vh ,V }

Z {vh ,V }

dist2 (z, {vh = V }) dz

dist2 (z, {vh = V }) dz ≤ Cε 2

From the two estimates above we get that lim sup h→0

(12.45)

1 h {v , V } = oh (1). 2 h

1 R |Error|3 h4 Ω 1

dx = oh (1). Consequently:

Z  1 1 h h (u ) ≤ Cε + lim sup E A(·, 0)−1 Q 3 g h4 h→0 2 Ω 1 
1 sym J2h + (J1h )T g(·, 0)−1 J1h A(·, 0)−1 dx. 2

(12.46)

3. Observe now that: 1 sym J2h + (J1h )T g(·, 0)−1 J1h 2 
∗
= − sym sym (∇y0 )T ∇vh A−1 ∂3 A (·, 0)     t2    + sym BT0 ∇wh , qh + tBT0 ∇d h , rh + BT0 ∇b2 , k 2       T  t 2 t2 1 T + ∇vh , d h ∇vh , d h + tsym ∇vh , d h B1 + BT1 B1 − ∂33 g(·, 0). 2 2 4 Using (12.44), it follows that the quantity: Z Ω1



−1

Q3 A(·, 0)

is bounded by:

sym J2h +


1 h T (J1 ) g(·, 0)−1 J1h A(·, 0)−1 dx 2

1/2

336

12 Limiting theories for prestressed films: von K´arm´an-like theory

Z


1 Q2 z, (∇y0 )T ∇wh + (∇vh )T ∇vh + t (∇y0 )T ∇d h + (∇vh )T ∇b1 2 Ω  1/2 t2 t4 + (∇b1 )T ∇b1 − ∂33 g(·, 0)tan dx 2 4
T h + ksym (∇y0 ) ∇v kL2 (ω) + k∆ rh kL2 (ω) .

The second term in the right hand side above converges to 0 by (12.45) and the third term also converges to 0, by assumption on the approximating fields {rh }h→0 . On the other hand, the first term can be split into the integral on the set {vh = V }, whose limit as h → 0 is estimated by I4,g (V, S), and the remaining integral is bounded by: Z

C

{vh ,V }×(− 12 , 21 )

1 + |∇wh |2 + |∇vh |4 + |∇2 vh |2 + |rh |3 dx ≤ Cε 2

1 |{vh , V }| +C h2

Z {vh ,V }

|∇vh |4 dz

≤ o(1) +C|{vh , V }|1/2 k∇vh k4L8 (ω) = oh (1). In conclusion, (12.46) becomes (with a constant C that does not depend on ε): lim sup h→0

1 h h E (u ) ≤ Cε + I4,g (V, S). h4 g

A diagonal argument applied to the indicated ε-recovery family {uh }h→0 , together with the already achieved lower bound in Theorem 12.6 complete the proof. As in section 11.4, we observe the easy consequences of Theorems 12.6, 12.7: Corollary 12.8. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz domain, ∞ ¯ R3 ) and b given in (12.3), and let g ∈ C ∞ (Ω¯ 1 , R3×3 1 pos,sym ). Assume that y0 ∈ C (ω, solve (12.2). Then, the functional I4,g attains its minimum among displacementstrain couples (V, S) ∈ V (y0 ) × B(y0 ). Moreover, there holds: 1 inf Egh = min I4,g . h→0 h4 lim

Corollary 12.9. In the context of Corollary 12.8, the following Γ -convergence holds, with respect to: H 1 (Ω 1 , R3 ) × H 1 (ω, R3 ) × L2 (ω, R2×2 ):  1 h Γ I4,g (V, S) if y = y0 , V ∈ V (y0 ) and S ∈ B(y0 ), F −→ 4 +∞ otherwise. h Here, we put uh (z,t) = y(z,t/h) ∈ H 1 (Ω h , R3 ) and define:
 h h Eg (u ) if V = V h [y] and S = h1 sym (∇y0 )T ∇V , F h (y,V, S)  +∞ otherwise.

12.4 Identification of von K´arm´an’s scaling regime and coercivity of von K´arm´an-like...

337

There is a one-to-one correspondence between limits of sequences of (global) approximate minimizers to the energies Egh and (global) minimizers of I4,g .

12.4 Identification of von K´arm´an’s scaling regime and coercivity of von K´arm´an-like energy for prestressed films In this section, we identify the equivalent conditions for inf Egh ' h4 , in terms of the curvatures in Riem(g). Analysis below is similar to that in section 11.5. The following formulas will be used in the sequel: Lemma 12.10. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz domain, ∞ ¯ R3 ) satisfy (12.2), and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume that y0 , b1 , b2 ∈ C (ω, (12.3) and (12.4). Then we have the following valid in ω:   Γi 1j Γi31 Γ331     ∂i j y0 , ∂i b1 , b2 = B0  Γi 2j Γi32 Γ332  (·, 0) for i, j = 1, 2. (12.47) 3 3 3 Γi j Γi3 Γ33 Consequently, for any vector field q ∈ C ∞ (ω, R3 ) there holds: h hD Ei
i 1 2 3 (∇y0 )T ∇ BT,−1 q = (∇q) − q, [Γ ,Γ ,Γ ](·, 0) tan ij ij ij 0 i, j=1,2

i, j=1,2

. (12.48)

Above, {Γi kj } are the Christoffel symbols of the metric g, given in (11.4). Proof. In view of (12.2) and recalling (11.42), we get: h∂i j y0 , b1 i =


1 ∂i g j3 + ∂ j gi3 − ∂3 gi j (·, 0) 2

for all i, j = 1, 2,

which easily results in: h∂i b1 , ∂ j y0 i =


1 ∂i g j3 − ∂ j gi3 + ∂3 gi j (·, 0) 2

and

1 h∂i b1 , b1 i = ∂i g33 (·, 0). 2

Thus (11.43) and the above allow for computing the coordinates in the basis ∂1 y0 , ∂2 y0 , b1 as claimed in (12.47). The formula (12.48) results from:







 ∂i y0 , ∂ j BT,−1 q = ∂i y0 , ∂ j BT,−1 q + ∂i y0 , BT,−1 ∂ jq 0 0 0




 = − ∂i y0 , BT,−1 ∂ j BT0 BT,−1 q + B−1 0 ∂i y0 , ∂ j q 0 0




 = − B−1 0 ∂ j B0 ei , q + ei , ∂ j q , in view of (12.47). The proof is done.

338

12 Limiting theories for prestressed films: von K´arm´an-like theory

We note in passing that the expression in the right hand side of (12.48) represents the tangential part of the covariant derivative of the (0, 1) tensor field q with respect to g. We are now ready to provide the identification of the last term in the dimensionally reduced energy functional I4,g : Lemma 12.11. In the context of Lemma 12.10, the following holds in ω:  
1 R1313 R1323 (∇b1 )T ∇b1 + sym (∇y0 )T ∇b2 − ∂33 g(·, 0)tan = (·, 0). (12.49) R1323 R2323 2 Above, Ri jkl are the components of the Riemann curvature tensor Riem(g) given in (11.4), and evaluated at the midplate ω × {0}. Proof. Using (12.4) and (12.2), we get, for all i, j = 1, 2: 1 ∂ j h∂i y0 , b2 i = ∂ j3 g(·, 0)i3 − ∂ j hb1 , ∂i b1 i = ∂ j3 g(·, 0)i3 − ∂i j g(·, 0)33 . 2 Recalling the formulas in (11.4), we arrive at:
h
i 1 sym (∇y0 )T ∇b2 = − h∂i j y0 , b2 i + ∂ j h∂i y0 , b2 i + ∂i h∂ j y0 , b2 i 2 i, j=1,2     1 = − h∂i j y0 , b2 i i, j=1,2 + ∂i3 g j3 + ∂ j3 gi3 − ∂i j g33 i, j=1,2 (·, 0) 2   1 = − h∂i j y0 , b2 i i, j=1,2 + ∂33 g(·, 0)tan 2 h
i (·, 0). + Ri3 j3 − gnp Γi3nΓj3p − Γi nj Γ33p i, j=1,2

Directly from (12.47) we now deduce: h∂i j y0 , b2 i = gnpΓi nj Γ33p (·, 0),

h∂i b1 , ∂ j b1 i = gnpΓi3nΓj3p (·, 0).

(12.50)

This ends the proof of (12.49) and the lemma. The above allows to restate the von K´arm´an-like energy for prestressed films: Corollary 12.12. In the context of Lemma (12.10), the energy functional (12.15) can be written as: Z
1 1 1 1 Q2,g z, S + (∇V )T ∇V + (∇b1 )T ∇b1 − ∂33 g(·, 0) dz 2 ω 2 24 48 Z
1 Q2,g z, (∇y0 )T ∇d + (∇b1 )T ∇V dz + 24 ω " # Z   1 R1313 R1323 + Q2,g z, (·, 0) dz, 1440 ω R1323 R2323

I4,g (V,S) 

339

12.4 Identification of von K´arm´an’s scaling regime and coercivity of von K´arm´an-like...

where d is related to V by means of (12.14), and where Ri jkl are the components of the Riemann curvature tensor Riem(g). The key result of this section identifies the equivalent conditions for the von K´arm´an scaling, extending the findings for the Kirchhoff scaling in Theorem 11.16: Theorem 12.13. Let ω ⊂ R2 be open, bounded, simply connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Then, the following are equivalent: (i) We have: lim

1

h→0 h4

inf Egh = 0.

¯ R3 ) (which are then automa(ii) There exists y0 , b1 , b2 , b3 ∈ C ∞ (ω, tically unique up to rigid motions), such that defining {Bi =  ∂1 bi , ∂2 bi , bi+1 }2i=1 and recalling B0 = ∂1 y0 , ∂2 y0 , b1 , there holds: 

2

tk

∑ k! Bk

k=0

T 

2

tk

∑ k! Bk



= g(·,t) + O(h3 )

on Ω h ,

as h → 0.

k=0

(iii) All the Riemann curvatures of the metric g vanish on the midplate: Ri jkl (·, 0) = 0

in ω

and all i, j, k, l = 1 . . . 3.

Proof. By Corollary 12.8, it suffices to determine the equivalent conditions for min I4 = 0. Clearly, min I4 = 0 implies the condition in (iii), in view of (11.45). Vice versa, if (iii) holds, then Lemma 12.11 yields:
1 1 1 (∇b1 )T ∇b1 − ∂33 g(·, 0) = − sym (∇y0 )T ∇b2 . 24 48 24
1 Taking V = p = 0 and S = 24 sym (∇y0 )T ∇b2 ∈ B(y0 ), we get I4,g (V, S) = 0, which is (i). To show that (i) ⇔ (ii), observe that (i) is equivalent to existence of y0 , b1 , b2 as in (12.3), (12.4), together with a new requirement that:
1 sym (∇y0 )T ∇b2 + (∇b1 )T ∇b1 = ∂33 g(·, 0)tan . 2

(12.51)

¯ R3 ) so that: However, this identity is equivalent to the existence of b3 ∈ C ∞ (ω, 1 sym(BT0 B2 ) + BT1 B1 = ∂33 g(·, 0). 2 The above with (12.3), (12.4), is jointly equivalent to (ii), as claimed. We further have the following counterpart of the essential uniqueness of the minimizing isometric immersion y0 statement in Theorem 11.16:

340

12 Limiting theories for prestressed films: von K´arm´an-like theory

Theorem 12.14. In context of Theorem 12.13, assume that any of its equivalent conditions are satisfied. Then I4,g (V, S) = 0 if and only if: V = Sy0 + c

and

 1
 1 S = sym (∇y0 )T ∇ S2 y0 + b2 2 12

on ω,

(12.52)

for some S ∈ so(3) and c ∈ R3 . Proof. 1. We first observe that the bending term in (12.15) is already symmetric: h i h∂i y0 , ∂ j di + h∂iV, ∂ j b1 i i, j=1,2 h h i
i = ∂ j h∂i y0 , di + h∂iV, b1 i − h∂i j y0 , di + h∂i jV, b1 i i, j=1,2 i, j=1,2 h i 2×2 = − h∂i j y0 , di + h∂i jV, b1 i ∈ Rsym , i, j=1,2

where we used the definition of d in (12.14). Recalling (12.49), we see that I4,g (V, S) = 0 if and only if:
1 1 S + (∇V )T ∇V − sym (∇y0 )T ∇b2 = 0, 2 24 (∇y0 )T ∇d + (∇V )T ∇b1 = 0.

(12.53)

  1 2. Consider the matrix field S = ∇V, d B−1 0 ∈ H (ω, so(3)). Note that:     −1 −1,T ¯i −1 −1 ∂i S = ∇∂iV, ∂i d B−1 S B0 0 − ∇V, d B0 (∂i B0 )B0 = B0     T where S¯i = BT0 ∇∂iV, ∂i d + ∇V, d (∂i B0 ) ∈ L2 (ω, so(3)),

(12.54)

for i = 1, 2. Then we have:

hS¯i e1 , e2 i = ∂i h∂2 y0 , ∂iV i + h∂2V, ∂i y0 i − h∂12 y0 , ∂iV i + h∂12V, ∂i y0 i = 0. Indeed, the first term in the right hand side above equals 0 in view of V ∈ V (y0 ), whereas the second term equals ∂2 h∂1 y0 , ∂1V i for i = 1 and ∂1 h∂2 y0 , ∂2V i for i = 2, both expression being null again in view of V ∈ V (y0 ). We now claim that {S¯i }i=1,2 = 0 is actually equivalent to the second condition in (12.53). It suffices to examine the only possibly nonzero components: hS¯i e3 , e j i = h∂ j y0 , ∂i di + h∂ jV, ∂i b1 i = (∇y0 )T ∇d + (∇V )T ∇b1


ij

for all i, j = 1, 2,

(12.55)

proving the claim that S¯1 = S¯2 = 0. Thus the second condition in (12.53) is equivalent to S being constant, to the effect that ∇V = ∇(Sy0 ), or equivalently that V − Sy0 is a constant vector. Then:

341

12.4 Identification of von K´arm´an’s scaling regime and coercivity of von K´arm´an-like...


1 1 S = (∇y0 )T ∇ S2 y0 ) + sym (∇y0 )T ∇b1 2 24  1 1
 = sym (∇y0 )T ∇ S2 y0 + b1 . 2 12 is equivalent to the first condition in (12.53), as (∇V )T ∇V = −(∇y0 )T S2 ∇y0 . This ends the proof of Theorem 12.14. We now deduce the quantitative version of the above result, which is a counterpart of Theorem 11.17 in the present von K´arm´an regime: Theorem 12.15. In context of Theorem 12.13, assume that any of its equivalent conditions are satisfied. Then for all V ∈ V (y0 ) there holds:    dist2H 2 (ω,R3 ) V, Sy0 + c; S ∈ so(3), c ∈ R3 Z (12.56)
≤ C Q2 z, (∇y0 )T ∇d + (∇V )T ∇b1 dz, ω

with a constant C > 0 that depends on g, ω and W but it is independent of V . We recall that d ∈ H 1 (ω, R3 ) is derived from V by means of (12.14). Proof. We argue by contradiction. Since Vlin  {Sy0 + c; S ∈ so(3), c ∈ R3 } is a linear subspace of V (y0 ) and likewise the bending expression in (12.15) is linear in V , with its kernel equal to Vlin in virtue of Theorem 12.14, it suffices to take a sequence {Vn ∈ V (y0 )}n→∞ such that: kVn kH 2 (ω,R3 ) = 1, and:

T

Vn ⊥H 2 (ω,R3 ) Vlin T

(∇y0 ) ∇dn + (∇Vn ) ∇b1 → 0

for all n, in L2 (ω, R2×2 ),

as n → ∞.

(12.57)

Passing to a subsequence if necessary and using (12.14), it follows that: Vn * V weakly in H 2 (ω, R3 ), dn * d weakly in H 1 (ω, R3 ). (12.58)   Clearly, BT0 ∇V, d ∈ L2 (ω, so(3)) so that V ∈ V (y0 ), but also (∇y0 )T ∇d+(∇V )T ∇b1 = 0. Thus, Theorem 12.14 and the perpendicularity assumption in (12.57) imply: V = p = 0. We will now show that: Vn → 0

strongly in H 2 (ω, R3 ),

(12.59)

which will contradict the first (normalisation) condition in (12.56). As in (12.54), the assumption Vn ∈ V (y0 ) implies that for each z ∈ ω and i = 1, 2, the following matrix (denoted previously by S¯i ) is skew-symmetric:    T BT0 ∇∂iVn , ∂i dn + ∇Vn , dn (∂i B0 ) ∈ so(3). Equating tangential entries and observing (12.57), yields for every i, j, k = 1, 2:

342

12 Limiting theories for prestressed films: von K´arm´an-like theory

  h∂ j y0 , ∂ikVn i + h∂k y0 , ∂i jVn i = − h∂ jVn , ∂ik y0 i + h∂kVn , ∂i j y0 i → 0

in L2 (ω, R).

Permuting i, j, k, we eventually get: h∂ j y0 , ∂ikVn i → 0

in L2 (ω)

for all i, j, k = 1, 2.

On the other hand, equating off-tangential entries, we get by (12.57) and (12.58) that for each i = 1, 2 there holds:


b1 , ∂i jVn = − (∇y0 )T ∇dn + (∇Vn )T ∇b1 i j − hdn , ∂i j y0 i → 0 in L2 (ω). Consequently, {BT0 ∂i jVn → 0}i, j=1,2 in L2 (ω, R3 ), which implies convergence (12.59) as claimed. This ends the proof of (12.56).

Remark 12.16. Although the kernel of the energy I4,g , displayed in Theorem 12.14, is finite dimensional, the full coercivity estimate of the form below is false:  2 min kV − (Sy0 + c)kW 2,2 (ω,R3 ) S∈so(3),c∈R3



1

2 1 + S − sym (∇y0 )T ∇ S2 y0 − b2 L2 (ω,R2×2 ) 2 12 ≤ CI4,g (V, S) for all (V, S) ∈ V (y0 ) × B(y0 ).

(12.60)

For a counterexample, consider the particular case of the classical von K´arm´an functional (12.16), specified in Remark 12.5. Clearly, I4 (v, w) = 0 if an only if v(x) = ha, xi + α and w(x) = β x⊥ − 21 ha, xia + γ, for some a ∈ R> 2 and α, β , γ ∈ R. Note that R R (12.56) reflects then the Poincar´e inequality: ω |∇v − ω ∇v|2 dz ≤ C ω |∇2 v|2 dz, whereas (12.60) takes the form: min

a∈R2

Z

2

|∇v − a| dz +

ω

 1 |sym∇w + a ⊗ a|2 dz ≤ CI4 (v, w). 2 ω

Z

(12.61)

Let ω = B1 (0). Given v ∈ H 2 (ω, R) such that det ∇2 v = 0, let w satisfy: sym∇w = − 21 ∇v ⊗ ∇v, which results in the vanishing of the first term in (12.16). Neglecting the first term in the left hand side of (12.61), leads in this context to the following weaker form, which we below disprove: Z

min

a∈R2 ω

|∇v ⊗ ∇v − a ⊗ a|2 dz ≤ C

Z

|∇2 v|2 dz.

(12.62)

ω

Take: vn (z) = n(z1 + z2 ) + 21 (z1 + z2 )2 for all z = (z1 , z2 ) ∈ ω. Then ∇vn = (n + z1 + z2 )(1, 1) and det ∇2 vn = 0. Minimization in (12.62) becomes: Z

min

a∈R2 ω

|(n + z1 + z2 )2 (1, 1) ⊗ (1, 1) − a ⊗ a|2 dz

12.5 Two examples

343

and an easy explicit calculation yields the necessary form of the minimizer: a = δ (1, 1). Thus, the same minimization can be equivalently written and estimated in: 4 · min

Z

δ ∈R ω

(n + z1 + z2 )2 − δ 2 2 dz ' 4n2 → ∞

as n → ∞.

However, |∇2 vn |2 ≡ 4 on ω. Therefore, the estimate (12.62) cannot hold.

12.5 Two examples To illustrate the results of this chapter, we compute the energy I4,g (V, S) in two particular cases, whose study has been initiated in section 11.9. g(z,t) = G(z) = diag(1, 1, λ (z))

or

g(z,t) = G(z) = λ (z)Id3 ,

corresponding to the case of nontrivial differential shrinking factor only in the normal direction, or to the isotropic prestress, respectively. Let d be as in (12.14). Writing: d = α 1 ∂1 y0 + α 2 ∂2 y0 + α 3 b1 , we obtain: G α 1, α 2, α 3


T


T = − h∂1V, b1 i, h∂2V, b1 i, 0 .

Consequently: d = −G1i h∂iV , b1 i∂1 y0 − G2i h∂iV , b1 i∂2 y0 − G3i h∂iV , b1 ib1 .

(12.63)

¯ R) be a positive function. Then the following hold Lemma 12.17. Let λ ∈ C ∞ (ω, for the metric g(z,t) = diag(1, 1, λ (z)) on Ω 1 : (i) g is immersible in R3 if and only if: Mλ  ∇2 λ −

1 ∇λ ⊗ ∇λ ≡ 0 2λ

in ω,

while the condition Mλ . 0 is equivalent to: ch4 ≤ inf Egh ≤ Ch4 . (ii) The Γ -limit energy functional I4,g becomes: I4,g (v, w) =


1 1 ∇λ ⊗ ∇λ dz Q2 sym∇w + ∇v ⊗ ∇v + 2 96λ ω Z Z √

1 1 λ ∇2 v + + Q2 Q2 Mλ dz, 24 ω 5760 ω

1 2

Z

for all w ∈ H 1 (ω, R2 ), v ∈ H 2 (ω, R), where Q2 = Q2,Id3 is independent of z. Proof. Part (i) of the assertion has been shown in Example 11.29. For (ii), note that: √ y0 = id2 and B0 = A(·, 0) = diag(1, 1, λ ).

344

12 Limiting theories for prestressed films: von K´arm´an-like theory

Consequently, Q2,g = Q2,Id3 , which we denote simply by Q2 . As in Remark 12.5, every admissible limiting strain S ∈ B(y0 ) has the form S = sym∇w for some w ∈ H 1 (ω, R2 ). Also, without loss of generality, every admissible limiting displacement V ∈ V (y0 ) is of the form: V = (0, 0, v) for some v ∈ H 2 (ω, R). By (12.63) we get: √ b1 =

1 b2 = − (∂1 λ , ∂2 λ , 0), 2

λ e3 ,

√ d = − λ (∂1 v, ∂2 v, 0).

Therefore: 1 1 ∇λ ⊗ ∇λ , (∇y0 )T ∇b2 = − ∇2 λ , 4λ 2 √ 1 1 (∇V )T ∇b1 = √ ∇v ⊗ ∇λ . (∇y0 )T ∇d = − √ ∇v ⊗ ∇λ − λ ∇2 v, 2 λ 2 λ (∇b1 )T ∇b1 =

This ends the proof of Lemma 12.17 in view of (12.15). ¯ R) be a positive function. Denote f = 21 log λ . Then Lemma 12.18. Let λ ∈ C ∞ (ω, the following hold for the metric g(z,t) = G(z) = diag(1, 1, λ (z)) on Ω 1 : (i) Condition (12.2) is equivalent to ∆ f = 0, which is also equivalent to the immersability of the metric Gtan in R2 . (ii) Under condition (12.2), condition (12.51) is equivalent to Ric(g) = 0 and therefore to the immersability of g. (iii) The Γ -limit energy functional has the following form: I4,g (V, S) =

1 2


1 1 e−2 f Q2 S + (∇V )T ∇V + e2 f ∇ f ⊗ ∇ f dz 2 24 ω Z
1 + Q2 2∇V3 ⊗ ∇ f − ∇2V3 − h∇V3 , ∇ f iId2 dz 24 ω Z
1 Q2 e f Ric(g)tan dz + 1440 ω

Z

where Q2 = Q2,Id3 and where Ric(g)tan denotes the tangential part of the Ricci curvature tensor of g on the midplate: # " R11 R12 (·, 0). Ric(g)tan  R12 R22 Proof. 1. The part (i) has been deduced in Example 11.30, and we recall that: Ric(g) = −(∇2 f − ∇ f ⊗ ∇ f )∗ − (∆ f + |∇ f |2 )Id3 .

(12.64)

In case (12.2) holds, by (i) the metric Gtan is immersable in R2 and in particular N = e3 . Writing V = (V1 ,V2 ,V3 ), the formula (12.63) yields: √ b1 =

λ e3 ,


b2 = − ∂1 f ∂1 y0 + ∂2 f ∂2 y0 ,


1 d = − √ ∂1V3 ∂1 y0 + ∂2V3 ∂2 y0 . λ

12.5 Two examples

345

(∇b1 )T ∇b1 = e2 f ∇ f ⊗ ∇ f ,

(∇V )T ∇b1 = e f ∇V3 ⊗ ∇ f .

Further, observe that: ∂i d = −(∂1i f ∂1 y0 + ∂2i f ∂2 y0 + ∂1 f ∂1i y0 + ∂2 f ∂2i y0 ), and so:
1 1 1 1 h∂1 y0 , ∂1 di = − λ ∂11 f + ∂1 λ ∂1 f + ∂2 λ ∂2 f = −(∂11 f + |∇ f |2 ). λ λ 2 2 In the same manner, we arrive at: 1 h∂2 y0 , ∂2 di = −(∂22 f + |∇ f |2 ), λ 1 1 h∂2 y0 , ∂1 di = −∂12 f , h∂1 y0 , ∂2 di = −∂21 f . λ λ Consequently, (∇y0 )T ∇d is already a symmetric matrix, and: (∇y0 )T ∇d = −e2 f (∇2 f + |∇ f |2 Id2 ). In particular, under condition ∆ f = 0, the formula (12.64) yields:
sym (∇y0 )t ∇d + (∇b1 )T ∇b1 = e2 f Ric(g)tan , which is equivalent to ∇ f = 0 and hence to Ric(g) = 0. This establishes (ii). 2. We now compute the remaining quantities in the expression of I4,g . Firstly: ∇d =

1 1 ∇y0 (∇V3 ⊗ ∇λ ) − √ ∇y0 ∇2V3 2λ 3/2 λ  1  − √ ∂1V3 (∂11 y0 , ∂12 y0 ) + ∂2V3 (∂12 y0 , ∂22 y0 ) . λ

Using the relations between h∂i j y0 , ∂k y0 i and ∂l G in (11.43), we obtain: (∇y0 )T ∇d =

1 1 Gtan ∇V3 ⊗ ∇λ − √ Gtan ∇2V3 3/2 2λ λ   1 h∇V3 , ∇λ i h∇V3 , ∇λ ⊥ i − √ , ⊥ 2 λ −h∇V3 , ∇λ i h∇V3 , ∇λ i

and therefore: √
√
√ sym (∇y0 )T ∇d = λ sym ∇V3 ⊗ ∇ f − λ ∇2V3 − λ h∇V3 , ∇λ iId2 . In a similar manner, it follows that:  
sym (∇y0 )t ∇b2 = −λ ∇2 f + |∇ f |2 Id2 . Since Q2,g = λ −1 Q2,Id3 , we finally get:

346

12 Limiting theories for prestressed films: von K´arm´an-like theory

I4,g (V, S) =


1 1 e−2 f Q2 S + (∇V )T ∇V + e2 f ∇ f ⊗ ∇ f dz 2 24 ω Z
1 + e−2 f Q2 2e f ∇V3 ⊗ ∇ f − e f ∇2V3 − e f h∇V3 , ∇ f iId2 dz 24 ω Z
1 e−2 f Q2 e2 f Ric(g)tan dz + 1440 ω

1 2

Z

which implies (iii). The proof is done.

12.6 Beyond von K´arm´an’s regime: an example According to Theorem 12.13, for any nonimmersable prestress metric g(z,t) = G(z) that is independent of t, there holds inf Egh ≥ ch2 for some c > 0. In chapter 13 we will analyze the higher order energy scalings ch2n which occur for thicknessdependent metrics g. Below, we anticipate this study and show that any scaling of even order ch2n is viable, for the conformal metrics of the form: g(z,t) = e2φ (t) Id3

for all x = (z,t) ∈ Ω h .

Take φ ∈ C ∞ ((− 21 , 12 ), R). Since G(·, 0)tan = e2φ (0) Id2 has a smooth isometric immersion y0 = eφ (0) id2 into R2 , Corollary 11.11 implies that: inf Egh ≤ Ch2 . By a direct calculation, the only possibly non-zero Christoffel symbols of g are: Γ113 = Γ223 = −φ 0 (t) and Γ131 = Γ232 = Γ333 = φ 0 (t), while the only possibly nonzero Riemann curvatures in Riem(g) are: R1212 = −φ 0 (t)2 e2φ (t) ,

R1313 = R2323 = −φ 00 (t)e2φ (t) .

(12.65)

Consequently, results of sections 11 and 12 provide the following hierarchy of the possible energy scalings of {Egh }h→0 at minimizers: (a) {ch2 ≤ inf Egh ≤ Ch2 }h→0 with c,C > 0. This scenario is equivalent to φ 0 (0) , 0. The functionals h12 E h exhibit compactness properties as in Theorem 11.7, and Γ -converge to the following energy I2,g defined on the set of deformations: {y ∈ H 2 (ω, R3 ); (∇y)T ∇y = Id2 }: I2,g (y) =

1 24

Z ω


Q2,Id3 Πy − φ 0 (0)Id2 dz.

Here Πy is the second fundamental form of the surface y(ω). (b) {ch4 ≤ inf Egh ≤ Ch4 }h→0 with c,C > 0. This scenario is equivalent to φ 0 (0) = 0 and φ 00 (0) , 0. The unique (up to rigid motions) minimizing isometric immersion is then y0 = id2 and the functionals h14 Egh have the compactness properties

12.6 Beyond von K´arm´an’s regime: an example

347

as in Theorems 12.6. The following Γ -limit of h14 Egh is defined on the set of displacements {(v, w) ∈ H 2 (ω, R) × H 1 (ω, R2 )} as in Remark 12.5: I4,g (v, w) =

1 2


1 1 Q2,Id3 sym∇w + ∇v ⊗ ∇v − φ 00 (0)Id2 dz 2 24 ω Z

1 1 2 00 Q2,Id3 ∇ v dz + φ (0)2 |ω|Q2,Id3 Id2 . + 24 ω 1440 Z

(c) {inf Egh ≤ Ch6 }h→0 with C > 0. This scenario is equivalent to φ 0 (0) = 0 and φ 00 (0) = 0 and in fact we have the following more precise result below.

Theorem 12.19. Let g(z,t) = e2φ (t) Id3 , where φ (k) (0) = 0 for k = 1 . . . n − 1 up to some n > 2. Then: inf Egh ≤ Ch2n and: lim

1

h→0 h2n

inf Egh ≥ cn φ (n) (0)2 |ω|Q2,Id3 (Id2 ),

(12.66)

where cn > 0. In particular, if φ (n) (0) , 0 then we have with c,C > 0: ch2n ≤ inf Egh ≤ Ch2n . Proof. 1. To show the the upper bound, we compute:
1Z
Egh eφ (0) id3 = W eφ (0)−φ (t) Id3 dx h h Ω Z tn
1 Q3 φ (n) (0) Id3 + O(h2n+2 ) dx = 2h Ω h n!  (n) 2  φ (0) 1 = h2n |ω|Q (Id ) + o(1) ≤ Ch2n , 3 3 (n!)2 (2n + 1)22n+1 n

where we used the fact that eφ (0)−φ (t) = 1 − φ (n) (0) tn! + O(|t|n+1 ). 2. To prove the lower bound (12.66), let {uh ∈ H 1 (Ω h , R3 )}h→0 be a family of deformations that satisfy Egh (uh ) ≤ Ch2n . Then there holds: Z
c ≥ dist2 ∇uh , eφ (t) SO(3) dx h Ωh Z Z
c c¯ φ (n) (0) t + O(hn+1 ) 2 dx, ≥ dist2 ∇uh , eφ (0) SO(3) dx − h Ωh h Ωh n!
R which yields: 1h Ω h dist2 e−φ (0) ∇uh , SO(3) dx ≤ Ch2n . As in Lemma 12.2 and Corollary 12.3, we get approximating rotation fields {Rh ∈ H 1 (ω, SO(3))}h→0 with:

Egh (uh )

1 h

Z Ωh

|∇uh − eφ (0) Rh |2 dx ≤ Ch2n ,

Z ω

|∇Rh |2 dx ≤ Ch2n−2 .

(12.67)

348

12 Limiting theories for prestressed films: von K´arm´an-like theory

As in the proof of Theorem 12.6, define the displacement and deformation fields: ?
uh ∈ H 1 (Ω , R3 ), yh (z,t) = (R¯ h )T uh (z, ht) − Ωh ? where R¯ h = PSO(3) e−φ (0) ∇uh (x) dx Ωh

V h (z) =

Z 1/2

1 hn−1

−1/2


yh (z,t) − eφ (0) id2 + hte3 dt ∈ H 1 (ω, R3 ).

From (12.67), we get then the convergences (up to a not relabelled subsequence): yh → eφ (0) id2

1 h ∂3 y → eφ (0) e3 h in H 1 (ω, R3 ),

in H 1 (ω, R3 ),

V h → V ∈ H 2 (ω, R3 ) 1 sym∇V h * sym∇w h

in L2 (ω, R3 ),

weakly in L2 (ω, R2×2 ).

The above allows to conclude the claimed lower bound: Z  1 1 tn  lim inf 2n E h (uh ) ≥ Q2,Id3 e−φ (0) sym∇w − te−φ (0) ∇2V 3 − φ (n) (0) Id2 dz h→0 h 2 ω n! Z   t n  tn 1 (n) (n) Q2,Id3 φ (0) Id2 − P1 φ (0) Id2 dz ≥ 2 ω n! n! Z 1/2 (n) 2
1 φ (0) 2 t n − P1 (t n ) dt · |ω|Q2,Id3 (Id2 ) · = 2 (n!)2 −1/2 = cn · φ (n) (0)2 |ω|Q2,Id3 (Id2 ). Here, P1 is the orthogonal projection onto the space of affine (in t) functions. Thus:   (n−1)2 for n odd 1 (2n+1)(n+2)2 cn = 2n+1 2 2 n  2 (n!) 2 for n even, (2n+1)(n+1)

similarly to (12.66), and consistently with the already obtained c2 =

1 1440 .

12.7 Bibliographical notes The results and the point of view followed in this section are taken from Lewicka et al. [2017b], Lewicka and Luci´c [2020] and Lewicka [2020]. Estimates similar to the ones developed here are also useful for the numerical analysis of thin elastic bodies, see Bonito et al. [2021]. The generalization of the first quantisation Theorem 12.1 and of Theorem 11.16, for the case of all dimensions N ≥ 3 and co-dimensions of the “midplate”, have been shown by Maor and Shachar [2019].

Chapter 13

Infinite hierarchy of limiting theories for prestressed films

In this chapter, we complete the scaling analysis of the non-Euclidean energies {Egh }h→0 of prestressed thin films and the derivation of the Γ -limits of {h−β Egh }h→0 , in the remaining scaling regime β > 4. We will show the energy quantisation, in the sense that only the even powers β = 2n of h are viable, all of them attained. Building on the results from chapter 11 for n = 1 and chapter 12 for n = 2, we now address the higher scaling order admissible exponents n > 3. For each such n, we identify conditions for the optimality of the scaling h2n , in terms of the vanishing of Riemann curvatures of g up to appropriate orders, and in terms of the matched isometry expansions. We also establish the asymptotics of the minimizing immersions. The outline of this chapter is as follows. In section 13.1 we state the main compactness and Γ -liminf results, whose proofs will be entwined and given by induction on n. We also introduce the limiting functionals {I2n,g }∞ n=3 , each acting on the space of the infinitesimal isometries V (y0 ) of the films’ midplate deformed by the unique isometric immersion y0 , representative of the kernel of the previous energy I2,g . These energies reduce to the linear elasticity functional Ilin studied in chapter 6, when g = Id3 . They each comprise of the bending term and of a completely new geometric term, measuring the squared L2 norm of the (n−2)nd covariant derivative ∇n−2 Riem(g) of the Riemann curvature tensor of g, restricted to the midplate. The projections of ∇n−2 Riem(g) on the limiting strain space B(y0 ) defined in section 12, and on its orthogonal complement, are assigned distinct weights, depending on the parity of n. The bending term, likewise, involves the interaction with the curvatures for n odd (this interaction was present β = 2 but not at β = 4). In section 13.2, we provide the approximation lemma, based on the FrieseckeJames-M¨uller’s inequality from chapter 4, and prove the compactness properties of the deformation sequence with low energy. We also develop a variational lower bound on the scaled energies of deformations, in terms of the displacements of various orders. How these displacement gradients can be combined into the aforementioned curvatures, yielding I2n,g as the Γ -limit, is shown in sections 13.3 and 13.4. The optimality of the obtained lower bound is deduced in section 13.5 by means of constructing a recover family. We then conclude the full Γ -convergence, convergence of the minima, and the coercivity results in section 13.6. © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_13

349

350

13 Infinite hierarchy of limiting theories for prestressed films

13.1 Energy quantisation and identification of scaling regimes for prestressed films beyond von K´arm´an’s regime Let ω ⊂ R2 be an open, bounded, connected and Lipschitz domain, and let g ∈ C ∞ (Ω¯ 1 , R3×3 a given Riemannian metric. We study the energies of thin films pos,sym ) be
h h h {Ω = ω × − 2 , 2 }h→0 prestressed by g, as in (11.10). In this section, we assume: lim

1

h→0 h4

inf Egh = 0.

Recall that by Theorem 12.13 the above condition is equivalent to Riem(g) ≡ 0 on ω × {0}, and further equivalent to the existence of the vector fields (unique up to a ¯ R3 ), satisfying: common rigid motion) y0 , b1 , b2 , b3 ∈ C ∞ (ω, BT0 B0 = g(·, 0)

with det B0 > 0, 1 sym(BT0 B1 ) = ∂3 g(·, 0), 2 1 T sym(B0 B2 ) + BT1 B1 = ∂33 g(·, 0), 2     where we denote: B0 = ∂1 y0 , ∂2 y0 , b1 and {Bk = ∂1 bk , ∂2 bk , bk+1 }2k=1 .

(13.1)

The following main result of this chapter generalizes the above statements to the even order powers 2n of h, in the infimum energy scaling, for n > 2. We will show that such scalings exhaust all possibilities in the regime: inf E h ' hβ with β > 4: Theorem 13.1. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz do3×3 main, and let g ∈ C ∞ (Ω¯ 1 , Rpos,sym ) be a Riemannian metric. The following statements are equivalent, for each integer n ≥ 2: (i) inf Egh ≤ Ch2(n+1) with C > 0 that depends on ω, g,W but not on h. (k)

(ii) R1212 (·, 0) = R1213 (·, 0) = R1223 (·, 0) = 0 in ω, and also ∂3 Ri3 j3 (·, 0) = 0 in ω for all k = 0 . . . n − 2 and all i, j = 1, 2.  ∞ ¯ R3 ) such that defining B = (iii) There exist vector fields y0 , {bk }n+1 k k=1 ∈ C (ω,    n  ∂1 bk , ∂2 bk , bk+1 k=1 , in addition to B0 = ∂1 y0 , ∂2 y0 , b1 which satisfies det B0 > 0, we have: 

t k T  n t k  ∑ k! Bk ∑ k! Bk = g(·,t) + O(hn+1 ) on Ω h , k=0 k=0 n

as h → 0.

(13.2)   m T (m) Equivalently: ∑ Bk Bm−k − ∂3 g(·, 0) = 0 for all m = 0 . . . n. k=0 k m

13.1 Energy quantisation and identification of scaling regimes for prestressed films beyond... 353

The proof will be given by induction on n, with the main arguments in sections 13.2, 13.3 and 13.4. The implication (iii) ⇒ (i) may be deduced directly, relying on a construction similar as in the of proof Theorem 12.1. In particular, from the statement below it already follows that limh→0 h14 inf Egh = 0 implies inf Egh ≤ Ch6 . Theorem 13.2. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume that condition (iii) in Theorem 13.1 holds, for some n ≥ 2. Then we have: inf Egh ≤ Ch2(n+1) . n+1 k t

Proof. Define uh (·,t)  y0 + ∑

k=1 n

∇uh (·,t) =

k!

tk

bk , so that:  t n+1  ∂1 bn+1 , ∂2 bn+1 , 0 .

∑ k! Bk + (n + 1)!

k=0

Thus, (∇uh )g−1/2 is positive definite for all small h, and modulo a rotation field, it equals the following matrix field on Ω h , where we used the assumption (13.2): q
T
q
(∇uh )g−1/2 (∇uh )g−1/2 = Id3 + g−1/2 (∇uh )T ∇uh − g g−1/2 q = Id3 + O(hn+1 ) = Id3 + O(hn+1 ). This implies: Egh (uh ) =

1 h

Z Ωh


W Id3 + O(hn+1 ) dx ≤ Ch2(n+1) , as claimed.

The next main result of this chapter concerns the compactness and the lower bound assertions, parallel to those given in Theorem 12.6 in section 12 for the quadratics energy scaling regime. We need to present these statements right away, as their proof (by induction on n) will be intertwined with the proof of Theorem 13.1. We start with defining the dimensionally reduced energy: Definition 13.3. Let ω ⊂ R2 be open, bounded, connected, Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). For every n ≥ 2 and V ∈ V (y0 ) as in Definition 12.4, we set: I2(n+1),g (V )  Z    (n−1)  1  Q2,g z, (∇y0 )T ∇d + (∇V )T ∇b1 + αn ∂3 Ri3 j3 i, j=1,2 dz 24 ω Z   (n−1) 
 + βn Q2,g z, PB(y0 )⊥ ∂3 Ri3 j3 i, j=1,2 dz Zω   (n−1) 
 + γn Q2,g z, PB(y0 ) ∂3 Ri3 j3 i, j=1,2 dz. ω

(13.3)

352

13 Infinite hierarchy of limiting theories for prestressed films

Above, the auxiliary vector field d is given in terms of V by (12.14). The projections PB(y0 ) and PB(y0 )⊥ denote, respectively, the orthogonal projections onto the subspace B(y0 ) and its orthogonal complement B(y0 )⊥ , in the weighted L2 space:
Eg  L2 (ω, R2×2 sym ), k · kQ2,g ,

kFkQ2,g 

Z

Q2,g (z, F(z)) dz

1/2

.

(13.4)

ω

The coefficients in (13.3) are:  for n odd 0 3 αn = for n even,  n 2 (n + 3)(n + 1)!  1 1 n2 βn =
2 ·  22n+3 (2n + 3) (n + 1)! (n + 3)2  (n + 1)2    1 (n + 2)2 γn =
2 · n2  2n+3 2 (2n + 3) (n + 1)!   (n + 3)2

for n odd for n even,

(13.5)

for n odd for n even.

Remark 13.4. (i) In the “flat” case of g ≡ Id3 , each functional in (13.3) reduces to the classical linear elasticity, in virtue of Remark 12.5 (iii). There holds: y0 = id2 , bZ1 = e3 , V = ve3 with v ∈ H 2 (ω, R), p = (−∇v, 0), and
1 I2(n+1) (V ) = Q2,Id3 z, ∇2 v dz yields the classical biharmonic energy, 24 ω defined in function of the out-of-plane scalar displacement v. Consequently, each dimensionally reduced energy {I2(n+1),g }n≥2 generalizes the linear elasticity Ilin derived in Theorem 6.9 in the context of shells. (ii) In the present nontrivial geometry context, the bending term is given by: (∇y0 )T ∇d + (∇V )T ∇b1 . It is of order hnt and it interacts with the curvature  (n−1)  ∂3 Ri3 j3 (·, 0) i, j=1,2 , which is of order t n+1 : The interaction occurs only when the two terms have the same parity in t, namely at even n, so that αn = 0 for all n odd. The two remaining terms in (13.3) measure the (squared) L2   (n−1) norm of ∂3 Ri3 j3 (·, 0) i, j=1,2 , with distinct weights assigned to the B(y0 )
⊥ and B(y0 ) projections, again according to the parity of n.

We are ready to state our next main result. Observe that the formula in (13.6) below relates the quantities appearing in conditions (ii) and (iii) of Theorem 13.1. The  (n−1)  Ri3 j3 (·, 0) i, j=1,2 is thus precisely the coefficient of the discrepancy curvature ∂3 of the order hn+1 in (13.2) at its tangential minor, scaled by (n + 1)!/2.

13.2 Higher order theories for prestressed films: compactness and preliminary lower bound 353

Theorem 13.5. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Fix n ≥ 2 and assume that any of the equivalent conditions in Theorem 13.1 holds. Let the family {uh ∈ H 1 (Ω h , R3 )}h→0 satisfy: Egh (uh ) ≤ Ch2(n+1) . Then, the following convergences hold, up to a subsequence: (i) there exists {R¯ h ∈ SO(3), ch ∈ R3 }h→0 such that the displacements in: V h [yh ](z) 

1 hn

?

 n k
 t (R¯ h )T uh (z,t) − ch − y0 (z) + ∑ bk (z) dt −h/2 k=1 k! h/2

converge as h → 0, strongly in H 1 (ω, R3 ), to some V ∈ V (y0 ), (ii) we have the lower bound, in terms of the energy in (13.3): lim inf h→0

1 h2(n+1)

E h (uh ) ≥ I2(n+1) (V ),

(iii) there holds on ω:  (n−1)  (n+1) 2 · ∂3 Ri3 j3 (·, 0) i, j=1,2 = −∂3 g(·, 0)tan   n
n+1 + 2sym (∇y0 )T ∇bn+1 + ∑ (∇bk )T ∇bn+1−k . k k=1

(13.6)

The proof will be partially given in next section. The full proof is postponed to section 13.4, as it will rely on the geometrical formulas developed in section 13.3.

13.2 Higher order theories for prestressed films: compactness and preliminary lower bound In this section, assuming condition (iii) of Theorem 13.1, we derive the compactness properties and (a version of) the lower bound in Theorem 13.5 (ii). We first develop the higher order rigidity estimates, parallel to Lemma 12.2: Lemma 13.6. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume condition (iii) in Theorem 13.1, for some n ≥ 2. Let V ⊂ ω be open, connected, Lipschitz and such that y0 is injective on V , and denote V h = V × (− 2h , h2 ). Then, for every u ∈ H 1 (V h , R3 ) there exists R¯ ∈ SO(3) with: 1 h

  Z n k
∇u − R¯ ∑ t Bk 2 dx ≤ C 1 W (∇u)g−1/2 dx + h2n+1 |V h | . h Vh Vh k=0 k!

Z

354

13 Infinite hierarchy of limiting theories for prestressed films

The constant C is uniform for all V h ⊂ Ω h that are bi-Lipschitz equivalent with controlled Lipschitz constants. Proof. Define Y = y0 +

n+1 k t

∑ k! bk , and observe that for h sufficiently small, Y

is a

k=1

smooth diffeomorphism of V h onto its image U h ⊂ R3 . Consider the change of variables vh = uh ◦Y −1 ∈ W 1,2 (U h , R3 ) and apply the geometric rigidity estimate (4.2), yielding existence of R¯ h ∈ SO(3) with: Z Uh

|∇v − R¯ h |2 dx ≤ C h

Z Uh


dist2 ∇vh , SO(3) dx

Changing variable in the left hand side gives: Z Uh

|∇vh − R¯ h |2 dx = Z

≥C

Vh

Z Vh

2 (det ∇Y ) · (∇uh )(∇Y )−1 − R¯ h dx

Z h ∇u − R¯ h ∇Y 2 dx = C

n

Vh

h ∇u − R¯ h

tk

∑ k! Bk


2 + O(h2(n+1) ) dx.

k=0

−1/2 ∈ Changing now variable
in the right hand side and using the fact that (∇Y )g n+1 SO(3) Id3 + O(h ) established in Lemma 13.2, results in:

Z Uh

Z

dist2 ∇vh , SO(3) dx = (det ∇Y ) · dist2 (∇uh )(∇Y )−1 , SO(3) dx Vh

≤C ≤C

Z Vh

Z Vh


dist2 (∇vh )g−1/2 , SO(3)(∇Y )g−1/2 dx
dist2 (∇vh )g−1/2 , SO(3) + O(h2(n+1) ) dx.

Combining the three displayed inequalities above proves the result. The approximation technique as in Theorem 4.8, Lemma 11.9 and Corollary 12.3, yield now the following estimate, whose proof we leave to the reader: Corollary 13.7. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume condition (iii) in Theorem 13.1, for some n ≥ 2. Then, given a family {uh ∈ H 1 (Ω h , R3 )}h→0 such that Egh (uh ) ≤ Ch2(n+1) , there exists a family of rotation-valued fields {Rh ∈ H 1 (ω, SO(3))}h→0 with: 1 h

n k h ∇u − Rh ∑ t Bk 2 dx ≤ Ch2(n+1) Ωh k=0 k!

Z

Z

and

|∇Rh |2 dz ≤ Ch2n .

ω

We are ready to prove the compactness part of Theorem 13.5: Lemma 13.8. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume condition (iii) in Theorem 13.1, for some n ≥ 2. Let the family of deformations {uh ∈ H 1 (Ω h , R3 )}h→0 satisfy:

13.2 Higher order theories for prestressed films: compactness and preliminary lower bound 355

Egh (uh ) ≤ Ch2(n+1) . Then, we have the following convergence statements, up to a subsequence: (i) {V h [yh ]}h→0 → V ∈ V (y0 ) in H 1 (ω, R3 ), as in Theorem 13.5 (i),
(ii) { 1h sym (∇y0 )T ∇V h [yh ] }h→0 * S ∈ B(y0 ) weakly in L2 (ω, R2×2 ). ? Proof. 1. For each h  1, define the rotation: R¯ h = PSO(3)

(∇uh ) Ωh



n

tk

∑ k! Bk

−1

dz.

k=0

In order to observe that the above definition is legitimate, we write for each z ∈ ω: ?

2  n t k −1  n t k −1  ? (∇uh ) ∑ Bk dx, SO(3) ≤ dx − Rh (z) (∇uh ) ∑ Bk h h k! k! Ω Ω k=0 k=0 ? ?  n t k −1 2
2 (∇uh ) ∑ Bk − Rh (z) dx + 2 Rh dz0 − Rh (z) , ≤2 ω Ωh k=0 k!

dist2

where {Rh }h→0 is the family of rotation-valued matrix fields provided by Corollary 13.7. Upon integrating dz on ω while noting Corollary 13.7, we get: dist2

?

(∇uh )



Ωh

n

 t k −1 dx, SO(3) ≤ Ch2(n+1) +Ch2n ≤ Ch2n . B ∑ k! k k=0

Consequently, there also follows: ?

2  n t k −1 dx − R¯ h ≤ Ch2n , (∇u ) ∑ Bk h k! Ω k=0

? |Rh − R¯ h |2 dz ≤ Ch2n .

h

Set now ch ∈ R3 , so that ∇V h [yh ] =

?

R

ωV

h [yh ]

dz = 0. We get:

 n tk    (R¯ h )T ∂1 uh , ∂2 uh − (R¯ h )T Rh ∑ Bk dt 3×2 −h/2 k=0 k! ? h/2  n k  t h +S Bk dt, ∑ 3×2 −h/2 k=0 k! 1 hn

(13.7)

ω

h/2

(13.8)

where we define the following matrix fields whose convergence (up to a subsequence) results from the second bound in (13.7) and from Corollary 13.7: Sh 

 1  ¯h T h *S ( R ) R − Id 3 hn

weakly in H 1 (ω, R3×3 ).

(13.9)

We also note that S ∈ so(3) a.e. in ω. Since the first term in the right hand side of (13.8) converges to 0 in L2 (ω, R3×2 ) in virtue of Corollary 13.7, we conclude the following convergence, up to a subsequence: ∇V h [yh ] → (SB0 )3×2 = S∇y0

strongly in L2 (ω, R3×2 ).

356

13 Infinite hierarchy of limiting theories for prestressed films

It also follows that the limit S∇y0 ∈ H 1 (ω, R3×2 ). A further application of the Poincar´e inequality to the mean-zero displacements in {V h [yh ]}h→0 , yields their convergence in H 1 (ω, R3 )) to some V ∈ H 2 (ω, R3 ) satisfying ∇V = (SB0 )3×2 . By the skew-symmetry of S, it follows that (∇y0 )T ∇V ∈ so(2), proving (i). 2. We observe that the first term in the right hand side of (13.8) has its L2 (ω) norm bounded by Ch2 , in view of the first estimate in Corollary 13.7. Consequently,
1 in the decomposition of sym (∇y0 )T ∇V h , as in (13.8), the corresponding first h term has a weakly converging subsequence. The remaining second term equals: 
 1 1 sym (∇y0 )T Sh ∇y0 + O(h2 ) = (∇y0 )T (sym Sh )∇y0 + O(h|Sh |). h h The L2 (ω) norm of the second term converges to 0, while the first term obeys: 1 hn−1 h T h sym Sh = − (S ) S → 0 h 2

in L2 (ω, R3×3 ).

(13.10)

This ends the proof of the claim and of the lemma. We are now ready to derive the lower bound on {h−2(n+1) Egh (uh )}h→0 , in terms of the expansion fields y0 , {bk }n+1 k=1 : Lemma 13.9. In the context of Lemma 13.8, there holds: lim inf h→0

≥

1 2

Z

1 E h (uh ) h2(n+1) g


Q2,g z, S − δn+1 (∇y0 )T ∇bn+1 + t (∇y0 )T ∇d + (∇V )T ∇b1 Ω1  n  n+1 t n+1  2(∇y0 )T ∇bn+1 + ∑ (∇bk )T ∇bn+1−k + 2(n + 1)! k k=1  (n+1) − ∂3 g(·, 0) dx,

with the coefficient δn+1 given by:  1  for n odd δn+1 = (n + 2)!2n+1 0 for n even.

(13.11)

Proof. 1. By Corollary 13.7, the following matrix fields {Z h ∈ L2 (Ω 1 , R3×3 )}h→0 have a converging subsequence, weakly in L2 (Ω 1 , R3×3 ): Z h (·,t) =

n hk t k  1  h h ∇u (·, ht) − R ∑ Bk * Z. hn+1 k=0 k!

(13.12)

13.2 Higher order theories for prestressed films: compactness and preliminary lower bound 357 n

We write: (Rh )T ∇uh (·, ht) = Egh (uh ) = ≥

Z Ω1

Z

hk t k ∑ k! Bk + hn+1 (Rh )T Z h and observe that: k=0


W (Rh )T ∇uh (·, ht)g(·, ht)−1/2 dx  q Id3 + g(z, ht)−1/2 J h g(·, ht)−1/2 dx, W

(13.13)

{|Z h |≤h−1/2 }

where the field J h has the following expansion, on the set {|Z h | ≤ h−1/2 } ⊂ Ω 1 : n

hk t k T  n hk t k  Bk ∑ Bk − g(·, ht) k=0 k! k=0 k!   n k k  T ht h T h B (R ) Z + h2(n+1) (Z h )T Z h + 2hn+1 sym ∑ k! k k=0    hn+1t n+1  n n + 1 (n+1) T = (B ) B − ∂ g(·, 0) k n+1−k ∑ k 3 (n + 1)! k=1
+ 2hn+1 sym BT0 (Rh )T Z h + o(hn+1 )

J h (·,t) =



∑

Consequently, we get from (13.13) and Taylor expanding W at Id3 : 1 E h (uh ) ≥ 8 h2(n+1) g 1

Z {|Z h |≤h−1/2 }

1

 Q3 g(·, ht)−1/2


h −1/2 J + o (1) g(·, ht) dx. h hn+1

Since {BT0 (Rh )T Z h }h→0 converges weakly in L2 (Ω 1 , R3×3 ), up to a subsequence, to BT0 R¯ T Z (with some rotation R¯ ∈ SO(3) which is an accumulation point of R¯ h in the proof of Lemma 13.8), the above yields: lim inf h→0

1 ≥ 2

1 Egh (uh ) 2(n+1) h

Z Ω1

Q3



t n+1 g(·, 0)−1/2 2(n + 1)! (13.14)  n n + 1   (n+1) T −1/2 (B ) B − ∂ g(·, 0) g(·, 0) k n+1−k ∑ k 3 k=1 
+ g(·, 0)−1/2 sym BT0 R¯ T Z g(·, 0)−1/2 dx


2. We need to identify the relevant 2×2 minor of the limiting term: sym BT0 R¯ T Z in (13.14). We apply the finite difference technique, by now familiar, and for each s  1 consider the fields { f s,h ∈ H 1 (Ω 1 , R3 )}h→0 in: ? f s,h (z,t)  0

s

n

hk (t + τ)k Bk (z) dτ e3 , k! k=0

h(R¯ h )T Z h (z,t + τ) + Sh ∑

358

13 Infinite hierarchy of limiting theories for prestressed films

where Sh is defined in (13.9). It follows that: f

s,h

  1 ? s n hk (t + τ)k h T h h ¯ (·,t) = n+1 (R ) u (·, h(t + s)) − u (·, ht) − n ∑ k! bk+1 dτ. h s h 0 k=0 1

Recall that, as shown in Lemma 13.8, there holds ∇V = (SB0 )3×2 and that S ∈ so(3) a.e. in ω. We further get that the vector field d must equal SB0 e3 = Sb1 . Consequently, using the definition of f s,h , we easily obtain: f s,h → Sb1 = d

strongly in L2 (Ω 1 , R3 ).

(13.15)

Using the second form of f s,h , we compute its in-plane derivatives for j = 1, 2: ∂ j f s,h (z,t)

? s n k
1 h (t + τ)k ¯ h )T ∂ j uh (z, h(t + s)) − ∂ j uh (z, ht) − 1 ( R ∂k bk+1 dτ ∑ n+1 n sh h 0 k=0 k!   1 = (R¯ h )T Z h (z,t + s) − Z h (z,t) e j s ? s n k n
1 hk h (t + τ)k 1 (t + s)k − t k Bk e j − n ∂ j bk+1 dτ. + n+1 (Id3 + hn Sh ) ∑ ∑ sh h 0 k=0 k! k=1 k! =


1 The first term in the right hand side above converges to R¯ T Z(z,t + s) − Z(z,t) e j , s weakly in L2 (Ω 1 , R3 ), whereas the last two terms may be rewritten as: n

1 1 n+1 hk hk k k n h (t + s) − t (t + s)k − t k ∂ j bk ∂ b − (Id + h S ) j 3 k ∑ ∑ n+1 n+1 sh sh k=1 k! k=1 k!

=



1 h n hk 1 S ∑ (t + s)k − t k ∂ j bk − (t + s)n+1 − t n+1 ∂ j bn+1 sh k=1 k! s(n + 1)!
1 (t + s)n+1 − t n+1 ∂ j bn+1 weakly in H 1 (Ω 1 , R3 ). * S∂ j b1 − s(n + 1)!

In conclusion, and recalling (13.15), we obtain the following convergence:
1 ∂ j f s,h (z,t) * R¯ T Z(z,t + s) − Z(z,t) e j s
1 (t + s)n+1 − t n+1 ∂ j bn+1 + S∂ j b1 − s(n + 1)! = ∂ j d,

weakly in H 1 (Ω 1 , R3 ).

We thus see that:  
R¯ T Z(z,t) − Z(z, 0) e j = t ∂ j d − S∂ j b1 +

1 t n+1 ∂ j bn+1 (n + 1)!

for j = 1, 2,

13.3 Identification of lower bound’s curvature term

which finally yields:     BT0 R¯ T Z(·,t) = BT0 R¯ T Z(·, 0) tan

tan

359

  + t (∇y0 )T ∇d + (∇V )T ∇b1 (13.16)

1 t n+1 (∇y0 )T ∇bn+1 . + (n + 1)!


3. We now compute the symmetric part of the trace term: sym BT0 R¯ T Z(·, 0) tan and finish the proof. From (13.8) and the definition of Z h in (13.12) we get: ∇V h [yh ] = h

Z 1/2 −1/2

h (R¯ h )T Z3×2 (·,t) dt + Sh ∇y0 + O(h2 )




In virtue of (13.12), (13.16) and (13.10), we obtain convergence, weakly in the weighted L2 space Eg defined in (13.4):  
1 sym (∇y0 )T ∇V h * sym (∇y0 )T R¯ T Z(·, 0)3×2 h   Z 1/2 t n+1
dt sym (∇y0 )T ∇bn+1 , + −1/2 (n + 1)! which allows to conclude, by Lemma 13.8 (ii):

sym BT0 R¯ T Z(z, 0) tan = S − δn+1 sym (∇y0 )T ∇bn+1 .

(13.17)

This ends the proof of lemma, in virtue of (13.14), (13.16) and (13.17).

13.3 Identification of lower bound’s curvature term In this section we show the relation between quantities appearing in conditions (ii) and (iii) of Theorem 13.1. Equivalence of (ii) and (iii) at n = 2 was shown in Theorem 12.13, whereas the proof of the general case will be carried out by induction on n ≥ 2. We start by introducing notation that allows for a systematic approach. h h 1 3 Definition 13.10. Let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ) be a metric on Ω = ω × (− 2 , 2 ) ⊂ R .

(i) We denote the matrix fields {Γa ∈ C ∞ (Ω¯ 1 , R3×3 )}a=1...3 whose coefficients (Γa )bc  Γacb are the Christoffel symbols of the metric g in (11.4). Since the LeviCivita connection is torsion-free, it follows that Γa eb = Γb ea for all a, b = 1 . . . 3 and also, the Riemann curvature tensor is expressed by, for all c, d = 1 . . . 3 by:
 a 
Rbcd a,b=1...3 = ∂cΓd + ΓcΓd − ∂d Γc + Γd Γc ,     Rabcd a,b=1...3 = g Rabcd a,b=1...3 .

360

13 Infinite hierarchy of limiting theories for prestressed films

(ii) Given a matrix field F : Ω 1 → R3×3 , we define: ∇a F = ∂a F + Γa F for each a = 1 . . . 3, so that (∇a F)eb coincides with the covariant derivative of vector fields: ∇a (Feb ) with respect to g. It also follows that:   ∇c ∇d F − ∇d ∇c F = Rabcd a,b=1...3 F and ∇a (F1 F2 ) = (∇a F1 )F2 + F1 ∂a F2 . For completeness of presentation, we now partially reprove the curvature statements at n = 1, 2 from sections 11, 12, using the above notation. Lemma 13.11. Let ω ⊂ R2 be open, bounded, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume ∞ (ω, 3 ) such that the matrix field: B = ¯ that there exist vector fields y , b ∈ C R 0 1 0   ∂1 y0 , ∂2 y0 , b1 has positive determinant and satisfies BT0 B0 = g(·, 0)


1 sym (∇y0 )T ∇b1 = ∂3 g(·, 0)tan 2

and

in ω.

Then, there holds: (i) ∂i B0 = B0Γi for all i = 1, 2, and in particular: ∂i b1 = B0Γ3 ei , (ii) Rabi j (·, 0) = Rabi j (·, 0) = 0 for all a, b = 1 . . . 3 and all i, j = 1, 2, ¯ R3 ) such that, defining B1 = (iii) there exists a unique vector field b2 ∈ C ∞ (ω,
  1 ∂1 b1 , ∂2 b1 , b2 , we have: sym BT0 B1 = ∂3 g(·, 0) on ω. Moreover: 2 B1 = B0Γ3

and

∂i b2 = B0 ∇iΓ3 e3

for all i = 1, 2.

Proof. 1. All the identities below are taken on ω × {0}. One easily calculates, by a repeated use of the assumption: h∂i y0 , ∂ j b1 i = ∂ j gi3 − h∂i j y0 , b1 i = ∂ j gi3 − ∂i g j3 + h∂ j y0 , ∂i b1 i and thus: ∂3 gi j = h∂i y0 , ∂ j b1 i + h∂ j y0 , ∂i b1 i = ∂ j gi3 − ∂i g j3 + 2h∂ j y0 , ∂i b1 i, for all i, j = 1, 2. Consequently: h∂ j y0 , ∂i b1 i =


1 ∂3 gi j + ∂i g j3 − ∂ j gi3 = gΓ3 ji 2

for all i, j = 1, 2.

(13.18)

Secondly: h∂ j y0 , ∂ik y0 i = ∂i g jk − h∂k y0 , ∂i j y0 i = ∂i g jk − ∂ j gik + h∂i y0 , ∂ jk y0 i = ∂i g jk − ∂ j gik + ∂k gi j − h∂ j y0 , ∂ik y0 i, which results in: h∂ j y0 , ∂ik y0 i =



1 ∂i g jk + ∂k gi j − ∂ j gik = gΓi jk 2

for all i, j, k = 1, 2.

Thirdly, from (13.18) we obtain: hb1 , ∂ik y0 i = ∂i gk3 − h∂i b1 , ∂k y0 i

1 = ∂i gk3 + ∂k gi3 − ∂3 gik = gΓi 3k for all i, k = 1, 2. 2
Finally: hb1 , ∂i b1 i = 12 ∂i g33 = gΓi 33 , so that the last two identities yield:

13.3 Identification of lower bound’s curvature term

BT0 ∂i B0 = gΓi

361

for all i = 1, 2.

This proves (i) and further: ∂i b1 = B0Γi e3 = B0Γ3 ei , as claimed. 2. Using (i) we compute:

0 = ∂i j B0 − ∂ ji B0 = ∂i B0Γj − ∂ j B0Γi
= B0ΓiΓj + B0 ∂iΓj − B0Γj Γi + B0 ∂ j Γi   = −B0 Rksi j (·, 0) k,s=1...3 for all i, j = 1, 2. which implies (ii). For (iii), uniqueness of b2 is obvious, while b2 = B0Γ3 e3 follows from the requested defining identity, in view of (13.18). The covariant derivative formula is a consequence of (i).

Lemma 13.12. Let ω ⊂ R2 be open, bounded, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). As∞ (ω, 3 ) such that the matrix ¯ sume that there exist vector fields y , b , b ∈ C R 0 1 2   field: B0 = ∂1y0 , ∂2 y0 , b1 has positive determinant and that together with B1 =  ∂1 b1 , ∂2 b1 , b2 , it satisfies:
1 sym BT0 B1 = ∂3 g(·, 0) 2
1 T T sym (∇y0 ) ∇b2 + (∇b1 ) ∇b1 = ∂33 g(·, 0)tan in ω. 2 BT0 B0 = g(·, 0)

and

Then, there holds: (i) Rabcd (·, 0) = 0 in ω for all a, b, c, d = 1 . . . 3, ¯ R3 ) such that defining B2 = (ii) there exists a unique vector field b3 ∈ C ∞ (ω,
  1 ∂1 b2 , ∂2 b2 , b3 , we have: sym BT0 B2 + BT1 B1 = ∂33 g(·, 0) on ω. Moreover: 2 B2 = B0 ∇3Γ3

and

∂i b3 = B0 ∇i ∇3Γ3 e3

for all i = 1, 2.

Proof. All identities below are taken on ω × {0}. Observe that for all a, b = 1 . . . 3:
h∂33 gea , eb i = ∂3 hgΓ3 ea , eb i + hgΓ3 eb , ea i (13.19) = h∇3Γa e3 , geb i + h∇3Γb e3 , gea i + 2hgΓ3 ea ,Γ3 eb i. By Lemma 13.11 (iii), the above gives equivalence of the last assumed condition to: 0 = hB0 ei , ∂ j b2 i + 2h∂i b1 , ∂ j b1 i + hB0 e j , ∂i b2 i − h∂33 gei , e j i = hgei , ∇ j Γ3 e3 i + 2hgΓ3 ei ,Γ3 e j i + hge j , ∇iΓ3 e3 i   − h∇3Γj e3 , gei i + h∇3Γi e3 , ge j i + 2hgΓ3 e j ,Γ3 e j i     = hgei , Ra3 j3 (·, 0) a=1...3 i + hge j , Ra3 j3 (·, 0) a=1...3 i = 2Ri3 j3 (·, 0) on ω,

for all i, j = 1, 2.

362

13 Infinite hierarchy of limiting theories for prestressed films

Thus we obtain (i), in virtue of Lemma 13.11 (ii) that guarantees Rabi j (·, 0) = 0 for all a, b = 1 . . . 3 and i, j = 1, 2. To show (ii), by Lemma 13.11 and by (i) we note: B2 ei = ∂i b2 = B0 ∇iΓ3 e3 = B0 ∇3Γi e3 = B0 ∇3Γ3 ei

for all i, j = 1, 2,

and also, in view of (13.19): 1 sym(BT0 B2 ) + BT1 B1 − ∂33 g(·, 0) 2 

 = sym(BT0 B2 ) + Γ3T gΓ3 − sym (g∇3Γ3 ) + Γ3T gΓ3 .


Hence, B2 = B0 ∇3Γ3 satisfies the defining relation. Finally, ∂i b3 = ∂i B0 ∇3Γ3 e3 = B0 ∇i ∇3Γ3 e3 results from Lemma 13.11 (i). The proof is done. Towards the generalization of Lemmas 13.11, 13.12 to n > 2, we need the following two useful observations: 3×3 Lemma 13.13. Let ω ⊂ R2 be open, bounded, and let g ∈ C ∞ (Ω¯ 1 , Rpos,sym ). For all n ≥ 0 there holds on ω × {0}:    n  n+1 (n+1) (n) (k−1)
T (n−k) ∂3 g = 2sym g∇3 Γ3 + ∑ ∇3 Γ3 g∇3 Γ3 . k k=1

Proof. The proof follows by induction. For n = 0, the statement is obviously true. Assume that it is true for some n − 1, then:   n−1   n (n+1) (n−1)
(k−1)
T (n−1−k) ∂3 g = ∂3 2 g∇3 Γ3 sym + ∑ ∇3 Γ3 g∇3 Γ3 k=1 k (n) (n)
T (n−1) (n−1)
T = g∇3 Γ3 + ∇3 Γ3 g + Γ3T g∇3 Γ3 + ∇3 Γ3 gΓ3  n−1   n (k−1)
T (n−k) (k)
T (n−k−1) +∑ ∇3 Γ3 g∇3 Γ3 + ∇3 Γ3 g∇3 Γ3 k=1 k (n−1) (n−1)
T (n) (n)
T Γ3 + ∇3 Γ3 gΓ3 = g∇3 Γ3 + ∇3 Γ3 g + Γ3T g∇3  n−1    n n (k−1)
T (n−k) +∑ + ∇3 Γ3 g∇3 Γ3 k k − 1 k=1     n T (n−1) n (n−1)
T − Γ g∇3 Γ3 + ∇3 Γ3 gΓ3 . 0 3 n−1

n
Collecting all the terms and recalling that nk + k−1 = n+1 implies the result. k

Lemma 13.14. Let ω ⊂ R2 be open, bounded, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Assume (k)

that R1212 (·, 0) = R1213 (·, 0) = R1223 (·, 0) = 0 and that ∂3 Ri3 j3 (·, 0) = 0 for all

13.3 Identification of lower bound’s curvature term

363

k = 0 . . . n, all i, j = 1, 2 on ω. Then, all the mixed partial derivatives of both Rabcd and Rabcd , of any order up to n, are zero on ω × {0}, for all a, b, c, d = 1 . . . 3. Proof. All identities below are taken on ω × {0}. We argue by induction on n. For n = 0 the result is obviously true. Assume that it is true for some n ≥ 0 and let the result assumption at n + 1 hold. Then: ∂ (k) Rabcd = ∂ (k) Rabcd = 0

for all k = 0 . . . n,

(n+1) ∂3 Ri3 j3

for all i, j = 1, 2,

=0

a, b, c, d = 1 . . . 3,

and we need to show that any partial derivatives of order n + 1, of the Riemann tensor’s components is zero on ω × {0}. This is certainly true for partial derivatives (n+1) containing ∂i for some i = 1, 2, so it suffices to prove the claim for ∂3 . Below, we consider various combinations of indices i, j = 1, 2 and a, b = 1 . . . 3. Firstly: (n+1) a Rbi j

∂3

(n)

(n)

= ∂3 ∇3 Rabi j = ∂3 (n)

= ∂3

− ∇i Rab j3 − ∇ j Rab31
− ∂i Rab3 − ∂ j Rab31 = 0,


(13.20)

where we used the induction assumption in the first and the third equalities and the second Bianchi identity in the second one. Secondly: (n+1)

∂3

   p  (n+1)  gap p=1...3 , Rbi Rabi j = ∂3 j p=1...3 

   (n+1) p  = gap p=1...3 , ∂3 Rbi j p=1...3 = 0,

(13.21)

where we used the induction assumption and (13.20) in the last equality. Thirdly: (n+1) a Rbi3

∂3

   (n+1)  ap  = ∂3 g p=1...3 , R pbi3 p=1...3

   (n+1)   = gap p=1...3 , ∂3 R pbi3 p=1...3 = 0,

(13.22)

by using (13.21) and the result assumption at n + 1, in the last equality. Finally: (n+1) ∂3 Rabcd = 0 by (13.21) and the result assumption. The proof is done. The following is the main new result of this section: 3×3 Lemma 13.15. Let ω ⊂ R2 be open and bounded, and let g ∈ C ∞ (Ω¯ 1 , Rpos,sym ). Fix n ∞ 3 ¯ R ) such that the matrix fields: n ≥ 2. Assume that there exist y0 , {bk }k=1 ∈ C (ω,     B0 = ∂1 y0 , ∂2 y0 , b1 with det B0 > 0 and {Bk = ∂1 bk , ∂2 bk , bk+1 }n−1 k=1 , satisfy: m

  m (m) ∑ k BTk Bm−k − ∂3 g(·, 0) = 0 for all m = 0 . . . n − 1, k=0  
n−1 n (n) T 2sym (∇y0 ) ∇bn + ∑ (∇bk )T ∇bn−k = ∂3 g(·, 0)tan k=1 k

on ω.

364

13 Infinite hierarchy of limiting theories for prestressed films

Then, we have the following assertions: (i) the condition in Theorem 13.1 (ii) holds. ∞ ¯ R3 ) such that defining B = (ii) there exists a unique vector field b n n+1∈ C (ω, n   n T (n) ∂1 bn , ∂2 bn , bn+1 , there holds: ∑ Bk Bn−k = ∂3 g(·, 0) on ω. Moreover: k=0 k (n−1)

Bn = B0 ∇3

Γ3

(n−1)

∂i bn+1 = B0 ∇i ∇3

and

for all i = 1, 2.

Γ3 e3

Proof. 1. The proof proceeds by induction. The statement at n = 2 has been shown in Lemma 13.12. We now assume it to be true for some n ≥ 2. By Lemma 13.14: All mixed partial derivatives up to order n − 2, of all components (13.23) of the Riemann curvature tensor, are 0 at ω × {0}.   Since Bn = ∂1 bn , ∂2 bn , bn+1 with bn+1 as in (ii), recalling Lemma 13.13, we get:  n 
n+1 (n+1) 2sym (∇y0 )T ∇bn+1 + ∑ (∇bk )T ∇bn+1−k − ∂3 g(·, 0)tan k k=1  n 
= 2sym (∇y0 )T ∇bn+1 + ∑ BTk Bn+1−k tan

k=1

   n  n+1 (n)
(k−1)
T (n−k) − 2sym g∇3 Γ3 + ∑ ∇3 Γ3 g∇3 Γ3 k tan k=1   (n) T = 2 sym (∇y0 ) ∇bn+1 − g∇3 Γ3 h i (n−1) (n) = 2 sym hgei , ∇ j ∇3 Γ3 e3 − ∇3 Γ3 e j i i, j=1,2 h i (n−1) (n) = 2sym hgei , ∇ j ∇3 Γ3 e3 − ∇3 Γj e3 i on ω × {0}.

(13.24)

i, j=1,2

By (13.23) we consecutively swap the order of the covariant derivatives on ω × {0}: (n−1)

∇ j ∇3

(n−2)

Γ3 = ∇3 ∇ j ∇3 (3)

(2)

(n−4)

= ∇3 ∇ j ∇3

(n−3)

Γ3 = ∇3 ∇ j ∇3

Γ3

(n−1)

Γ3 = (. . .) = ∇3

∇ j Γ3 ,

so that: (n−1)

∇ j ∇3

(n)

(n−1)


∇ j Γ3 − ∇3Γj  (n−1)  a = ∇3 Rb j3 (·, 0) a,b=1...3 .

Γ3 − ∇3 Γj = ∇3

In conclusion, using (13.23) again, the formula in (13.24) becomes:

(13.25)

13.4 Higher order theories for prestressed films: lower bound

365

n

 
n+1 (n+1) 2sym (∇y0 )T ∇bn+1 + ∑ (∇bk )T ∇bn+1−k − ∂3 g(·, 0)tan k k=1 h i  (n−1) a 0   (n−1) a  (13.26) = hgei , ∂3 R3 j3 (· , 0) a=1...3 i + hge j , ∂3 R3i3 (·, 0) a=1...3 i i, j=1,2 h i (n−1) = 2 ∂3 Ri3 j3 (·, 0) on ω, i, j=1,2

proving (i) in view of the second assumption at n + 1. 2. For (ii), observe that Bn+1 is indeed uniquely defined, by choosing bn+2 = Bn+1 e3 that satisfies the condition: n+1 

∑ k=0

 n+1 T (n+1) Bk Bn+1−k = ∂3 g(·, 0) k

on ω,

as the principal 2 × 2 minors of both sides in the above formula coincide by assumption. Further, by (13.25) and the already established (i) at n + 1, we get:   (n−1) (n−1) Bn+1 ei = ∂i bn+1 = ∂i B0 ∇3 Γ3 e3 = B0 ∇i ∇3 Γ3 e3   (n) (n−1) a (n) = B0 ∇3 Γi e3 + ∇3 R3i3 (·, 0) a=1...3 = B0 ∇3 Γi e3 (n)

= B0 ∇3 Γ3 ei

for all i = 1, 2

on ω.

(n)

Hence, there must be bn+1 = B0 ∇3 Γ3 , as claimed. The proof is complete. We note that the argument in the proof above leading to (13.26), automatically gives: Corollary 13.16. In the context of Theorem 13.1, condition in (iii) implies the formula (13.6), for any n ≥ 1.

13.4 Higher order theories for prestressed films: lower bound In this section, we complete proofs of Theorems 13.1 Theorem 13.5. The following statement yields Theorem 13.5, assuming (iii) of Theorem 13.1: Lemma 13.17. In the context of Lemma 13.8, there holds: lim inf h→0

1 h2(n+1)

E h (un ) ≥ I2(n+1),g (V ),

for the dimensionally reduced energy I2(n+1),g in Definition 13.3. Proof. By Lemma 13.9 and Corollary 13.16, we get:

366

13 Infinite hierarchy of limiting theories for prestressed films

lim inf h→0

1 2

≥

1 Egh (uh ) 2(n+1) h



Q2 z, S − δn+1 sym (∇y0 )T ∇bn+1 + t (∇y0 )T ∇d + (∇V )T ∇b1 Ω1   t n+1  (n−1) ∂3 Ri3 j3 (z, 0) i, j=1,2 dx + (n + 1)!

Z

Denoting the z-dependent components at different powers of t in the integrand above by I, II and III, and recalling (13.11), the right hand side becomes: 1 2

Z

=

Ω1

1 2


Q2,g z, I + tII + t n+1 III dx

Z

Z  Q2,g z, I +

1/2


 t n+1 dt III dz

−1/2

ω

Z 1/2
 Q2,g z, II + 12 t n+2 dt III dz ω −1/2  Z 1/2 Z 1/2 Z  Z 1/2
2

2
1 t n+2 dt Q2,g z, III t n+1 dt − 12 t 2n+2 dt − + 2 ω −1/2 −1/2 −1/2   Z  
1 (n−1) = Ri3 j3 (z, 0) i, j=1,2 dz Q2,g z, S − δn+1 sym (∇y0 )T ∇bn+1 + δn+1 ∂3 2 ω   Z  (n−1)  1 Q2,g z, (∇y0 )T ∇d + (∇V )T ∇b1 + αn ∂3 Ri3 j3 (z, 0) i, j=1,2 dz + 24 ω   Z  (n−1)  + γn Q2,g z, ∂3 Ri3 j3 (z, 0) i, j=1,2 dz,

1 + 24

Z



ω

where by a direct calculation one easily checks that the numerical coefficients αn
and γn have the form (13.5). Further, since S − δn+1 sym (∇y0 )T ∇bn+1 ∈ B(y0 ), the first term in the right hand side above is bounded from below by:    (n−1)  1 Ri3 j3 (·, 0) i, j=1,2 , B(y0 ) dist2Q2,g δn+1 ∂3 2    δ2 (n−1) = n+1 dist2Q2,g ∂3 Ri3 j3 (·, 0) i, j=1,2 , B(y0 ) 2 2  (n−1) 
δn+1

2

= Ri3 j3 (·, 0) i, j=1,2 .

PB(y0 )⊥ ∂3 2 Q2,g Decomposing the third term into:

 (n−1) 


2

Ri3 j3 (·, 0) i, j=1,2 γn PB(y0 )⊥ ∂3

Q2,g

 (n−1) 


2 Ri3 j3 (·, 0) i, j=1,2 + γn PB(y0 ) ∂3

Q2,g

,

13.5 Higher order theories for prestressed films: recovery family

the claim follows by checking that:

367

2 δn+1 + γn = βn in (13.5). 2

We are now ready to give: A proof of Theorem 13.1. The proof is carried out by induction on n ≥ 2. When n = 2, then (ii) is equivalent with (iii) by Theorem 12.13. Condition (iii) implies (i) by Lemma 13.2, whereas (i) implies (ii) again in view of Theorem 12.13. Assume now the equivalence of the three conditions at n ≥ 2. We want to show the equivalence at n + 1. Condition (ii) implies (iii) by Corollary 13.16. Condition (iii) implies (i) by Lemma 13.2. Finally, assuming (i) at n + 1 allows to write: 1 1 0 = lim 2(n+1) inf Egh = lim 2(n+1) Egh (uh ) ≥ I2(n+1),g (V ) h→0 h h→0 h



2 

(n−1)

≥ γn · ∂3 Ri3 j3 (·, 0) i, j=1,2 , Q2,g

for some infimizing sequence {uh ∈ H 1 (Ω h , R3 )}h→0 and a resulting V from Theorem 13.5. This establishes (ii) at n + 1, in view of the inductive assumption. The proof is complete.

13.5 Higher order theories for prestressed films: recovery family Our next result proves the variational upper bound on the energies {h−2(n+1) Egh }h→0 , which is consistent with Theorem 13.1 and yields the Γ -convergence to the dimensionally reduced limits I2(n+1)g in (13.3): Theorem 13.18. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Fix n ≥ 2 and assume that any of the equivalent conditions in Theorem 13.1 hold. Then for every V ∈ V (y0 ) in Definition 12.4, there exists a sequence {uh ∈ H 1 (Ω h , R3 )}h→0 so that, as h → 0: 1 hn

?

  n k t uh (z,t) − y0 + ∑ bk dt → V −h/2 k=1 k! h/2

1 Egh (uh ) 2(n+1) h→0 h

and that: lim

in H 1 (ω, R3 ),

(13.27)

= I2(n+1),g (V ), where the limiting energy func-

tional is as in (13.3).

n+1 k t

Proof. 1. Denote Y (z,t) = y0 + ∑

k=1

k!

bk and define:

368

13 Infinite hierarchy of limiting theories for prestressed films

uh (·,t) = Y (z,t) + hn vh + hn+1 wh + hntd h + hn+1tqh +

t2 t n+2 k + hn rh (n + 2)! 2

(13.28)

for all (z,t) ∈ Ω h .

We now introduce terms in the above expansion. For a fixed small ε > 0, the truncated sequence {vh ∈ W 2,∞ (ω, R3 )}h→0 is chosen according to the second order truncation result in Theorem 5.12, in a way that: vh → V

in H 2 (ω, R3 ) as h → 0,

hn kvh kW 2,∞ (ω,R3 ) ≤ ε

and

lim

1 {z ∈ ω; vh (z) , V (z)} = 0.

(13.29)

h→0 h2n

The sequence {d h ∈ W 1,∞ (ω, R3 )}h→0 is defined by:    ∗    −(∇vh )T b1 T h B0 d = with sym BT0 ∇vh , dh = sym (∇y0 )T ∇vh . (13.30) 0 ¯ R3 )}h→0 is such that recalling (13.4): The sequence {wh ∈ C ∞ (ω,      (n−1) in Eg Ri3 j3 i, j=1,2 sym (∇y0 )T ∇wh → −δn+1 PB(y0 ) ∂3 lim h1/2 kwh kW 2,∞ (ω,R3 ) = 0.

(13.31)

h→0

¯ R3 ) and {qh ∈ C ∞ (ω, ¯ R3 )}h→0 are defined by: Finally, k ∈ C ∞ (ω,    n  n+1 (n+1) 2BT0 k = c ·, 2(∇y0 )T ∇bn+1 + ∑ (∇bk )T ∇bn+1−k − ∂3 g(·, 0)tan k k=1   n   n+1 T # (∇bn+1−k ) ∇bk+1  " (n+1) 2 ∑ k 2∂3 g(·, 0)31,32   k=0  − n  , + (n+1)   n+1 T ∂3 g(·, 0)33 (∇b ) ∇b k+1 n+2−k ∑ k k=1 " #   (∇wh )T b1 T h T h B0 q = c ·, (∇y0 ) ∇w − , 0 ¯ R3 )}h→0 is chosen so that: while {rh ∈ C ∞ (ω, # "   (∇vh )T b2 T h h T T h →0 B0 r − c ·, (∇y0 ) ∇d + (∇v ) ∇b1 + hd h , b2 i h1/2 krh kW 1,∞ (ω,R3 ) → 0

in L2 (ω, R3 ),

as h → 0.

2. By the definitions in step 1, we easily deduce (13.27). Compute now, for all rescaled variables (z,t) ∈ Ω 1 :

13.5 Higher order theories for prestressed films: recovery family

369

n    hk t k hn+1 xt n+1  ∇uh (z, ht) = hn ∇vh , d h + ∑ ∂1 bn+1 , ∂2 bn+1 , k Bk + (n + 1)! k=0 k!  h h  h h n+1 n+1 + h t ∇d , r + h ∇w , q
n+2 h + O(h ) 1 + |∇d | + |∇rh | .

Consequently, it follows that for h  1 we have:

dist (∇uh )g−1/2 , SO(3) ≤ C |∇uh − B0 | + h ≤ Cε, which justifies writing, by Taylor’s expansion of W and taking ε  1: q 

W (∇uh )g−1/2 = W Id3 + g−1/2 (∇uh )T ∇uh − g g−1/2 

1 = W Id3 + g−1/2 (∇uh )T ∇uh − g g−1/2 + O |(∇uh )T ∇uh − g|2 2 
1 = W Id3 + g(·, 0)−1/2 (∇uh )T ∇uh − g g(·, 0)−1/2 2

 + O h|(∇uh )T ∇uh − g| + O |(∇uh )T ∇uh − g|2 
1  = Q3 g(·, 0)−1/2 (∇uh )T ∇uh − g g(·, 0)−1/2 8

+ O h|(∇uh )T ∇uh − g|2 + oh (1) |(∇uh )T ∇uh − g|2 . The above implies that: 1 E h (uh ) h2n+2 g Z  1 
1 Q3 n+1 g(·, 0)−1/2 (∇uh )T ∇uh (·, ht) − (·, ht) g(·, 0)−1/2 dx = 8 Ω1 h (13.32) Z
1 h T h 2 + O h|(∇u ) ∇u − g| 2n+2 Ω1 h
1 + 2n+2 oh (1) |(∇uh )T ∇uh − g|3 dx h We thus need to compute, for all x = (z,t) ∈ Ω 1 , the difference expression in: (∇uh )T ∇uh (z, ht) − g(z, ht). Denote the error quantity: Error =o1 (hn+1 ) + O(hn+2 ) |∇vh | + |∇2 vh |




+ O(h2n )|∇2 vh |2 + O(h2n+2 )|∇2 vh |2 . Then, there holds:

370

13 Infinite hierarchy of limiting theories for prestressed films

(∇uh )T ∇uh (·, ht) − g(·, ht)
∗ = 2hn sym (∇y0 )T ∇vh    hn+1t n+1 n n + 1 T + ∑ k Bk Bn+1−k (n + 1)! k=1   
(n+1) + 2sym ∂1 bn+1 , ∂2 bn+1 , k − ∂3 g(·, 0)  
 
+ 2hn+1tsym BT0 ∇d h , rh + 2hn+1tsym BT1 ∇vh , d h  
+ 2hn+1 sym BT0 ∇wh , qh + Error. BT0

3. We now estimate the two last terms in the right hand side of (13.32). Since: |(∇uh )T ∇uh − g| =
= O(hn+1 ) 1 + |∇vh | + |∇wh | + |d h | + |∇d h | + |qh | + |rh |
+ Error + O(hn )|sym (∇y0 )T ∇vh |

= O(hn+1 ) 1 + |∇vh | + |∇2 vh | + h−1/2 oh (1) + O(hn )|sym (∇y0 )T ∇vh | + O(h2n )|∇vh |2 + O(h2n+2 )|∇2 vh |2 , where we have repeatedly used (13.30), (13.31), we note that: 1 h2

Z


sym (∇y0 )T ∇vh 2 dx

ω


C ≤ 2 k∇vh k2L∞ + k∇2 vh k2L∞ h Cε 2 ≤ 2n+2 h ≤

Z {vh ,V }

dist2 (z, {vh = V }) dz

2 Cε 2 dist (z, {v = V }) dz ≤ 2n+2 {vh , V } h h {v ,V }

Z

2

(13.33)

h

Cε 2 4n h · oh (1) → 0 h2n+2

as h → 0.

This completes the convergence analysis of the second error term in (13.32). Likewise for the first term, we get: 1 h2n+2

O h|(∇uh )T ∇uh − g|2





= O(h) 1 + |∇vh |2 + |∇2 vh |2 + h−1 oh (1) + O(h2n−1 )|∇vh |4

1 + O(h2n+3 )|∇2 vh |4 + O sym| (∇y0 )T ∇vh |2 . h As before, the first three terms converge to 0 in L1 (ω), whereas convergence of the 1 Error converges to 0 in last term follows by (13.33). Concluding, and since hn+1 2 1 L (Ω ), the limit in (13.32) becomes:

13.6 Convergence of minima and coercivity of linear elasticity-like energies

371

1 Egh (uh ) lim h→0 h2n+2 Z  1 
1 Q3 n+1 g(·, 0)−1/2 (∇uh )T ∇uh (·, ht) − g(·, ht) g(·, 0)−1/2 dx h→0 8 Ω 1 h Z   1 = lim Q3 g(·, 0)−1/2 K h (x)g(·, 0)−1/2 dx, h→0 8 Ω 1

= lim

where for a.e. x = (z,t) ∈ Ω 1 we define:
∗ 2 K h (·,t) = sym (∇y0 )T ∇vh h  n   t n+1 n+1 T + ∑ k Bk Bn+1−k (n + 1)! k=1  
(n+1) ∂1 bn+1 , ∂2 bn+1 , k − ∂3 g(·, 0) + 2sym      
 
+ 2t BT0 ∇d h , rh + sym BT1 ∇vh , d h + 2sym BT0 ∇wh , qh . BT0



In view of (13.33), the compatibility in the definitions in step 1 now yields: 1 Egh (uh ) 2n+2 h→0 h lim

1 = lim h→0 2

 t n+1 2(∇y0 )T ∇bn+1 2(n + 1)! Ω1   n  n+1 (n+1) T +∑ (∇bk ) ∇bn+1−k − ∂3 g(·, 0)tan k k=1   
+ t (∇y0 )T ∇d h + (∇vh )T ∇b1 + (∇y0 )T ∇wh dz.

Z

 Q2,g z,

(13.34)

Decomposing the integrand above as in the proof of Lemma 13.17 and recalling convergences in (13.29), (13.31), we conclude that the right hand side of (13.34) equals I2(n+1),g (V ), as claimed. The proof is done.

13.6 Convergence of minima and coercivity of linear elasticity-like energies As in section 6.1, it is worth observing the following direct consequences of Theorems 13.5 and 13.18, regarding the convergence of minima and the Γ -convergence: Corollary 13.19. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let g ∈ C ∞ (Ω¯ 1 , R3×3 pos,sym ). Fix n ≥ 2 and assume that any of the equivalent conditions in Theorem 13.1 holds. Then we have:

372

13 Infinite hierarchy of limiting theories for prestressed films

(i) The functional I2(n+1),g in (13.3) attains its infimum on the space V (y0 ) defined with respect to y0 as in (12.2) and we have: lim

1

h→0 h2(n+1)

inf Egh = min I2(n+1),g .

(ii) The quantity in the right hand side above is the square of an appropriate  weighted L2 norm of ∇(n−1) Riem(g) restricted to ω × {0}, as it is quadratic  (n−1)  in ∂3 Ri3 j3 (·, 0) i, j=1,2 and:

 (n−1)  min I2(n+1) ≥ γn · ∂3 Ri3 j3 (·, 0) i, j=1,2 k2Q2,g ,
 (n−1)  1 2 min I2(n+1) ≤ I2(n+1) (0) ≤ Ri3 j3 (·, 0) i, j=1,2 k2Q2,g . αn + βn · ∂3 24 We note that:


−1 1 2 α + βn = 22n+3 (2n + 3)((n + 1)!)2 . 24 n

Corollary 13.20. In the context of Corollary 13.19, the following Γ -convergence holds, with respect to convergence in H 1 (Ω 1 , R3 ) × H 1 (ω, R3 ), and for every n ≥ 2:  1 I2(n+1),g (V ) if y = y0 , V ∈ V (y0 ), h Γ F −→ 2(n+1) +∞ otherwise. h Here, we put uh (z,t) = y(z,t/h) ∈ H 1 (Ω h , R3 ) and define:  h h Eg (u ) if V = V h [y], F h (y)  +∞ otherwise. There is a one-to-one correspondence between limits of sequences of (global) approximate minimizers to the energies Egh and (global) minimizers of I2(n+1),g . Regarding the coercivity of I2(n+1),g , Theorem 12.15 yields: Corollary 13.21. In the context of Corollary 13.19, we have:  Ker I2(n+1),g = Sy0 + c; S ∈ so(3), c ∈ R3 , together with the following coercivity estimate:
dist2H 2 (ω,R3 ) V, Ker I2(n+1),g ≤ CI2(n+1),g (V ) for all V ∈ V (y0 ), with a constant C > 0 that depends on g, ω and W but is independent of V .  (m)  Finally, we point out that if the metric g that satisfies ∂3 Ri3 j3 (·, 0) i, j=1,2 = 0  (n−1)  Ri3 j3 (·, 0) i, j=1,2 . 0, for some n ≥ 2, then: for all m = 0 . . . n − 2, but ∂3

13.7 Bibliographical notes

373

ch2(n+1) ≤ inf Egh ≤ Ch2(n+1) , for some c,C > 0. This is consistent with the analysis in section 12.6 in chapter 12, for the conformal class of metrics g. We gather the findings about the infinite hierarchy of limiting models of prestressed films in Figure 13.1 below. This table should be compared with the finite hierarchy of elastic plate models in Figure 8.1.

β

2

4

asymptotic expansion y(z)  3d : y(z) + tb(z)

y0 (z) + hV (z) + h2 wh (z)

constraint / regularity y ∈ H2 (∇y)T ∇y

= g(·, 0)tan

R1212 , R1213 , R1223 (·, 0) = 0
sym (∇y0 )T ∇V = 0,
sym (∇y0 )T ∇wh → S V ∈ H 2 , wh ∈ H 1

6 .. .

y0 (z) + h2V (z)

y0 (z) + hn−1V (z)  tk 2n 3d : y0 + ∑n−1 k=1 k! bk (z) .. +hn−1V (z) . +hn−1 d(z)

Γ -limiting energy Iβ ,g ck(∇y)T ∇b − 12 ∂3 g(·, 0)tan k2Q2,g   ∂1 y, ∂2 y, b ∈ SO(3)g(·, 0)1/2 1 c1 k 12 (∇V )T ∇V + S + 24 (∇b1 )T ∇b1 1 − 48 ∂33 g(·, 0)tan k2Q2,g

+ c2 k(∇y0 )T ∇d + (∇V )T ∇b1 k2Q2,g   + c3 k Ri3 j3 (·, 0) i, j=1,2 k2Q2,g

sym (∇y0 )T ∇V = 0, V ∈ H 2

  c2 k(∇y0 )T ∇d + (∇V )T ∇b1 + α ∂3 R k2Q2     2 + c3 kPB(y0 )⊥ ∂3 R kQ2,g + c4 kPB(y0 ) ∂3 R k2Q2,g

Rabcd (·, 0) = 0  (k)  ∂3 R (·, 0) = 0 ∀k ≤ n − 3
sym (∇y0 )T ∇V = 0, V ∈ H 2

 (n−2)  2 R kQ2,g c2 k(∇y0 )T ∇d + (∇V )T ∇b1 + α ∂3  (n−2)  2 R kQ2,g + c3 kPB(y0 )⊥ ∂3  (n−2)  2 + c4 kPB(y0 ) ∂3 R kQ2,g

Rabcd (·, 0) = 0


Fig. 13.1 The infinite hierarchy of Γ -limits for prestressed films (β ≥ 2). For simplicity, we use R the notation kFk2Q2,g = ω Q2,g (F) dz and R = Riem(g), while ci denote constants indicated in particular Γ -convergence results in previous sections.

13.7 Bibliographical notes The results in this section are taken from Lewicka [2020]. Extensions of results in sections 12 and 13 pertaining to the case of the so-called oscillatory prestress appeared in Lewicka and Luci´c [2020]. This work encompasses a general class of incompatibilities, where the transversal dependence of the lower order terms is not necessarily linear in t and extends the analysis in Agostiniani et al. [2019], Schmidt [2007] to arbitrary metrics and higher order scalings. The connection between the aforementioned non-oscillatory and oscillatory cases, re-

374

13 Infinite hierarchy of limiting theories for prestressed films

lies on projections of appropriate curvature forms on the polynomial tensor spaces, and the dimensionally reduced energies contain now four independents terms: the stretching, the bending, the curvature, and the new excess energy term quantifying the distance of the general prestress from its closest non-oscillatory counterpart. We also mention that the paper by Acharya et al. [2016] proposes another generalization of the basic non-Euclidean energy Eg studied in this monograph, arising naturally in the context of the inverse design problems in solid mechanics: Eg,g˜ (u) =

Z


W g˜1/2 (∇u)g−1/2 dx

for all u ∈ H 1 (ω, R3 ).

Ω

Given two Riemannian metrics g, g˜ on the referential configuration Ω ⊂ R3 , the zero-minimizer u of Eg,g˜ above solves the system g = (∇u)T g(∇u), ˜ which reduces to the classical problem of equi-dimensional embeddability of Riemannian manifolds when g˜ = Id3 . In this general new context, necessary and sufficient conditions for existence of solutions were derived, through a system of total differential equations. For t-independent metrics g, g, ˜ we also displayed a Γ -convergence result for dimension reduction, leading to the new Kirchhoff-like energy functional: I2,g,g˜ (y) =

1 24

Z


Q2,g z, (∇y)T g˜ ∇b dz,

ω

posed on the set of compatible H 2 deformations y in (∇y)T g(∇y) ˜ = gtan , and (g, g)˜ induced Cosserat vector b. The above formulation is ripe for future analysis. Finally, we remark that well-posedness results for an evolutionary stress-assisted diffusion system appeared in in Lewicka and Mucha [2016]. The studied system:      utt − div ∂F W (φ , ∇u) = 0     φt = ∆ ∂φ W (φ , ∇u) . consisted of a balance of linear momentum in the deformation field u : R3 × R+ → R3 , and the diffusion law of the scalar field φ : R3 × R+ → R representing the inhomogeneity factor, interpreted as the local swelling/shrinkage rate in morphogenesis at polymerization, or the localized conformation in liquid crystal elastomers. The related direction of research, including control problems, is also widely open.

Chapter 14

Limiting theories for weakly prestressed films

We now turn to analyzing the dimension reduction problem in presence of the prestress that is a perturbation of identity, depending on the film’s thickness h. Thus, instead of a single Riemannian metric g with curvature of order one and the corresponding energies {Egh }h→0 posed on the thin films Ω h as in chapters 11 - 13, below we consider the family of weak prestress metrics {gh }h→0 with curvatures of the order given by positive powers of h. This leads to the non-Euclidean elastic energies h } {Eweak h→0 , differentiating between the stretching and bending-related components of the prestress which come with the potentially distinct scaling orders. The outline of this chapter is as follows. In section 14.1 we use a result on convex integration for the Monge-Amp´er´e equation to give a general bound on the h } asymptotic behaviour of {inf Eweak h→0 . The purpose of section 14.2 is to investigate different scaling orders in the bending and stretching-related components of the prestress, pertaining to the von K´arm´an and linear elasticity-like energies obtained h } as Γ -limits of the appropriately scaled energies {Eweak h→0 . The compactness and lower bounds are proved, in a unified manner, in section 14.3, while the upper bound is shown in section 14.4 by means of constructing recovery families. In section 14.5 we restate these Γ -convergence results by eliminating the in-plane displacements, now treated as auxiliary fields whose explicit stretching-generated terms in the limiting energies are replaced by the negative Sobolev norms of the out-of-plane displacements. Section 14.6 contains the equivalent conditions for the specific scalings h } of {inf Eweak h→0 , discussed before, to hold. These are given in terms of the vanishing of combinations of components in the prestress, further identified as appropriate coefficients in the expansion of the Riemannian curvatures of {gh }h→0 in h. In section 14.7 we turn to the case of the linearised Kirchhoff-like energy for weakly prestressed films. As in the case of the parallel analysis for shallow shells in section 7.7, we prove the compactness and the Γ -liminf statement for the full range of scaling exponents, while the Γ -limsup estimate is only shown to hold for sufficiently low energies. This bound can be improved under the matching property for isometries relative to the weak convex prestress, which we exhibit in section 14.8. Section 14.9 contains results on the multiplicity of minimizers to the Kirchhoff-like dimensionally reduced energy, while in section 14.10 we discuss its critical points. © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7_14

375

376

14 Limiting theories for weakly prestressed films

14.1 Weakly prestressed films h ) Consider a family (W h , gh , Ah , Eweak h>0 given as in section 11, but where all quantities depend on the film’s thickness parameter h. In analogy to the case of “shallow shells” in section 7.6, here we treat p the “weak prestress” case, i.e. we assume that the imposed incompatibilities Ah ⌘ gh consist of perturbation of Id3 of the order that is a power of the thickness. To this end, define:

Ah (z,t) ⌘ Id3 + ha/2 S(z) + hg/2tB(z) h h , , 2 2

for all x = (z,t) 2 W h = w ⇥

(14.1)

¯ R3⇥3 with S, B 2 C • (w, sym ) and a, g > 0. Note that we explicitly distinguish between the stretching-generating leading order prestress S and the bending-related coefficients in B. The open, bounded set w ⇢ R2 with Lipschitz boundary, is viewed as the midplate of the film W h , on which we pose the energy of elastic deformations: h Eweak (uh ) =

1 h

Z

Wh

W (—uh )(Ah )

1

dx

for all uh 2 H 1 (W h , R3 ).

(14.2)

h . Observe the By a simple calculation we now find the first estimate of inf Eweak expansion of the prestress inverse:

Ah (z, ht)

1

= Id3

ha/2 S

h1+g/2tB + O(ha + h2+g ).

(14.3)

Consequently, we get: h h inf Eweak  Eweak (id3 ) =

C

Z

 Ch

Z

W1

W (Ah (z, ht))

1

dx

ha |S|2 + h2+g |B|2 + O(ha + h2+g )2 dz

w a^(2+g)

(14.4)

for all h ⌧ 1.

For a 4, g 2 the above bound is optimal, while the result may be completed with the G -convergence statement as shown in section 14.2 below. On the other hand, for a < 4 the bound (14.4) may be improved, via a counterpart of De Lellis-InauenSzekelyhidi’s theorem in section 11.2, where the isometric immersion problem is replaced by the weak version of the classical Monge-Amp´er´e equation: Theorem. [Cao-Szekelyhidi’s theorem] 2 ¯ R2⇥2 Let T 2 C 2 (w, sym ) be a matrix field on an open, bounded domain w ⇢ R with ¯ R) C 1,1 -regular boundary. Then, for every a 2 (0, 15 ) there exist fields v 2 C 1,a (w, ¯ R2 ), satisfying: and w 2 C 1,a (w, 1 —v ⌦ —v + sym —w = T 2

in w.

(14.5)

14.1 Weakly prestressed films

377

Similarly to the proof of Theorem 11.4, constructing deformations {uh }h→0 through the Kirchhoff-Love extension as in the proof of Lemma 14.4, for the out-of-plane and the in-plane displacements (v, w) in (14.5), yields energy scaling bounds: Theorem 14.1. Let α ∈ (0, 4) and γ > 0 be the exponents in the prestress family {Ah }h→0 in (14.1), given on thin films with the midplate ω ⊂ R2 that is open, bounded, simply connected, with C 1,1 boundary. Then, the following hold: h ≤ Ch2+γ , (i) if α ∈ [ 74 , 4) and 56 α + 32 > 2 + γ, then inf Eweak h (ii) if α ∈ [ 47 , 4) and 56 α + 23 ≤ 2 + γ, then inf Eweak ≤ Chδ for every exponent 2 5 δ ∈ (0, 6 α + 3 ), h (iii) if α ∈ (0, 74 ), then inf Eweak ≤ Ch2α . h In case when B = 0, then either inf Eweak ≤ Chδ for every δ ∈ (0, 65 α + 23 ) or h 2α inf Eweak ≤ Ch , depending on whether α ∈ [ 47 , 4] or α ∈ (0, 74 ).

Proof. 1. We apply Cao-Szekelyhidi’s theorem to T = Stan and get v, w as in (14.5). ¯ R), wε ∈ C ∞ (ω, ¯ R2 )}ε→0 , by We now define the regularized families {vε ∈ C ∞ (ω, −2 means of the standard convolution kernels {φε (z) = ε φ (z/ε)}ε→0 : vε = v ∗ φε ,

wε = w ∗ φε

and

ε = ht .

where ε is a power of h to be chosen later. The following bound, results from the commutator estimate in Lemma 11.5:

1

∇vε ⊗ ∇vε + sym ∇wε − Stan 0 C (ω) 2

1 ≤ ∇vε ⊗ ∇vε + sym ∇wε − Stan ∗ φε C 0 (ω) 2 + kStan ∗ φε − Stan kC 0 (ω)


1 = ∇vε ⊗ ∇vε − ∇v ⊗ ∇v ∗ φε C 0 (ω) + kStan ∗ φε − Stan kC 0 (ω) 2 ≤ Cε 2a +Cε 2 ≤ Cε 2a ,

(14.6)

where the Cε 2 bound follows by Taylor expanding Stan up to second order terms at each base point z ∈ ω. Further, we get the uniform bounds: k∇vε kC 0 (ω) + k∇wε kC 0 (ω) ≤ C, k∇2 vε kC 0 (ω) + k∇2 wε kC 0 (ω) ≤ Cε a−1 .

(14.7)

Let δ = α2 . We denote s = (S31 , S32 ) and define {uh ∈ C ∞ (Ω¯ h , R3 )}h→0 by:

378

14 Limiting theories for weakly prestressed films





  0 δ wε u (·,t)  id3 + h +h vε 0       2s −∇vε 3δ /2 + h b , + t hδ /2 + hδ 0 S33 − 12 |∇vε |2 δ /2

h

with the higher order smooth correction vector field on ω:

 T 1 b  − S33 ∇vε + |∇vε |2 ∇vε + (∇wε )T ∇vε 2 ∇vε , s . 2 Thus, the deformation gradients defined for (z,t) ∈ Ω 1 have the form: " # " # 0 ∇wε −∇vε 2s h δ /2 δ ∇u (·, ht) = Id3 + h +h ∇vε 0 0 S33 − 12 |∇vε |2 # "   −∇2 vε 0 1+δ /2 3δ /2 t +h 0 0 b +h 0 0
+ O h1+δ + h1+3δ /2 (1 + |∇2 vε | + |∇2 wε |). Recalling (14.3), we obtain: " h

h −1

h

δ

∇u (A ) (·, ht) = Id3 + P + h "

∇vε ⊗ s +h −Stan ∇vε "  −∇2 vε + t h1+δ /2 0 3δ /2

∇wε − Stan 0

0 1 − 2 |∇vε |2

1 2 2 |∇vε | ∇vε

+ (∇wε )T ∇vε h∇vε , si

0 0

#

 − h1+γ/2 B


+ O h2δ + h1+δ /2+γ/2 + h2+γ
+ O h1+3δ /2 + h2+δ /2+γ/2 (|∇2 vε | + |∇2 wε |). Above, we used the following skew-symmetric matrix field:   0 ph h P = , where ph = −hδ /2 ∇vε + hδ s. −ph 0 ¯ SO(3)), defined by: 2. Consider the rotation fields qh ∈ C ∞ (ω, 1 1 qh  exp(−Ph ) = Id3 − Ph + (Ph )2 − (Ph )3 + O(h2δ ). 2 6 Then we get:

#

#

14.1 Weakly prestressed films

379

" qh ∇uh (Ah )−1 (·, ht) = Id3 + hδ

1 ⊗2 + ∇w ε 2 (∇vε )

− Stan

0 0

0 #

"

#

1 skew(∇vε ⊗ s) (∇wε )T ∇vε + (Ph )3 3 −(∇wε )T ∇vε 0 " #   2v −∇ 0 ε + t h1+δ /2 − h1+γ/2 B 0 0

+ O h1+δ + h2δ + h1+δ /2+γ/2 + h2+γ + O h1+δ (|∇2 vε | + |∇2 wε |) . 3δ /2

+h

¯ SO(3)): Finally, we apply a further rotation field q¯h ∈ C ∞ (ω, q¯h  exp(−P¯ h ) = Id3 − P¯ h + O(h2δ ). # "
3δ /2 T ∇v δ ∇w + h3δ /2 ∇v ⊗ s 1 h (∇w ) skew h ε ε ε ε h + (Ph )3 , where P¯ = 3δ /2 T 3 −h 0 (∇wε ) ∇vε As before, it follows that: " h h

h

h −1

δ

q¯ q ∇u (A ) (·, ht) = Id3 + h  + t h1+δ /2

"

−∇2 vε 0

0 0

#

1 ⊗2 + sym∇w ε 2 (∇vε )

− Stan

0

0 0

#

 − h1+γ/2 B



+ O h1+δ + h2δ + h1+δ /2+γ/2 + h2+γ + O h1+δ (|∇2 vε | + |∇2 wε |) . 3. We thus obtain the energy bound, valid provided that we may
use Taylor’s expansion of W , which here requires h1+δ /2 (k∇2 vε kC 0 + k∇2 wε kC 0 → 0 as h → 0: Z   h h inf Eweak ≤ Eweak (uh ) = W q¯h qh ∇uh (Ah )−1 (·, ht) dx Ω1 Z  2 1 ≤C h2δ ∇vε ⊗ ∇vε + sym∇wε − Stan + h2+δ (k∇2 vε k2C 0 + k∇2 wε k2C 0 ) 2 Ω1  + h2+γ + h4δ dx.

Recalling (14.6), (14.7), this leads to: h inf Eweak ≤ C h2δ ε 4a + h2+δ ε 2a−2 + h2+γ + h4δ





= C h2δ +4at + h2+δ +(2a−2)t + h2+γ + h4δ . Minimizing the right hand side above is equivalent to maximizing the minimal of the four displayed exponents. We hence choose t in ε = ht so that 2δ + 4at = 2−δ 2 + δ + (2a − 2)t, namely t = 2a+2 . In conclusion:

380

14 Limiting theories for weakly prestressed films h inf Eweak ≤ C h2

δ +2a a+1



+ h2+γ + h4δ ≤ C h(5δ /3+2/3)− + h2+γ + h4δ

upon recalling that a ∈ (0, 15 ). The result follows by a direct inspection. Figure 14.1 shows a diagram depicting various cases and the corresponding scaling exponents in Theorem 14.1. If we had v ∈ H 2 and w ∈ H 1 satisfying h the same equation in (14.5) with T = Stan , then inf Eweak might be further decreased. Indeed, in section 14.7 we will show that existence of v ∈ H 2 (ω, R) with h det ∇2 v = −curl curl Stan yields inf Eweak ≤ Chα/2+2 , for any α ∈ (2, 4) and γ = α/4. Naturally, this bound is superior in any of the cases (i)-(iii) in Theorem 14.1.

h Fig. 14.1 Bounding exponents of inf Eweak in Theorem 14.1 (i), (ii), (iii).

14.2 Von K´arm´an and linear elasticity-like theories for weakly prestressed films In this section, we start the dimension reduction analysis for the energies (14.2) and prestress (14.1), in various regimes of the scaling powers α, γ. As in chapters 11, 12, 13, and Part II of this monograph devoted to elasticity without prestress, our results comprise the compactness statements and the matching Γ -liminf and Γ limsup bounds. The results and proofs will be given in a unified manner and only specified to different prestress scalings in the theorems statements below, expressed in terms of Γ -convergence. By now, the reader should be well familiar with the type of results that are available from the dimension reduction in our context. Recall the quadratic form Q2 , derived from Q3 = D2W (Id3 ) in Definition 5.6:  ˜ F˜ ∈ R3×3 with F˜tan = F Q2 (F) = min Q3 (F); for all F ∈ R2×2 . (14.8) In the scaling regimes consistent with the preliminary bound in (14.4), we have:

14.2 Von K´arm´an and linear elasticity-like theories for weakly prestressed films

381

Theorem 14.2. Let α ≥ 4, γ ≥ 2 be the exponents of the prestress family {Ah }h→0 in (14.1), given on thin films with open, bounded, connected midplate ω ⊂ R2 with C 1,1 boundary. Then, we have the energy scaling and Γ -limits: h h (i) When γ = 2 then inf Eweak ≤ Ch4 , and h−4 Eweak −→ I where: Γ

I (v, w) =

1 24

Z


Q2 ∇2 v + Btan dz

ω

   1 0 for α > 4 Q2 sym∇w + ∇v ⊗ ∇v − dz. Stan for α = 4 2 ω

1 + 2

Z

h h (ii) When γ ∈ (2, α − 2] then inf Eweak ≤ Ch2+γ , and h−(2+γ) Eweak −→ I : Γ

1 24

Z


Q2 ∇2 v + Btan dz ω  Z   1 0 for γ < α − 2 Q2 sym∇w − dz. + Stan for γ = α − 2 2 ω

I (v, w) =

h h ≤ Chα , and h−α Eweak (iii) When γ > α − 2 then inf Eweak −→ I where: Γ

I (v, w) =

1 24

Z


Q2 ∇2 v dz

ω

1 + 2

  0 Q2 sym ∇w + 1

Z

 for α > 4 − Stan dz 2 ∇v ⊗ ∇v for α = 4

ω

All Γ -limit functionals I above are defined on the scalar out-of-plane displacements v ∈ H 2 (ω, R) and the in-plane displacements w ∈ H 1 (ω, R2 ). The Γ -convergence statements are with respect to the following compactness properties (with convergence up to a subsequence that we do not relabel): in H 1 (ω, R3 ) for some R¯ h ∈ SO(3), ch ∈ R3 ,

yh (z,t)  (R¯ h )T uh (z, ht) − ch → z ? h

h

V [y ]  h −1 h

h

−δ /2

1/2

−1/2

h V [y ]tan * w

yh (·,t) − id2 dt → (0, 0, v) in H 1 (ω, R3 ),

(14.9)

in H 1 (ω, R2 ),

where δ = 2 in case (i), δ = γ in case (ii), and δ = α − 2 in case (iii). Remark 14.3. The case α = 4, γ = 2 is of particular interest. The limiting energy I in (i) above is then a generalization of the von K´arm´an’s energy for plates, whose Euler-Lagrange equations were derived in section 6.6. Reproducing the steps in the proof of Theorem 6.20 and in Remark 6.21, we obtain the following version of the von K´arm´an equations in presence of prestress.

382

14 Limiting theories for weakly prestressed films

Assume additionally that ω is simply connected, and that W is isotropic as in (5.5) with the Lam´e constant µ, λ satisfying: µ > 0 and 3λ + µ > 0. Similarly to (6.66), we define the Airy stress potential Φ ∈ H 2 (ω, R) by: cof ∇2 Φ =

   1 1 λµ  div w + |∇v|2 − tr Stan Id2 + µ sym ∇w + ∇v ⊗ ∇v − Stan . λ +µ 2 2

The Euler-Lagrange equations are written in terms of Φ and the out-of-plane displacement v ∈ H 2 (ω, R), using the Airy bracket [v1 , v2 ]  h∇2 v1 : cof ∇2 v2 i in:  µ(µ + 3λ )  1 [v, v] + λg , µ + 2λ 2   2 2λ µ + µ ∆ 2 v + Λg = [v, Φ] in ω, 12(λ + µ)

∆ 2Φ = −

(14.10)

The additional quantities above are prestress-related: λg = curl curl Stan = ∂22 S11 + ∂11 S22 − 2∂12 S12 ,   Λg = div div Btan + p cof Btan

= ∂11 B11 + pB22 + ∂22 B22 + pB11 + 2(1 − p)∂12 B12 , where p =

λ . 2λ + µ

In fact, p may be interpreted as the Poisson’s ratio, the first prefactor (14.10) is Young’s modulus, and the second prefactor ness, while

1 2 [v, v]

2λ µ+µ 2 12(λ +µ)

µ(µ+3λ ) µ+2λ

in

is the bending stiff-

is the Gaussian curvature in the problem.

Denoting by η the outward unit normal to ∂ ω and by τ the unit tangent vector, the free boundary conditions complementing (14.10) are: Φ = ∂η Φ = 0 on ∂ ω, D

E ∇2 v + Btan : η ⊗ η + pτ ⊗ τ = 0, E D (1 − p)∂τ ∇2 v + Btan ) : η ⊗ τ D

E + div ∇2 v + p cof ∇2 v + div Btan + p cof Btan , η = 0.

(14.11)

In the particular case when Btan = ∂η Btan = 0 on ∂ ω, the last two equations in (14.11) coincide with the previously obtained system (6.70). Before giving the proof of Theorem 14.2, we state a parallel result concerning the special case when the leading order in-plane prestress vanishes. The scaling of the bending prestress component hγ/2tB is as in Theorem 14.2 (i.e. at least h2 ), but we allow for the stretching hα/2 S in the out-of-plane directions to be of order h:

14.2 Von K´arm´an and linear elasticity-like theories for weakly prestressed films

383

Theorem 14.4. Let α, γ ≥ 2 be the exponents in the prestress family {Ah }h→0 given in (14.1), which is defined on thin films with open, bounded, connected midplate ω ⊂ R2 with C 1,1 boundary. We further assume that: Stan ≡ 0

in ω.

Then, we have the following energy scaling and Γ -convergence results: h h (i) when α = 2 then inf Eweak ≤ Ch4 , and h−4 Eweak −→ I where:  Z  1 0 for γ > 2 I (v, w) = Q2 ∇2 v − 2 sym∇(S31 , S32 ) + dz Btan for γ = 2 24 ω Z   1 1 1 + Q2 sym∇w + ∇v ⊗ ∇v − (S31 , S32 )⊗2 dz, 2 ω 2 2 Γ

h (ii) when γ = 2 < α then inf Eweak ≤ Ch4 , and h−4 I h −→ I where: Γ

I (v, w) =

1 24

Z  
1 1 Q2 ∇2 v + Btan dz + Q2 sym ∇w + (∇v)⊗2 dz, 2 ω 2 ω

Z

h h ≤ Ch2+α∧γ , and h−(2+α∧γ) Eweak (iii) when α, γ > 2 then inf Eweak −→ I : Γ

Z
1 Q2 sym ∇w dz 2 ω   Z   1 0 for γ < α 0 for α < γ 2 dz. Q2 ∇ v − + + 2 sym∇(S31 , S32 ) for α ≤ γ Btan for γ ≤ α 24 ω

I (v, w) =

All Γ -limits I above are defined on the scalar out-of-plane displacement fields v ∈ H 2 (ω, R) and the in-plane displacements w ∈ H 1 (ω, R2 ). The Γ convergences are with respect to the compactness statements in (14.9), which are valid with δ = 2 in case (i) and (ii), and δ = α ∧ γ in case (iii). Remark 14.5. We note that in cases (i) and (iii) of Theorem 14.4, the non-zero stretching components in S of Ah contribute to the bending term and are mixed with the original bending components in B. We point out that the presence of similar terms also occurred in section 6.4 in the context of shells with varying thickness. One further checks that the above results are stronger than the general energy bounds in Theorem 14.1, which are however valid for any nonzero Stan . Since h ≤ Ch(5α/6+2/3)− whenever α < 4. This leads to: α, γ ≥ 2, it follows that inf Etan 7/3− h in case (i), which is a bound indeed inferior to h4 . Likewise, in case (iii) we (5α/6+2/3)− get: h  h2+α∧γ . We will prove the convergence properties (14.9) and the lower bounds of Theorems 14.2 and 14.4 in section 14.3. Section 14.4 will provide constructions of recovery families and the upper bounds portion of the Γ -limit statements.

384

14 Limiting theories for weakly prestressed films

14.3 Compactness and lower bounds In this section, we begin the proofs of Theorems 14.2 and 14.4. The arguments below are written in a unified manner and are only specified to the two assumed prestress scalings in the last steps, which are Corollaries 14.10 and 14.11. We start with the approximation statement based on the geometric rigidity estimate (4.2). The proof is exactly the same as in in Theorem 4.8, Lemma 11.9 or Corollary 12.3, so we leave its details to the reader: Lemma 14.6. Let ω ⊂ R2 be an open, bounded, connected and Lipschitz domain, and let {Ah }h→0 satisfy (14.1). Assume that {uh ∈ H 1 (Ω h , R3 )}h→0 satisfies: h Eweak (uh ) ≤ Ch2+δ ,

(14.12)

for some δ > 0. Then there exists {Rh ∈ H 1 (ω, SO(3))}h→0 such that: 1 h

Z Ωh

|∇uh − Rh Ah |2 dx ≤ Ch2+δ ∧α∧γ ,

Z

|∇Rh |2 dz ≤ Chδ ∧α∧γ .

(14.13)

ω

Before proving proving the compactness properties in (14.9), let us motivate the h (uh ), equal to: choice of exponent δ . The scaled integrands in h−(2+δ ) Eweak
1 1 W (Rh )T ∇uh (Ah )−1 = D2W (Id3 )(Z h )⊗2 + o(|Z h |2 ) 2 h2+δ

as h → 0, (14.14)

in virtue of frame invariance and by Taylor expanding W . Above, the strains {Z h ∈ L2 (Ω 1 , R3×3 )}h→0 ate accordingly defined by: Z h (z,t) =

1



h1+δ /2

Rh (z)T ∇uh (z, ht)Ah (z, ht)−1 −Id3



1 1
for all z ∈ ω, t ∈ − , . 2 2

h (uh ) In view of the expansion (14.14), it is natural to expect that the limit of h−(2+δ ) Eweak h quantifies the limit of {Z }h→0 . By (14.13), a sufficient condition to get a subsequential convergence in:

Zh * Z

weakly in L2 (Ω 1 , R3×3 ),

as h → 0

(14.15)

is thus: 1 + (δ ∧ α ∧ γ)/2 ≥ 1 + δ /2, or equivalently: δ ≤ α ∧ γ. Lemma 14.7. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let {Ah }h→0 be as in (14.1). Assume that {uh ∈ H 1 (Ω h , R3 )}h→0 satisfies (14.12) with: δ ∈ (0, α ∧ γ].

14.3 Compactness and lower bounds

385

Then, there exist {R¯ h ∈ SO(3), ch ∈ R3 }h→0 such that the following holds for the rescaled deformations yh (z,t)  (R¯ h )T uh (z, ht) − ch ∈ H 1 (Ω 1 , R3 ). All convergences are up to a subsequence that we do not relabel, as h → 0: (i) yh → π in H 1 (Ω 1 , R3 ), > 1/2 (ii) V h [yh ] = h−δ /2 −1/2 yh (·,t) − id2 dy → V in H 1 (ω, R3 ). The limit V has higher regularity V ∈ H 2 (ω, R3 ) and it satisfies:  0 for δ < α sym ∇V = Stan for δ = α. In particular, for δ < α or when Stan ≡ 0 in ω, we get that V = (0, 0, v) for an out-of-plane displacement v ∈ H 2 (ω, R). The case δ = α is viable only when curl curl Stan ≡ 0 in ω. (iii) The in-plane part of the limiting strain Z in (14.15) satisfies:  
0 for δ < α 0 for δ < γ 2
∂3 Z tan = −∇ V3 + − 2 ∇(S31 , S32 ) tan for δ = α Btan for δ = γ. Hence Z(·, 0)tan is well defined, and it equals: Proof. 1. Using Lemma 14.6, we define: ? h h ˆh h ¯ ˜ ˜ R = R R where: R = PSO(3) Rh dz,

> 1/2 −1/2

Z(·,t)tan dt ∈ L2 (ω, R2×2 ). ?

Rˆ h = PSO(3)

ω

(R˜ h )T ∇uh dx, Ωh

where the uniqueness of the above projections follows by (14.13), together with: k(R¯ h )T Rh − Id3 kH 1 (ω) ≤ Chδ /2 .

(14.16)

> > By choosing ch so that Ω 1 yh dx = ω z dz, we obtain the following bounds that imply convergence in (i): ? ? h 2 δ |(∇y − Id3 )3×2 | dx ≤ Ch , |∂3 yh |2 dx ≤ Ch2 . Ω1

Ω1

2. Fix s  1. From (14.16), (14.15) we get the weak convergence in L2 (Ω 1 , R3×3 ):
1 Z(·,t + s) − Z(·,t) tan s  1  ¯h T h h (14.17) ( (R ) R Z (·,t + s)Ah (·, ht + hs) − Z h (z,t)Ah (z, ht) s tan   1 1 ∇yh (z,t + s) − ∇yh (z,t) − hγ/2−δ /2 Btan . = s h1+δ /2 tan In order to identify the limit of the right hand side above, we expand
first term in> the s h−(1+δ /2) 1s yh (·,t + s) − yh (·,t) = h−(1+δ /2) 0 ∂3 yh (·,t + τ) dτ. Namely, we write:

386

14 Limiting theories for weakly prestressed films


1  ¯h T 1 h h h = ( R ) ∂ u (z, ht) − e ± A (z, ht)e ∂ y (z,t) − he 3 3 3 3 3 h1+δ /2 hδ /2   1  1 ¯h T  h = δ /2 (R ) ∇u (z, ht) − Rh Ah (z, ht) e3 + δ /2 (R¯ h )T Rh (z) − Id3 Ah (z, ht)e3 h h + hα/2−δ /2 S(z)e3 + hγ/2+1−δ /2tB(z)e3 . The first term in the right hand side above is bounded by Ch from (14.13), so it converges to 0. For the second term, we first denote the subsequential limit:  1  ¯h T h ( R ) R − Id 3 *P hδ /2

weakly in H 1 (ω, R3×3 )

as h → 0,

(14.18)

whose existence is implied by (14.16) and where the limiting field satisfies: P ∈ H 1 (ω, so(3)). Recalling (14.17) we hence obtain: 
0
for δ < α Z(·,t)tan − Z(·, 0)tan = t ∇(Pe3 ) tan + t ∇(Se3 ) tan for δ = α  0 for δ < γ −t Btan for δ = γ. 3. We now identify the entries P31 , P32 of the limiting skew field in (14.18). Since: ∇V h =

?

1

h/2

(R¯ h )T hδ /2

−h/2

α/2−δ /2

¯h T

+h

∇uh − Rh Ah


3×2

dt +

1 hδ /2

(R¯ h )T Rh − Id3


3×2

(14.19)

h

(R ) R S3×2 ,

another application of (14.18) yields the subsequential convergence:  0 for δ < α h h ∇V [y ] → P3×2 + strongly in L2 (ω, R3×2 ) as h → 0. S3×2 for δ = α > As ω V h [yh ] dz = 0, this proves the convergence statement in (ii), with ∇V given by the right hand side above. Equivalently, we record the useful formula:  0 for δ < α P3×2 = ∇V − (14.20) S3×2 for δ = α. > Note now that the definitions of R˜ h and Rˆ h imply skew ω (∇V [yh ])tan dz = 0, as: ?

(∇V h [yh ])tan dz = ω

?  1  ˆh T ) (R˜ h )T ∇uh − Rˆ h dx ∈ R2×2 ( R sym . δ /2 h tan h Ω

(14.21)

> There also holds: ω V dz = 0. Consequently, in case δ < α or Stan = 0 it follows that the tangential component of V must have gradient 0 and mean 0, and hence be equal to 0. This completes the proof of (ii), while from (14.20) we get:

14.3 Compactness and lower bounds

387


(Pe3 )1 , (Pe3 )2 = −(P31 , P32 ) = −(∇V3 )tan +



0 for δ < α (S31 , S32 ) for δ = α.

(14.22)

The formula in (iii) is now a consequence of step 2. We now give the first bound from below of the scaled energies: Corollary 14.8. In the context of Lemma 14.7, there holds: lim inf h→0

1

1 E h (uh ) ≥ 2 h2+δ weak

Z

1 Q2 sym Z(z, 0)tan dz + Q2 sym ∂3 Ztan dz, 24 ω ω

Z

Proof. Recalling (14.14) and the boundedness of {Z h }h→0 in L2 (Ω 1 , R3 ), we write: 1

h

E h (uh ) ≥ 2+δ weak =

Z

1 Q3 (Z h ) + o(1)|Z h |2 dx 2
Q3 1{hδ /2 |Z h |≤1} |Z h | dx + oh (1).

{hδ /2 |Z h |≤1}

1 2

Z Ω1

Further, since the argument in Q3 in the right hand side above converges, weakly in L2 (Ω 1 , R3×3 ), to Z by (14.15), the weak lowersemicontinuity argument implies: lim inf h→0

1

1 h Q3 (Z) dx Eweak (uh ) ≥ 2+δ 2 Ω1 h Z Z 1 1 Q3 (sym Z) dx ≥ Q2 (sym Ztan ) dx. ≥ 2 Ω1 2 Ω1 Z

Here, we also used that Q3 (Z) = Q3 (sym Z) and the definition of Q2 . By Lemma 14.7 (iii) we get: Z(z,t)tan = Z(z, 0)tan +t∂3 Z(z, ·)tan , which concludes the proof. We now identify the integrand in the lower bond obtained in Corollary 14.8: Lemma 14.9. In the context of Lemma 14.7, we have the following subsequential convergence, weakly in L2 (ω, R2×2 ) as h → 0:

1 1 sym Z(·, 0)tan ( sym ∇V h tan − 1+δ /2 sym (R¯ h )T Rh − Id3 tan h h
− hα/2−1−δ /2 sym (R¯ h )T Rh S tan . Proof. We decompose 

> 1/2

(R¯ h )T Rh − Id3  + (R¯ h )T ·  + (R¯ h )T ·

Z h (·,t) −1/2 tan


T

(R¯ h )T ·

1 h1+δ /2 1 h1+δ /2

?

dt as the sum below:

1 h1+δ /2

?

h/2

(∇uh − Rh Ah ) dt · (Ah )−1

h/2

−h/2 h/2

−h/2



−h/2

(∇uh − Rh Ah ) dt ?

(14.23)

 tan

(∇uh − Rh Ah ) dt · (Ah )−1 − Id3


 tan

.

tan

388

14 Limiting theories for weakly prestressed films

The first and the third terms in the right hand side above converge to 0, in L2 (ω, R2×2 ) as h → 0, because of (14.16) and (14.13). We rewrite the second term as:


1 1 ∇V h tan − 1+δ /2 (R¯ h )T Rh − Id3 tan − hα/2−1−δ /2 (R¯ h )T Rh S tan . h h > 1/2 This implies the result, because of the subsequential convergence of −1/2 Z h (·,t)tan dt > 1/2 to −1/2 Z(·,t)tan dt = Z(·, 0)tan , weakly in L2 (ω, R2×2 ). We are now ready to conclude the lower bound in Theorems 14.2 and 14.4: Corollary 14.10. In the context of Lemma 14.7, assume further that: δ ≥2

and

α ≥ 2+δ.

Then, {h−1V h [yh ]tan }h→0 converges to some in-plane displacement w, up to a subsequence weakly in H 1 (ω, R2 ). Moreover, there holds:   0 for δ > 2 0 for α > 2 + δ − sym Z(·, 0)tan = sym ∇w + 1 ⊗2 for δ = 2 Stan for α = 2 + δ . (∇v) 2 h (uh ) is bounded from below by: Consequently, lim infh→0 h−(2+δ ) Eweak  Z   1 0 for δ < γ I (v, w) = Q2 ∇2 v + dz Btan for δ = γ 24 ω   Z   1 0 for δ > 2 0 for α > 2 + δ + Q2 sym ∇w + 1 dz, − ⊗2 for δ = 2 S2×2 for α = 2 + δ 2 ω 2 (∇v)

which coincides with the functionals in Theorem 14.2 in each indicated case. Proof. The second term in the right hand side of (14.23) can be rewritten using: −sym (R¯ h )T Rh − Id3



1 ¯ h )T Rh − Id3 T (R¯ h )T Rh − Id3 . ( R sym = tan 2 tan




By (14.18) and since δ ≥ 1 + δ /2, we conclude the following subsequential convergence, weakly in L2 (ω, R2×2 ): 
1 0 for δ > 2 h T h ¯ − 1+δ /2 sym (R ) R − Id3 tan * 1 T (P P) for δ = 2. h tan 2 When α ≥ 2 + δ , then the third term in (14.23) converges, by (14.16), to 0 for α > 2 + δ and to −Stan for α = 2 + δ . In this case, we also have V = (0, 0, v) and (14.20) yields that (PT P)tan = ∇v ⊗ ∇v. This proves the formula for Z(·, 0)tan , since
the resulting weak subsequential convergence of the first term { 1h sym ∇V h [yh ] tan }h→0 in the right hand side of (14.23) is equivalent with the weak convergence of {h−1V h [yh ]tan }h→0 to some limiting in-plane displacement field w ∈ H 1 (ω, R2 ).

14.3 Compactness and lower bounds

389

h Finally, the lower bound on lim infh→0 h−(2+δ ) Eweak is derived from Corollary 14.8, upon recalling Lemma 14.7 (iii). The optimal values of the exponent δ given in Theorem 14.2 in function of α ≥ 4 and γ ≥ 2, follow by a direct inspection.

Corollary 14.11. In the context of Lemma 14.7, assume further that: δ ≥2

and

Stan ≡ 0

in ω.

Then, {h−1V h [yh ]tan }h→0 converges to some in-plane displacement w, up to a subsequence weakly in H 1 (ω, R2 ). Moreover, there holds:

sym Z(·, 0)tan = sym∇w +

 0

for δ > 2 for δ = 2 < α



for δ = α = 2.

1 ⊗2 2 (∇v) 1 ⊗2 − 1 (S , S )⊗2 2 (∇v) 2 31 32

(14.24)

h (uh ) is bounded from below by: Consequently, lim infh→0 h−(2+δ ) Eweak

I (v, w) = =

   0 for δ < α 0 for
Q2 ∇2 v − + 2 sym ∇(S31 , S32 ) 2×2 for δ = α Btan for ω   Z  1 0 for δ > 2 0 for + Q2 sym ∇w + 1 1 ⊗2 for δ = 2 − ⊗2 for (∇v) (S , S ) 2 ω 31 32 2 2 1 24

Z

δ 2  dz, α =2

which coincides with the functionals in Theorem 14.4 in each of the indicated cases. Proof. Convergence of the second term in the right hand side of (14.23) follows as in the proof of Corollary 14.10. To show (14.24), observe that the third term in (14.23) can be now written as: −hα/2−1 sym

 (R¯ h )T Rh − Id 3 hδ /2

·S

 tan

,

(14.25)

and we may identify its limit when α ≥ 2, in virtue of (14.18). Namely, when α > 2 this limit is 0 and we recover the first two cases of (14.24). When α = 2 then automatically δ = 2 as well, and (14.25) converges subsequentially, weakly in L2 (ω, R2×2 ) as h → 0 to:  

−sym PS tan = sym ∇v − (S31 , S32 ) ⊗ (S31 , S32 )
= sym ∇v ⊗ (S31 , S32 ) − (S31 , S32 )⊗2
⊗2 where we used (14.20). At the same time: 21 (PT P)tan = 21 ∇v − (S31 , S32 ) , which concludes the proof of the last case in (14.24). Finally, the asserted lower bound follows from Corollary 14.8, Lemma 14.7 (iii) and the identities in (14.24). The optimal values of δ in function of α, γ ≥ 2 are obtained by direct inspection.

390

14 Limiting theories for weakly prestressed films

14.4 Von K´arm´an and linear elasticity-like theories for weakly prestressed films: recovery family In this section, we construct families of deformations {uh }h→0 with the desired h (uh ), completing the Γ -convergence results given asymptotics of the energy Eweak in Theorems 14.2 and 14.4. We present the unified constructions for the general form of the limiting functional I given in Corollaries 14.10 and 14.11. End of proof of Theorem 14.2 ¯ R), Given α ≥ 4 and γ ≥ 2, let δ ∈ [2, γ] and α ≥ 2 + δ . Assume that v ∈ C ∞ (ω, ¯ R2 ), and d, d¯ ∈ C ∞ (ω, ¯ R3 ). We define uh ∈ C ∞ (Ω¯ h , R3 ) by: w ∈ C ∞ (ω,       1 ∇v h δ /2 0 1+δ /2 w δ /2 ¯ (14.26) u = id3 + h +h −h t + h1+δ /2td + hδ /2t 2 d. v 0 0 2 ¯ SO(3)): Consider the rotation fields qh ∈ C ∞ (ω,    0 ∇v qh = exp hδ /2 −∇v 0     3 1 (∇v)⊗2 0 0 ∇v + O(h 2 δ ). = Id3 + hδ /2 − hδ 2 −∇v 0 0 |∇v| 2 Above, the constants in O(·) depend on k∇vkL∞ (ω) . Thus, we get for all (z,t) ∈ Ω 1 :     1 δ (∇v)⊗2 0 1+δ /2 ∇w 0 +h q ∇u (z, ht) = Id3 + h 0 0 0 |∇v|2 2  2    ∇ v0 + h1+δ /2 0 d + t d¯ + O(h2+δ /2 ), − h1+δ /2t 0 0 h

h

where the bound in O(·) depends on the L∞ (ω) norms of: ∇v, ∇2 v, ∇w, d, d,¯ ∇d and ∇d.¯ Recalling (14.3), we further obtain: h

h

h −1

q (∇u )(A ) +h




α/2

(·, ht) = Id3 − h

1+δ /2



∇w 0

S−h

 d

1+γ/2

−h

1+δ /2

 t

1 tB + hδ 2 ∇2 v 0

"

0 (∇v)⊗2 0 |∇v|2  −d¯ + O(h2+δ /2 ).

#


It follows that the integrand W qh (∇uh )(Ah )−1 equals: # "    1 δ (∇v)⊗2 0 ∇w 1+δ /2 α/2 Q3 − h S+ h + h d 0 2 0 |∇v|2  
∇2 v − h1+γ/2tB − h1+δ /2t + o h2+δ , −d¯ 0

14.4 Von K´arm´an and linear elasticity-like theories for weakly prestressed films: recovery... 391

which yields:  2   ∇ v γ/2−δ /2 ¯ Q3 h B+ −d dz 0 ω     Z  1 1 δ /2−1 (∇v)⊗2 0 ∇w α/2−(1+δ /2) + lim Q3 h S− h − d dz. 0 0 |∇v|2 2 h→0 ω 2 1

h lim Eweak (uh ) = h→0 h2+δ

1 lim 24 h→0

Z

Setting d and d¯ to be affine functions of ∇w, (∇v)⊗2 , ∇2 v and S, B, so that Q3 above gets replaced by Q2 evaluated on the principal 2 × 2 minors of the respective arguments, we obtain the claimed convergence to the energy in Corollary 14.10: lim

1

h→0 h2+δ

h Eweak (uh ) = I (v, w).

(14.27)

Finally, we observe that given v ∈ H 2 (ω, R) and w ∈ H 1 (ω, R2 ), one can first find their smooth approximations {vn , wn }n→∞ , and construct the recovery family {uh }h→0 as the diagonal sequence given by formula in (14.26) with v, w replaced by vn , wn . Taking n = n(h) → ∞ as h → 0 at a sufficiently slow rate, guarantees the same limit as in (14.27). End of proof of Theorem 14.4 Assume that Stan ≡ 0 in ω, α, γ ≥ 2, and let δ ∈ [2, α ∧γ]. Given smooth displace¯ R), w ∈ C ∞ (ω, ¯ R2 ), d, d¯ ∈ C ∞ (ω, ¯ R3 ), and denoting s = (S31 , S32 ), ments v ∈ C ∞ (ω, h ∞ h 3 we define the recovery family {u ∈ C (Ω¯ , R )}h→0 by:     2s 0 uh = id3 + hα/2t + hδ /2 S33 v     (14.28) 1 w ∇v ¯ + h1+δ /2 − hδ /2t + h1+δ /2td + hδ /2t 2 d. 0 0 2 As in the previous construction of the    recovery family for Theorem 14.2, we apply  0 ∇v rotations qh = exp hδ /2 and obtain the following expansion on Ω 1 : −∇v 0 " #   1 δ (∇v)⊗2 0 ∇w 0 h h 1+δ /2 q ∇u (z, ht) = Id3 + h +h 0 0 2 0 |∇v|2 " # " # " # ∇2 v 0 2s 2∇s 0 1+δ /2 α/2 1+α/2 −h t +h 0 +h t 0 0 S33 ∇S33 0 " #   S33 ∇v − hδ /2+α/2 0 + h1+δ /2 0 d + t d¯ + O(h2+δ /2 ), −2hs, ∇vi where the bound in O(·) depends on the L∞ (ω) norms ∇2 v, ∇w, d, d,¯ ∇d,  of: ∇v,   0 −s ∇d.¯ We now apply a further rotation q¯h = exp hα/2 to get: s 0

392

14 Limiting theories for weakly prestressed films

"

# " # ⊗2 1 (∇v) ∇w 0 0 + h1+δ /2 q¯h qh ∇uh (z, ht) = Id3 + hδ 2 0 |∇v|2 0 0 " # # " 1 α/2 2s⊗2 −2S33 s ∇2 v 0 α/2 1+δ /2 +h S+ h −h t 2 0 0 0 3|s|2 " # # " 2∇s 0 S ∇v 33 + h1+α/2t + hδ /2+α/2 0 ∇S33 0 −2hs, ∇vi   + h1+δ /2 0 d + t d¯ + O(h2+δ /2 ), By the improved (with respect to (14.3)) second order expansion of the inverse: (Ah (z, ht))−1 = Id3 − hα/2 S − h1+γ/2tB + hα S2 + O(h1+δ ), we obtain: " #
1 δ (∇v)⊗2 0 h h h h −1 1+γ/2 q¯ q (∇u )(A ) (z, ht) = Id3 − h tB + h 2 0 |∇v|2 # " # # " " 1 α s⊗2 2S33 s 2∇s 0 0 S33 ∇v 1+α/2 δ /2+α/2 − h +h t +h 2 2 0 −2S33 ∇S33 0 0 −2hs, ∇vi # " # " ∇2 v ∇w d − h1+δ /2t −d¯ + O(h2+δ /2 ). + h1+δ /2 0 0
It follows that the integrand W q¯h qh (∇uh )(Ah )−1 equals: " # " #  1 1 S33 s S33 ∇v s⊗2 0 α δ /2+α/2 2 Q3 − h +h 1 2 2 S33 s −2S33 2 S33 ∇v −2hs, ∇vi # " # " 1 δ (∇v)⊗2 ∇w 0 + h1+δ /2 d + h 2 0 |∇v|2 0 # " 2 sym∇s 12 ∇S33 1+α/2 +h t 1 0 2 ∇S33 " #  2v
−∇ − h1+γ/2tB + h1+δ /2t d¯ + o h2+δ , 0 1 h (uh ) as the sum of two limit terms. The first term Eweak allowing to write limh→0 h2+δ carries the induced bending terms and it has the form: # " Z  1 2 sym∇s 12 ∇S33 α/2−δ /2 lim Q3 h 1 24 h→0 ω 0 2 ∇S33 " #  −∇2 v γ/2−δ /2 ¯ +h B+ dz, d 0

14.5 Elimination of out-of-plane displacements

393

while the second, stretching-related term, is: " " # # Z  1 1 1 δ /2−1 (∇v)⊗2 S33 s 0 s⊗2 α−(1+δ /2) lim Q3 − h + h 2 2 h→0 ω 2 2 S33 s −2S33 0 |∇v|2 # " # "  1 ∇w 0 2 S33 ∇v + dz. d + hα/2−1 1 0 2 S33 ∇v −2hs, ∇vi In each of the cases of ordering α, γ and δ , we may set d and d¯ to be affine functions of ∇w, ∇v⊗∇v, ∇2 v and S, B, so that Q3 above gets replaced by Q2 evaluated on the principal 2×2 minors of the respective arguments. Thus, we obtain the claimed convergence to the energy functional in Corollary 14.11 in case when the displacements v and w are smooth. The general case v ∈ H 2 (ω, R) and w ∈ H 1 (ω, R2 ) follows by a diagonal argument as before.

14.5 Elimination of out-of-plane displacements When S = B ≡ 0 in ω then all three cases in Theorem 14.2 reduce to the von K´arm´an and linear theories in classical nonlinear elasticity, derived in chapter 5. In case (iii) and when α = 4, the minimization of I amounts to finding the displacement v whose combined magnitude of the total induced curvature ∇2 v and the deviation of the Gaussian curvature det ∇2 v from the given −curl curl Stan , is the smallest. In this line, we further remark that the in-plane displacement w is always slaved to S, B and v, and as such can be omitted all together. To make this statement more precise, we first present two decomposition results for symmetric matrix fields below: Lemma 14.12. Let ω ⊂ R2 be an open, bounded, simply connected domain with 2 Lipschitz boundary. For every F ∈ L2 (ω, R2×2 sym ) there exist the unique v ∈ H (ω, R) 1 2 (up to linear maps) and φ ∈ H0 (ω, R ) satisfying: F = ∇2 v + cof sym ∇φ . Moreover, there holds the following equivalences with curl F (taken row-wise):    kF − ∇2 vkL2 (ω) = distL2 (ω) F, ∇2 r; r ∈ H 2 (ω, R)    ' kcurl FkH −1 (ω) = distL2 (ω) F, ∇w; w ∈ H 1 (ω, R2 ) , where the congruency symbol a ' b means that a ≤ Cb and b ≤ Ca with a constant C depending only on ω. Proof. Since the linear space {cof sym ∇φ ; φ ∈ H01 (ω, R2 )} is a closed subspace of L2 (ω, R2×2 ), the following minimization problem has the unique solution:

394

14 Limiting theories for weakly prestressed films

minimize

nZ

o |F − cof sym ∇φ |2 dz; φ ∈ H01 (ω, R2 ) ,

ω

identified as the solution to the Euler-Lagrange equation: Z

hF : cof sym ∇αi dz =

ω

Z

hsym ∇φ : sym ∇αi dz

ω

(14.29)

for all α ∈ H01 (ω, R2 ). By Korn’s inequality, the right hand side above is a scalar product on H01 (ω, R2 ). The Riesz representation theorem then yields existence of the unique φ , and: ksym ∇φ kL2 (ω) nZ o = sup hF : cof sym ∇αi dz; α ∈ H01 (ω, R2 ), ksym∇αkL2 (ω) ≤ 1 . ω

  ⊥

∇ (−α2 )  and using Observing that hF : cof sym ∇αi = hF : cof ∇αi = F : ∇⊥ α1 Korn’s inequality again, we get: ksym ∇φ kL2 (ω) n Z D  ∇⊥ α  E o 1 1 2 ' sup F: ), k∇αk ≤ 1 dz; α ∈ H (ω, R 2 0 L (ω) ∇⊥ α2 ω o nZ

 (F11 , F12 ), ∇⊥ α1 ; α1 ∈ H01 (ω, R), k∇α1 kL2 (ω) ≤ 1 ' sup ω o nZ

 (F21 , F22 ), ∇⊥ α2 ; α2 ∈ H01 (ω, R), k∇α2 kL2 (ω) ≤ 1 + sup ω

' kcurl (F11 , F12 )kH −1 (ω) + kcurl (F21 , F22 )kH −1 (ω)
' kcurl FkH −1 (ω) ' distL2 (ω) F, {∇w; w ∈ H 1 (ω, R2 )} . Finally, from (14.29) we deduce that:  ⊥ E Z D ∇ α1 F − cof sym ∇φ : dz = 0 ∇⊥ α2 ω

for all α1 , α2 ∈ H01 (ω, R).

Hence, by de Rham’s theorem there must be: F − cof sym ∇φ = ∇w for some w ∈ H 1 (ω, R2 ). Since F − cof sym ∇φ is symmetric, we get that ∇w = ∇2 v for some v ∈ H 2 (ω, R), as claimed. The dual decomposition is as follows: Lemma 14.1. Let ω ⊂ R2 be open, bounded, simply connected, with Lipschitz 2 boundary. For every F ∈ L2 (ω, R2×2 sym ), there exist the unique r ∈ H0 (ω, R) and 1 2 w ∈ H (ω, R ) (up to linear maps with skew-symmetric gradient), such that: F = cof ∇2 r + sym ∇w.

14.5 Elimination of out-of-plane displacements

395

Moreover, there hold the following equivalences with the scalar field curl curl F:    kF − sym ∇wkL2 (ω) = distL2 (ω) F, sym ∇φ ; φ ∈ H 1 (ω, R2 ) ' kcurlT curl FkH −2 (ω) . Proof. Similarly to the proof of Lemma 14.12, we consider the problem: nZ o minimize |F − cof ∇2 r|2 dz; r ∈ H02 (ω, R) , ω

whose unique solution is given through the orthogonal projection on the closed subspace {cof ∇2 r; r ∈ H02 (ω, R)} of L2 (ω, R2×2 sym ). Equivalently, the solution r satisfies: Z

hF : cof ∇2 αi dz =

ω

Z

h∇2 r : ∇2 αi dz for all α ∈ H02 (ω, R),

ω

and we get: k∇rk2L2 (ω) = sup

nZ ω

o hF : cof ∇2 αi dz; α ∈ H02 (ω, R), k∇2 αkL2 (ω) ≤ 1

= kcurl curl FkH −2 (ω) . The last equality above follows by observing that for all α ∈ Cc∞ (ω) there holds: Z

Z
curl curl F α dz = − hcurl F, ∇⊥ αi dz ω

ω

Z

  (F21 , F22 ), ∇⊥ (∂1 α) dz (F11 , F12 ), ∇ (−∂2 α) dz +

=

Z

=

Zω

⊥

(14.30)

ω

 F : cof ∇2 α dz.

ω

Denoting F¯ = F − cof ∇2 r ∈ L2 (ω, R2×2 sym ), it thus follows that: Z

 F¯ : cof ∇2 α dz = 0

for all α ∈ H02 (ω, R).

ω

As in (14.30), we deduce that: curl curl F¯ = 0 in distributions. Hence: # " v 0 , curl F¯ = ∇v = curl (skew2 v), where: skew2 v  −v 0
for some v ∈ L2 (ω, R). Consequently curl F¯ + skew2 v = 0, and thus further F¯ + skew2 v = ∇w for some w ∈ H 1 (ω, R2 ). Since F¯ is symmetric, this yields: F¯ = sym ∇w. The proof is done.

396

14 Limiting theories for weakly prestressed films

A direct consequence of the decomposition Lemmas 14.12 and 14.1 is the following result in which Γ -limits in Theorem 14.2 are replaced by functionals given solely in terms of the scalar out-of-plane displacement v: Theorem 14.13. In the context of Theorem 14.2, assume additionally that ω is simply connected. Then each stretching term  may be replaced by the following squared distance from the space B(ω) = sym∇w; w ∈ H 1 (ω, R2 ) , where δ = 2 in case (i), δ = γ in case (ii), and δ = α − 2 in case (iii):   Z   1 0 for δ > 2 0 for α > 2 + δ Q2 F + 1 min dz. − ⊗2 S2×2 for α = 2 + δ for δ = 2 2 F∈B(ω) ω 2 (∇v) Consequently, we have the equivalences below, where the congruency symbol a ' b means that a ≤ Cb and b ≤ Ca with a constant C depending only on ω: Γ h (i) If γ = 2, α > 4 then h−4 Eweak −→ I¯, where: I¯(v) ' k∇2 v+Btan k2L2 (ω) + h k det ∇2 vk2H −2 (ω) . If γ = 2, α = 4 then: h−4 Eweak −→ I¯, where Γ

I¯(v) ' k∇2 v + Btan k2L2 (ω) + k det ∇2 v + curl curl Stan k2H −2 (ω) . Γ h (ii) If γ ∈ (2, α − 2] then h−(2+γ) Eweak −→ I¯, with: I¯(v) ' k∇2 v + Btan k2L2 (ω) + kcurl curl Stan k2H −2 (ω) . Alternatively, the same rescaled energies Γ -converge to the constant limit:

I¯ ≡ min I¯ ' kcurl Btan k2H −1 (ω) + kcurl curl Stan k2H −2 (ω) . Γ h (iii) If α = 4, γ > 2 then h−4 Eweak −→ I¯, where: I¯(v) ' k∇2 vk2L2 (ω) + Γ k det ∇2 v + curl curl S k2 . If γ > α − 2 > 2 then h−α E h −→ I¯, tan H −2 (ω)

weak

where: I¯ ≡ min I ' kcurl curl Stan k2H −2 (ω) . All Γ -limit functionals I¯(v) are defined on the scalar out-of-plane displacements v ∈ H 2 (ω, R). We now observe the bound on the infimum of the limiting functional corresponding to γ = 2, α = 4 and cases (i) and (iii) in Theorem 14.13. The bound is consistent with the optimality conditions in Theorem 14.16 in section 14.6. Lemma 14.14. Let ω ⊂ R2 be open, bounded, simply connected. Denote B¯ = Btan , S¯ = Stan , and for v ∈ H 2 (ω, R) define: ¯ 2 −2 . ¯ 2 2 + k det ∇2 v + curl, curl Sk I¯0 (v)  k∇2 v + Bk L (ω) H (ω) Then, with some constants c,C > 0 depending only on ω we have:

14.5 Elimination of out-of-plane displacements

397

  ¯ 2 −1 ¯ 2 ∞ + kcurl Bk ¯ 2 −1 inf I¯0 ≤ C kcurl Bk 1 + k Bk L (ω) H (ω) H (ω) ¯ 2 −2 , +C k det B¯ + curl curl Sk H (ω) c 2 ¯ 2 −2 , ¯ −1 + k det B¯ + curl curl Sk inf I¯0 ≥ c kcurl Bk H (ω) H 1∨a ¯ 2 −2 +kcurl Bk ¯ 2 −1 ). ¯ ¯ 2 −1 (1+kBk ¯ 2L∞ +kcurl Bk ¯ 2L∞ +k det B+curl curl Sk where a = kBk H H H Proof. 1. For v ∈ H 2 (ω, R) we write: ¯ + det B¯ − hcofB¯ : ∇2 v + Bi. ¯ det ∇2 v = det(∇2 v + B) It now easily follows that: ¯ H −2 (ω) ≤ k det B¯ + curlT curl Sk ¯ H −2 (ω) k det ∇2 v + curlT curl Sk ¯ H −2 (ω) + khcofB¯ : ∇2 v + Bik ¯ H −2 (ω) . + k det(∇2 v + B)k Observe further that: ¯ H −2 (ω) ≤ Ck∇2 v + Bk ¯ 22 , k det(∇2 v + B)k L (ω) ¯ H −2 (ω) ≤ CkBk ¯ L∞ (ω) · k∇2 v + Bk ¯ L2 (ω) , khcof B¯ : ∇2 v + Bik

(14.31)

which implies:   2 ¯ 2 ¯ 22 ¯ 2 I¯0 (v) ≤ Ck∇2 v + Bk L (ω) 1 + kBkL∞ (ω) + k∇ v + BkL2 (ω) ¯ 2 −2 . +Ck det B¯ + curlT curl Sk H (ω) The upper bound on inf I¯0 follows now by infimizing the right hand side expression with respect to v, and applying Lemma 14.12. 2. For the lower bound, we use (14.31) to get: ¯ 2 −2 ≥ 1 k det B¯ + curl curl Sk ¯ H −2 (ω) k det ∇2 v + curl curl Sk H (ω) 2
2 ¯ 22 ¯ 2 ¯ 2 − ck∇2 v + Bk L (ω) kBkL∞ (ω) + k∇ v + BkL2 (ω) . The established upper bound yields along a minimizing sequence {vn }n→∞ of I¯0 :
¯ 2 2 ≤ Ckcurl Bk ¯ 2 −1 ¯ 2 ¯ 2 k∇2 vn + Bk L (ω) H (ω) 1 + kBkL∞ (ω) + kcurl BkH −1 (ω) ¯ 2 −2 , +Ck det B¯ + curl curl Sk H (ω) so consequently, for every ε ≤ 1 there holds:

398

14 Limiting theories for weakly prestressed films

inf I¯0 = lim I¯0 (vn ) n→∞

 ¯ 2 ¯ 22 ¯ 2 ¯ ≥ k∇2 vn + Bk L (ω) 1 − ε kBkL∞ (ω) + k det B + curl curl SkH −2 (ω) ¯ 2 −1 ¯ 2 ¯ 2 + kcurl Bk H (ω) 1 + kBkL∞ (ω) + kcurl BkH −1 (ω)





¯ 2 −2 , + cεk det B¯ + curl curl Sk H (ω) The result now follows by invoking Lemma 14.12 and taking 2ε to be the minimum T ¯ 2 −2 + ¯ 2 ∞ +k det B+curl ¯ curl Sk of 1 and the inverse of the expression: a1 = kBk L (ω) H (ω)
¯ 2 −1 ¯ 2 ∞ + kcurl Bk ¯ 2 −1 kcurl Bk 1 + kBk . This completes the proof. H

(ω)

L (ω)

H

(ω)

Similarly, as in Theorem 14.13 we have the following observation for the case where Stan ≡ 0 in ω. Cases (i) and (ii) below enjoy the bound as in Lemma 14.14. Theorem 14.15. In the context of Theorem 14.4, assume additionally that ω is simply connected. Then each stretching term  may be replaced by the following squared distance from the space B(ω) = sym∇w; w ∈ H 1 (ω, R2 ) , where δ = 2 in cases (i), (ii) and δ = α ∧ γ in case (iii):   Z  1 0 for δ > 2 0 for α > 2  Q2 F + 1 min − dz. 1 ⊗2 for δ = 2 ⊗2 for α = 2 2 F∈B(ω) ω 2 (∇v) 2 (S31 , S32 ) Consequently, we have the equivalences of the following Γ -limits I¯(v) defined on the scalar out-of-plane displacements v ∈ H 2 (ω, R): Γ h (i) If α = 2, γ > 2 then h−4 Eweak −→ I¯, where:

1 I¯(v) ' k∇2 v−2 sym∇(S31 , S32 )k2L2 (ω) +k det ∇2 v− curl curl (S31 , S32 )⊗2 k2H −2 . 2 Γ h If α = γ = 2 then h−4 Eweak −→ I¯, where:

I¯(v) ' k∇2 v − 2 sym∇(S31 , S32 ) + Btan k2L2 (ω) 1 + k det ∇2 v − curlT curl(S31 , S32 )⊗2 k2H −2 (ω) . 2 Γ h (ii) If γ = 2 < α then h−4 Eweak −→ I¯, where: I¯(v) ' k∇2 v + Btan k2L2 (ω) +

k det ∇2 vk2H −2 (ω) . Γ h (iii) If α, γ > 2 then h−(2+α∧γ) Eweak −→ I¯, where:  kcurl Btan k2H −1 (ω)    I¯ ≡ min I ' k∇curl(S31 , S32 )k2H −1 (ω)    kcurl B − ∇curl(S , S )k2 tan

31

32

H −1 (ω)

for γ < α for γ > α for γ = α.

14.6 Identification of scaling regimes

399

14.6 Identification of scaling regimes In this section, we derive a set of results concerning the optimality of the energy scalings implied by Theorems 14.2 and 14.4 and their connection to curvature of the prestress {Ah }h→0 . We first note the following consequences of the two decomposition lemmas in section 14.5: Theorem 14.16. In the setting of Theorem 14.2, assume additionally that the midplate ω is simply connected. Then we have: h (i) When γ = 2 then ch4 ≤ inf Eweak ≤ Ch4 with c > 0, if and only if:  0 for α > 4 curl Btan . 0, or det Btan + . 0 in ω. curl curl Stan for α = 4 h (ii) When γ ∈ (2, α − 2] then ch2+γ ≤ inf Eweak ≤ Ch2+γ with c > 0, iff:

curl Btan . 0,

or

γ = α − 2 and curl curl Stan . 0

in ω.

h ≤ Chα with c > 0, if and only if: (iii) When γ > α − 2 then chα ≤ inf Eweak

curl curl Stan . 0

in ω.

To further identify the optimality conditions in all three cases above, consider now the underlying Riemann metrics on Ω h : gh = (Ah )T Ah = Id3 + 2hα/2 S + hα S2
+ t 2hγ/2 B + 2h(α+γ)/2 sym(SB) + t 2 hγ B2 .

(14.32)

Lemma 14.17. The lowest order terms of Riem(gh ) in (14.32) on ω × {0}, are: R1212 ' −hα/2 curl curl Stan − hγ det Btan , (R1213 , R1223 ) ' −hα/2 ∇curl (S13 , S23 ) + hγ/2 curl Btan ,   Ri3 j3 i, j=1,2 ' −hα/2 ∇2 S33 + 2 sym∇(B13 , B23 ),

(14.33)

where ∇ denotes the tangential gradient on ω × {0}. Remark 14.18. (i) In all cases indicated in Theorem 14.16, both the optimality conditions and the energy scaling orders can be read from the first three curvatures in (14.33). These are: R1212 corresponding to stretching, and R1213 , R1223 corresponding to bending. Indeed, when γ = 2, α = 4 we have: R1212 ' −h2 (curl curl Stan + det B2×2 )

and

(R1213 , R1223 ) ' h curl Btan ,

400

14 Limiting theories for weakly prestressed films

which are the two quantities displayed in case (i). When γ = 2, α > 4 the only h difference is that R1212 ' −h2 det B2×2 , and in both cases inf Eweak is of order 4 equal to the square of the stretching order, i.e. h . When γ > 2, α = γ + 2 we have: R1212 ' −hγ/2+1 curl curl Stan

and

(R1213 , R1223 ) ' hγ/2 curl Btan ,

in agreement with case (ii). For γ > 2, α < γ +2 the stretching-related curvature α R1212 has order hγ∧ 2 which is strictly less than the bending-compatible order 1+γ/2 h , so this term becomes irrelevant. The energy order in both cases equals the square of the compatible stretching order, i.e. hγ+2 . Finally, when γ > α − 2 we have: R1212 ' −hα/2 curl curl Stan

and

γ

α

(R1213 , R1223 ) ' h 2 ∧ 2 .

This last order is strictly less than the stretching-compatible hα/2−1 . Thus, the bending tensor is discarded on the basis of not contributing towards the energy, which is in agreement with case (iii). Similarly, the energy order equals the square of the stretching order i.e. hα . (ii) We note that the above observations are precisely the “small-curvature” regime counterparts of the findings in Theorem 11.16 in chapter 11. There, in case of h-independent prestress A = A(z,t), the optimal energy scaling Ch2 was superseded if and only if R1212 = R1213 = R1223 ≡ 0 on ω. The next viable scaling was then Ch4 , corresponding to the three remaining curvatures R1313 , R1323 , R2323 whose vanishing, in turn, implied the energy scaling order Ch6 according to Theorem 12.13.

Proof of Lemma 14.17. 1. We will use notation for the matrix fields {Γ a ∈ C ∞ (Ω¯ 1 , R3×3 )}a=1...3 , gathering the Christoffel symbols of metrics gh as in Definition 13.10. To alleviate notation, we suppress the dependence on h. Our goal is to find the Riemann curvatures {R··,cd }c,d=1...3 at t = 0, where we denote R··,cd = [Rabcd ]a,b=1...3 and R··,cd = [Rabcd ]a,b=1...3 , and recall: We directly compute for all a, b, c = 1, 2:

Γacb = hα/2 ∂a Sbc + ∂c Sab − ∂b Sac + hγ/2t ∂a Bbc + ∂c Bab − ∂b Bac + 2hγ tBb3 Bac + e,

Γac3 = hα/2 ∂a Sc3 + ∂c Sa3 − hγ/2 Bac + hγ/2t ∂a Bc3 + ∂c Ba3
− hγ t (B2 )ac − 2B33 Bac + e,

Γa3b = hα/2 ∂a S3b − ∂b Sa3 + hγ/2 Bab + hγ/2t ∂a Bb3 − ∂b Ba3
− hγ t (B2 )ab − 2Bb3 Ba3 + e,
Γa33 = hα/2 ∂a S33 + hγ/2 ∂a B33 − hα t (B2 )a3 − B33 Ba3 + e,

14.6 Identification of scaling regimes

401

Γ33b = 2hγ/2 Bb3 − hα/2 ∂b S33 − hγ/2t∂b B33 − 2hγ t (B2 )b3 − B33 Bb3 + e,
Γ333 = hγ/2 B33 − hγ t 3(B2 )33 − 2B233 + e,
where e denotes the error terms of order O hα∧(α+γ)/2 + h(α+γ)/2t + hγ t 2 .


2. We thus obtain the skew matrix fields at t = 0, with the lowest order terms:   0 −hγ/2 curlT curl Stan −hα/2 ∂1 curl(S13 , S23 ) + hγ/2 (curlBtan )1 ∂1Γ2 − ∂2Γ1 '  · 0 −hα/2 ∂2 curl(S13 , S23 ) + hγ/2 (curlBtan )2  , · · 0   γ 0 −h det Btan 0 0 0 . Γ1Γ2 − Γ2Γ1 '  · · · 0 Denoting: ∇ = (∂1 , ∂2 ), the lowest order terms of the curvatures at t = 0 are: " # P −hα/2 ∇curl (S13 , S23 ) + hγ/2 curl Btan · R·,12 ' , · 0 "
# 0 −hα/2 curlT curl S2×2 − hγ det B2×2 where P = . · 0 3. For every a = 1, 2 we have, at t = 0 and with the same notation as above: " #
B13 Ba1 B13 Ba2 γ ∂aΓ3 − ∂3Γa 2×2 ' −2h B23 Ba1 B23 Ba2 "
# 0 −hα/2 ∂a curl(S13 , S23 ) + hγ/2 curl Btan a + , · 0



∂aΓ3 − ∂3Γa 13 , ∂aΓ3 − ∂3Γa 23 ' hγ ((B2 )a1 , (B2 )a2 ) − 2Ba3 (B13 , B23 ) ,
− hα/2 ∂a ∇S33 + hγ/2 ∇Ba3 + ∂a (B13 , B23 ) ,



∂aΓ3 − ∂3Γa 31 , ∂aΓ3 − ∂3Γa 32 ' hγ ((B2 )a1 , (B2 )a2 ) − 2B33 (Ba1 , Ba2 ) ,
+ hα/2 ∂a ∇S33 − hγ/2 ∇Ba3 + ∂a (B13 , B23 )

∂aΓ3 − ∂3Γa 33 ' 2hγ B13 Ba1 + B23 Ba2 . We further have: # B B B B 13 a2 13 a1 , ΓaΓ3 − Γ3Γa tan ' 2hγ B23 Ba1 B23 Ba2





ΓaΓ3 − Γ3Γa 13 , ΓaΓ3 − Γ3Γa 23 = ΓaΓ3 − Γ3Γa 31 , ΓaΓ3 − Γ3Γa 32
' hγ B33 (Ba1 , Ba2 ) − Btan (Ba1 , Ba2 )

ΓaΓ3 − Γ3Γa 33 ' −2hγ B13 Ba1 + B23 Ba2 . "




402

14 Limiting theories for weakly prestressed films

We thus obtain the corresponding lowest order terms of curvatures at t = 0:
   0 −hα/2 ∂a curl Stan + hγ/2 curl Btan a v , R··,a3 '  · 0 · 0
where v = −hα/2 ∇∂a S33 + hγ/2 ∇0 Ba3 + ∂a (B13 , B23 ) . Finally, we observe that Rab,cd ' Rabcd at t = 0. We now turn to the optimality of Theorem14.4. The following result is a counterpart of Theorem 14.16, following from the decomposition lemmas in section 14.5: Theorem 14.19. In the setting of Theorem 14.4, assume additionally that ω is simply connected. Then we have: h (i) When α = 2 then ch4 ≤ inf Eweak ≤ Ch4 with c > 0, if and only if:

when γ > 2 :

∇ curl (S31 , S32 ) . 0

or

det ∇(S31 , S32 ) . 0

in ω,

curl Btan − ∇ curl (S31 , S32 ) . 0
1 or det Btan − 2 sym ∇(S31 , S32 ) + curl curl(S31 , S32 )⊗2 . 0. 2

when γ = 2 :

Equivalently, the last condition above may be rewritten as: 

3 det ∇(S31 , S32 ) − ∇ curl(S31 , S32 ), (S31 , S32 )⊥

 − 2 cof Btan : ∇(S31 , S32 ) + det Btan . 0

in ω.

h ≤ Ch4 with c > 0, if and only if: (ii) When γ = 2 < α then ch4 ≤ inf Eweak

curl Btan . 0

or

det Btan . 0

in ω.

h (iii) When α, γ > 2 then ch2+α∧γ ≤ inf Eweak ≤ Ch2+α∧γ with c > 0, iff:

curl Btan . 0 and γ < α, or

or

∇0 curl(S31 , S32 ) . 0 and γ > α,

curl Btan − ∇ curl(S31 , S32 ) . 0 and γ = α.

Above, by ∇ we denote the tangential derivatives on the midplate ω × {0}. Observe that (ii) above coincides with the statement of Theorem 14.16 (i) when S2×2 ≡ 0, while (iii) gives more precise information than Theorem 14.16 (ii) and (iii). As in Lemma 14.17, we now compute: Lemma 14.20. Assume that Stan ≡ 0 on ω. Then, the lowest order terms in the Riemann curvatures of the metric gh in (14.32), at t = 0, are given by:

14.6 Identification of scaling regimes



R1212 ' hα S13 ∂2 curl(S31 , S32 ) − S32 ∂1 curl(S31 , S32 ) − 3 det ∇(S31 , S32 )

 − hγ det Btan + 2h(α+γ)/2 Btan : cof ∇(S31 , S32 ) ,

403



(14.34)

(R1213 , R1323 ) ' −hα/2 ∇curl(S31 , S32 ) + hγ/2 curl Btan ,   Ri3 j3 i, j=1,2 ' −hα/2 ∇2 S33 + 2 hγ/2 sym∇(B13 , B23 ). Remark 14.21. The optimality conditions and the energy scalings in all cases of Theorem 14.19, can be read from the first three curvatures in (14.34). When α = 2, γ > 2, then:


R1212 ' h2 − 3 det ∇(S31 , S32 ) + ∇ curl(S31 , S32 ), (S31 , S32 )⊥ , (R1213 , R1223 ) ' −h∇curl(S31 , S32 ) and the fact that at least one of these expressions is nonzero is equivalent to the condition displayed in case (i). When α = γ = 2, then we get the full expressions:


R1212 ' h2 − 3 det ∇(S31 , S32 ) + ∇ curl(S31 , S32 ), (S31 , S32 )⊥


− det Btan + 2 Btan : cof ∇(S31 , S32 ) ,
(R12,13 , R12,23 ) ' h curl Btan − ∇curl(S31 , S32 ) h is of order equal to the which are again consistent with (i). In both cases inf Eweak 4 square of the stretching order, i.e. Ch . For γ = 2, α > 2 we get: R1212 ' −h2 det Btan and (R1213 , R1223 ) ' h curl Btan , in agreement with case (ii). Finally, when α, γ > 2 then (R1213 , R1223 ) are of the order h(α∧γ)/2 , with the corresponding coefficients indicated in (iii). The compatible stretching order in that case is Ch1+(α∧γ)/2 which is strictly larger than the order Chα∧γ∧(α+γ)/2 of R1212 , as computed in (14.34). Thus, stretching-related curvature does not contribute towards h equals twice the bending order the energy, and the exponent of the order of inf Eweak 2+α∧γ plus 1, which yields the order Ch .

Proof of Lemma 14.20. 1. Similarly as in the proof of Lemma 14.17, we compute for all a, b, c = 1, 2: 1

 1 3 S3c (∂a S3b − ∂b S3a ) + S3a (∂c S3b − ∂b S3c ) − S3b (∂a S3c + ∂c S3a ) 2 2 2
(α+γ)/2 γ/2 + 2h Sb3 Bac + h t ∂a Bbc + ∂c Bab − ∂b Bac + 2hγ tBb3 Bac  + h(α+γ)/2t ∂a sym(SB)bc + ∂c sym(SB)ab − ∂b sym(SB)ac  − 2Sb3 (∂a B3c + ∂c Ba3 ) − 2Bb3 (∂a S3c + ∂c Sa3 ) + e,

Γacb = hα

404

14 Limiting theories for weakly prestressed films

 3 S3c ∂a S33 + S3a ∂c S33 − S33 (∂a S3c + ∂c S3a ) 2 2
γ/2 (α+γ)/2 − sym(SB)ac + B33 Bac − h Bac + h

γ/2 + h t ∂ + aB3c + ∂c B3a + hγ t − (B2 )ac + B33 Bac  + h(α+γ)/2t ∂a sym(SB)3c + ∂c sym(SB)3a − 2(S∂a B)3c − 2(A∂c B)3a  + 2h(S31 , S32 ), ∇Bac i − 2B33 (∂a S3c + ∂c S3a ) + e,
Γa3b = hα/2 ∂a S3b − ∂b Sa3 + hγ/2 Bab + h(α+γ)/2 sym(SB)ab 1  3 1 + hα S33 (∂a Sb3 − ∂b Sa3 ) − Sb3 ∂a S33 − Sa3 ∂b S33 2 2 2

γ/2 γ 2 + h t ∂a Bb3 − ∂b Ba3 − h t (B )ab − 2Bb3 Ba3  + h(α+γ)/2t ∂a sym(SB)b3 − ∂b sym(SB)a3  − 2(S∂a B)b3 − 2Bb3 ∂a S33 + 2h(Bb1 , Bb2 ), ∇Sa3 i + e,  1 3 Γa33 = hα/2 ∂a S33 + hα ((∂a S)S)33 − (S∂a S)33 + 2h(S31 , S32 ), ∂ Sa3 i 2 2
+ 2h(α+γ)/2 − (SB)3a + S33 B3a
+ hγ/2t∂a B33 + 2hγ t − (B2 )3a + B33 B3a  + h(α+γ)/2t ∂a sym(SB)33 − 2(S∂a B)33 − 2(B∂a S)33  + 2h(S31 , S32 ), ∇Ba3 i + h(B31 , B32 ), ∇Sa3 i + e,
Γac3 = hα/2 ∂a Sc3 + ∂c Sa3 + hα

1


1 Γ33b = 2hγ/2 Bb3 − hα/2 ∂b S33 − hα ∂b (S2 )33 + 2h(α+γ)/2 sym(SB)b3 − Sb3 B33 2
− hγ/2t∂b B33 − 2hγ t (B2 )b3 − B33 Bb3   + h(α+γ)/2t − ∂b sym(SB)33 + h(B31 , B32 ), ∇S33 i + e, Γ333 = hγ/2 B33 + 2hα h(S31 , S32 ), ∇S33 i  
+ h(α+γ)/2 sym(SB)33 − 4(SB)33 + 2S33 B33 + hγ t − 3(B2 )33 − 2B233   + 2h(α+γ)/2t h(S31 , S32 ), ∇B33 i + h(B31 , B32 ), ∇S33 i + e,
where e denotes the error terms of order: O hα+(α∧γ)/2 + h(α+γ)/2+(α∧γ)/2t + hγ t 2 . 2. We consequently obtain the following expressions for the lowest order terms of tangental curvatures at t = 0, where we denote s = (S31 , S32 ):




 R1212 ' hα − 3 det ∇s + ∇ curl s, s⊥ − hγ det Btan + 2h(α+γ)/2 Btan : cof ∇s , (R1312 , R2312 ) ' −hα/2 ∇curl s + hγ/2 curl Btan . It is also instructive to directly check that:

14.7 Linearised Kirchhoff-like theory for weakly prestressed films

405




1 4 det sym ∇s + curl curl(S31 , S32 )⊗2 = 3 det ∇s − ∇ curl s, s⊥ , 2 which justifies the equivalence of the two conditions in Theorem 14.19 (i) when α = γ = 2. 3. The lowest order terms of the remaining curvatures are contained in the following skew-symmetric matrix field at t = 0, where a = 1, 2:   
 0 −hα/2 ∂a curl(S31 , S32 ) + hγ/2 curl Btan a v   · 0 R··,a3 '  , · 0
where: v = −hα/2 ∇∂a S33 + hγ/2 ∇Ba3 + ∂a (B13 , B23 ) . The proof is complete.

14.7 Linearised Kirchhoff-like theory for weakly prestressed films In this section, we consider the case of weak prestress as in (14.1), assuming that α ∈ (0, 4) and γ = α2 . As we shall see, this case and its conclusions are parallel to the dimension reduction analysis for shallow shells in section 7.7. Define: Ah (z,t) = Id3 + hα/2 S(z) + hα/4tB(z),

(14.35)

where S, B ∈ C ∞ (ω, R3×3 sym ). Following the analysis in section 14.3, it is not hard to observe that the smallness of the energy as in Theorem 14.23 below, relative to the scaling in (14.35), induces the thickness-averaged deformations to have the form: ω 3 z 7→ z + hα/4 v(z)e3 + higher order terms,

(14.36)

given in terms of the scalar out-of-plane displacement v ∈ H 2 (ω, R). Additionally, v must satisfy the Monge-Amp´er´e-like constraint in the spirit of (7.50) (or the weak form in (7.13)), with the right hand side dictated by Stan in (14.38). More precisely: Remark 14.22. The meaning of the constraint (14.38) is that the Gaussian curvatures κ of the image surface obtained by displacing the midplate ω by the out-ofplane displacement hα/4 ve3 as in (14.36), and of the metric Id2 + 2hα/2 Stan on ω, must coincide at their highest order in the expansion in h. Indeed, there holds:
κ Id2 + 2ε 2 Stan = −ε 2 curl curl Stan + O(ε 4 ) (14.37)
κ ∇(id 2 + εve3 )T ∇(id 2 + εve3 ) = ε 2 det ∇2 v + O(ε 4 ). The limiting functional IB in (14.39) measures the L2 difference between the full second fundamental forms, at their highest orders in the expansion in h. These are:

406

14 Limiting theories for weakly prestressed films

hα/4 Btan deduced from the metric gh = (Ah )2 and −∇2 v deduced from the aforementioned surface. See Remark 11.8 (ii) for a similar heuristics in the context of Kirchhoff’s energy of prestressed films.

Theorem 14.23. Let ω ⊂ R2 be open, bounded, connected and Lipschitz, and let {Ah }h→0 be given as in (14.35), with the exponent α in the range: 0 < α < 4. Assume that a family of deformations {uh ∈ H 1 (Ω h , R3 }h→0 satisfies: h Eweak (uh ) ≤ Chα/2+2

for all h  1.

Then, there exist {R¯ h ∈ SO(3), ch ∈ R3 }h→0 such that for the normalized deformations: yh (z,t)  (R¯ h )T uh (z, ht)−ch ∈ H 1 (Ω 1 , R3 ) the following holds, up to a subsequence that we do not relabel: (i) yh → π in H 1 (Ω 1 , R3 ). > 1/2 1 yh (·,t)−id2 dt → (0, 0, v) in H 1 (ω, R3 ). The only non(ii) V h [yh ] = hα/4 −1/2 zero out-of-plane scalar component v satisfies: v ∈ H 2 (ω, R) and: det ∇2 v = −curl curl Stan

in ω.

(14.38)

(iii) Moreover: lim inf h→0

1

1 E h (uh ) ≥ IB (v)  24 hα/2+2 weak

Z


Q2 ∇2 v + Btan dz (14.39)

Ω

Proof. We use the notation and proofs of section 14.3. By the approximation Lemma 14.6, we obtain the family {Rh ∈ H 1 (ω, SO(3))}h→0 with the properties: 1 h

Z Ωh

|∇uh − Rh Ah |2 dx ≤ Ch2+α/2 ,

Z

|∇Rh |2 dz ≤ Chα/2 .

(14.40)

ω

Convergences in (i) and (ii) follow directly from Lemma 14.7. For the constraint (14.38), we recall (14.19) where δ = α/2. Further scaling by hδ /2 = hα/4 yields: ? h/2 
 h T h h h ¯ sym∇V = sym ( R ) ∇u − R A dt hα/4 hα/2 −h/2

1 + α/2 sym (R¯ h )T Rh − Id3 tan + sym (R¯ h )T Rh S tan . h 1

h

1

The first term in the right hand side above converges to 0 in L2 (ω, R2×2 ), because of the first bound in (14.40) and the assumption α < 4. The third term converges to Stan because (R¯ h )T Rh → Id3 . The second term satisfies:

14.7 Linearised Kirchhoff-like theory for weakly prestressed films

1

sym (R¯ h )T Rh − Id3 hα/2


tan

→−

407

1 T
1 P P tan = − ∇v ⊗ ∇v, 2 2

where the L2 (ω, R2×2 ) convergence follows by (14.18), (14.20), in view of: 1

h

¯h T

h

sym (R ) R − Id3 α/2




1 (R¯ h )T Rh − Id3 =− · 2 hα/4


T ·

(R¯ h )T Rh − Id3 . hα/4

Consequently, there follows the convergence: 1 hα/4

sym∇V h → sym∇w

in L2 (ω, R2×2 )

for some in-plane-displacement w ∈ H 1 (ω, R2 ), satisfying the identity: 1 sym∇w = − ∇v ⊗ ∇v + Stan . 2 Thus, condition (14.38) follows in virtue of Lemma 6.19. Finally, to show the lower bound in (iii), we use Corollary 14.8 and Lemma 14.7 (iii). We now turn to the optimality of the energy bound in Theorem 14.23. As in Theorem 7.14, the general upper bound can be deduced in the restricted range of α: Theorem 14.24. Let ω ⊂ R2 be open, bounded, simply connected and Lipschitz, and let {Ah }h→0 be as in (14.35), with the exponent α in the range: 2 < α < 4. Then, for every v ∈ H 2 (ω, R) satisfying (14.38), there exists deformations {uh ∈ H 1 (Ω h , R3 )}h→0 such that for {yh (z,t) = uh (z, ht)}h→0 there holds: (i) yh → π in H 1 (Ω 1 , R3 ), (ii) {V h [yh ]}h→0 defined in Theorem 14.23 (ii), converge to ve3 in H 1 (ω, R3 ). 1 h (uh ) → I (v), where I is given in (14.39), (iii) hα/2+2 Eweak B B Proof. The proof mimics arguments in the proof of Theorem 7.14, so we only indicate the necessary changes. Naturally, we put: 1 α sym∇w = − ∇v ⊗ ∇v + Stan , β = + 2, 2 2 1 h h h h s = sym∇w − Stan + ∇v ⊗ ∇v , 2 where {vh , wh }h→0 are the truncated families of displacements on ω. Then, the recovery family is given by the formula:

408

14 Limiting theories for weakly prestressed films

uh (·,t) = id2 +



 1 hβ −2 wh −hβ /2−1 ∇vh + hβ −2td 0,h + hβ /2−1t 2 d 1,h , + t β /2−1 h 1 h v 2 



where the auxiliary fields {d 0,h , d 1,h ∈ W 1,∞ (ω, R3 )}h→0 are as follows:
T 1 d 0,h = 2S13 , 2S23 , S33 − |∇vh |2 , h1/4 kd 1,h kW 1,∞ (ω) → 0 as h → 0, 2 d 1,h → d 1  2Be3 − B33 e3 − c(Btan + ∇2 v) in L2 (ω, R3 ). Writing (Ah )−1 = Id3 − hβ −2 S − hβ /2−1tB + O(hβ ), we arrive at:
(∇uh )(Ah )−1 = Id3 + hβ −2 (∇wh )∗ − S + d 0,h ⊗ e3 # "
0 −(∇vh )T β /2−1 + thβ /2−1 − B − (∇2 vh )∗ + d 1,h ⊗ e3 +h h ∇v 0 + O(h3β /2−3 )(1 + |∇vh |) + O(h3β /2−2 )(|∇2 vh | + |d 1,h |) + O(h2β −4 )(|∇wh | + |d 0,h |) + O(hβ −1 )|∇d 0,h | + O(hβ /2+1 )|∇d 1,h |. Further, there holds: (Ah )−1,T (∇uh )T (Ah )−1 (∇uh )(Ah )−1 = Id3 + F h + Error,
where: F h = hβ −2 (2sh )∗ + 2thβ /2−1 − B − (∇2 vh )∗ + sym(d 1,h ⊗ e3 ) , and where the quantity Error is exactly as in (7.57). Constructing a recovery family for the full range including α ∈ (0, 2), requires the matching property for isometries on weakly prestressed films, which, similarly as in Theorem 7.16, will be shown under an additional convexity assumption. This result, stated and proved in the next section, will be the key building block for: Theorem 14.25. In the context of Theorem 14.23, assume additionally that ω is star-shaped with respect to a ball, and that −curl curl Stan ≡ c0 > 0 in ω. Then, for every α ∈ (0, 4) and for every v ∈ H 2 (ω, R) satisfying det ∇2 v = c0 in ω, there exist deformations {uh ∈ H 1 (Ω h , R3 )}h→0 such that (i), (ii) and (iii) of Theorem 14.24 hold. The proof of Theorem 14.25 will be given in the next section.

14.8 Matching isometries on weakly prestressed films We now follow the approach outlined in section 7.8 in the context of shallow shells. We first prove a version of Theorem 7.16 about matching property for infinitesimal

14.8 Matching isometries on weakly prestressed films

409

isometries on convex weakly prestressed films, and then deduce Theorem 14.25. Note that in the statement below, S is only a R2×2 -valued matrix field on ω, playing the role of the tangential minor of the stretching component of Ah in (14.35). Theorem 14.26. Let ω ⊂ R2 be open, bounded, Lipschitz and simply con¯ R2×2 nected. Let S ∈ C ∞ (ω, sym ) be such that −curl curl S ≥ c0 > 0. For a fixed 2,β ¯ R) satisfy: β ∈ (0, 1), let v ∈ C (ω, det ∇2 v = −curl curl S

in ω,

¯ R2×2 and let {sε ∈ C ∞ (ω, sym }ε→0 be a given family of smooth symmetric matrix fields, equibounded in C 1,β (ω). Then there exists an equibounded family ¯ R3 )}ε→0 , such that: {wε ∈ C 2,β (ω, ∇(id 2 + εve3 +ε 2 wε )T ∇(id 2 + εve3 + ε 2 wε ) = Id2 + 2ε 2 S + ε 3 sε

for all ε  1.

(14.41)

Proof. 1. The proof mimics the arguments in the proof of Theorem 7.16, so we only indicate the necessary changes. We decompose each unknown vector field wε into its tangential and normal components: wε = wε,tan +

ζε e3 , ε

¯ R2 ) and ζε = εhwε , e3 i ∈ C 2,β (ω, ¯ R). Then, the equation where wε,tan ∈ C 2,β (ω, (14.41) is equivalent to: ∇(id 2 + ε 2 wε,tan )T ∇(id 2 + ε 2 wε,tan ) = gε (ζε ), where: gε (ζ ) = Pε − ε 2 (∇v + ∇ζ ) ⊗ (∇v + ∇ζ ) + ε 3 sε ,

(14.42)

2

Pε = [Pi j ]i, j=1,2 = Id2 + 2ε S Observe that the Christoffel symbols, the determinant and the inverse of Pε , satisfy: k 2 Γε,i j = 1 + O(ε ),

det Pε = 1 + O(ε 2 ),

(Pε )−1 = [Pεi j ]i, j=1,2 = Id2 + O(ε 2 ).

(14.43)

As in Lemma 7.17 and in virtue of [Han and Hong, 2006, Lemma 2.1.2], we find the formula for the Gaussian curvature of gε (ζε ). Denoting v1 = v + ζε , we have:
κ(gε (ζε )) = κ Pε − ε 2 ∇v1 ⊗ ∇v1 =

ε 2 det(∇2 v1 − [Γi kj ∂k v1 ]i j ) − .
2
4 1 − ε 2 (Pεi j ∂i v1 ∂ j v1 ) 1 − ε 2 (Pεi j ∂i v1 ∂ j v1 ) det Pε κ(Pε )

(14.44)

410

14 Limiting theories for weakly prestressed films

It follows that κ(gε (ζε )) = 0 if and only if Φ(ε, ζε ) = 0, where:
1 κ(Pε ) ε2
k − det ∇2 v + ∇2 ζ − [Γε,i j ∂k (v + ζ )]i, j=1,2 .

Φ(ε, ζ ) = 1 − ε 2 Pεi j ∂i (v + ζ )∂ j (v + ζ )


2

det Pε

2,β

¯ R) → C 0,β (ω, ¯ R) Hence, we consider the above mapping Φ : (−ε0 , ε0 ) × C0 (ω, 2,β ¯ R) satisfying Φ(ε, ζε ) = 0. By using (14.37) to approxiand look for ζε ∈ C0 (ω, mate κ(Pε ) and recalling (14.43), we get:
2
Φ(ε, ζ ) = − 1 + O(ε 2 )|∇v + ∇ζ |2 (1 + O(ε 2 )) curl curl Stan + O(ε 2 )
− det ∇2 v + ∇2 ζ + O(ε 2 )|∇v + ∇ζ | . It follows that: Φ(0, 0) = −curl curl Stan − det ∇2 v = 0, and that the partial derivative 2,β ¯ R) → C 0,β (ω, ¯ R) is a linear continuous operator in: L = ∂ Φ/∂ z(0, 0) : C0 (ω, 1 L (z) = lim Φ(0,tζ ) = −hcof∇2 v : ∇2 ζ i. t→0 t Clearly, L above is invertible to a continuous linear operator, because of the uniform ellipticity of ∇2 v, implied by det ∇2 v being strictly positive. By the implicit function theorem there exists hence the solution operator: Z : (−ε0 , ε0 ) → 2,β ¯ R) such that ζε = Z (ε) satisfies Φ(ε, ζε ) = 0. Moreover: Z 0 (0) = 0. We C0 (ω, also obtain: khwε , e3 ikC 2,β (ω) = ε1 kζε kC 2,β (ω) → 0 as ε → 0. 2. Thanks to Mardare [2004], for each small ε there exists exactly one (up to ¯ R2 ) of gε (ζε ): rotations) orientation preserving isometric immersion φε ∈ C 2 (ω, ∇φεT ∇φε = gε (zε ) and

det ∇φε > 0.

(14.45)

We now sketch the argument to the effect that: φε = id 2 + ε 2 wε,tan with {wε,tan }ε→0 k ∂ φ ] ¯ R2 ). Firstly, (14.45) is equivalent to: ∇2 φε −[Γ˜ε,i equibounded in C 2,β (ω, j k ε i, j=1,2 = k are the Christoffel symbols of the metric g (ζ ) in (14.42). By 0, where Γ˜ε,i ε ε j k , it follows that: kφ k (14.45) and the boundedness of Γ˜ε,i 2,β ε ¯ 2 ) ≤ C. Further, j C (ω,R k 2 ˜ kΓε,i j kC 0,β (ω) = O(ε ) which implies that: k∇φε − Bε kC 1,β (ω) ≤ Cε 2

for some Bε ∈ R2×2 .

It then follows that dist(Bε , SO(3)) ≤ Cε 2 , so without loss of generality: k∇φε − Id3 kC 1,β (ω) ≤ Cε 2 and therefore: kφε − id2 kC 2,β (ω) ≤ Cε 2 . The proof is done. We now sketch the proof of the Γ -limsup counterpart of the result in Theorem 14.23, in the full leading exponent scaling range α ∈ (0, 4), but valid only for the weak prestress with constant and positive leading order curvature. The complete calculations are similar to the proof of Theorem 7.15 in section 7.10.

14.8 Matching isometries on weakly prestressed films

411

Proof of Theorem 14.25. In view of the the density result in Theorem 7.30, it is enough to assume that ¯ R) and that it satisfies det ∇2 v = c0 . In the general case of v ∈ H 2 (ω, R) v ∈ C 2,β (ω, under the same constraint, the result follows by a diagonal argument. By Theorem 14.26 used with ε = hα/4 and sε = ε(S2 )tan , there exists an equi¯ R3 )}h→0 such that the deformations uh  id2 + bounded family {wh ∈ C 2,β (ω, α/4 α/2 h ve3 + h wh are isometrically equivalent to the metric in: (∇uh )T ∇uh = Id2 + 2hα/2 Stan + hα (S2 )tan

for all h  1.

Define the recovery family {uh ∈ C 1,β (Ω h , R3 )}h→0 by the formula: uh (·,t) = uh + tbh +


t 2 α/4 h d − (2B13 , 2B23 , B33 )T , h 2

(14.46)

where the Cosserat-like vector fields {bh : ω → R3 }h→0 are given by: 

∂1 uh ∂2 uh bh

T 

 ∂1 uh ∂2 uh bh = (Ah )2 (·, 0)

in ω,

¯ R3 ) approximate the effective warping d ∈ C 0,β (ω, ¯ R3 ) in and d h ∈ C 1,β (ω, hα/4 kd h kC 1,β (ω) ≤ C

lim kd h − dkL∞ (ω) = 0,
d = −c(∇2 v + Btan ) + 2B13 , 2B23 , B33 . and

h→0

The remaining calculations are as in the proofs of Theorems 7.15 and 7.14.

Remark 14.27. We may follow the heuristics in section 8.1 and conjecture the following infinite hierarchy of the dimensionally reduced models of weakly prestressed 4 , with α1 = ∞, α2 = 4 and α∞ = 0. We say that the n-tuple of films. Define αi = i−1 displacements (V1 , . . . ,Vn ) : ω → (R3 )n is an n-th order infinitesimal isometry of the prestressed film, when the metrics induced by {uh  id 2 + ∑ni=1 hkα/4Vi }h→0 differ from the prescribed metrics (Ah )2 by terms of order at most O(h(n+1)α/4 ). In this framework, several regimes can be distinguished: (i) When αn+1 < α < αn , we expect the limiting energy to be a linearised bending model with the nth-order isometry constraint. (ii) At the critical values α = αn , the isometry constraint of the limit model should be of order n, but in addition to the bending energy term, the limiting energy will also contain the n + 1-th order stretching term. (iii) Whenever the structure of the stretching-correlated prestress term S allows for it, any n-th order isometry can be matched to a higher order isometry of some order m > n. In that case, the theories in the range αm+1 < α < αn are expected to collapse to the same theory (with the n-th order isometry constraint).

412

14 Limiting theories for weakly prestressed films

The results in sections 14.7 and 14.8 can be now interpreted as follows. In Theorem 14.24 we derived the correct model under the second order isometry constraint (14.38), corresponding to the values of α between α3 = 2 and α2 = 4. The Monge-Amp´er´e constraint (14.38) is naturally valid for the full range 0 < α < α2 as shown in Theorem 14.23) but this information is not enough for characterizing the limiting model when α ≤ α3 . Theorem 14.26 and the corresponding density result provides the tools to let all the expected higher order constraints for the full range 0 = α∞ < α ≤ α2 = 1 be derived from the second order constraint (14.38). This leads to Theorem 14.25. For other instances where such matching properties have been proved and exploited to a similar purpose see chapter 8.

14.9 Uniqueness of minimizers to linearised Kirchhoff-like energy for prestressed films In this section, we discuss the multiplicity of minimizers to the limiting problem (14.39), (14.38). Given an open, bounded, simply connected and Lipschitz domain ω ⊂ R2 , and a function f ∈ L1 (ω, R), we consider the variational problem: minimize: I (v) =

Z

|∇2 v|2 dz,

(14.47)

ω

subject to the constraint: v ∈ A f  {v ∈ H 2 (Ω ); det ∇2 v = f }. Here, we assumed that Btan = 0 and that Q2 (F) = |sym, F|2 for every F ∈ R2×2 , which is consistent with Definition 5.6 and its particular case in (14.8), when W (F) = 12 dist2 (F, SO(3)) for F ∈ R3×3 close to SO(3). This scenario corresponds also to the isotropic elastic energy density with the Lam´e coefficients λ = 0, µ = 1; see Examples 5.3 and 5.8. The function f replaces the prestress-related curvature −curl curl Stan . We observe given f ∈ L p (ω, R), we may extend it to f ∈ L p (ω1 , R) on an open smooth ω1 ⊃ ω and solve the Poisson’s equation: −∆ λ = f in ω1 ,

λ = 0 on ∂ ω1 ,

to the effect that −curl curl (λ Id2 ) = f . We start by noting that the minimization problem (14.47) may indeed have multiple or unique solutions, depending on the choice of f . Example 14.28. (i) Let ω = B(0, 1) ⊂ R2 and f ≡ −1. Then (14.47) has a nontrivial one-parameter family of absolute minimizers: vθ (z1 , z2 ) = (cos θ )

z21 − z22 + (sin θ )(z1 z2 ), 2

for θ ∈ R.

14.9 Uniqueness of minimizers to linearised Kirchhoff-like energy for prestressed films

413

This is because for any v ∈ A−1 , the quantity |∇2 v|2 = (tr ∇2 v)2 − 2 det ∇2 v = (tr ∇2 v)2 + 2 is minimized when tr ∇2v = ∆ v = 0. This condition holds for each  cos θ sin θ vθ , because: ∇2 vθ = . sin θ − cos θ (ii) For ω = B(0, 1) and f ≡ 1, the problem (14.47) has a unique minimizer: v(z1 , z2 ) =

z21 + z22 . 2

This is because for v ∈ A1 we have: |∇2 v|2 = (tr ∇2 v)2 − 2 = (λ1 + λ2 )2 − 2, where λ1 , λ2 are the eigenvalues of ∇2 v. This quantity achieves its minimum, under the constraint λ1 λ2 = 1, precisely when λ1 = λ2 = 1. Example 14.29. (i) An argument as in Example 14.28 (i), allows for constructing ¯ R) satisfies: a one-parameter family of minimizers to (14.47) when f ∈ C ∞ (ω, f ≤ c0 < 0

and ∆ (log | f |) = 0 in ω. (14.48) p To this end, observe that λ  | f | is positive, smooth and satisfies ∆ (log λ ) = ¯ Hence there exists φ ∈ C ∞ (ω) ¯ such that the function (log λ + iφ ) is 0 in ω. holomorphic in ω ⊂ C. Trivially, for every constant θ ∈ R, the function z 7→ (log λ + i(φ + θ )) is holomorphic, as is its exponential: ω 3 z 7→ exp(log λ + i(φ + θ )) = λ cos(φ + θ ) + iλ sin(φ + θ ). The Cauchy-Riemann equations of the above mapping are equivalent to the vanishing of the curl of the symmetric matrix field in the left hand side of:   λ cos(φ + θ ) −λ sin(φ + θ ) = ∇2 vθ . (14.49) −λ sin(φ + θ ) −λ cos(φ + θ ) ¯ R) as in Since ω is simply connected, for each θ ∈ R there exists vθ ∈ C ∞ (ω, (14.49). We see that: ∆ vθ = 0

and

det ∇2 vθ = −λ 2 = −| f | = f ,

(14.50)

which proves the claim. (ii) For completeness, we prove that (14.48) √ is in fact equivalent to the existence of v satisfying (14.50). Denote λ = f and let r1 , r2 : Ω → R3 be the (unitlength) eigenvectors fields of ∇2 v corresponding to the eigenvalues  λ and −λ . cos φ − sin φ Since hr1 , r2 i = 0, we may write: [r1 , r2 ] = Rφ = ∈ SO(2), for sin φ cos φ some φ ∈ C ∞ (ω → (0, 2π). The fact that the range of φ may be taken in (0, 2π) follows from the simply-connectedness of ω. We then note that:   λ cos(−2φ ) −λ sin(−2φ ) ∇2 v = Rφ diag{λ , −λ } RTφ = . −λ sin(−2φ ) −λ cos(−2φ )

414

14 Limiting theories for weakly prestressed films

Since curl of the matrix field in the right hand side above vanishes in ω, we reason as in (14.49) to conclude that the (nonzero) function λ exp(−2iφ ) satisfies the Cauchy-Riemann equations, and hence it is holomorphic in ω ⊂ C. Further, its logarithm: (log λ − 2iφ ) is well defined and holomorphic as well. Consequently: ∆ (log λ ) = 0, which yields (14.48) for v. In what follows, we derive conditions for uniqueness of minimizers to (14.47). In this context, it is useful to consider the relaxed constraint: A f∗  {v ∈ H 2 (ω); det ∇2 v ≥ f }.

(14.51)

We will denote by I f and I f∗ the restrictions of I in (14.47) to A f and A f∗ , respectively. The next two results assert existence and uniqueness of minimizers to the relaxed problem. We remark that there clearly holds: inf I f∗ ≤ inf I f . Lemma 14.30. Let ω ⊂ R2 be open, bounded, connected and Lipschitz. Assume / Then I f (respectively I f∗ ) admits a minimizer. that A f , 0/ (respectively A f∗ , 0). Moreover, there must be f ∈ L1 log L1 (ω), namely: Z

| f log(2 + f )| dz < ∞,

ω1

for every subset ω1 compactly contained in ω. Proof. Take a minimizing sequence {vn ∈ A f }n→∞ , there holds: k∇2 vn kL2 (ω) ≤ C. > > Modifying vn by v and ( ∇v)z, in view of Poincar´e’s inequality it follows that: kvn kH 2 (ω) ≤ C. Therefore, up to a subsequence that we do not relabel, vn * v weakly in H 2 (ω, R), implying that I (v) ≤ lim inf I (vn ). It follows that v is a minimizer of I f (respectively I f∗ ) if only v satisfies the appropriate constraint. Since ∇vn * ∇v weakly in H 1 (ω), then the same convergence is also valid strongly in any L p (ω) for p ∈ [1, ∞). Hence ∇vn ⊗ ∇vn → ∇v ⊗ ∇v in L2 (ω). Applying curl curl, this yields the following convergence, in the sense of distributions: 1 1 det ∇2 vn = − curl curl (∇vn ⊗ ∇vn ) → − curl curl (∇v ⊗ ∇v) = det ∇2 v, 2 2 and we see that if vn ∈ A f then v ∈ A f as well (likewise, if vn ∈ A f∗ then v ∈ A f∗ ). The final assertion follows from the result in M¨uller [1990]: If u ∈ H 1 (ω, RN ) on ω ⊂ RN satisfies det ∇u ≥ 0 then det ∇u ∈ L1 log L1 (ω). Lemma 14.31. In the context of Lemma 14.30, assume that f ≥ c > 0 in ω. Let v1 , v2 ∈ A f∗ be two minimizers of I f∗ . Then ∇2 v1 = ∇2 v2 , i.e. v1 − v2 is an affine function and det ∇2 v1 ∈ L1 log L1 (ω).

14.9 Uniqueness of minimizers to linearised Kirchhoff-like energy for prestressed films

415

Proof. By Theorem 7.18, without loss of generality and possibly replacing vi by −vi , we may assume that ∇2 v1 and ∇2 v2 are strictly positive definite a.e. in ω. For each λ ∈ [0, 1], consider vλ  λ v1 + (1 − λ )v2 . We claim that vλ ∈ A f∗ . This follows by the Brunn-Minkowski inequality: (det ∇2 vλ )1/2 ≥ λ (det ∇2 v1 )1/2 + (1 − λ )(det ∇2 v2 )1/2 p p p f. ≥ λ f + (1 − λ ) f = Since I (vλ ) ≤ λ I (v1 ) + (1 − λ )I (v2 ) = min I f∗ , this inequality is in fact an equality. By the strict convexity of the L2 norm, we get ∇2 v1 = ∇2 v2 , as claimed. Remark 14.32. Consider the related functional: I∆ (v) =

Z

|∆ v|2 dz,

ω

constrained to A f or A f∗ in (14.47) and (14.51), which we then respectively denote I∆ , f and I∆∗, f . Since |∇2 v|2 = |∆ v|2 −2 det ∇2 v, any minimizing sequence {vn }n→∞ of I∆ , f or I∆∗, f , satisfies k∇2 vn kL2 (ω) ≤ C. Arguing as in the proof of Lemma 14.30 we obtain existence of minimizers to both problems. On the other hand, there is no uniqueness as in Lemma 14.31, in the sense that the two minimizers of I∆∗, f may differ by a non-affine harmonic function. We now observe that if min I f = min I f∗ , then min I∆ , f = min I∆∗, f . Indeed, let v0 ∈ A f be the common minimizer of I f and I f∗ . Then: I∆ , f (v) = I (v) + 2

Z

det ∇2 v dz ≥ I (v0 ) + 2

ω

Z

f dz = I∆ , f (v0 ) for all v ∈ A f∗ ,

ω

hence v0 is also the common minimizer of I∆ , f and I∆∗, f . To prove uniqueness of minimizers to the problem (14.47) rather than to its relaxation, we restrict to the radial case on a ball. The following is our main result: Theorem 14.33. Let f ∈ L2 (B(0, 1), R), given on the ball B(0, 1) ⊂ R2 be radiR ally symmetric i.e.: f = f (r), and such that 01 r f 2 (r) dr < ∞. Assume moreover that f ≥ c > 0, and that f is a.e. non-increasing, i.e.: f (r) ≤ f (s)

for a.e. r ∈ [0, 1] and a.e. s ∈ [0, r].

Then the functional I (v) = B(0,1) |∇2 v|2 dz, restricted to the constraint set A f in (14.47), has a unique (up to an affine map) minimizer, which is radially symmetric and given by v f in: R

v f (r) =

Z r Z s 0

0

2t f (t) dt

1/2

ds.

(14.52)

416

14 Limiting theories for weakly prestressed films

The proof of Theorem 14.33 will be a direct consequence of Corollary 14.35 and the sequence of observations below, where we effectively show the relaxed and the original problems have a common minimizer. Naturally, for a radial function f = f (r), uniqueness is tied to the radial symmetry of minimizers: Lemma 14.34. Assume that ω = B(0, 1) ⊂ R2 and that: f = f (r) ≥ c > 0 is a radial function such that f ∈ L1 (ω, R), i.e.: 01 r f (r) dr < ∞. If a radial function v = v(r) ∈ H 2 (ω, R) satisfies det ∇2 v = f , then there holds: R

0

2

|v (r)| =

Z r

for all r ∈ (0, 1).

2s f (s) ds

(14.53)

0

Proof. Let v = v(r) be as in the statement. Writing ∂r v = v0 , the gradient of v in polar coordinates has the form: ∇v(r, θ ) = (v0 (r) cos θ , v0 (r) sin θ )T , and also:
0 1 1 det ∇2 v = v0 v00 = |v0 |2 . r 2r Hence, there must be: |v0 (r)|2 =

Z r

2s f (s) ds +C,

(14.54)

0

for some C ≥ 0. Since v ∈ H 2 (ω, R), we get: ∆ v = v00 + 1r v0 ∈ L2 , or equivalently: Z

|v00 |2 +

ω

1 0 2 2 0 00 |v | + v v dz < ∞. r2 r

Note that the last term above equals 2 f ∈ L1 (ω, R), and thus there must be L1 (ω, R). By (14.54) this allows to conclude: Z 1 2πC 0

r

dr < 2π

Z 1 Z 1 0 |v (r)|2 dr = 0

r

Ω

1 0 2 |v | r2

∈

1 02 |v | dr < ∞, r2

implying C = 0. The proof is done. Corollary 14.35. In the context of Lemma 14.34, an equivalent condition for existence of a radial solution v = v(r) ∈ H 2 (ω, R) to det ∇2 v = f ≥ c > 0 is: Z 1

Z 1

r| log r| f (r) dr < ∞ 0

and 0

r3 f (r)2 dr < ∞. 0 s f (s) ds

Rr

(14.55)

The unique (up to a constant) solution v f is then given by (14.52). In particular, (14.55) is satisfied when f ∈ L2 (ω, R), so then A f , 0. / ¯ R) by (14.52), it remains to check when ∇2 v f ∈ L2 (ω, R). Proof. Since ∇v f ∈ C 1 (ω, We compute:

14.9 Uniqueness of minimizers to linearised Kirchhoff-like energy for prestressed films

Z

|v0f |2 1 0 2 00 2 | + r|v | dz = 2π |v dr f f r2 r 0 ω Z 1 Z 1 r3 f (r)2 Rr = 2π 2r| log r| f (r) dr, dr + 2π 0 0 2s f (s) ds 0

|∇2 v f |2 dz =

ω

Z

|v00f |2 +

proving the first claim. In case f ∈ L2 (ω, R), we obtain Z 1

Z 1

r| log r| f (r) dr ≤

0

Z 1 0

r3 f (r)2 dr ≤ 0 s f (s) ds

Rr

417

Z 1

Z 0

r| log r|2 dr

0 1 r 3 f (r)2

c

Rr 0

s ds

R1 0

1


1/2 Z

(14.56)

r f 2 (r) dr < ∞, and so:

r f 2 dr


1/2

R) be a minimizer of I f , which we modify (if needed) so that: v(0) = 0 and ∇v dz = 0. For any θ ∈ [0, 2π) let   cos θ − sin θ Rθ = be the planar rotation by angle θ . Note that: sin θ cos θ
∇2 (v ◦ Rθ ) = RTθ (∇2 v) ◦ Rθ Rθ , which implies det ∇2 (v ◦ Rθ ) = (det ∇2 v) ◦ Rθ . In view of radial symmetry of f , if follows that v ◦ Rθ ∈ A f∗ and I (v ◦ Rθ ) = I (v). Therefore, by uniqueness, v = v ◦ Rθ is radially symmetric and so the result follows from Corollary 14.35. We are now ready to give: Proof of Theorem 14.33. We will show that the minimization problems for I f and I f∗ have a unique (up to an affine map) solution, which is common to both problems, necessarily radially symmetric and given by v f in (14.52). Define ψ = det ∇2 (argminI f∗ ). By Lemma 14.36, the radial function vψ is the unique minimizer of I f∗ . Consider v f given by (14.52). We will prove that I (v f ) ≤ I (vψ ). This will imply that v f ∈ H 2 (ω, R) and hence, by uniqueness of minimizers there must be: v f = vψ , as claimed. To this end, recall that ψ ≥ f and R that 0r 2s f (s) ds ≥ r2 f (r) in view of (14.53). As in (14.56), we compute:

418

14 Limiting theories for weakly prestressed films

Z

|∇2 vψ |2 dz −

Ω

Z

|∇2 v f |2 dz =

Ω

r3 f (r)2 r3 ψ(r)2 − Rr dr + 2π 0 2sψ(s) ds 0 2s f (s) ds

Z 1

= 2π 0

≥ −2π

Rr

Z 1 0

Z 1 Rr 0 2s(ψ − f ) ds

r3 f 2 0r 2s(ψ − f ) ds R dr + 2π ( 0 2sψ(s) ds)( 0r 2s f (s) ds) R

Rr

r

0

dr

Z 1 Rr 0 2s(ψ − f ) ds

dr

r

0

Z 1 3 2Rr Z 1 Rr r f 0 2s(ψ − f ) ds 0 2s(ψ − f ) ds Rr ≥ −2π dr + 2π dr 2 0

≥ −2π

(

0

2s f (s) ds)

Z 1 3 2Rr r f 0 2s(ψ − f ) ds 0

(r2 f (r))2

r

0

dr + 2π

Z 1 Rr 0 2s(ψ − f ) ds

r

0

dr = 0.

The proof is done in view of Corollary 14.35 and Lemma 14.36. Our final observation is that, in general, v f is not a minimizer of the relaxed problem I f∗ , as shown by the following example: Example 14.37. Consider the family { fε (r) = ε χ(0,1/2] + χ(1/2,1] }ε→0 . Since fε ≤ R ψ ≡ 1, we get Ω |∇2 vψ |2 = 2π where vψ = 21 r2 . But v fε ∈ H 2 (ω, R) and, by (14.56): Z

|∇2 v fε |2 dz

Ω 1 1 r3 dr ≥ c dr ε 1 1−ε 2 2 1/2 r − 4 1/2 4 + (r − 4 ) 0 √ √  1−ε
1− 1−ε
≥ c (log 1 − − log 2 √ 2 √ 1 + 1 − ε
 1−ε
→ ∞ as ε → 0. + log − log 1 + 2 2

≥ 2π

Z 1

r|v00 (r)|2 dr ≥ 2π

Z 1

Z

Therefore I (vψ ) < I (v fε ) for all small ε. A standard approximation argument leads to similar counter-examples with smooth f .

14.10 Critical points in radially symmetric case The Euler-Lagrange equations for (14.47) are related to the, in general unknown, structure of the tangent space to the constraint set A f . Consider instead the functional, defined for all v ∈ H 2 (ω, R) and λ ∈ L∞ (ω, R): Z

Λ (v, λ )  ω

|∇2 v|2 dz +

Z

λ (det ∇2 v − f ) dz.

ω

We first record a few properties tied to the minimization of Λ .

(14.57)

14.10 Critical points in radially symmetric case

419

Lemma 14.38. Let ω ⊂ R2 be open, bounded, connected and Lipschitz. If (v, λ ) is a critical point for Λ in (14.57) then v is a critical point for (14.47). Proof. Let w be a tangent vector to A f at a given v ∈ A f , so that there exists a continuous curve φ : [0, 1] → A f with φ (0) = v such that φ 0 (0) = w. Note that φ (ε) = v + εw + o(ε) ∈ A f . Expanding the determinant, we obtain: f = det ∇2 φ (ε) = det(∇2 v + ε∇2 w + o(ε)) = det ∇2 v + εhcof∇2 w : ∇2 vi + o(ε), which implies that: hcof ∇2 w : ∇2 vi = 0

a.e. in ω.

(14.58)

Let (v, λ ) be a critical point of Λ . Taking variation µ in λ we get: f ) dz = 0, thus v ∈ A f . Taking a variation w in v we obtain: Z

h∇2 v : ∇2 wi dz +

2 ω

Z

λ hcof∇2 v : ∇2 wi dz = 0

R ω

µ(det ∇2 v −

for all w ∈ H 2 (ω, R). (14.59)

ω

In particular, for every w satisfying (14.58), the above reduces to h∇2 v : ∇2 wi dz = 0 which is the variation of I . Hence v must indeed be a critical point of (14.47). R

Lemma 14.39. In the context of Lemma 14.38, the Euler-Lagrange equations of Λ and their natural boundary conditions are: 2∆ 2 v + hcof∇2 v : ∇2 λ i = 0, det ∇2 v = f in ω, D E   ∂τ 2∇2 v + λ cof ∇2 v : (τ ⊗ η) + 2∇∆ v + (cof ∇2 v)∇λ η = 0, D E 2∇2 v + λ cof ∇2 v : (η ⊗ η) = 0 on ∂ ω.

(14.60)

Here, η is the outward unit normal and τ is the unit tangent vector to ∂ ω. Proof. Assuming enough regularity on v, λ , integration by parts gives: Z

h∇2 v : ∇2 wi dz = 2

2

Z

ω

Z

w∆ 2 v dz + 2

ω

λ hcof ∇2 v : ∇2 wi dz =

ω

Z

(∇2 v∇w)η − w(∇∆ v)η dσ (z)

∂ω

Z

whcof ∇2 v : ∇2 λ i dz

ω

Z

+

λ ((cof ∇2 v)∇w)η − w((cof ∇2 v)∇λ )η dσ (z)

∂ω

In view of (14.59), the above calculations yield the first two equations in (14.60) and the following identity, valid for all w ∈ H 2 (ω, R): Z     (2∇2 v + λ cof∇2 v)∇w η − w 2∇∆ v + (cof∇2 v)∇λ n dσ (z) = 0. ∂ω

Writing now ∇w = (∂τ w)τ + (∂η w)η, we get:

420

14 Limiting theories for weakly prestressed films

Z

D E   (∂τ w) 2∇2 v + λ cof ∇2 v : (τ ⊗ η) − w 2∇∆ v + (cof ∇2 v)∇λ η dσ (z) ∂ω Z D E + (∂η w) 2∇2 v + λ cof ∇2 v : (η ⊗ η) dσ (z) = 0. ∂ω

Integrating by parts on the boundary in the first integral above, we deduce the last two equations in (14.60). The proof is done. The following is the main result of this section: ¯ 1), R) is radially symmetric i.e. f = Theorem 14.40. Assume that f ∈ C ∞ (B(0, f (r), and that f ≥ c > 0. Then, the radially symmetric v = v(r) ∈ A f must be R a critical point of I (v) = B(0,1) |∇2 v|2 dz restricted to A f . ¯ 1), R) Proof. 1. We will show that there exists a radial function λ = λ (r) ∈ C ∞ (B(0, such that (v, λ ) is a critical point for Λ in (14.57). By Theorem 7.18, it follows that v ∈ C ∞ (B(0, 1), R). On the other hand, by radial symmetry, v = v f given in (14.52), ¯ 1), R). In particular, v ∈ C ∞ ([0, 1], R) and v0 (0) = (∆ v)0 (0) = 0. so v ∈ C ∞ (B(0, Let Rθ denote the planar rotation by angle θ . In polar coordinates, we have: " # 00 v 0 ∇v(r, θ ) = v0 (r)Rθ e1 = v0 (r)η, ∇2 v(r, θ ) = Rθ RTθ , 0 0 vr and also note that: cof(Rθ ARTθ ) = Rθ (cofA)RTθ . We now rewrite (14.60) using the ansatz λ = λ (r) and assuming sufficient regularity. The equations in ω become: 0 (∆ v)0
1 00 0 0 00 00 , where we used that ∆ v = v00 + vr . Equivar (v λ + v λ ) = −2 (∆ v)
+ r 0 lently: (λ 0 v0 )0 = −2 r(∆ v)0 , which yields: λ 0 (r) = −2

r v0 (r)

(∆ v)0

in (0, 1).

(14.61)

Note that this is consistent with λ 0 (0) = 0, because: 2 0r s f (s) ds
1/2 (v0 (r))2
1/2 v0 (r) = lim = lim r→0 r→0 r→0 r r2 r2 2r f (r)
1/2 p = f (0) , 0. = lim r→0 2r R

lim

(14.62)

2. We now examine the boundary equations in (14.60). We have: D E D E 2∇2 v + λ cof ∇2 v : (τ ⊗ η) = Rθ A(r)RTθ : (τ ⊗ η) D E

 = A(r) : (RTθ τ ⊗ RTθ η) = A(r) : (e2 ⊗ e1 ) , for a matrix field A depending only on r, and hence:

14.11 Bibliographical notes

421

D

E

∂τ 2∇2 v + λ cof ∇2 v : (τ ⊗ η) = 0. Also, in view of (14.61) there holds:   D 2∇∆ v + (cof∇2 v)∇λ η = 2(∆ v)0 + λ 0 = 2(∆ v)0 +

"

v0 r

0

0 v00

# RTθ η, RTθ η

E

v0 0 λ = 0, r

so that the first boundary equation in (14.60) is automatically satisfied. Similarly: D E 2∇2 v + λ cof∇2 v : (η ⊗ η) = # # " 0 " D E v v00 0 0 r = 2 : (e1 ⊗ e1 ) = 2v00 + λ v0 , +λ v0 00 0 r 0 v so that the second boundary equation in (14.60) is satisfied if and only if: 2v00 (1) + λ (1)v0 (1) = 0.

(14.63)

3. Let now λ ∈ C 1 ([0, 1], R) be the solution of the initial value problem (14.61) (14.63). We remark that λ possesses the following limits: (∆ v)0 λ 0 (r) = −2 lim 0 = (∆ v)00 (0), r→0 v r→0 r

lim λ 00 (r) = lim

r→0

so there must be λ = λ (r) ∈ W 2,∞ (B(0, 1), R). In fact, λ is a solution of (14.60) so in view of the elliptic regularity: λ ∈ C ∞ (B(0, 1), R). The proof is done.

14.11 Bibliographical notes Cao-Szekelyhidi’s theorem in section 14.1 appeared in [Cao and Sz´ekelyhidi, 2019, Theorem 1.1]. It improved the result of Lewicka and Pakzad [2017] where the exponent a belonged to the smaller range (0, 17 ), to the range a ∈ (0, 15 ) by taking advantage of the conformal structure of the plane. The full version of the quoted re¯ R3 ) such that sult is actually much stronger; it states that every pair (v0 , w0 ) ∈ C 1 (ω, 1 ¯ 2 (∇v0 ) ⊗ (∇v0 ) + sym∇w − T is positive definite at each x ∈ ω, may be approxi¯ R3 )}n→∞ each mated in C 0 by a sequence of displacement pairs {(vn , wn ) ∈ C 1,a (ω, satisfying the equation in (14.5). Since (14.5) can be seen as the very weak form of the Monge-Amp´ere equation, we obtain the density of the C 1,a -regular solutions ¯ R). In paper Lewicka to det ∇2 v = f (for any fixed a ∈ (0, 51 ), in the space C 0 (ω, and Pakzad [2017], both the flexibility (up to the aforementioned exponent 17 ) and rigidity (for a ∈ ( 23 , 1)) results were obtained, consistently with the application of

422

14 Limiting theories for weakly prestressed films

convex integration techniques in the classical context of isometric immersions of Riemannian metrics, of which the context of rigidity and flexibility of solutions to the Monge-Amp´er´e equation can be seen as the small-slope variation. One may consider a generalization of (14.5) to problems posed on higher-dimensional domains W ⇢ RN , in connection to dimension reduction and isometry matching. Indeed, the set {sym—w; w 2 H 1 (w, RN )} ⇥is the kernel of the operator Curl2 ⇤ 2 2 which for T 2 L2 (w, RN⇥N ) returns Curl (T ) = Curl (T )abcd a,b,c,d=1...N given in: ⇥ Curl2 (T ) = ∂a ∂c Tbd + ∂b ∂d Tac

∂a ∂d Tbc

∂b ∂c Tad

⇤

a,b,c,d=1...N

,

by the application of two exterior derivatives. Similarly to (14.37), there holds: Riem(IdN + e 2 T )abcd =

e2 Curl2 (T )abcd + o(e 2 ), 2

and also, for a possibly higher-dimensional displacement field v : w ! Rk so that (—v)T —v = —v ⌦ —v when k = 1, one gets: ⇥ ⇤ Curl2 ((—v)T —v) = 2 h∂a ∂c v, ∂b ∂d vi h∂a ∂d v, ∂b ∂c vi a,b,c,d=1...N ⌘ Det(—2 v).

Hence v can be matched by an auxiliary field w : w ! RN , so that the given weak prestress metric IdN + h2 T and the pull-back of IdN+k via the maps idN + h[0, v] + h2 [w, 0] : w ! RN+k coincide at the lowest order expansion terms, if and only if: Det(—2 v) = Curl2 (T ). It is worth mentioning that flexibility via convex integration for the weak version of the above system, generalizing the system in (14.5) for k 1: 1 (—v)T —v + sym—w = T, 2

(14.64)

¯ RN+k ) has been proved in Lewicka [2022]. Namely, solution pairs (v, w) 2 C 1,a (w, are dense within the set of subsolution pairs to (14.64), in the space of continuous 1 functions and with any exponent a < 1+(N+N 2 )/k . This statement is consistent with the aforementioned results by Lewicka and Pakzad [2017] and Conti et al. [2010]. Theorems 14.1, 14.4 and 14.2 are taken from Jimen´ez-Bola˜nos and Lewicka [2021]. This last result encompasses several cases studied before. The case g = 2, a = 4 in (i) has been covered in Lewicka et al. [2011a], and the case a = 2g, g > 2 in (ii) was analyzed in Jimen´ez-Bola˜nos and Zemlyanova [2020]. We also note that the present limiting functionals are of the same types as those derived in chapter 7 in case of shallow shells. There, we assumed that the reference domains W h were configured around the mid-surfaces {z + hv0 (z)e3 ; z 2 w} rather than the flat mid-plate w. Taking (S31 , S32 ) = —v0 and Stan = —v0 ⌦ —v0 leads to the energies in which both bending and stretching are relative to the matching order terms derived from v0 .

14.11 Bibliographical notes

423

We also refer to the paper by de Benito Delgado and Schmidt [2021], where the 1 authors considered prestress of the type: Ah (t) = Id3 + ha 1 b ht with a > 2. In our notation and for smooth b, this is equivalent to having: a = 2a 2 > 2, g = a 2, and S = b(0), B = b0 (0) constant. When a = 3 this leads to a subcase of (i), while when a > 3 to a subcase of (ii) in which I (v, w) can be always minimized to h 0. Thus, the optimal scaling of Eweak in that case must have scaling exponent larger than 2a 2 = a, studied in that paper. We remark however that the main contribution there, was allowing the energy density W to depend on ht and b : ( 12 , 12 ) ! R3⇥3 sym to have regularity L• . The von K´arm´an’s equations in Remark 14.3 appeared first in Liang and Mahadevan [2009] in the context of describing the growth of a long leaf. The same equations, together with the associated free boundary conditions were rigorously derived in Lewicka et al. [2011a]. If additionally ∂ w is a polygonal, then equations (14.11) simplify to [Liang and Mahadevan, 2009, equation (5)]. Results in sections 14.5 and 14.6 are taken from Jimen´ez-Bola˜nos and Lewicka [2021]. In agreement with Remark 14.18, we conjecture that the next viable energy scalings after those proved in Theorem 14.16 are: h6 in case (i), h(4+g)^2g in case (ii) for g = a 2, h(4+g)^2g^a in case (ii) for g < a 2, and h(2+g)^(2+a) in case (iii). By analogy, vanishing of the lowest order terms of R1313 , R1323 , R2323 given in (14.33) should then be responsible for even higher energy scalings. Results in sections 14.7, 14.8 and 14.9 appeared in Lewicka et al. [2017a]. A combination of results displayed in section 14.2 and pertaining to the fixed scaling exponents a = 4, g = 2 in Theorem 14.2, with the analysis of elastic shallow shells in section 7.6 of chapter 7, leading to four scaling and energy-limiting scenarios for prestressed shallow shells has been presented in Lewicka et al. [2014].

References

A. Acharya, M. Lewicka, and R. Pakzad. A note on the metric-restricted inverse design problem. Nonlinearity, 29:1769–1797, 2016. G. Acosta and R.G. Dur´an. Divergence operator and related inequalities. Springer New York, NY, 2017. R. Agemi. Global existence of nonlinear elastic waves. Invent. Math., 142(2):225– 250, 2000. S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math., 17:35–92, 1964. V. Agostiniani, A. Lucantonio, and D. Luci´c. Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets. ESAIM: COCV, 25, 2019. G. Alberti. A variational convergence result for Ginzburg-Landau functionals in any dimension. Boll. Un. Mat. Ital., 8(4):289–310, 2001. R. Alicandro and M. Cicalese. A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal., 36(1):1–37, 2006. M. Amar and A. Braides. A characterization of variational convergence for segmentation problems. Discr. Cont. Dynam. Syst., 1:347–369, 1995. L. Ambrosio and V.M. Tortorelli. Approximation of functionals depending on jumps by elliptic functionals via convergence. Comm. Pure Appl. Math., 43:999–1036, 1990. G. Andrews. On the existence of solutions to the equation utt = uxxt + σ (ux )x . J. Diff. Eqs., 35:200–231, 1980. S. Antman. Nonlinear problems of elasticity. Springer New York, 1995. S. Antman and R. Malek-Madani. Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media. Quart. Appl. Math., 46:77–93, 1988. M. Arroyo and A. De Simone. Shape control of active surfaces inspired by the movement of euglenids. J. Mech. Phys. Solids, 62:99–112, 2014. B. Audoly and A. Boudaoud. Self-similar structures near boundaries in strained systems. Phys. Rev. Lett., 91:086105, 2004. © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7

425

426

References

D. Azagra, E. Le Gruyer, and C. Mudarra. Kirszbraun’s theorem via an explicit formula. Canadian Mathematical Bulletin, 64(1):142–153, 2021. J.M. Ball. Strict convexity, strong ellipticity and regularity in the calculus of variations. Math. Proc. Camb. Phil. Soc., 87, 1980. J.M. Ball. Minimizers and the Euler-Lagrange equations. Proc. ISIMM conference, 1983. B. Barker, M. Lewicka, and K. Zumbrun. Existence and stability of viscoelastic shock profiles. Arch. Rational Mech. Anal., 200(2):491–532, 2011. S. Bartels, A. Bonito, and R.H. Nochetto. Bilayer plates: model reduction, Γ convergent finite element approximation and discrete gradient flow. Comm. Pure Appl. Math., 70(3):547–589, 2017. P. Bella and R.V. Kohn. Metric-induced wrinkling of a thin elastic sheet. J. Nonlinear Sci., 24:1147–1176, 2014. P. Bella and R.V. Kohn. The coarsening of folds in hanging drapes. Comm. Pure Appl. Math., 70(5):978–1021, 2017. M. Ben Amar, M.M. M¨uller, and M. Trejo. Petal shapes of sympetalous flowers: the interplay between growth, geometry and elasticity. New J. Physics, 14:085014, 2012. H. Ben Belgacem, S. Conti, A. De Simone, and S. M¨uller. Rigorous bounds for the F¨oppl-von K´arm´an theory of isotropically compressed plates. J. Nonlinear Sci., 10:661–683, 2000. H. Ben Belgacem, S. Conti, A. De Simone, and S. M¨uller. Energy scaling of compressed elastic films - three-dimensional elasticity and reduced theories. Arch. Ration. Mech. Anal., 164(1):1–37, 2002. Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis, Vol. I. American Math. Soc., 2000. K. Bhattacharya, M. Lewicka, and M. Schaffner. Plates with incompatible prestrain. Arch. Rational Mech. Anal., 221(1):143–181, 2016. A. Bonito, R.H. Nochetto, and D. Ntogkas. DG approach to large bending plate deformations with isometry constraint. M3AS, 31(1):133–175, 2021. Y.F. Borisov. C1,α -isometric immersions of Riemannian spaces. Doklady, 163: 869–871, 1965. A. Braides. Gamma-convergence for beginners. Oxford Lecture Series in Mathematics and Its Applications, Clarendon Press, 2002. A. Braides and A. Defrancheschi. Homogenization of multiple integrals. Oxford Lecture Series in Mathematics and Its Applications, Clarendon Press, 1999. A. Braides, A. Defranceschi, and E. Vitali. Homogenization of free discontinuity problems. Arch. Ration. Mech. Anal., 135:297–356, 1996. D. Breit and L. Diening. Sharp conditions for Korn inequalities in Orlicz spaces. Journal of Mathematical Fluid Mechanics, 14(3):565–573, 2012. D. Breit, A. Cianchi, and L. Diening. Trace-free Korn inequalities in Orlicz spaces. SIAM Journal on Mathematical Analysis, 49(4):2496–2526, 2017. S.C. Brenner. Korn’s inequalities for piecewise H 1 vector fields. Mathematics of Computation, 73(247):1067–1087, 2004.

References

427

H. Brezis and S. Wainger. A note on limiting cases of Sobolev imbeddings. Comm P.D.E., 5:773789, 1980. L.A. Caffarelli. Interior W 2,p estimates for solutions of the Monge-Amp`ere equation. Ann. of Math. (2), 131(1):135–150, 1990. L.A. Caffarelli and X. Cabr´e. Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, 43, 1995. E. Calabi and P. Hartman. On the smoothness of isometries. Duke Math. J., 37: 741–750, 1970. A. Calder´on and A. Zygmund. On singular integrals. Amer. J. Math., 78:289–309, 1956. W. Cao and L. Sz´ekelyhidi. Very weak solutions to the two-dimensional MongeAmp`ere equation. Science China Mathematics, 62:1041–1056, 2019. W. Cao and L. Sz´ekelyhidi. Global Nash-Kuiper theorem for compact manifolds. J. Diff. Geom., 2022. E. Cerda and L. Mahadevan. Conical surfaces and crescent singularities in crumpled sheets. Physical Review Letters, 80(11):2358, 1998. E. Cerda and L. Mahadevan. Confined developable elastic surfaces: cylinders, cones and the elastica. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2055):671–700, 2005. E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan. Conical dislocations in crumpling. Nature, 401(6748):46–49, 1999. E. Cerda, L. Mahadevan, and J. Passini. The elements of draping. Proceedings of the National Academy of Sciences (USA), 101(7):1806–1810, 2004. A. Chambolle, S. Conti, and G. Francfort. Korn-Poincar´e inequalities for functions with a small jump set. Indiana Univ. Math. J., 65(4):1373–1399, 2016. N. Chaudhuri and S. M¨uller. Rigidity estimate for two incompatible wells. Calc. Var., 19:379–390, 2004. N. Chaudhuri and S. M¨uller. Scaling of the energy for thin martensitic films. SIAM J. Math. Anal., 38(2):468–477, 2006. W. Chen and J. Jost. A Riemannian version of Korn’s inequality. Calc. Var., 14: 517–530, 2002. M. Chermisi and S. Conti. Multiwell rigidity in nonlinear elasticity. SIAM J. Math. Anal., 42(5):1986–2012, 2010. D. Choi. On geometrical rigidity of surfaces. Application to the theory of thin linear elastic shells. Math. Models Methods Appl. Sci., 7:507–555, 1997. A. Cianchi. Korn type inequalities in Orlicz spaces. Journal of Functional Analysis, 267(7):2313–2353, 2014. P.G. Ciarlet. A justification of the von K´arm´an equations. Arch. Rational Mech. Anal., 73:349–389, 1980. P.G. Ciarlet. Mathematical Elasticity, Vol I-III. North-Holland, Amsterdam, 2000. S. Conti and G. Dolzmann. Gamma-convergence for incompressible elastic plates. Calc. Var. Partial Differ. Equ., 34:531–551, 2009. S. Conti and A. Garroni. Sharp rigidity estimates for incompatible fields as consequence of the Bourgain-Brezis div-curl result. C. R. Math. Acad. Sci. Paris, 359(2):155–160, 2021.

428

References

S. Conti and F. Maggi. Confining thin elastic sheets and folding paper. Archive for Rational Mechanics and Analysis, 187(1):1–48, 2008. S. Conti, I. Fonseca, and G. Leoni. A Γ -convergence result for the two-gradient theory of phase transitions. Comm. Pure Appl. Math., 55:857–936, 2002. S. Conti, D. Faraco, and F. Maggi. A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions. Archive for Rational Mechanics and Analysis, 175(2): 287–300, 2005. S. Conti, F. Maggi, and S. M¨uller. Rigorous derivation of F¨oppl’s theory for clamped elastic membranes leads to relaxation. SIAM J. Math. Anal., 38(2):657–680, 2006. S. Conti, C. De Lellis, and L. Sz´ekelyhidi. h-principle and rigidity for C1,α isometric embeddings. Proceedings of the Abel Symposium, 2010. S. Conti, G. Dolzmann, and S. M¨uller. Korn’s second inequality and geometric rigidity with mixed growth conditions. Calculus of Variations and Partial Differential Equations, 50(1-2):437–454, 2014. M. Costabel and M. Dauge. On the inequalities of Babuˇska-Aziz, Friedrichs and Horgan-Payne. Arch. Ration. Mech. Anal., 217(3):873898, 2015. S. Dain. Generalized Korn’s inequality and conformal Killing vectors. Calculus of Variations and Partial Differential Equations, 25:535–540, 2006. G. Dal Maso. An introduction to Γ -convergence. Progress in Nonlinear Differential Equations and their Applications (8), Birkh¨auser, 1993. M. de Benito Delgado and B. Schmidt. A hierarchy of multilayered plate models. ESAIM: COCV, 27, 2021. E. De Giorgi and T. Franzoni. Su un tipo di convergenca variazionale. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. Rend., 58(6):842–850, 1975. C. De Lellis and D. Inauen. C1,α isometric embeddings of polar caps. Advances in Mathematics, 363:106996, 2020. C. De Lellis and J.L. Sz´ekelyhidi. Simple proof of two well rigidity. C. R. Math. Acad. Sci. Paris, 343(5):367–370, 2006. C. De Lellis, D. Inauen, and L. Sz´ekelyhidi. A Nash-Kuiper theorem for C1,1/5−δ immersions of surfaces in 3 dimensions. Revista Matematica Iberoamericana, 34:1119–1152, 2018. S. Demoulini. Weak solutions for a class of nonlinear systems of viscoelasticity. Arch. Rational Mech. Anal., 155(4):299–334, 2000. J. Dervaux and M. Ben Amar. Morphogenesis of growing soft tissues. Phys. Rev. Lett., 101:068101, 2008. J. Dervaux, P. Ciarletta, and M. Ben Amar. Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the F¨oppl-von K´arm´an limit. J. Mech. Phys. Solids, 57:458–471, 2009. L. Desvillettes and C. Villani. On a variant of Korn’s inequality arising in statistical mechanics. ESAIM Optim. Contr. Calc. Var, 8:603–619, 2002. M. Dias, J. Hanna, and C. Santangelo. Programmed buckling by controlled lateral swelling in a thin elastic sheet. Phys. Rev. E, 84:036603, 2011.

References

429

L. Diening, M. Ruzicka, and K. Schumacher. A decomposition technique for John domains. Annales Academiae Scientiarum Fennicae Mathematica, 35(1):87–114, 2010. G. Duvaut and J.L Lions. Les inequations en mecanique et en physique. Dunod, Paris, 1972. D.G. Ebin. Global solutions of the equations of elastodynamics of incompressible neo-Hookean materials. Proc. Nat. Acad. Sci. U.S.A., 90(9):38023805, 1993. D.G. Ebin. Global solutions of the equations of elastodynamics for incompressible material. Electron. Res. Announc. Amer. Math. Soc., 2(1):50–59, 1996. E. Efrati, E. Sharon, and R. Kupferman. Elastic theory of unconstrained nonEuclidean plates. J. Mech. Phys. Solids, 57:762–775, 2009. L.C. Evans and R.F. Gariepy. Measure theory and fine properties of functions. CRC Press, 1992. D. Faraco and X. Zhong. Geometric rigidity of conformal matrices. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 5(4):557–585, 2005. A. F¨oppl. Vorlesungen uber technische Mechanik 5. Leipzig, 1907. D. Fox, A. Raoult, and J.C. Simo. A justification of nonlinear properly invariant plate theories. Arch. Rational Mech. Anal., 124:157199, 1993. M. Friedrich. A piecewise Korn inequality in SBD and applications to embedding and density results. SIAM Journal on Mathematical Analysis, 50(4):3842–3918, 2018. G. Friesecke and G. Dolzmann. Implicit time discretization and global existence for a quasilinear evolution equation with nonconvex energy. SIAM J. Math. Anal., (28):363–380, 1997. G. Friesecke, R. James, and S. M¨uller. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure. Appl. Math., 55:1461–1506, 2002. G. Friesecke, R. James, M. Mora, and S. M¨uller. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by gammaconvergence. C. R. Math. Acad. Sci. Paris, 336(8):697–702, 2003. G. Friesecke, R. James, and S. M¨uller. A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Ration. Mech. Anal., 180(2): 183–236, 2006. M. Fuchs. Korn inequalities in Orlicz spaces. Irish Mathematical Society Bulletin, 65:5–9, 2010. M. Fuchs and O. Schirra. An application of a new coercive inequality to variational problems studied in general relativity and in Cosserat elasticity giving the smoothness of minimizers. Archiv der Mathematik, 93(6):587–596, 2009. J. Gawinecki and P. Kacprzyk. Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity. Appl. Math. (Warsaw), 35(2):193– 208, 2008. F. Gehring. Rings and quasiconformal mappings in space. Trans. Amer. Math. Soc., 103:353–893, 1962.

430

References

G. Gemmer, E. Sharon, T. Shearman, and S. Venkataramani. Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets. EPL, 114:24003, 2016. J. Gemmer and S. Venkataramani. Shape selection in non-Euclidean plates. Physica D: Nonlinear Phenomena, pages 1536–1552, 2011. J. Gemmer and S. Venkataramani. Shape transitions in hyperbolic non-Euclidean plates. Soft Matter, pages 8151–8161, 2013. G. Geymonat and E. Sanchez-Palencia. On the rigidity of certain surfaces with folds and applications to shell theory. Arch. Ration. Mech. Anal., 129(1):11–45, 1995. D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin, 2001. A. Gladman, E. Matsumoto, R. Nuzzo, L. Mahadevan, and J. Lewis. Biomimetic 4D printing. Nature Materials, 15:413–418, 2016. F. Gmeineder and D. Spector. On Korn-Maxwell-Sobolev inequalities. Journal of Mathematical Analysis and Applications, 502(1), 2021. F. Gmeineder, P. Lewintan, and P. Neff. Korn-Maxwell-Sobolev inequalities for general incompatibilities. Arxiv:2212.13227, 2022a. F. Gmeineder, P. Lewintan, and P. Neff. Optimal incompatible Korn-MaxwellSobolev inequalities in all dimensions. Arxiv:2206.10373, 2022b. Y. Grabovsky and D. Harutyunyan. Exact scaling exponents in Korn and Korntype inequalities for cylindrical shells. SIAM Journal on Mathematical Analysis, 46(5):3277–3295, 2014. Y. Grabovsky and D. Harutyunyan. Korn inequalities for shells with zero Gaussian curvature. Annales de lInstitut Henri Poincar`e. Analyse Non Lin`eaire, 35(1):267– 282, 2018. G. Griso. Decompositions of displacements of thin structures. J. Math. Pures Appl., 89:199–223, 2008. M. Gromov. Partial differential relations. Springer-Verlag, Berin-Heidelberg, 1986. P. Guan and Y. Li. The Weyl problem with nonnegative Gauss curvature. J. Diff. Geometry, 39:331–342, 1994. M. Gurtin. Topics in finite elasticity. CBMS-NSF, 1981. Q. Han and J.X. Hong. Isometric embedding of Riemannian manifolds in Euclidean spaces. Mathematical Surveys and Monographs (130), American Mathematical Society, Providence, RI, 2006. G.H. Hardy, J.E. Littlewood, and G. Polya. Inequalities. Cambridge Mathematical Library. Cambridge University Press. Reprint of the 1952 edition, 1988. P.J. Harris. Carbon nanotubes and related structures - new materials for the twentyfirst century. Cambridge University Press, Cambridge, 1999. P. Hartman and L. Nirenberg. On spherical image maps whose Jacobians do not change sign. Amer. J. Math., 81:901–920, 1959. D. Harutyunyan. Sharp weighted Korn and Korn-like inequalities and an application to washers. Journal of Elasticity, 127(1):59–77, 2017a. D. Harutyunyan. Gaussian curvature as an identifier of shell rigidity. Archive for Rational Mechanics and Analysis, 226(2):743–766, 2017b.

References

431

D. Harutyunyan. On the Korn interpolation and second inequalities in thin domains. SIAM J. Math. Anal, 50(5):4964–4982, 2018. D. Harutyunyan and H. Mikayelyan. On the fractional Korn inequality in bounded domains: counterexamples to the case ps < 1. J. Elast., 147:59–82, 2021. J. Heinonen, P. Koskela, N. Shanmugalingam, and J.T. Tyson. Sobolev spaces on metric measure spaces: an approach based on upper gradients. Cambridge University Press, New Mathematical Monographs, Series Number 27, 2015. J.X. Hong and C. Zuily. Isometric embedding of the 2-sphere with nonnegative curvature in R3 . Math Z., 219:323–334, 1995. J. Horak, G.J. Lord, and M.A. Peletier. Cylinder buckling: the mountain pass as an organizing center. SIAM J. Appl. Math., 66(5):1793–1824, 2006. C.O. Horgan. Korn’s inequalities and their applications in continuum mechanics. SIAM Rev., 37(4):491–511, 1995. P. Hornung. Fine level set structure of flat isometric immersions. Arch. Rational Mech. Anal., 199:934–1014, 2011a. P. Hornung. Approximation of flat W 2,2 isometric immersions by smooth ones. Arch. Rational Mech. Anal., 199:1015–1067, 2011b. P. Hornung, M. Lewicka, and R. Pakzad. Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells. Journal of Elasticity, 111(1), 2013. T. Hughes, T. Kato, and J. Marsden. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal., 63(3):273294, 1997. J.A. Iaia. Isometric embeddings of surfaces with nonnegative curvature in R3 . Duke Math. J., 67:423–459, 1992. D. Ies¸an. Prestressed bodies. Pitman Research Notes in Mathematical Series, 195, 1988. D. Iftimie, G. Raugel, and G.R. Sell. Navier-Stokes equations in thin 3D domains with Navier boundary conditions. Indiana Univ. Math. J., 56(3):1083–1156, 2007. T. Iwaniec. p-harmonic tensors and quasiregular mappings. Ann. of Math. (2), 136(3):389–624, 1992. T Iwaniec and G. Martin. Quasiregular mappings in even dimensions. Acta Math., 170(1):29–81, 1993. ˇ ak. On mappings with integrable dilatation. Proc. Amer. T Iwaniec and V. Sver´ Math. Soc., 118:181–188, 1993. K. Jensen, W. Mickelson, A. Kis, and A. Zettl. Buckling and kinking force measurements on individual multiwalled carbon nanotubes. Phys. Rev. B, 76:195436, 2007. R. Jerrard and A. Lorent. On multiwell Liouville theorems in higher dimension. Adv. Calc. Var., 6:247–298, 2013. R. Jiang and A. Kauranen. Korn’s inequality and John domains. Calc. Var. Partial Differ. Equ., 56.109:1–18, 2017. X. Jie and Q. Tiehu. Almost global existence for the initial value problem of nonlinear elastodynamic system. J. Math. Anal. Appl., 339(1):517529, 2008.

432

References

S. Jimen´ez-Bola˜nos and M. Lewicka. Dimension reduction for thin films prestrained by shallow curvature. Proceeding of the Royal Society A, 2021. S. Jimen´ez-Bola˜nos and A. Zemlyanova. Relative bending energy for weakly prestrained shells. Rocky Mountains Journal of Mathematics,, 50(3):1001–1020, 2020. W. Jin and P. Sternberg. Energy estimates for the von K´arm´an model of thin-film blistering. Journal of Mathematical Physics, 42, 2001. F. John. Rotation and strain. Comm. Pure Appl. Math., 14:391–413, 1961. F. John. Bounds for deformations in terms of average strains. in O. Shisha (ed.), Inequalities III, pages 129–144, 1972. F. John. Formation of singularities in elastic waves. Trends and applications of pure mathematics to mechanics (Palaiseau, 1983), 195 of Lecture Notes in Phys.:194– 210, 1984. F. John. Almost global existence of elastic waves of finite amplitude arising from small initial disturbances. Comm. Pure Appl. Math., 41(5):615–666, 1988. J. Jost. Harmonic maps between surfaces. Lecture Notes in Mathematics, 1062. Springer-Verlag, Berlin, 1984. R. Kempaiah and Z. Nie. From nature to synthetic systems: shape transformation in soft materials. J. Mater. Chem. B, 2:2357–2368, 2014. J. Kim, J. Hanna, M. Byun, C. Santangelo, and R. Hayward. Designing responsive buckled surfaces by halftone gel lithography. Science, 335:12011205, 2012. B. Kirchheim. Geometry and rigidity of microstructures. Habilitation Thesis, Leipzig, 2001. G. Kirchhoff. Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math., 40:51–88, 1850. ¨ M.D. Kirszbraun. Uber die zusammenziehende und Lipschitzsche Transformationen. Fund. Math., 22:77–108, 1934. S. Klainerman and T.C. Sideris. On almost global existence for nonrelativistic wave equations in 3D. Comm. Pure Appl. Math., 49(3):307–321, 1996. Y. Klein, E. Efrati, and E. Sharon. Shaping of elastic sheets by prescription of nonEuclidean metrics. Science, 315:11161120, 2007. Y. Klein, S. Venkataramani, and E. Sharon. Experimental study of shape transitions and energy scaling in thin non-Euclidean plates. PRL, 106:118303, 2011. R.J. Knops. Computable constants for Korn’s inequalities on Riemannian manifolds. Adv. Nonlinear Anal., 12, 2023. R.J. Knops and L.E. Payne. Nonexistence of global solutions in nonlinear Cauchy elastodynamics. Arch. Rational Mech. Anal., 70(2):125133, 1979. S. Kobayashi and K. Nomizu. Foundations of differential geometry. Interscience Publishers, 1963. R.V. Kohn and E. O’Brien. On the bending and twisting of rods with misfit. J. Elasticity, 130(1):115–143, 2018a. R.V. Kohn and E. O’Brien. The wrinkling of a twisted ribbon. Journal of Nonlinear Science, 28(4):1221–1249, 2018b. R.V. Kohn and M. Vogelius. A new model for thin plates with rapidly varying thickness. II: A convergence proof. Quart. Appl. Math., 43:1–22, 1985.

References

433

V. Kondratiev and O. Oleinik. On Korn’s inequalities. C.R. Acad. Sci. Paris, 308: 483–487, 1989. A. Korn. Solution g´en´erale du probl`eme d’´equilibre dans la th´eorie de l’´elasticit´e dans le cas o`u les efforts sont donn´es a` la surface. Ann. Fac. Sci. Toulouse, ser. 2., 10:165–269, 1908. ¨ A. Korn. Uber einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen. Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat., pages 705–724, 1909. N.H. Kuiper. On C1 isometric embeddings, I, II. Indag. Math., 17:545–556, 1955. R. Kupferman and C. Maor. A Riemannian approach to the membrane limit in non-Euclidean elasticity. Comm. Contemp. Math., 16(5):1350052, 2014. R. Kupferman and J.P. Solomon. A Riemannian approach to reduced plate, shell, and rod theories. Journal of Functional Analysis, 226:2989–3039, 2014. R. Kupferman, C. Maor, and A. Shachar. Reshetnyak rigidity for Riemannian manifolds. Archive for Rational Mechanics and Analysis, 231:367–408, 2019. G. Lauteri and S. Luckhaus. Geometric rigidity estimates for incompatible fields in dimension ≥ 3. Arxiv:1703.03288, 2017. H. Le Dret and A. Raoult. The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl., 73:549–578, 1995. H. Le Dret and A. Raoult. The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci., 6:58–84, 1996. M. Lecumberry and S. M¨uller. Stability of slender bodies under compression and validity of the von K´arm´an theory. Archive for Rational Mechanics and Analysis, 193:255–310, 2009. Z. Lei. Global well-posedness of incompressible elastodynamics in two dimensions. Comm. Pure Appl. Math., 69(11):2072–2106, 2016. M. Lewicka. A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry. ESAIM: Control, Optimisation and Calculus of Variations, 17:493–505, 2011. M. Lewicka. Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models. Arch. Ration. Mech. Anal., 236:1677–1707, 2020. M. Lewicka. The Monge-Amp`ere system: convex integration in arbitrary dimension and codimension. Arxiv:2210.04363, 2022. M. Lewicka and H. Li. Convergence of equilibria for incompressible elastic plates in the von K´arm´an regime. Communications on Pure and Applied Analysis, 14(1): 143–166, 2015. M. Lewicka and D. Luci´c. Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case. Communications on Pure and Applied Mathematics, 73(9):1880–1932, 2020. M. Lewicka and L. Mahadevan. Geometry, analysis and morphogenesis: problems and prospects. Bulletin of the AMS, 2022. M. Lewicka and P.B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations and Control Theory, 2(2):337–353, 2013.

434

References

M. Lewicka and P.B. Mucha. A local and global well-posedness results for the general stress-assisted diffusion systems. Journal of Elasticity, 123(1):19–41, 2016. M. Lewicka and S. M¨uller. The uniform Korn-Poincar´e inequality in thin domains. Annales de l’Institut Henri Poincar´e (C) Non Linear Analysis, 28(3):443–469, 2011. M. Lewicka and S. M¨uller. On the optimal constants in Korn’s and geometric rigidity estimates in bounded and unbounded domains, under Neumann boundary conditions. Indiana Univ. Math. J., 65(2):377–397, 2016. M. Lewicka and R. Pakzad. The infinite hierarchy of elastic shell models: some recent results and a conjecture. Fields Institute Communications, 2010. M. Lewicka and R. Pakzad. Scaling laws for non-euclidean plates and the W 2,2 isometric immersions of Riemannian metrics. ESAIM: Control, Optimisation and Calc. Var., 17(4):1158–1173, 2011. M. Lewicka and R. Pakzad. Convex integration for the Monge-Amp`ere equation in two dimensions. Analysis and PDE, 10(3):695–727, 2017. M. Lewicka, M. Mora, and R. Pakzad. A nonlinear theory for shells with slowly varying thickness. Comptes Rendus Mathematique, 347(3-4):211–216, 2009. M. Lewicka, M. Mora, and R. Pakzad. Shell theories arising as low energy Γ -limit of 3d nonlinear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5(9):1–43, 2010. M. Lewicka, L. Mahadevan, and R. Pakzad. The F¨oppl-von K´arm´an equations for plates with incompatible strains. Proceedings of the Royal Society A, 467:402– 426, 2011a. M. Lewicka, M. Mora, and R. Pakzad. The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. Arch. Rational Mech. Anal., 200(3):1023–1050, 2011b. M. Lewicka, L. Mahadevan, and R. Pakzad. Models for elastic shells with incompatible strains. Proceedings of the Royal Society A, 470:20130604, 2014. M. Lewicka, L. Mahadevan, and R. Pakzad. The Monge-Amp`ere constraint: matching of isometries, density and regularity and elastic theories of shallow shells. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 34(1):45–67, 2017a. M. Lewicka, A. Raoult, and D. Ricciotti. Plates with incompatible prestrain of higher order. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 34(7): 1883–1912, 2017b. P. Lewintan and P. Neff. Ne˘cas-Lions lemma revisited: An L p -version of the generalized Korn inequality for incompatible tensor fields. Mathematical Methods in the Applied Sciences, 2021. P. Lewintan, S. M¨uller, and P. Neff. Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy. Calc. Var. PDE, 60:150, 2021. H. Li and M. Chermisi. The von K´arm´an theory for incompressible elastic shells. Calc. Var. Partial Differ. Equ., 48:185–209, 2013. H. Liang and L. Mahadevan. The shape of a long leaf. Proceedings of the National Academy of Sciences, 106, 2009.

References

435

J. Liouville. Th´eor`eme sur l’´equation dx2 + dy2 + dz2 = λ (dα 2 + dβ 2 + dγ 2 ). J.Math. Pures Appl., 15(1), 1850. F.C. Liu. A Lusin property of Sobolev functions. Indiana U. Math. J., 26:645651, 1977. F. L´opez-Garc´ıa. Weighted generalized Korn inequalities on John domains. Mathematical Methods in the Applied Sciences, 41(17):8003–8018, 2018. A.E.H. Love. A treatise on the mathematical theory of elasticity. Cambridge University Press, Cambridge, 1927. L. Mahadevan, A. Vaziri, and M. Das. Persistence of a pinch in a pipe. EPL, 77: 40003, 2007. J. Mal´y and O. Martio. Lusin’s condition (N) and mappings of the class W 1,n . J. Reine Angew. Math., 458:19–36, 1995. C. Maor and A. Shachar. On the role of curvature in the elastic energy of nonEuclidean thin bodies. Journal of Elasticity, 134:149–173, 2019. S. Mardare. On isometric immersions of a Riemannian space with little regularity. Anal. Appl. (Singap.), 2(3):193–226, 2004. T. Mengesha and J. Scott. A fractional Korn-type inequality. Discrete and Continuous Dynamical Systems, 39(6):3315–3343, 2019. C.D. Modes, K. Bhattacharya, and M. Warner. Disclination-mediated thermooptical response in nematic glass sheets. Phys. Rev. E, 81, 2010. C.D. Modes, K. Bhattacharya, and M. Warner. Gaussian curvature from flat elastica sheets. Proc. Roy. Soc. A, 467:1121–1140, 2011. R. Monneau. Justification of nonlinear Kirchhoff-Love theory of plates as the application of a new singular inverse method. Arch. Rational Mech. Anal., 169:1–34, 2003. C.B. Morrey. Multiple integrals in the calculus of variations. Springer, 1996. S. M¨uller. Higher integrability of determinants and weak convergence in L1 . J. Reine Angew. Math., 412:20–34, 1990. S. M¨uller and H. Olbermann. Conical singularities in thin elastic sheets. Calculus of Variations and Partial Differential Equations, 49(3-2):1177–1186, 2014. S. M¨uller and R. Pakzad. Regularity properties of isometric immersions. Math. Z., 251(2):313–331, 2005. S. M¨uller and R. Pakzad. Convergence of equilibria of thin elastic plates the von K´arm´an case. Comm. Partial Differential Equations, 33(4-6):1018–1032, 2008. S. M¨uller, L. Scardia, and C.I. Zeppieri. Geometric rigidity for incompatible fields, and an application to strain-gradient plasticity. Indiana Univ. Math. J., 63(5): 1365–1396, 2014. P. Neff. On Korn’s first inequality with non-constant coefficients. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 132(1):221–243, 2022. R. Neufeld. The deformation of the catenoid into the helicoid. Blog: The Almost Mathematician, WordPress, 2011. URL https://rkneufeld.wordpress.com/2010/11/29/associated-family-of-surfacesof-the-catenoid/. L. Nirenberg. The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math., 6:337–394, 1953.

436

References

J.A. Nitsche. On Korn’s second inequality. RAIRO Anal. Numer., 15(3):237–248, 1981. H. Olbermann. Energy scaling law for the regular cone. Journal of Nonlinear Science, 26(2):287–314, 2016. H. Olbermann. On a boundary value problem for conically deformed thin elastic sheets. Anal. PDE, 12(1):245–258, 2019. D. Ornstein. A non-equality for differential operators in the L1 norm. Archive for Rational Mechanics and Analysis, 11:40–49, 1962. R. Pakzad. On the Sobolev space of isometric immersions. J. Differential Geom., 66(1):47–69, 2004. L.E. Payne and H.F. Weinberger. On Korn’s inequality. Arch. Rational Mech. Anal., 8:89–98, 1961. R. Pego. Phase transitions in one-dimensional nonlinear viscoelasticity. Arch. Rational Mech. Anal., 97:353–394, 1987. P. Petersen. Riemannian geometry. Springer, 2nd edition, 2006. A. Piatnitski and E. Remy. Bilayer plates: model reduction, Γ -convergent finite element approximation and discrete gradient flow. Homogenization of elliptic difference operators, 33:53–83, 2001. A.V. Pogorelov. Surfaces with bounded extrinsic curvature (Russian). Kharkov, 1956. A.V. Pogorelov. An example of a two-dimensional Riemannian metric which does not admit a local realization in E3 . Dokl. Akad. Nauk. SSSR (N.S.), 198:42–43, 1971. A.V. Pogorelov. Extrinsic geometry of convex surfaces. Translation of mathematical monographs vol. 35, American Math. Soc., 1973. W. Pompe. Korn’s first inequality with variable coefficients and its generalization. Commentationes Mathematicae Universitatis Carolinae, 44(1):57–70, 2003. Q. Quang. Sharp regularity results on second derivatives of solutions to the MongeAmp`ere equation with VMO type data. Comm. Pure Appl. Math., 62(5):677–705, 2009. G. Raugel and G.R. Sell. Navier-Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions. J. Amer. Math. Soc., 6:503–568, 1993. J.G. Reshetnyak. Liouville’s conformal mapping theorem under minimal regularity hypotheses. Sibirsk. Mat. Z., 8:835–840, 1967. Yu.G. Reshetnyak. Stability theorems in geometry and analysis. Kluwer Academic Publishers (Mathematics and Its Applications, 304), 1994. F. Rindler. Calculus of variations. Springer Universitext, 2018. P.E.K. Rodriguez, A. Hoger, and A. McCulloch. Stress-dependent finite growth in finite soft elatic tissues. J. Biomechanics, 27:455–467, 1994. P. Rybka. Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions. Proc. Roy. Soc. Edin. A, 121:101–138, 1992. M. Sabl´e-Tougeron. Existence pour un probl´eme de l’elastodynamique Neumann non lin´eaire en dimension 2. Arch. Rational Mech. Anal., 101(3):261292, 1988.

References

437

´ Sanchez-Palencia. Statique et dynamique des coques minces. II. Cas de flexion E. pure inhibe. Approximation membranaire. C. R. Acad. Sci. Paris S´er. I Math, 309(7):531–537, 1989. B. Schmidt. Plate theory for stressed heterogeneous multilayers of finite bending energy. J. Math. Pures Appl., 88:107–122, 2007. E. Sharon, B. Roman, and H.L. Swinney. Geometrically driven wrinkling observed in free plastic sheets and leaves. Phys. Rev. E, 75:046211, 2007. Y.C. Shu. Heteregeneous thin films of martensitic materials. Arch. Rational Mech. Anal., 153:39–90, 2000. T.C. Sideris. The null condition and global existence of nonlinear elastic waves. Invent. Math., 123(2):323–342, 1996. T.C. Sideris. Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. of Math. (2), 152(2):849–874, 2000a. T.C. Sideris. The null condition and global existence of nonlinear elastic waves. Differential equations and mathematical physics (Birmingham, AL, 1999), 16 of AMS/IP Stud. Adv. Math.:339–345, 2000b. T.C. Sideris and B. Thomases. Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit. Comm. Pure Appl. Math., 58(6):750–788, 2005. T.C. Sideris and B. Thomases. Global existence for three-dimensional incompressible isotropic elastodynamics. Comm. Pure Appl. Math., 60(12):1707–1730, 2007. D.E. Spector and S.J. Spector. BMO and elasticity: Korn’s inequality; local uniqueness in tension. Journal of Elasticity, 143:85–109, 2021. M. Spivak. A comprehensive introduction to differential geometry, 3rd edition. Publish or Perish Inc., 2nd edition, 1999. I. Tobasco. Curvature-driven wrinkling of thin elastic shells. Arch. Ration. Mech. Anal., 239(3):1211–1325, 2021. N.S. Trudinger and X.J. Wang. The Monge-Amp`ere equation and its geometric applications. Handbook of Geometric Analysis, International Press, Vol I, 467524, 2008. C. Truesdell and W. Noll. The non-linear field theories of mechanics. Third edition. Edited and with a preface by Stuart S. Antman. Springer-Verlag, Berlin, 2004. S. Venkaratarami. Lower bounds for the energy in a crumpled elastic sheet - a minimal ridge. Nonlinearity, 17(1):301–312, 2004. S.K. Vodopyanov and V.M. Goldstein. Quasiconformal mappings and spaces with generalized derivatives. Siberian Math. J., 17:399–411, 1976. T. von K´arm´an. Festigkeitsprobleme im Maschinenbau. Encyclopadie der Mathematischen Wissenschaften, Leipzig, IV/4:311–385, 1910. ˇ ak. Regularity properties of deformations with finite energy. Arch. Rational V. Sver´ Mech. Anal., 100:105–127, 1988. ˇ ak. On regularity for the Monge-Amp`ere equation without convexity asV. Sver´ sumptions. preprint, Heriot-Watt University, 1991.

438

References

W. Wang. Korn’s inequality and Donati’s theorem for the conformal Killing operator on pseudo-Euclidean space. Journal of Mathematical Analysis and Applications, 345(2):777–782, 2008. M. Warner and E. Terentjev. Liquid crystal elastomers. Oxford University Press, 2003. Z. Wei, J. Jia, J. Athas, C. Wang, S. Raghavan, T. Li, and Z. Nie. Hybrid hydrogel sheets that undergo pre-programmed shape transformations. Soft Matter, 10: 8157–8162, 2014. P.F. Yao. Optimal exponentials of thickness in Korn’s inequalities for parabolic and elliptic shells. Annali di Matematica Pura ed Applicata, 200(2), 2020. K. Zhang. Quasiconvex functions, SO(n) and two elastic wells. Annales de l’Institut Henri Poincar´e (C) Non Linear Analysis, 14(6):759–785, 1997. W. Ziemer. Weakly differentiable functions. Springer-Verlag: New York, 1989.

Terminology and notation

Numbers, sets a ∧ b = min{a, b}, a ∨ b = max{a, b} respectively: the minimum and maximum of two numbers a, b ∈ R ¯ = [−∞, ∞] extended real line R C, c strictly positive constants that are independent from the parameters of the problem and whose values may differ from line to line; C usually is linked with upper bounds and c with lower bounds h  1, x  1 notation for “h is a sufficiently small positive number” and “x is a sufficiently large positive number” O(·), o(·) Landau’s symbols “big O” and “little o” describing the limiting behavior of a function when the argument tends towards a particular value or infinity; we write f (x) = O(g(x)) as x → ∞, instead of | f (x)| ≤ Cg(x) for all x  1; we write f h = o(gh ) as h → 0, instead of limh→0 f h /gh = 0 ' equality up to leading order terms Ω ⊂ RN open, bounded, connected domain in RN , with sufficiently smooth boundary (usually given as a collection of finitely many Lipschitz curves) Ω1 b Ω2 notation for the closure of the set Ω1 contained in the interior of Ω2 |Ω | Lebesgue measure of Ω ⊂ RN Br (x) open ball of radius r centered at x dist(x, Ω ) = infy∈Ω |x − y| distance of a point x from a set Ω Vectors, matrices e1 , . . . , eN canonical basis of RN v⊥ = (−v2 , v1 ) ∈ R2 rotated two-dimensional vector v = (v1 , v2 ) ∈ R2 © Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7

439

440

|v| =

Terminology and notation

q

∑Ni=1 v2i norm of a vector v = (v1 , . . . , vN ) ∈ RN

hv, wi = ∑Ni=1 vi wi scalar product of two vectors v, w ∈ RN F = [Fi j ]i=1...N, j=1...M ∈ RN×M real-valued matrix with N rows and M columns, its coefficients are denoted by Fi j IdN ∈ RN×N identity matrix in RN v ⊗ w = vwT = [vi w j ]i, j=1...N ∈ RN×N tensor product of two vectors v, w ∈ RN v⊗2 = v ⊗ v ∈ RN×N tensor product of a vector v ∈ RN with itself tr F = ∑Ni=1 Fii trace of a matrix F ∈ RN×N F T = [Fji ] ∈ RM×N transpose of a matrix F = [Fi j ] ∈ RN×M ¯ = ∑i, j=1...N Fi j F¯i j = tr (F T F) ¯ scalar product of two matrices F, F¯ ∈ RN×N hF : Fi p |F| = hF : Fi Frobenius norm of a matrix F ∈ RN×N det F determinant of a matrix F ∈ RN×N F −1 inverse of a matrix F ∈ RN×N cof F ∈ RN×N cofactor matrix of F ∈ RN×N given by (cof F)i j = (−1)i+ j det Fˆ ji , where Fˆ ji ∈ R(N−1)×(N−1) is the matrix obtained by removing the j-th row and the i-th column from F; for invertible F there holds cof F = (det F)F −1,T Ftan = F2×2 ∈ R2×2 the principal 2 × 2 minor of F ∈ RN×M ; we similarly denote F3×2 ∈ R3×2 the principal 3 × 2 minor of F, etc  so(N) = A ∈ RN×N ; AT = −A linear subspace of RN×N , consisting of all skew symmetric matrices of dimension N  N×N ; F T = F linear subspace of RN×N , consisting of all symRN×N sym = F ∈ R metric matrices of dimension N RN×N sym,pos set of all symmetric, positive definite matrices of dimension N  SO(N) = R ∈ RN×N ; RT R = IdN , det R = 1 special orthogonal group consisting of all rotations in RN N×N sym F = 21 (F + F T ) ∈ RN×N sym symmetric part of a matrix F ∈ R

skew F = 12 (F − F T ) ∈ so(N) skew-symmetric part of a matrix F ∈ RN×N Functions, operators, convergence idN : Ω → RN inclusion (or identity) function on Ω ⊂ RN 1E : Ω → R characteristic function of the set E ⊂ Ω , given by (1E )|E ≡ 1 and (1E )|(Ω \E) ≡ 0

Terminology and notation

441

f (x) dx integral of a function f on Ω ⊂ RN with respect to the Lebesgue measure dx, in the variable x ∈ Ω R

Ω

R

S f (z) dσ (z) surface integral of a function f on the surface S with respect to the surface Lebesgue measure dσ , in the variable z ∈ S > R f dx = |Ω1 | Ω f dx mean value of the function f defined on Ω such that |Ω | > 0 Ω

∇ f (x) = [∂ j f i (x)]i=1...M, j=1...N ∈ RM×N gradient of a function (vector field) f = ( f 1 , . . . , f M ) : Ω → RM defined on Ω ⊂ RN , at a point x ∈ Ω div f = ∑Ni=1 ∂i f i = tr ∇ f : Ω → R divergence of f = ( f 1 , . . . , f N ) : Ω → RN defined on Ω ⊂ RN ; if f : Ω → RN×N is a matrix field then its divergence is taken row-wise, returning a vector field curl f = ∂2 f 1 − ∂1 f 2 : ω → R curl of a two-dimensional vector field f = ( f 1 , f 2 ) : ω → R2 defined on ω ⊂ R2 ; if f : ω → R2×2 is a matrix field then its curl is taken row-wise, returning a vector field   e1 e2 e3 curl f = det  ∂1 ∂2 ∂3  : Ω → R3 curl of a vector field f = ( f 1 , f 2 , f 3 ) : Ω → R3 f1 f2 f3 defined on a three-dimensional domain Ω ⊂ R3 ∆ f = ∑Ni=1 ∂ii f : Ω → R Laplacian of f : Ω → R defined on Ω ⊂ RN ; if f is a vector field then its Laplacian is taken component-wise, returning a vector field ( f ∗ g)(x) =

R

RN

f (y)g(x − y) dy convolution of two functions f , g ∈ L1 (RN , R)

{ fn }n→∞ sequence of functions indexed by natural numbers n → ∞ { fh }h→0 family of functions indexed by positive real numbers h → 0 fn → f sequence { fn } converges to f in the appropriate strong topology, as n → ∞ fn * f sequence { fn } converges weakly to f , as n → ∞ Γ

fn −→ f sequence of functions Γ -converges to f , with respect to appropriate topology or metric, as n → ∞ Function spaces and norms L2 (Ω , R), L p (ω, RN ), L∞ (Ω , RN×M ) various Lebesgue spaces of functions (vector fields, matrix fields) defined on the domains Ω or ω, with values in the target spaces R, RN or RN×M , and with the indicated integrability exponent p ∈ [1, ∞]; the corresponding norms are denoted k · kL2 (Ω ) , k · kL p (ω) , k · kL∞ (Ω ) , etc W 1,2 (Ω , R),W 2,p (ω, RN ),W k,∞ (Ω , RN×M ) Sobolev spaces of functions (vector fields, matrix fields) defined on the domains Ω or ω, with values in the respective target spaces R, RN or RN×M , and whose gradients up to order k are integrable with the exponent p ∈ [1, ∞]; the corresponding norms are denoted k · kW 1,2 (Ω ) , k · kW 2,p (ω) , k · kW 1,∞ (Ω ) , etc

442

Terminology and notation

H k (Ω , RN ) = W k,2 (Ω , RN ), H k (Ω , RN×M ) = W k,2 (Ω , RN×M ) Sobolev spaces of functions (vector fields, matrix fields) in the square-integrable case

p = 2; the corresponding norms are denoted k · kH K (Ω ) , the scalar product is f , giH k (Ω ) = R k i i Ω ∑i=0 h∇ f , ∇ gi dx H0k (Ω , RN ), W0k,p (ω, RN×M ) Sobolev spaces of functions (vector fields, matrix fields) that vanish on the boundary; they are defined as closures of the spaces of compactly supported smooth functions C0∞ (Ω , RN ), C0∞ (Ω , RN×M ) etc, in the corresponding norms k · kH k (Ω ) , k · kW k,p (ω) , etc k (Ω , RN ), W k,p (ω, RN×M ) local Sobolev spaces of functions (vector fields, maHloc loc trix fields); they consist of f such that each point x in their domain Ω , ω has a neighbourhood Ωx , ωx on which f belongs to H k (Ωx , RN ), W k,p (ωx , RN×M ), etc

C (Ω , R), C k (ω, RN ), C ∞ (ω, RN×M ) spaces of continuous functions (vector fields, matrix fields) with continuous gradients up to order k ∈ {0, 1, . . .}; the corresponding (supremum) norms are denoted k · kC (Ω ) , k · kC k (ω) , etc C 0,α (Ω , R), C k,α (ω, RN ) spaces of H¨older continuous functions (vector fields, matrix fields) whose gradients up to order k ∈ {0, 1, . . .} are continuous and whose k-th order gradient has the H¨older exponent α ∈ (0, 1); the corresponding norms are
f (y)| denoted k f kC k,α (Ω ) = ∑ki=1 k∇i f kC 0 (Ω ) + supx,y∈Ω | f (x)− |x−y|α Geometry of surfaces S ⊂ RN smooth, connected, oriented, N − 1 dimensional surface which is either closed (compact, boundary-less) or with boundary ∂ S given by finitely many Lipschitz curves n, Tz S unit normal vector and the tangent plane at z ∈ S to a surface S as above π(z + tn) = z projection defined on a small neighbourhood of the surface S Π (z) = ∇n(z) : Tz S → Tz S shape operator (which is the negative second fundamen2×2 and, with a tal form) on S; when S ⊂ R3 we identify Π (z) with a matrix in Rsym 3×3 slight abuse of notation, we often write Π (z) ∈ Rsym where Π (z)n = 0 ∂τ v(z) directional derivative of a vector field v ∈ H 1 (S, RN ) in the tangent direction τ ∈ Tz S at a point z ∈ S sym∇v(z) symmetric part of the (covariant) gradient of the tangent component of a vector field v ∈ H 1 (S, RN ); see page 46 I (S) Killing vector fields on a surface S; see page 48 vtan ∈ Tz S projection of a vector v ∈ R3 on the tangent space Tz S to the twodimensional surface S, at an indicated point z ∈ S Ftan ∈ Tz S × Tz S principal “tangential” minor of F ∈ R3×3 or R3×2 associated with a point z on a two-dimensional surface S; see page 101

Terminology and notation

443

V (S), B(S) spaces of first order infinitesimal isometries and finite strains on a surface S; we denote V ∈ V (S) and B ∈ B(S), see pages 127, 136 {Vn (S)}∞ n=1 spaces of infinitesimal isometries of various orders; see page 226 Matching property m 7→ n infinitesimal isometry matching from m-th order to n-th order on a surface S; see page 225 Differential geometry N g ∈ C ∞ (Ω , RN×N sym,pos ) Riemannian metric given on a domain Ω ⊂ R

|g| = det g determinant of a Riemannian metric g as above Γi kj Christoffel symbols of a metric g; occasionally we use the convenient matrix notation Γi = [(Γi )k j  Γi kj ]k, j=1...N ; see page 359 Riem(g) = [Ri jkl ]1, j,k,l=1...N Riemann curvature tensor of a metric g Ric(g) = [Ri j ]i, j=1...3 Ricci curvature tensor of a three-dimensional metric g κ(g) Gaussian curvature of a two-dimensional metric g p ∆g f = √1 ∂i ( |g|gi j ∂ j f ) : Ω → R Laplace-Beltrami operator with respect to a |g|

given Riemannian metric g, applied on the scalar field f : Ω → R Einstein’s summation convention implies summation over a set of indexed terms in a formula when the same index appears twice in a single term in both an upper and a lower position, e.g. gik f i  ∑Ni=1 gik f i Classical elasticity E , J elastic and total energies of a deformation or displacement W : R3×3 → [0, ∞] elastic energy density function, satisfying the basic assumptions listed on page 96 Q3 = D2W (Id3 ) quadratic form derived from the energy density W as; see page 97 L3 : R3×3 → R3×3 linear map associated to the quadratic form Q3 , derived from the elastic energy density W ; see page 97 Q2 (z, ·), L2 (z, ·), c(z, ·) restricted quadratic forms, their associated linear maps and linear recovery vector maps, defined for points z on the midplate or midsurface, in the context of dimension reduction on plates and shells; see page 101 ω midplate of a thin plate; we usually assume that ω ⊂ R2 is open, bounded, connected (or simply connected) and Lipschitz Ω h thin plate with the midplate ω and thickness of order h > 0, which may be either uniform or given by a profile function; the related variables are denoted x = (z,t) ∈ Ω h , with z ∈ ω, |t| < 2h

444

Terminology and notation

S midsurface of a thin shell; we usually assume that S ⊂ R3 has co-dimension 1, is smooth, bounded, connected and with boundary given by finitely many Lipschitz curves (or boundary-less) Sh thin shell obtained by thickening its midsurface S in the direction of the normal vector n either by uniform thickness h > 0 or non-uniformly; the variables are denoted by x = z + tn ∈ Sh with z ∈ S, |t| < 2h Sh midsurface of a thin shallow shell; see page 198 (Sh )h thin shallow shell with the shallow midsurface Sh ; see page 198 vh displacement of a plate Ω h or shell Sh uh deformation of a plate Ω h or shell Sh w, v in-plane displacement and out-of-plane displacement on a midplate ω Φ Airy’s stress potential on a midplate ω; see page 169 E h , J h elastic and total energies of a deformation or displacement on a thin plate or shell, scaled per unit thickness { f h }h→0 family of external forces acting on thin plates or shells, parametrised by the vanishing shell’s thickness h α, β = α ∧ (2α − 2) scaling exponents of the external forces { f h }h→0 , and of the elastic energies {E h }h→0 yh rescaled deformation of a thin plate Ω h or shell Sh ; there holds yh ∈ H 1 (Sh0 , R3 ) where h0 = 1 or h0  1 is a fixed referential thickness ∇h yh rescaling of the deformation gradient in the dimension reduction analysis on thin (prestrained) shells or plates; see page 100 V h [yh ] averaged displacement in the dimension reduction analysis on thin (prestrained) shells or plates; see page 129 {Rh }h→0 , {Z h }h→0 approximating rotation fields (defined on the midplate ω or midsurface S) and rescaled strains (defined on the referential shell Sh0 ) in the dimension reduction analysis {Qh ∈ SO(3), ch ∈ R3 }h→0 fixed rotations and translations for the normalizations of the deformation families {uh ∈ H 1 (Sh , R3 )}h→0 with prescribed energy level, in the dimension reduction analysis {d 0,h }h→0 , {d 1,h }h→0 first and second order warping fields in the construction of recovery families IK , IlinK , IvK , Ilin respectively: Kirchhoff’s energy, linearised Kirchhoff’s energy, von K´arm´an’s energy, linear elastic energy of plates or shells; these energies g1 ,g2 g1 ,g2 possess various modifications, e.g. Ilin , Ilin for shells of non-constant thickv0 ness, or IlinK for shallow shells, etc

Terminology and notation

445

Prestressed elasticity 3 g ∈ C ∞ (Ω , R3×3 sym,pos ) prestress metric given on the referential domain Ω ⊂ R ; its square root is usually denoted by A  g1/2 ; when posed on Ω h = ω × (− h2 , h2 ) then the tangential minor of g restricted to the midplate ω is denoted by G  (gtan )|ω×{0} ∈ C ∞ (ω, R2×2 sym,pos )

Eg elastic energy of deformations in presence of the prestress g Egh elastic energy of deformations on a thin film Ω h prestressed by the metric g, scaled per unit thickness h gh , Ah ∈ C ∞ (Ω h , R3×3 sym,pos ), Eweak weak prestress metric, its square root and the corresponding elastic energy of deformations on the thin film Ω h ; see page 376

α, γ, β scaling exponents in the weak prestress metric corresponding to the stretching and bending-inducing components; β corresponds to the scaling of the infima h } of elastic energies {Eweak h→0 Q2,g (z, ·), L2,g (z, ·), cg (z, ·) restricted quadratic forms, their associated linear maps and linear recovery vector maps, defined for points z on the midplate ω of a thin film Ω h prestressed by g; see page 278 Πy ∈ L2 (ω, R2×2 sym ) shape operator of the surface y(ω) given as the image of the midplate ω via an isometric immersion y ∈ H 2 (ω, R3 ) of the restricted prestress G y0 , b1 ∈ C ∞ (ω, R3 ) compatible isometric immersion and the associated Cosserat vector beyond the non-Euclidean energy scaling h2 ; see page 291 V (y0 ), B(y0 ) spaces of first order infinitesimal isometries and finite strains with respect to the compatible isometric immersion y0 ; we denote V ∈ V (y0 ) and S ∈ B(y0 ); see page 323 {bi ∈ C ∞ (ω, R3 }∞ i=1 compatible Cosserat vectors of higher orders; see page 350 I2,g , I4,g , {I2n,g }∞ arm´an-like enn=3 respectively: Kirchhoff-like energy, von K´ ergy and the infinite hierarchy of the linear elasticity-like energies of thin films prestressed by a single metric g Ker I2n,g kernel i.e. the zero-level set, of the dimensionally reduced energy I2n,g

Index

approximate robustness, 228 approximation theorem linear, 42 nonlinear, 80 prestressed, 280, 320, 354 Brezis-Wainger’s inequality, 189 Cao-Szekelyhidi’s theorem, 376 chain rule, 255 Christoffel symbols, 271, 359 Codazzi-Mainardi’s equations, 292 coercivity, 152, 296, 341, 342, 372 commutator estimate, 275 conformal-anticonformal decomposition, 85 convexity, 210, 234 Cosserat vector, 279, 290, 305, 318, 338, 350 curl curl operator, 167 Darboux frame, 253 deformation, 96 Delellis-Inauen-Szekelyhidi’s theorem, 275 developability, 106, 192, 252, 254 displacement, 12 higher order, 226 in-plane, 141 out-of-plane, 141 elasticity linearised, 12, 155 nonlinear, 96 prestressed, 270, 312 weakly prestressed, 376 energy density, 96, 97, 101, 278 elastic, 100, 198 isotropic, 98, 167, 300

linear elastic, 12 prestressed, 271 total, 12, 97, 100, 198 weakly prestressed, 376 energy quantisation, 318, 339, 350 Euler-Lagrange equations, 18, 161, 169, 381, 419 F¨oppl-von K´arm´an energy, 224 finite strains, 136, 323 frame invariance, 96 Friesecke-James-M¨uller’s constant in the plane, 83 thin shells, 82 Friesecke-James-M¨uller’s inequality conformal, 91 general, 66 in the plane, 83 local, 73 prestressed, 319, 353 fundamental theorem of Riemannian manifolds, 271 Gamma-convergence, 9, 123, 156, 194, 230, 250, 264, 288, 336, 372 Gauss’s equation, 292 Hardy’s inequality, 34 helicoid-catenoid family, 289 hierarchy of theories finite, 225 infinite, 226, 373, 411 infinitesimal isometries, 127, 323 integrable dilatation, 210 ˇ ak’s theorem, 210 Iwaniec-Sver´ Killing fields

© Springer Nature Switzerland AG 2023 M. Lewicka, Calculus of Variations on Thin Prestressed Films, Progress in Nonlinear Differential Equations and Their Applications 101, https://doi.org/10.1007/978-3-031-17495-7

447

448 conformal, 89 on surfaces, 48 Kirchhoff energy liquid crystal glass, 305 plates, 142 prestressed, 279 shells, 105 total, 123 Kirchhoff-Love ansatz, 17, 224 Korn’s constant blowup, 51 tangential boundary conditions, 40, 51 thin shells, 44 uniformity, 53 Korn’s inequality conformal, 89 first Korn, 24 general, 22 homogeneous, 26 Korn-Poincar´e, 24 mixed boundary conditions, 27 on hypersurfaces, 47, 49 on star-shaped domains, 36, 37 thin shells, 51 variant of, 51, 238 Lam´e constants, 99 linear elasticity plates, 12, 156 prestressed, 351 shells, 155, 158 weakly prestressed, 381, 383, 396, 398 linearised Kirchhoff energy plates, 176, 180 prestressed, 406 shallow shells, 201 shells, 175 Liouville’s theorem, 67, 91 liquid crystal glass, 303

Index matching property convex surfaces, 241 developable surfaces, 256 general, 226 plates, 181, 182 prestressed, 409 revolution surfaces, 228 shallows shells, 205 midsurface, 100 multiplicative decomposition, 270 recovery family, 9, 119, 142, 155, 176, 181, 188, 193, 202, 203, 217, 228, 247, 263, 285, 332, 367, 390, 407 Ricci curvature, 308 Riemann curvature, 271, 359 star-shaped domains, 32, 202 thin films prestressed, 274 shallow shells, 198 shells, 100 variable thickness, 157 truncation theorem first order, 68 second order, 110 Vitali’s covering theorem, 68 Von K´arm´an energy conformal, 347 equations, 169, 381 plates, 169 prestressed, 323, 338, 343 shells, 140, 148, 158 weakly prestressed, 381, 383, 396, 398 wrinkling, 126, 138