Geometric Flows on Planar Lattices (Pathways in Mathematics) 3030699161, 9783030699161

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Pathways in Mathematics

Andrea Braides Margherita Solci

Geometric Flows on Planar Lattices

Pathways in Mathematics Series Editors T. Hibi, Toyonaka, Japan W. König, Berlin, Germany J. Zimmer, Bath, UK

Each “Pathways in Mathematics” book offers a roadmap to a currently well developing mathematical research field and is a first-hand information and inspiration for further study, aimed both at students and researchers. It is written in an educational style, i.e., in a way that is accessible for advanced undergraduate and graduate students. It also serves as an introduction to and survey of the field for researchers who want to be quickly informed about the state of the art. The point of departure is typically a bachelor/masters level background, from which the reader is expeditiously guided to the frontiers. This is achieved by focusing on ideas and concepts underlying the development of the subject while keeping technicalities to a minimum. Each volume contains an extensive annotated bibliography as well as a discussion of open problems and future research directions as recommendations for starting new projects. Titles from this series are indexed by Scopus.

More information about this series at http://www.springer.com/series/15133

Andrea Braides • Margherita Solci

Geometric Flows on Planar Lattices

Andrea Braides Dipartimento di Matematica Università di Roma Tor Vergata Roma, Italy

Margherita Solci Dipartimento di Architettura Università di Sassari Alghero, Italy

ISSN 2367-3451 ISSN 2367-346X (electronic) Pathways in Mathematics ISBN 978-3-030-69916-1 ISBN 978-3-030-69917-8 (eBook) https://doi.org/10.1007/978-3-030-69917-8 Mathematics Subject Classification: 49J45, 49Q20, 53C44, 74Q10 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This 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

To Andrea Ses fluctuat nec mergitur C’était pas d’la littérature. . . Georges Brassens, Les copains d’abord

Preface

This book stems from a course given by the first author at the Institute for Advanced Study, Technische Universität München, as part of the Summer School “Multiscale Phenomena in Geometry and Dynamics” organized by M. Cicalese and C. Kühn from July 22 to 26, 2019. On the one hand, the course was focused on a variational approach to geometric flow on lattice as a prototypical example where we can tackle a simplified version of the general problem of evolution in heterogeneous media, referred to by De Giorgi as a “hard nut to crack,” and on the other hand as a subject where the audience might get in touch with various advanced topics in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, -convergence tools, applications of geometric measure theory, and properties of interfacial energies. The present book is a substantial enlargement and a completion of the content of the course, including more theoretical issues on minimizing movements, the detailed treatment of examples only hinted at in the course and of new ones, and some additional results on the convergence of lattice energies. We acknowledge the perfect organization and the stimulating environment of the summer school, and the very kind interest of Johannes Zimmer, who encouraged putting the content of the course into a book. A special thank goes to Andrea "il Tenero" Causin, to whom this book is dedicated, for the many stimulating discussions, for producing all the figures, and for the support provided during the lockdown period when this book was drafted. Roma, Italy Alghero, Italy December 2020

Andrea Braides Margherita Solci

vii

Contents

1 Introduction: Motion on Lattices . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 6

2 Variational Evolution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Discrete Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Discrete Orbits at a Given Time Scale τ . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Passage to the Limit as τ → 0 in Discrete Orbits .. . . . . . . . . . . . 2.2 The Minimizing-Movement Approach .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Discrete-to-Continuum Limit for Lattice Energies .. . . . . . . . . . . 2.2.2 Minimizing Movements Along a Sequence . . . . . . . . . . . . . . . . . . . 2.3 Some Notes on Minimizing Movements on Metric Spaces . . . . . . . . . . . 2.3.1 An Existence Result . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Minimizing Movements and Curves of Maximal Slope . . . . . . 2.3.3 The Colombo-Gobbino Condition . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7 7 8 11 14 15 17 21 21 22 25 29

3 Discrete-to-Continuum Limits of Planar Lattice Energies . . . . . . . . . . . . . . 3.1 Energies on Sets of Finite Perimeter . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Limits of Homogeneous Energies in a Square Lattice . . . . . . . . . . . . . . . . 3.2.1 The Prototype: Homogeneous Nearest Neighbours .. . . . . . . . . . 3.2.2 Next-to-Nearest Neighbour Interactions .. .. . . . . . . . . . . . . . . . . . . . 3.2.3 Directional Nearest-Neighbour Interactions .. . . . . . . . . . . . . . . . . . 3.2.4 General Form of the Limits of Homogeneous Ferromagnetic Energies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Limits of Inhomogeneous Energies in a Square Lattice .. . . . . . . . . . . . . . 3.3.1 Layered Interactions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 Alternating Nearest Neighbours (‘Hard Inclusions’) . . . . . . . . . 3.3.3 Homogenization and Design of Networks .. . . . . . . . . . . . . . . . . . . . 3.4 Limits in General Planar Lattices by Reduction to the Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

31 31 35 37 38 41 42 44 44 45 46 48 50

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Contents

4 Evolution of Planar Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 53 4.1 Flat Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 53 4.1.1 Flat Flow for the Square Perimeter . . . . . . . .. . . . . . . . . . . . . . . . . . . . 54 4.1.2 Motion of a Rectangle . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 55 4.1.3 Motion of a General Set . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 58 4.1.4 An Example with Varying Initial Data. . . . .. . . . . . . . . . . . . . . . . . . . 60 4.1.5 Flat Flow for an ‘Octagonal’ Perimeter. . . .. . . . . . . . . . . . . . . . . . . . 63 4.2 Discrete-to-Continuum Geometric Evolution on the Square Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 69 4.2.1 A Model Case: Nearest-Neighbour Homogeneous Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71 4.2.2 Next-to-Nearest-Neighbour Homogeneous Energies . . . . . . . . . 77 4.2.3 Evolutions Avoiding Hard Inclusions .. . . . .. . . . . . . . . . . . . . . . . . . . 83 4.2.4 Asymmetric Motion . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 87 4.2.5 Homogenized Motion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 88 4.2.6 Motions with an Oscillating Forcing Term . . . . . . . . . . . . . . . . . . . . 91 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 5 Perspectives: Evolutions with Microstructure . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 High-Contrast Ferromagnetic Media: Mushy Layers . . . . . . . . . . . . . . . . . 5.2 Some Evolutions for Antiferromagnetic Systems .. . . . . . . . . . . . . . . . . . . . 5.2.1 Nearest-Neighbour Antiferromagnetic Interactions: Nucleation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Next-to-Nearest Neighbour Antiferromagnetic Interactions: The Effect of Corner Defects . . . . . . . . . . . . . . . . . . . . 5.3 More Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

103 103 109 110 113 118 118

A -Limits in General Lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119 B A Non-trivial Example with Trivial Minimizing Movements . . . . . . . . . . . 125 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133

Chapter 1

Introduction: Motion on Lattices

The scope of these notes is to present a model case –but already complex enough– of motion in heterogeneous media. Even though this analysis will be performed in the relatively simplified setting of a periodic lattice, where the heterogeneous structure is somewhat built-in in the environment itself, this must be thought of as a case study for a large class of inhomogeneous media. In such a lattice setting we consider the simplest order parameter –obtained by labelling the nodes of the lattice with zeros or ones–, and define an energy that favours constant values of such an order parameter and penalizes the creation of ‘discrete interfaces’. In such a way we expect an overall geometric motion driven by surface minimization such as mean-curvature flow. The very discrete structure of the environment provides an obstruction to this motion, with energy barriers that contrast this evolution and in our mind are a prototype of the effect of local minima in a general energy-driven motion in a heterogeneous structure. The seemingly unsolvable contrast between an overall tendency towards motion and a microscopic pinning by local minima can be overcome by resorting to a notion of ‘homogenized motion’ obtained with a balance between minimization of a scaled energy and a scaled dissipation at proper time and space scales, adapting the minimizing-movement approach that has led to a general approach to gradient flow type evolutions in the last 20 years. We now provide a schematic description of the ingredients of our evolution recipe. We will restrict to a two-dimensional setting, which contains the main geometric features beyond the (already interesting) one-dimensional setting. Many arguments carry on to any dimensions, but the description of crystalline geometric motions, which will be our comparison overall motions, is still an object of active research in higher dimensions and necessitates a background beyond the scopes of these notes. Geometrical Environment In what follows, the environment will be given by a lattice, whose elements i ∈ L will be called nodes. We will restrict to the square lattice L = Z2 . This is mainly due to the simplicity of the notation in that case, but with little effort many results can be extended to any Bravais lattice or multi-lattice © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Braides, M. Solci, Geometric Flows on Planar Lattices, Pathways in Mathematics, https://doi.org/10.1007/978-3-030-69917-8_1

1

2

1 Introduction: Motion on Lattices

Fig. 1.1 Representation of a spin function (black circles corresponding to the value 1, white circles corresponding to the value 0)

L ⊂ R2 (for instance, the triangular lattice L = T or the non-Bravais hexagonal lattice L = H), or other periodic lattices. Some of the static results also hold for random lattices. Order Parameter We consider functions L  i → ui which label the nodes of the lattice (such a function may model, for instance, an ensemble of magnetic spins, or a crystalline solid with phases labelled with the values of ui , a set of data labeled on L, etc.). The simplest case is ui ∈ {0, 1}; the system is then described by a (scalar) spin function u ∈ {0, 1}L (see Fig. 1.1). A lattice described by such a parameter is usually called an Ising system. Goal Our scope is to describe evolutions in time (motions) of the order parameter within the framework of such an Ising system. Since we expect these evolutions to be governed by a ‘collective behaviour of the nodes’ we will use some homogenized description; i.e., we will not trace the behaviour of each single node, but their average behaviour when the number of involved nodes is large. Working Assumption We assume the existence of an underlying variational structure; that is, the geometrical patterns described by u tend to optimize (minimize) an energy. Energies We will only consider energies which are the sum of pair interactions; i.e., sum of functions fij (ui , uj ) that take into account only the value of u at pairs (i, j ) of points in L.1 The simplest case is that of nearest-neighbour (NN) interactions; that is, when only interactions between i and j such that i − j  = 1 are taken into account.2 There are two possibilities:

1 Even when limiting or analysis to spin functions, we may consider energies on lattices of a more complex form, taking into account ‘many-body interactions’; e.g., three-body interactions on a triangular or hexagonal lattice, or even an arbitrary dependence assuming the interaction of the site i ∈ L with all other sites of the lattice. 2 One of the main reasons why we use L = Z2 is precisely the easier notation for nearest neighbours.

1 Introduction: Motion on Lattices

3

Fig. 1.2 The set corresponding to the function pictured in Fig. 1.1

1. the minimum is when ui = uj (the energy favours uniform states or ‘aggregations’: this is the ferromagnetic case, with a terminology derived from Statistical Physics); 2. the minimum is when ui = uj (the energy favours ‘mixtures’: the antiferromagnetic case). The simplest ferromagnetic energy3 is obtained with fij (ui , uj ) = |ui −uj |; i.e., E(u) =



|ui − uj |,

i,j

where i, j is a shorthand for NN interactions; that is, the sum is performed over indices i, j ∈ L such that i − j  = 1. This will be the prototype of our lattice energies. Each u ∈ {0, 1}L can be identified with the subset of R2 defined by Au =

 i∈L:ui =1



 1 1 2 − , +i , 2 2

so that E can be interpreted as an energy defined on sets (see Fig. 1.2). With the identification above, E(u) equals the perimeter of Au or the length of ∂Au , which we denote by H 1 (∂Au ). With this interpretation of E in mind, the evolution of the order parameter u can be linked to evolution of sets driven by minimization of the perimeter.

that in Statistical Physics, the order parameter ui is equivalently taken in {−1, 1} and the  ferromagnetic energy is written as E(u) = − i,j ui uj . In both cases we have pair interactions minimized when ui = uj .

3 Note

4

1 Introduction: Motion on Lattices

We can (and we will) consider more general ferromagnetic energies given by E(u) =



aij |ui − uj |,

(1.1)

i,j ∈L

where aij are non-negative coefficients that satisfy some suitable decay condition (as i − j  gets large). If there exists C > 1 such that aij = 0 if i − j  ≥ C then we talk of finite-range interactions, in which case decay conditions are automatically satisfied. If aij > 0 for i −j  arbitrarily large we talk of long-range interactions. If aij depends only on i −j , that is aij = αi−j , we say that the energy is homogeneous or translations invariant; otherwise, the energy depends on the location and we say that it is heterogeneous. An important class of heterogeneous energies are heterogeneous periodic energies, for which there exist a period N ∈ N such that a(i+Ne1 ) (j +Ne1 ) = a(i+Ne2 ) (j +Ne2 ) = aij . Note that homogeneous energies can also be viewed as periodic of period 1. Gradient-Flow Type Evolutions In the description of an energy-driven evolution problem for u, we would like to be able to define a ‘motion’ u(t) = {ui (t)}i∈L satisfying a gradient-flow equation; that is, an equation of the form ∂u = −∇E(u(t)). ∂t In the discrete setting, when u(t) ∈ {0, 1}L , the terms in this equation make little sense as we do not have a linear structure neither on the domain or in the co-domain. Moreover, any point u is isolated, and hence it is a local minimum for E, so that any sensible generalization of ∇E(u) will likely give the value 0. If that is so we only have trivial motions, for which u is constant. In order to overcome this drawback we will make use of a generalized definition of gradient flow in metric spaces, combined with a ‘multiscale approach’. This idea will lead to the definition of time-discrete evolution. In this setting, we introduce a ‘time-scale’ τ > 0 and define a family of states uk = {uki }i∈L parameterized with k ∈ N as an approximation of u(t). We can move from a state uk to the following by minimizing the energy subject to a ‘dissipation constraint’, which allows to explore the energy landscape at a ‘distance’ governed by a ‘time-scale’ τ > 0; i.e., we may compare uk = {uki } with a family of allowed variations v = {vi } such that #{i ∈ L : vi = uki } ≤ C(τ ). Moreover, the minimization also takes into account this set of indices where vi = uki through a dissipation, which favours the minimization of such indices. Homogenized Evolution In order to have an averaged description from this discretetime evolution, we will at the same time approximate a continuum-evolution and

1 Introduction: Motion on Lattices

5

perform a homogenization procedure by scaling the dimensions of the lattice. This is done traditionally by introducing a small parameter ε, and scale the energies so that they are defined on the lattice εL. In a sense, these lattices ‘approximate’ the continuum environment R2 so that by letting ε → 0 and at the same time scale the discrete-time evolution we obtain an evolution on the continuum. Geometric Motions The identification of spin functions u as sets Au carries through in this limit scheme, so that the outcome can be interpreted as a geometric motion of sets in the plane. The prototype of such motions is obtained on the continuum as a gradient-flow type evolution for some perimeter energy, and may be described as a motion by curvature, where the boundary of the set A(t) moves according to its curvature. Of course, the two-dimensional setting is particularly suited for this as the boundary reduces to a set of curves. Motion by Crystalline Curvature We will see that the final motion may depend on the way the parameters τ and ε scale, highlighting the multiscale nature of the problem. Anyhow, a relevant ‘comparison motion’ is obtained by computing the discrete-to-continuum -limit of the energies obtained by scaling the energy E. This is a perimeter energy of crystalline type; that is, minimal configurations with fixed measure, called Wulff shapes, are convex polyhedra (for example, for the nearest-neighbour ferromagnetic energy, such configurations are squares with sides parallel to the coordinate axes). The related geometric motions is called a motion by cristalline curvature, and in two dimensions can be often reduced to studying motions of classes of polyhedral set, which are described by a system of ODEs governing the motion of each edge. Besides the ferromagnetic case, we may also examine energies which favor microstructure. The prototype antiferromagnetic energy can be written as E(u) = −



|ui − uj |,

i,j

or, better, adding the constant 1 in order to have non-negative energy functions,4 E(u) =



(1 − |ui − uj |) .

i,j

Note that in Z2 this energy favours alternating states, whose pattern is a twodimensional checkerboard. Contrary to the ferromagnetic case, the form of minimizers of such E is lattice dependent; for example, if we consider the same energy for the triangular lattice, we have what is called frustration; i.e., that it is not possible to find u which minimize separately all pair interactions, and the pattern of minimizers are more complex and, to some extent, arbitrary (the only restriction being that not

4 In

the usual notation of Statistical Physics we would have E(u) =



i,j ui uj .

6

1 Introduction: Motion on Lattices

all values of ui be equal on the nodes of each triangle). Moreover, we may consider energies of longer range or E(u) =



aij |ui − uj |,

i,j

with non-positive aij , upon possible normalizations in order to have non-negative energies, for which we have an analogous terminology for the corresponding (longrange, heterogeneous, etc.) versions as in the ferromagnetic case.5 These energies in general do not have an interpretation as a perimeter or surface energy, but may have minimizers with complex patterns. Even though a general evolution for such energies is not available, in some cases we will be able to follow a discrete-time minimization scheme as in the ferromagnetic case, obtaining geometric evolutions involving surface and bulk microstructure, and boundary defects. The notes are meant to be relatively self-contained, but at the same time to stimulate the interest towards some research issues in which areas the interested reader may already find some guiding text. In particular, the standard reference for variational evolution is [1], a recent introduction to sets of finite perimeter is the book [8], the two reference texts for -convergence are [3] and [6] (see also [2]), the notes of a course on local minimization and variational evolution issues related to convergence is [4], and a presentation of variational problems for discrete systems can be found in the review paper [7].

References 1. L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2008) 2. A. Braides, A handbook of -convergence, in Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 3, ed. by M. Chipot, P. Quittner (Elsevier, Amsterdam, 2006), pp. 101–213 3. A. Braides, -Convergence for Beginners (Oxford University Press, Oxford, 2002) 4. A. Braides, Local Minimization, Variational Evolution and -convergence. Lecture Notes in Mathematics, vol. 2094 (Springer, Berlin, 2014) 5. A. Braides, M. Cicalese, Interfaces, modulated phases and textures in lattice systems. Arch. Ration. Mech. Anal. 223, 977–1017 (2017) 6. G. Dal Maso, An Introduction to -Convergence (Birkhäuser, Boston, 1993) 7. A. Braides, Discrete-to-continuum variational methods for lattice systems, in Proceedings of the International Congress of Mathematicians August, Seoul, 13–21, 2014, vol. IV, ed. by S. Jang, Y. Kim, D. Lee, I. Yie (Kyung Moon Sa, Seoul, 2014), pp. 997–1015 8. F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory (Cambridge University Press, Cambridge, 2012)

5 We

can also have aij with changing sign, with or without frustration. For a zoo of possible behaviors in that case we refer to the paper [5].

Chapter 2

Variational Evolution

The first subject of our course will be the definition of a variational evolution that retains a gradient-flow type character, in that it is driven by an energy-minimizing ansatz.1 While keeping in mind the application to a lattice environment, we consider the issue of the definition of a variational evolution in a general setting, not necessarily for a discrete energy. We first consider evolution for a single fixed energy; in the following we will treat the homogenization process characteristic of heterogeneous environments, which will make it necessary to take into account ε-depending energies (where ε is a typical length scale, in the case of a lattice its spacing) and will make it possible to overcome some (shallow) local minima in the evolution.

2.1 Discrete Orbits In this section we introduce the fundamental objects of our analysis: discrete-in-time orbits with a fixed time increment. In the next sections we will let the time scale tend to 0 and pass to the limit; however, rather than to concentrate on the technicalities of the passage to the time-continuous limit, a crucial point will be to understand the (approximate) behaviour at this discrete-time level, in order to describe the fine effects that characterize the limit motion.

1 The content of this chapter is a necessarily partial and incomplete introduction to a subject which has proved to be a very effective field of research and applications. We refer to the book by Ambrosio, Gigli and Savaré [2] for a detailed and rigorous treatment of the subject. Some aspects of that theory of variational evolution in metric spaces are also treated in Sect. 2.3.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Braides, M. Solci, Geometric Flows on Planar Lattices, Pathways in Mathematics, https://doi.org/10.1007/978-3-030-69917-8_2

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8

2 Variational Evolution

The ingredients necessary to start our analysis will be • an energy E : X → (−∞; +∞] defined on a set X; • a dissipation D : X × X → [0; +∞); • a time scale τ . Note that we do not make any topological assumptions on X, E and D, even though they are used in existence arguments (see Remark 2.1(1) below). Such assumptions will not be needed in many of our arguments since existence will be achieved by direct computations made easy by the discrete nature of our environment.

2.1.1 Discrete Orbits at a Given Time Scale τ An orbit will be a sequence of points in X defined iteratively. The passage from one point to the following will be determined by a minimization procedure involving the energy E and the dissipation ‘from the previous point’ scaled by the time scale. Definition 2.1 (Time-Discrete Orbit) The set {uτk : k ∈ N} is a time-discrete orbit for E with initial datum u0 ∈ X with respect to the dissipation D and at a time-scale τ if uτ0 = u0 and uτk+1 is a minimizer of v → E(v) +

1 D(v, uτk ). τ

(2.1)

Remark 2.1 1. (Existence) We will not consider in detail conditions under which time-discrete orbits for E exist. In general, they are easily obtained by using the direct method of the Calculus of Variations, assuming that X be a topological space, E and D be lower semicontinuous and problems be coercive (i.e., we may find converging minimizing sequences). 2. (Metric-space case) An important particular case is when the energy is defined on a complete metric space (X, d). In that case we may take D = 12 d 2 ; i.e., {uτk : k ∈ N} is a time-discrete orbit for E with initial datum u0 ∈ X with respect to the distance d if we have uτ0 = u0 and uτk+1 is a minimizer of v → E(v) +

1 2 d (v, uτk ). 2τ

(2.2)

In general, in a metric setting, on the dissipation D we require that it is a function D : X × X → [0, +∞) such that D(v, u) = 0 if and only if v = u and the function v → D(v, uτk ) is d-continuous for any k on sets of v with equi-bounded E(v) + τ1 D(v, uτk ).

2.1 Discrete Orbits

9

3. (Implicit Euler scheme) If X is an Hilbert space and E is (Fréchet) differentiable, then the Euler-Lagrange equation of (2.2) gives uτk+1 − uτk τ

= −∇E(uτk+1 ),

(2.3) ♦

which implicitly defines uτk+1 given uτk . Example 2.1 Let X = R with d(u, v) = |u − v| and E(u) = 1 1 E(v) + 2τ |v − uτk |2 we get uτk+1 = 1+τ uτk ; hence, uτk =

1 2 2u .

Minimizing

1 u0 (1 + τ )k

for all k ∈ N.

♦

We now give an example which will be further elaborated along the notes, and will be extended to a prototypical geometric motion. In this case the dissipation is not directly given through a distance squared. Example 2.2 (Motion of Squares) We consider the energy defined on bounded subsets of R2 with piecewise-C 1 boundary by E(A) = (|γ1 (t)| + |γ2 (t)|) dt, γ

where γ = (γ1 , γ2 ) is a parameterization of ∂A, and the dissipation given by2 D(A, B) =

dist(x, ∂B) dx. AB

The energy E(A) is the integral of the L1 -norm of the tangent to ∂A, whose minimizers with given measure are squares. The dissipation D can be interpreted as an L2 -distance of the boundary of A from the boundary of B. This is pictorially explained in Fig. 2.1 in the case when the boundaries can be locally parameterized as the graphs of two functions f and g: for a small displacement of A from B the dissipation can be written (in local coordinate) as an integral on (a set parameterizing) ∂B of the integral of the vertical 2 For

every p ≥ 1 we define x − yp = (|x1 − y1 |p + |x2 − y2 |p )1/p ,

and x − y∞ = max{|x1 − y1 |, |x2 − y2 |}, and correspondingly distp (x, A) = inf{x − yp : y ∈ A}. If p = 2 we obtain the Euclidean distance, for which we remove the index 2 in the notation.

10

2 Variational Evolution

Fig. 2.1 Interpretation of D as an L2 distance of graphs

coordinate (minus f ) between the values of f and g, which gives an integral of 1 2 2 |f − g| . If we consider the square A0 = [−R0 , R0 ]2 as initial datum we may easily show that the minimizer Aτ1 is still a square. Indeed, we may sketch the argument as follows. • If a competitor A is not contained in A0 then A ∩ A0 is a competitor with strictly less energy and dissipation. • If a connected component of a competitor A is not a rectangle with sides parallel to the coordinate axes then replacing this component with the largest such rectangles containing that component of A we obtain a competitor with strictly less dissipation and not larger energy. • If a rectangle is not centered in the origin then shifting it towards the origin strictly decreases the dissipation maintaining the same energy. This argument shows that we have a single rectangle. • A direct computation then gives that the minimizer is A = [−R1 , R1 ]2 , where R1 solves the Euler-Lagrange equation, which simply reads 1−

R0τ

1 (R0 − R1 )R0 = 0 . τ

The argument above can be iterated, giving a family Aτk = [−Rkτ , Rkτ ]2 with = R0 and

Rkτ

=

⎧ τ ⎨Rk−1 − ⎩

0

τ τ Rk−1

τ if Rk−1 −

otherwise.

τ τ Rk−1

>0

2.1 Discrete Orbits

11

Note that the (half)-side lengths of this family of square satisfy the difference equation τ Rkτ − Rk−1

τ

=−

1 τ Rk−1

as long as Rkτ > 0. In particular, Rkτ is decreasing with k, and Rkτ ≤ R0τ − so that that indeed Rkτ = 0 after a finite number of steps.

(2.4) τ R0τ

k, ♦

Example 2.3 (A Wiggly Energy) Let X = R with D(u, v) = 12 |u − v|2 ,

1 1 + min{|u − z|2 : z ∈ Z}. E(u) = − u + 2 2 We note that −u−1 ≤ E(u) ≤ −u+1, from which we deduce that |uτk−1 −uτk −τ | ≤ √ √ 2 τ , so that uτk−1 ≥ uτk + τ − 2 τ , and the sequence is increasing with a positive velocity if τ > 2. Conversely, if τ is small, then xkτ may be ‘trapped’ is an energy well. For example, if τ < 1 and u0 = 0 then uτk = 0 for all k. The interested reader may explicitly describe the orbits in dependence of τ and u0 as an exercise. ♦ Example 2.4 (Discrete Environment) We may give a discrete-in-space version of the previous energy, by choosing X = Z with d(u, v) = |u − v|, D(u, v) = 12 |u − v|2 , and E(u) = −au (note that this is the restriction to Z of the energy in item 2 above if a = 1). We may compute the discrete orbit iteratively. Minimization gives

1 1 if aτ + ∈  Z uτk = uτk−1 + aτ + 2 2 1 uτk ∈ {uτk−1 + m, uτk−1 + m + 1} if aτ + = m ∈ Z. 2 Note that if aτ < 12 then the motion is trivial; i.e., uτk = u0 for all k, while we have non-uniqueness when aτ + 12 ∈ Z. ♦ Note that in the last example above the space X is composed of isolated points; nevertheless, for some values of τ it is possible to define some type of (non-trivial) energy-driven time-discrete evolution. This example should be kept in mind as a parallel for the lattice environment, where X = {0, 1}L .

2.1.2 Passage to the Limit as τ → 0 in Discrete Orbits In this section we show that we often may define a continuum-time limit from discrete-time orbits. If E is differentiable, this is connected to the passage to the limit in the implicit Euler scheme in (2.3). Since the left-hand side therein can be

12

2 Variational Evolution

seen as a discretization of the gradient of the function uτ defined by uτ (t) = uτt /τ  ,

(2.5)

we can interpret discrete orbits as a discrete version of the gradient flow for E. The following example shows in a simple case that the definition of discrete orbits may indeed give an approximation by discretization of the gradient flow. Example 2.5 Let {uτk } be as in Example 2.1. The corresponding piecewise-constant function uτ is then given by uτ (t) =

1 u0 . (1 + τ )t /τ 

Letting τ → 0, we have pointwise convergence to the function u(t) = u0 e−t , which is the solution of the gradient flow u = −∇E(u) with initial datum u(0) = u0 . ♦ Note that the limit process uτ → u as in the example above can be repeated for a general E, and gives an existence result for Cauchy problems for ODEs in the case that ∇E be a continuous function. The following example deals with a case when the dissipation is not the distance squared. Example 2.6 (Flat Flow of Squares) Let {Aτk } be as in Example 2.2, and Aτ (t) = Aτt /τ . Then the limit of Aτ as τ → 0 is A(t) = [−R(t), R(t)]2 , where R(t) = (see Fig. 2.2). Fig. 2.2 Evolution of the side length

⎧ ⎨ R 2 − 2t ⎩0

0

if 0 ≤ t ≤ R02 /2 otherwise ♦

2.1 Discrete Orbits

13

The existence, up to subsequences, of the limit of functions (2.5) can be taken as an alternative definition of a gradient flow-type evolutions in metric spaces.3 Even though the theory for minimizing movements is usually set in a complete metric space (X, d), for the following definition we only need a notion of convergence in X. Definition 2.2 ((Generalized) Minimizing Movements) Let X be a topological space, let E : X → (−∞, +∞], D : X × X → [0; +∞) and u0 ∈ X. A function u : [0, +∞) → X is a (generalized) minimizing movement for the energy E with respect to the dissipation D and with initial datum u0 if there exist τj → 0 such that uτj (t) → u(t) pointwise as j → +∞, where uτ (t) = uτt /τ  and {uτk }k∈N is a time-discrete orbit for E with respect to D and with initial datum u0 . Remark 2.2 (Dependence on the Initial Data) The definition of (generalized) minimizing movement can also be given with a family of initial data {uτ0 } depending on τ and converging to u0 ∈ X; that is, we may define the orbit at each fixed τ by starting from uτ0 . We note that the minimizing movement obtained in this way is not necessarily equal to the minimizing movement with initial point u0 , as shown in the following example. Let E : R → R be defined by E(u) = −(u)+ , where (·)+ denotes the positive part, and D(u, v) = 12 |u − v|2 . Let uτ0 < 0 for any τ be such that uτ0 → 0 as τ → 0. We compute the orbits obtaining that: • if uτ0 < − τ2 , then uτ1 = uτ0 , and iterating the minimization scheme uτ (t) ≡ uτ0 ; the minimizing movement is then u(t) ≡ 0; • if uτ0 > − τ2 , then uτ1 = uτ0 + τ , and iterating the minimization scheme uτ (t) ≡ uτ0 +

t  τ; τ

the minimizing movement is then u(t) = t, and in this case it coincides with the minimizing movement with initial datum u0 = 0; • if uτ0 = − τ2 , we have a bifurcation, in the sense that at the first step in the construction of the orbit we can choose either uτ1 = uτ0 or uτ1 = uτ0 + τ ; this leads to a non-uniqueness phenomenon. Note that, if we consider the ‘descending slope’ of the function E as defined in Definition 2.5 below, which measures the derivative in the direction of the maximal

3 The interested reader may find several interesting issues and results connected to this theory, for which we refer e.g. to [1, 2, 4, 6, 8].

14

2 Variational Evolution

(negative) slope, we note that at uτ0 this slope vanishes, while at the limit point 0 it is equal to 1. ♦ We will not be concerned with proving general existence theorems of minimizing movements, which can be established under very mild conditions. An important property is that they are often curves of maximal slope, which is a generalization of the concept of gradient flow. For these issues we refer to Sect. 2.3.

2.2 The Minimizing-Movement Approach In the previous sections we have seen that non-trivial energy-driven time-discrete orbits can be obtained also in environments where it is not possible –or not meaningful– to directly define a gradient-flow evolution, as in Example 2.4. Now, we would like to define a time-continuous evolution in a way suitable to a discrete lattice environment in the same spirit as in the minimizing-movement scheme described above. In particular, we would like to be able to define a meaningful evolution for energies of the form E(u) =



aij |ui − uj |

(2.6)

i,j ∈L

as considered in the Introduction. The problem we face is twofold, in that • the passage to the limit in the discrete orbits needs τ → 0; • the definition of non-trivial discrete orbits needs τ ‘large’ with respect to the lattice dimension. In order to comply to both these seemingly opposite requirements, we scale the lattice dimension, so that “τ large with respect to the lattice dimension” is not in contradiction with τ → 0. To this end, we proceed as follows • we introduce a small parameter ε > 0 (the space scale) so that “τ large” means “τ large with respect to ε”. This allows to have non-trivial discrete orbits which are defined starting form suitable scaled versions of the energies and the dissipations; • we pass to the limit as ε, τ → 0: this gives a minimizing movement-type evolution. The introduction of a small space scale ε is characteristic of problems with microstructure, for which we want an averaged description without resolving the details at the microstructure level (homogenization). The process outlined above entails that in the passage to the limit varying Eε and Dε must be considered, which are often obtained by scaling arguments from some reference energy and dissipation. In the following sections we first outline how scaled energies and dissipations can be introduced for lattice energies. Accordingly, the definition of minimizing

2.2 The Minimizing-Movement Approach

15

movements will be extended to a general definition of a ‘minimizing movement along a sequence’.

2.2.1 Discrete-to-Continuum Limit for Lattice Energies In the case of ferromagnetic lattice energies, we identify functions defined on a lattice with suitable characteristic sets. In view of the compactness properties of sets of equibounded perimeter which will be used in the following, we may take X = {A : A set of finite perimeter}4 with the distance d(A, B) = |AB|. The domain of the energies will be the subsets of X corresponding to interpolations of discrete functions. Now, given ε > 0, we define the scaled quantities at this space scale as follows. • Scaled environment. The scaled parameters will be u ∈ {0, 1}εL; that is, functions u : εL → {0, 1}. In the following we fix the lattice L = Z2 , for which identification of functions with sets is particularly straightforward5 and for i ∈ Z2 we define ui = u(εi). Setting Q = [− 12 , 12 ]2 , we identify u with the union of εQ + εi for i such that ui = 1; that is, 

u ∼ Aεu :=

(εQ + εi)

i:ui =1

(see Figs. 1.1 and 1.2). Correspondingly, we define the family of sets Xε = {Aεu : u ∈ {0, 1}εL }. • Scaled energies (at the surface scaling). Given aij coefficients satisfying suitable positiveness and decay conditions,6 starting from energies E of the form (2.6), for u ∈ {0, 1}εL we set Eε (u) =



ε aij |ui − uj |.

(2.7)

i,j ∈L

In this way, Eε are defined in Xε . We extend such energies with the value +∞ on X \ Xε , so that they are defined on the common space X.

4 See

Chap. 3 for a fast review on sets of finite perimeter. lattice, we may use Voronoi cells centered at points of the lattice in place of squares. 6 We may also (and we will) consider ε-depending coefficients a ε , which may model some ij multiscale properties of the lattice environment. 5 For a general

16

2 Variational Evolution

• Definition of the scaled dissipations. We introduce the (scaled) dissipation Dε : Xε × Xε → [0, +∞) by setting Dε (u, v) =



ε3 d∞ (i; {j : vj = 0})

i:ui =0,vi =1



+

ε3 d∞ (i; {j : vj = 1}),

(2.8)

i:ui =1,vi =0

where d∞ is the discrete L∞ -distance; that is, for i ∈ Z2 and A ⊂ Z2 d∞ (i, A) = min{i − j ∞ : j ∈ A} with (z1 , z2 )∞ = max{z1 , z2 }. In the same way as for the energies, we extend Dε to +∞ outside Xε × Xε . This definition is the discrete analog of the dissipation D(A, B) = AB

dist∞ (x, ∂B) dx

introduced by Almgren and Taylor for the description of the evolution of the socalled 1-crystalline perimeter (see Sect. 4.1.1). Note that    1 d∞ (i; A) = dist∞ i, ∂ (Q + j ) + 2 j ∈A

(this is simply due to the fact that a point in Z2 is at distance 12 from the unit squares centered at neighboring points). This equality allows to compare Dε (u, v) with D(Aεu , Aεv ). Remark 2.3 1. (Existence) Given Eε , Dε and initial data uε0 ∈ {0, 1}εL such that Eε (uε0 ) < +∞, for any τ > 0 we may define the discrete orbit {uτk }k∈N and the corresponding piecewise-constant extension uτ (t) = uτt /τ  (see Definition 2.1). Indeed, note that by the finiteness of Eε (uε0 ) we may suppose that uε0 (εi) = 0 for i outside a cube QR with side length R. Otherwise, uε0 (εi) = 1 outside such a cube and we proceed analogously up to exchanging the role of 0 and 1. Reasoning as in Example 2.2, we can show that if u ∈ {0, 1}εL is such that ui = 0 outside QR , then in the minimization of the functional v → Eε (v) +

1 Dε (v, u), v ∈ {0, 1}εL τ

2.2 The Minimizing-Movement Approach

17

we can assume that v satisfies the same property, since for any v ∈ {0, 1}εL the discrete function vR = χQR v has lower energy and dissipation. This implies that at any step in the construction of the discrete orbit the minimization is carried out over the finite set of functions {0, 1}ε(L∩QR ) , thus we obtain the existence of {uτk }k∈N . 2. In general, the orbit is non trivial as long as τ is large enough with respect to ε. If we rewrite Example 2.4 in the scaled lattice εZ with the (scaled) dissipation given by 12 ε2 |u − v|2 , the computation of the discrete orbits gives 1 aτ 1 if + ∈ Z ε 2 ε 2 1 aτ uτk ∈ {uτk−1 + m, uτk−1 + m + 1} if + = m ∈ Z. ε 2 uτk = uτk−1 +

aτ

Hence, the motion is non trivial if τ >

+

♦

ε 2a .

In the following section we formalize the passage to the limit as ε, τ → 0 in a general definition, which will be subsequently applied to the lattice environment.

2.2.2 Minimizing Movements Along a Sequence Before addressing specifically the case of lattice energies, in this section we consider limits u of discrete orbits for general energies Eε and dissipations Dε as ε, τ → 0. We first show that even for simple energies on the real line, it is not reasonable to expect that such a limit u be a minimizing movement for an ‘effective energy’ F and a dissipation D (according to the previous Definition 2.2 of minimizing movement for a functional) independent on the way ε, τ → 0. This justifies a new definition of minimizing movements along a sequence. With in mind the fact that minimizing movements for a single differentiable energy are a way to define a solution to the corresponding gradient flow, in the following example we analyze the behaviour of gradient flows in dependence of a parameter ε. Example 2.7 Let X = R and d(u, v) = |u − v|. For any ε > 0 we define Eε (u) = −u + ε sin

u ε

.

Note that, after setting E0 (u) = −u, Eε → E0 uniformly as ε → 0. If we consider the gradient flow of the limit E0 (with initial datum u0 ), the gradient flow evolution (which is also the unique minimizing movement for E0 with

18

2 Variational Evolution

dissipation D(u, v) = 12 (u − v)2 ) is given by the solution of  u = −E0 (u) = 1 u(0) = u0 ; that is, u(t) = t + u0 . On the other hand, we may easily compute the limit of the gradient flows at a fixed ε > 0 (i.e., of the minimizing movements uε for Eε ) with initial datum uε0 . Indeed, such gradient flows are described by the initial-value problems ⎧  ε ⎨(uε ) = −E  (uε ) = 1 − cos u ε ε ⎩ ε ε u (0) = u0 . The stationary solutions of these equations are u = z with cos( εz ) = 1; that is, u ∈ 2επZ. This implies that |uε (t) − uε0 | < 2επ for all t, so that any solution of the gradient flow is confined in a neighbourhood of the initial value of size of order ε. If we assume that there exists u0 = lim uε0 , then we get that ε→0

lim uε (t) = u0 for all t;

ε→0+

that is, the minimizing movements of Eε tend to a constant as ε → 0. This limit is different from any minimizing movement for E0 . These computations show that we have at least two possible definitions for the ‘effective behaviour’ of discrete orbits for Eε with dissipation Dε (u, v) = 12 (u−v)2 , and suggests that we may have more by ‘interpolating’ those two. ♦ In complete analogy with Definition 2.2 we introduce the following notion of minimizing movement along a sequence, which generalizes the previous one when energies and dissipations do not depend on ε. Definition 2.3 (Minimizing Movement Along a Sequence Eε with Dissipations Dε at Time-Scale τ = τε with Initial Data uε0 ) Let X be a topological space, Eε : X → (−∞, +∞], Dε : X × X → [0; +∞), uε0 ∈ X. A function u : [0, +∞) → X is a minimizing movement along the sequence Eε with dissipations Dε at time-scale τ = τε with initial data uε0 if, up to extraction of subsequences τj , εj , u is a pointwise limit of uτ,ε as ε → 0 ε where {uτ,ε k }k∈N is a time-discrete orbit for Eε with initial datum u0 ∈ X with respect to the dissipation Dε and at a time-scale τ ; that is, τ,ε ε uτ,ε 0 = u0 and uk+1 is a minimizer of v → E(v) +

and uτ,ε (t) = uτ,ε t /τ  .

1 D(v, uτ,ε k ), τ

2.2 The Minimizing-Movement Approach

19

We note that the limit u in general indeed depends on εj , τj . Our focus in the following will be on the characterization of u and on its subtle dependence on Eε and Dε through τj , εj . Again, we will not be concerned with the problem of proving existence results for such minimizing movements, which in our case will be directly obtained by an analysis of discrete orbits. We refer to Sect. 2.3 for some comments on the existence of minimizing movements when (X, d) is a complete metric space, and the possibility in some special cases to obtain a ‘trivial’ characterization of u independent of εj and τj . Example 2.7 suggests that in general even a strong convergence of the energies Eε to some E0 does not imply a relation between a given minimizing movement along Eε and the minimizing movements for E0 . Nevertheless, in the case of an equicoercive family of energies Eε , minimizing movements along Eε are indeed minimizing movements for E0 in some ε-τ regimes. This is important since it suggests that minimizing movements for E0 can be used as a comparison test for general minimizing movements along Eε , and in particular the description of the former gives an environment in which the latter can be expected. The following result shows that the two possibilities of the previous example give ‘extreme’ minimizing movements along Eε . In order to compare minimizing movements along Eε and minimizing movements for E0 it is sufficient to assume that7 minimum problems for Eε + Gε tend to minimum problems for E0 + G if Gε → G continuously

(2.9)

whenever the family Eε + Gε is equicoercive (i.e., it admits converging subsequences of minimizing sequences uε , such that Eε (uε ) + Gε (uε ) = inf{Eε + Gε } + o(1)). By condition (2.9) we mean that the minimum values (or infima) for Eε + Gε converge to the minimum value of E0 + G, and cluster points of the corresponding minimizing sequences are minimizers of E0 + G. If we assume the -convergence of Eε to E0 (which can always be assumed since -convergence is compact if we are e.g. in a separable metric space) then condition (2.9) is automatically satisfied.8 Theorem 2.1 (‘Extreme’ Cases) Let (X, d) be a metric space and let D be such that D(·, v) is continuous for any v ∈ X. Let u be a minimizing movement along Eε at time-scale τ = τε with dissipation Dε = D and initial data uε0 such that uε0 → u0 and {|Eε (uε0 )|} is equibounded. ε → G continuously if Gε (uε ) → G(u) whenever uε → u. This holds in particular if Gε = G and G is continuous. 8 Actually, in metric spaces condition (2.9) can be taken as a definition of -convergence of E to ε E0 . A more usual and operative definition is that for all u the following two conditions hold: 7 We say that G

(i) (liminf inequality) for all uε → u we have E0 (u) ≤ lim infε→0 Eε (uε ); (ii) (existence of recovery sequences) uε → u exists such that E0 (u) = limε→0 Eε (uε ). We will recall and specialize this definition for our lattice energies. We refer to [3] for an introduction to -convergence.

20

2 Variational Evolution

(a) Assume that the (piecewise-constant) orbits uε,τ are pre-compact with respect to the uniform convergence on compact sets as ε, τ small enough.9 Then, there exists τ ε such that if τε < τ ε then u is a limit of minimizing movements for Eε (with fixed ε) with dissipation D and initial datum uε0 . (b) Assume that (2.9) holds for Eε and a functional E0 ; then, there exists τ ε such that if τε > τ ε then u is a minimizing movement for E0 with dissipation D and initial datum u0 . Proof (a) For any fixed ε, by the uniform convergence on compact sets (up to subsequences) of uτ,ε to a minimizing movement uε for Eε with initial datum uε0 , we have that such uε are a precompact family. Then, the sequence uε converges uniformly (up to subsequences), and a diagonal argument proves the claim uε → u (by choosing τε small enough with respect to ε). k (b) Fixed τ > 0, for k ≥ 1 we define Gkε = τ1 D(u, uτ,ε k−1 ) and G (u) = 1 τ τ τ D(u, uk−1 ), where {uk }k∈N is a discrete orbit of E0 with initial datum u0 . By using iteratively (2.9) and the continuity hypothesis on D, we get the continuous convergence Gkε → Gk , and then (up to subsequences) the convergence of the τ minimizers uτ,ε k → uk . Since the sequence of the piecewise-constant extensions τ of uk converges to a minimizing movement for E0 , the claim follows, again by using a diagonal argument.   Remark 2.4 In the theorem above, we can also take Dε instead of D as long as uniform convergence is assumed in (a) and the sequence Gkε (u) = τ1 Dε (u, uτ,ε k−1 ) ♦ continuously converges to Gk (u) = τ1 D(u, uτk−1 ) in (b). Remark 2.5 We note that the scales τ ε and τ ε depend on the initial data uε0 . In the following, however, we will treat families of initial data uε0 with respect to which they can be chosen independent. The situation (a)=(b); that is, the coincidence of the ‘extreme’ behaviours, is ‘rare’. An analysis on conditions under which this happens (mainly, the convexity of Eε ) is presented in Theorem 2.4 of Sect. 2.3.3. In that case, in the -convergence of the functionals Eε (·) + τ1ε D(·, v ε ) (with converging v ε ) we may decouple the effect of the two terms. Conversely, it is likely that the condition (a)=(b) be not sufficient in order that all minimizing movements be equal to either of the extreme cases. The interested reader is encouraged to find an example. ♦ Theorem 2.1 has two important consequences. First, thanks to the remark above, it states that (in the more common case when “(a)=(b)”) there are one or more ‘critical scales’ at which the behaviour of the minimizing movements along Eε are neither of the two extreme cases. These are the most interesting regimes, which

9 Note

that this is ensured if D(u, v) = 12 d 2 (u, v).

2.3 Some Notes on Minimizing Movements on Metric Spaces

21

moreover in general we expect to describe also the ‘extreme regimes’ as a limit case. Secondly, case (b) gives a suggestion on the environment in which we may expect some or all the minimizing movements along Eε to be set; i.e., the one described by the minimizing movements of E0 . A starting point of the analysis is then to compute the -limit of the energies Eε since this implies (and indeed is often equivalent to) condition (2.9).

2.3 Some Notes on Minimizing Movements on Metric Spaces This section is devoted to functionals defined on a metric space coupled with the ‘natural’ dissipation given by the squared distance. We first outline a classical existence theory for a single energy E, which also ensures that minimizing movements are curves of maximal slope. The analysis of the proof of this result highlights that for varying energies Eε the passage to the limit is more complex.

2.3.1 An Existence Result We only recall an existence result when (X, d) is a complete metric space, Eε is a family of energies defined on X, and Dε = D : X × X → [0, +∞) is given by D(u, v) =

1 2 d (u, v). 2

(2.10)

In this case not only we have pointwise convergence, but the continuous-time interpolations uτ also converge uniformly. This allows in particular to conclude the validity of claim (a) in Theorem 2.1. Let Eε : X → (−∞, +∞] be such that (H1) Eε is lower semicontinuous; (H2) (equiboundeness) there exist C ∗ > 0 and τ ∗ > 0 such that for any ε > 0 we have   1 inf Eε (v) + ∗ d 2 (v, u∗ ) : v ∈ X ≥ C ∗ ; 2τ (H3) (equicoercivity) there exists u∗ ∈ X such that for all C > 0 there exists a compact set K such that {u : d 2 (u, u∗ ) ≤ C, |Eε (u)| ≤ C} ⊂ K for any ε > 0. Theorem 2.2 (Existence of Minimizing Movements) Let (X, d) be a complete metric space. Let uε0 , u0 ∈ X be such that uε0 → u0 . If Eε : X → (−∞, +∞] satisfy (H1)–(H3) for any ε > 0, Dε = D is given by (2.10) and {Eε (uε0 )} is bounded, then for all τ = τε → 0 there exists a minimizing movement along the sequence Eε with dissipation Dε at time-scale τ = τε with initial data uε0 according to Definition 2.3.

22

2 Variational Evolution

Moreover, the functions uτ,ε defined therein uniformly converge on compact sets of [0, +∞) (up to subsequences). Proof We only sketch the main steps of the proof. For the sake of simplicity we suppose Eε ≥ 0. The existence of discrete orbits {uτ,ε k } comes from the direct method of the Calculus of Variations. Using uτ,ε as a test function in the problem defining uτ,ε k−1 k we get Eε (uτ,ε k )+

1 2 τ,ε τ,ε d (uk , uk−1 ) ≤ Eε (uτ,ε k−1 ), 2τ

from which d(uτ,ε (t), uτ,ε (s)) ≤

  2|Eε (uε0 )| |t − s| + τ ,

so that the family {uτ,ε } is equibounded and (almost) equicontinuous10 as τ → 0, and we can apply a slight variant11 of the Ascoli-Arzelà Theorem to get the claim.  

2.3.2 Minimizing Movements and Curves of Maximal Slope Even though in the previous sections we have motivated our limit process as a passage to the limit to obtain some gradient-flow evolution, the definition of minimizing movement is extremely general. It is worth then explaining how it can be seen as a ‘curve of maximal slope’, which is a concept generalizing a gradient flow when no linear structure is present. Assume first that E be differentiable and X be a Hilbert space. If u satisfies the gradient-flow equation for E; that is, u (t) = −∇(E(u(t))), then for any t 0 = |u (t) + ∇E(u(t))|2 = |u (t)|2 + |∇E(u(t))|2 + 2 ∇E(u(t)), u (t) . Hence, since the inequality − ∇E(u(t)), u (t) ≤

1  2 1 |u (t)| + |∇E(u(t))|2 2 2

say that a family fτ is (almost) equicontinuous as τ → 0 if there exists a non-negative function ω continuous in 0 with ω(0) = 0 such that for all ε > 0 there exists δ > 0 such that |fτ (x) − fτ (y)| < ε for all τ and for all x, y with ω(τ ) ≤ d(x, y) ≤ δ. 11 Actually, the usual proof of Ascoli-Arzelà’s Theorem works unchanged under this weaker condition of (almost) equicontinuity. 10 We

2.3 Some Notes on Minimizing Movements on Metric Spaces

23

is always true, such a gradient flow is characterized by the inequality E(u(s)) − E(u(t)) ≥

1 2



t

|u (r)|2 dr +

s

1 2



t

|∇E(u(r))|2 dr, s ≤ t

(2.11)

s

obtained by integrating on (s, t). Now, a meaning to such an inequality can be given also when we are in a space without a linear structure and with a more general (i.e., not differentiable) functional E. Indeed, this is done using the definitions of metric derivative of a function u with values in a metric space X and of descending slope of a functional E defined on X, generalizing the modulus of the derivative |u | and the modulus of the gradient |∇E(u)|, respectively. Definition 2.4 (Metric Derivative) Let u : [0, T ] → X. We say that u belongs to AC 2 ([0, T ]; X) if there exists A ∈ L2 (0, T ) such that

t

d(u(s), u(t)) ≤

A(r) dr

for any 0 ≤ s ≤ t ≤ T .

s

The smallest such A is the metric derivative of u and it is denoted by |u |. The metric derivative satisfies |u |(t) = lim s→t

d(u(s), u(t)) |s − t|

for L1 -almost every t ∈ (0, T ).

Definition 2.5 ((Local) Descending Slope) Let E : X → (−∞, +∞]. We define the (local) descending slope of E as |∂E|(u) = lim sup v→u

(E(u) − E(v))+ d(u, v)

if E(u) ∈ R and u is not isolated. The slope at an isolated u in the domain of E is defined as 0, and as equal to +∞ at a point u such that E(u) = +∞. The descending slope is a weak upper gradient for E; that is, for any u ∈ AC([0, T ]; X) such that • |∂E|(u)|u | ∈ L1 (0, T ); • there exists ϕ : (0, T ) → R with finite pointwise variation such that E ◦ u = ϕ L1 -almost everywhere in (0, T ) we have |ϕ  |(t) ≤ |∂E|(u(t))|u |(t) for L1 -almost every t ∈ (0, T ). The following definition generalizes condition (2.11).

24

2 Variational Evolution

Definition 2.6 (Curves of Maximal Slope in a Metric Space) A curve of maximal slope for E is a function u ∈ AC 2 ([0, T ]; X) such that there exists a non-increasing ϕ : [0, T ] → R such that ϕ(s) − ϕ(t) ≥

1 2



t

|u |2 (r) dr +

s

1 2



t

|∂E|2 (u(r)) dr

s

(2.12)

for any 0 ≤ s ≤ t ≤ T

and a L1 -negligible set A such that E(u(t)) = ϕ(t) in [0, T ] \ A. The following theorem is a fundamental result, which is often used to prove the existence of curves of maximal slope for a functional E. We state it in a slightly weaker form than in the literature, and give a sketch of its proof, which will be useful to understand some issues in the case of varying functionals Eε .12 Theorem 2.3 (Minimizing Movements and Curves of Maximal Slope) Let hypotheses (H1)–(H3) be satisfied by Eε = E and let u0 ∈ X be such that E(u0 ) < +∞. If the local descending slope of E is lower semicontinuous, then a (generalized) minimizing movement for E with initial datum u0 is a curve of maximal slope. Proof We only outline the main steps in the proof. Let u be a (generalized) minimizing movement for E as in Definition 2.2, that is (up to extracting a subsequence) u(t) = limτ →0 uτ (t), where uτ is the piecewiseconstant function given by uτ (t) = uτ t  τ

and {uτk }k is a time-discrete orbit for E with initial datum u0 , as in Definition 2.1. A variational-interpolation argument due to De Giorgi, gives another (not piecewise-constant) approximation of the minimizing movement u allowing to obtain sharper estimates. More precisely, we define u˜ τ : [0, +∞) as any interpolation of the function defined in τ N by the values uτk satisfying  u˜ τ (t) ∈ arg min E(u) +

 1 d 2 (u, uτk ) for any t ∈ (kτ, (k + 1)τ ]. 2(t − kτ ) (2.13)

It can be shown that u˜ τ is close to uτ , in the sense that for any fixed T > 0 there exists a constant C > 0 depending only on the bounds on E and the initial point u0 such that for τ small enough we have d 2 (u˜ τ (t), uτ (t)) ≤ Cτ 12 For

in [0, T ].

(2.14)

a stronger version of this theorem and the details of the proof see Theorem 2.3.1 in [2].

2.3 Some Notes on Minimizing Movements on Metric Spaces

25

Estimate (2.14) allows to use u˜ τ to characterize the properties of the limit of uτ . Indeed, for any t, s with s ≤ t we can prove that E(u˜ τ (s)) − E(u˜ τ (t)) ≥



1 2 +

1 2

t s

|(uτ ) |2 (r) dr t

s

(2.15) |∂E|2(u˜ τ (r)) dr + o(1)τ →0 ,

d(uτk+1 ,uτk )

where |(uτ ) | = in each interval (kτ, (k + 1)τ ) and with the remainder τ depending only on the bounds on E and the initial point u0 . Using monotonicity properties and Helly’s Theorem, we may show that E ◦ u˜ τ converges to E ◦ u almost everywhere as τ → 0. Moreover, E ◦ u˜ τ converges everywhere to a monotone ϕ. As for the right-hand side of (2.15), the lower semicontinuity of the L2 -norm and the lower semicontinuity of the slope |∂E|, using Fatou’s Lemma, allow to conclude that u is a curve of maximal slope for E as in Definition 2.6.  

2.3.3 The Colombo-Gobbino Condition Now, we consider a sequence of functionals Eε , and follow the arguments of the proof of Theorem 2.3 in order to understand if and under which hypotheses it is possible to conclude that a minimizing movement along Eε is a curve of maximal slope for some ‘effective energy’ E. We first assume that the hypotheses of Theorem 2.3 be satisfied uniformly; i.e., that Eε satisfy (H1)–(H3). We also require that the initial data be equibounded and with equibounded energy; that is, that d(uε0 , u∗ ) and Eε (uε0 ) be equibounded. We let τ = τε , and uτ,ε be the piecewise-constant function with pointwise (and uniform on compact sets) limit a minimizing movement u. For each ε > 0 we let u˜ τ,ε denote the interpolation as in (2.13). A priori estimates analog to the case independent of ε hold for u˜ τ,ε , so that, in particular, u˜ τ,ε converge to u, and Eε (u˜ τ,ε (s)) − Eε (u˜ τ,ε (t)) ≥

1 2 +

1 2

t s

|(uτ,ε ) |2 (r) dr t

(2.16) |∂Eε |2 (u˜ τ,ε (r)) dr + o(1)ε→0

s

holds, with the remainder depending only on the uniform bounds on Eε and the initial points uε0 . Now we have to pass to the limit in (2.16), aiming at proving that (2.12) holds for some E. The passage to the limit is easy for the term depending on the derivative |(uτ,ε ) | thanks to the weak convergence. As for the left-hand side, from the convergence of u˜ τ,ε and Helly’s theorem, we know that an increasing ϕ exists

26

2 Variational Evolution

such that Eε (u˜ τ,ε (t)) → ϕ(t) (up to subsequences). A first requirement is that we may deduce that u˜ τ,ε (t) → u(t) ⇒ Eε (u˜ τ,ε (t)) → E(u(t))

(2.17)

for almost all t. As the term regarding the slopes is concerned, Fatou’s lemma ensures that

t

lim inf ε→0



t

|∂Eε |2 (u˜ τ,ε (r)) dr ≥

s

s

lim inf |∂Eε |2 (u˜ τ,ε (r)) dr ε→0

(so that the slopes are equibounded for almost all t). In order to obtain (2.12), a further assumption is needed ensuring that (under equiboundedness assumptions) u˜ τ,ε (t) → u(t) ⇒ lim inf |∂Eε |(u˜ τ,ε (t)) ≥ |∂E|(u(t)). ε→0

(2.18)

We can include conditions (2.17) and (2.18) in a common assumption, and state the following result, which ensures that minimizing movements are curves of maximal slope for an effective energy. Its proof is the observation that, upon extracting further subsequences, we may pass to the limit in (2.16) obtaining (2.12) for E. Theorem 2.4 (A ‘Commutativity’ Result) Let Eε , E : X → (−∞, +∞] be such that (H1)–(H3) are satisfied. Let uε0 be such that d(uε0 , u∗ ) and Eε (uε0 ) are equibounded. Then, a sufficient condition for any minimizing movement along a sequence Eε with initial data uε0 to be a curve of maximal slope for E is that Eε and E satisfy condition (H) given by13 (H)

for any vε → v such that sup{|Eε (vε )| + |∂Eε |(vε )} < +∞, we have lim Eε (vε ) = E(v) and lim inf |∂Eε |(vε ) ≥ |∂E|(v).

ε→0

ε→0

Remark 2.6 (Local Minima and Condition (H)) We note that condition (H) is in a sense in contrast with having many local minima. As a limit case, consider the situation when local minima for Eε are asymptotically dense in X; i.e., for each v there exists a family of local minima vε for Eε converging to v. In this case |∂Eε |(vε ) = 0 so that (H) would imply that |∂E|(v) = 0 for all v and hence (upon some mild assumptions) that E is constant. ♦ Remark 2.7 (Slope-Cone Property, Convexity and Condition (H)) Assume that the functionals Eε satisfy (H1)–(H3), and the following property (slope-cone property)

13 This condition has been stated in [7] in the context of curves of maximal slope, and extended to minimizing movements in [5].

2.3 Some Notes on Minimizing Movements on Metric Spaces

27

holds: if u ∈ X is such that both Eε (u) and |∂Eε |(u) are finite, then Eε (v) ≥ Eε (u) − d(u, v)|∂Eε |(u) for any v ∈ X.

(2.19)

Moreover, let E be the -limit of Eε . Then condition (H) holds, and by Theorem 2.4 any minimizing movement along the sequence Eε is a curve of maximal slope for E. Note that condition (2.19) holds in particular if the functionals Eε are convex. ♦ Remark 2.8 (Condition (H) and -Convergence) Let E and Eε satisfy condition (H). If u ∈ X is such that |∂E|(u) < +∞ and the descending slopes of Eε are equibounded in a neighborhood of u, then E(u) = - lim Eε (u). ε→0

Indeed, choosing uε = u we get supε {Eε (u) + |∂Eε |(u)} < +∞ and condition (H) gives lim sup Eε (u) = E(u) ε→0

showing the upper inequality for the -limit. As for the lower bound, we consider a sequence uε → u such that supε Eε (uε ) < +∞; the boundedness hypothesis on the slopes gives sup{Eε (uε ) + |∂Eε |(uε )} < +∞. ε

Then by condition (H) we deduce lim inf Eε (uε ) = E(u) ε→0

♦

concluding the proof.

We note that, in general, condition (H) is not sufficient to ensure -convergence, not even at points where the slope is finite, as the following example shows. Example 2.8 Let X = [0, 1] with d(u, v) = |u − v|; we set  En (u) =

0

if u ≤ 1 −

n − 1 − nu

if u ≥ 1 −

1 n 1 n

E(u) ≡ 0. (Fig. 2.3). Condition (H) is satisfied at u = 1; indeed, if we compute the descending slopes of En , we obtain that  1 , |∂En |(u) = 0 if u ∈ 0, 1 − n

28

2 Variational Evolution

Fig. 2.3 Graph of the functionals En

and the same holds if u = 1 since |∂En |(1) = lim sup v→1

(n(v − 1))+ = 0. 1−v

Moreover, |∂En |(u) = lim sup v→u

 1  (n(v − u))+ = n if u ∈ 1 − , 1 . |u − v| n

Now, we fix a sequence {un } such that un → 1 and   sup En (un ) + |∂En |(un ) < +∞. n

Then, there are only finitely many points un belonging to the interval (1 − n1 , 1), and this implies that for any other point both the slope |∂En | and the functional En vanish, and condition (H) is satisfied. However, the sequence En does not -converge to E at u = 1; for instance, if 1 we choose un = 1 − 2n we have lim inf En (un ) = −

n→+∞

1 < 0 = E(1), 2

and the lower-bound inequality does not hold.

♦

Remark 2.9 (Condition (H) Is Not Quite Necessary) Let X = [0, +∞) with the distance d(u, v) = |u − v|. We construct an example of Eε , E : X → R such that condition (H) is not satisfied at 0 but any minimizing movement along Eε with initial datum 0 at any scale τ is the minimizing movement for E with initial datum 0. We set E(u) = −u,

Eε (u) = −(u)1+ε ,

References

29

u0 = 0, and uε0 = 0 for each ε > 0. We have |∂E|(0) = lim sup v→0

(E(0) − E(v))+ −E(v) = lim sup =1 |v| v v→0

|∂Eε |(0) = lim sup v→0

(Eε (0) − Eε (v))+ −Eε (v) = lim sup = lim sup(v)ε = 0. |v| v v→0 v→0

Hence, • Eε and E satisfy hypotheses (H1)–(H3); • setting uε0 = 0 we have that sup{Eε (uε0 ) + |∂Eε (uε0 )|} = 0 (so that it is bounded), the limit of Eε (uε0 ) is E(0), but lim inf |∂Eε |(uε0 ) = 0 < 1 = |∂E|(0). ε→0

This shows that condition (H) is not satisfied; nevertheless, we have the following proposition. Proposition 2.1 For every time-scale τ the minimizing movement along Eε with initial data 0 is the minimizing movement of E with initial datum 0; that is, u(t) = t. The interested reader may find the proof of this result in Appendix B. Note that the fact that the minimizing movements are the same at all scales is not an immediate consequence of the equality of the extreme cases, and needs non-trivial monotonicity arguments. ♦

References 1. F. Almgren, J.E. Taylor, L. Wang, Curvature driven flows: a variational approach. SIAM J. Control Optim. 50, 387–438 (1983) 2. L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2008) 3. A. Braides, -Convergence for Beginners (Oxford University Press, Oxford, 2002) 4. A. Braides, Local Minimization, Variational Evolution and -Convergence. Lecture Notes in Mathematics, vol. 2094 (Springer, Berlin, 2014) 5. A. Braides, M. Colombo, M. Gobbino, M. Solci, Minimizing movements along a sequence of functionals and curves of maximal slope. C. R. Acad. Sci. Paris Ser. I 354, 685–689 (2016) 6. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, vol. 5 (North-Holland Publishing, Amsterdam, 1973) 7. M. Colombo, M. Gobbino, Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow. Math. Models Methods Appl. Sci. 22(8), 1250017 (2012) 8. E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications, Masson, ed. by C. Baiocchi, J. L. Lions (1993), pp. 81–98

Chapter 3

Discrete-to-Continuum Limits of Planar Lattice Energies

Before dealing with their evolution, in this chapter we examine the ‘static’ limit of families of energies on lattices with vanishing spacing. This preliminary analysis is suggested by Theorem 2.1, which implies that a proper environment for the study of minimizing movements along sequences of functionals Eε may be provided by the computation of their -limit F , which we may always suppose exists up to subsequences. With this scope in mind, this chapter is devoted to the study of the -limit of energies defined on lattices at a space scaling which gives a surface energy in the limit. The corresponding minimizing movement for F will provide a reference geometric motion (motion by crystalline curvature), which will be defined and analyzed in the next chapter.

3.1 Energies on Sets of Finite Perimeter We will recall some classical results in Geometric Measure Theory, which allow to give a description of the continuum limit of discrete energies as perimeter functionals.1 Since our framework will be two-dimensional, we will specialize the description to that setting, even though most of the results of this chapter extend to arbitrary dimension with the due changes. Definition 3.1 If P is a polyhedron in R2 ; i.e., the boundary of P is a (disjoint) union of segments, then the perimeter of P is defined in an elementary way as the sum of the lengths of these segments and is denoted by Per(P ).

1 For proofs of the relevant results in this chapter and more information on sets of finite perimeter see e.g. [1].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Braides, M. Solci, Geometric Flows on Planar Lattices, Pathways in Mathematics, https://doi.org/10.1007/978-3-030-69917-8_3

31

32

3 Discrete-to-Continuum Limits of Planar Lattice Energies

A subset A in R2 is a set of finite perimeter if the perimeter of A, defined by   Per(A) = inf lim inf Per(Pj ) : |Pj A| → 0, Pj polyhedra j

is finite.2 Note that if ∂A is the disjoint union of piecewise-C 1 curves, then the perimeter of A is the sum of the lengths of these curves. In particular, for polyhedral sets the perimeter coincides with the perimeter elementary defined. The following theorem describes the compactness properties of sets of finite perimeter. Note that the convergence is local because of possible translations to infinity (e.g., we have to take into account of families Aε = A + rε with rε ∈ R2 and rε  → +∞, which converge locally to the empty set, but not on R2 ). Theorem 3.1 (Compactness) Let {Aε } be a family of subsets of R2 with Per(Aε ) ≤ C < +∞, then, up to extracting a subsequence, there exists a set A with finite perimeter such that Aε → A locally in R2 (i.e., for all R > 0 |(Aε A) ∩ BR (0)| → 0) and Per(A) ≤ lim inf Per(Aε ). ε→0

Now we can give a definition of convergence for spin function. Definition 3.2 (Convergence of Spin Functions to a Set) If ε > 0 and v : εZ2 → {0, 1} is a spin function defined on εZ2 then we define Aεv =

 i∈Z2 :v

(εQ + εi) .

(3.1)

i =1

For all ε > 0 let uε be a spin function defined on εZ2 and let A be a subset of R2 . We say that uε converge to A (and write uε → A) if the sets Aεuε defined by (3.1) converge to A; i.e., |Aεuε A| → 0, or, equivalently, χAεuε converge in L1 (R2 ) to χA . We can now specialize the definition of -convergence to the lattice setting as follows.

2 An

alternative way to define the perimeter is in a measure-theoretical way as   Per(A) = sup div ϕ dx : ϕ ∈ C01 (R2 ; R2 ), ϕ∞ ≤ 1 , A

which states that the characteristic function of A is a function of bounded variation. This characterization is very handy to prove functional-analytic properties of the space of sets of finite perimeter.

3.1 Energies on Sets of Finite Perimeter

33

Definition 3.3 (-Convergence on Lattices) The sequence Eε -converges to F at A if and only if (i) (liminf inequality) for all uε converging to A we have F (A) ≤ lim inf Eε (uε ); ε→0

(ii) (existence of a recovery sequence) there exist uε converging to A such that3 F (A) = lim Eε (uε ). ε→0

A family {uε } as in (ii) is called a recovery sequence for F (A). Now we recall some results concerning the structure of sets of finite perimeter, which will be necessary to better describe the functionals F obtained as -limits of lattice energies. Theorem 3.2 (Structure of Sets of Finite Perimeter) Let A be a set of finite perimeter. Then the following properties hold. (i) There exists a set denoted by ∂ ∗ A (the reduced boundary of A) such that Per(A) = H 1 (∂ ∗ A), where H 1 is the one-dimensional Hausdorff measure;4 (ii) (1-rectifiability of ∂ ∗ A) There exists a set N with H 1 (N) = 0 such that ∂ ∗A = N ∪

+∞ 

Kj ,

j =1

where each Kj is a compact subset of a C 1 -curve. Hence (by the implicitfunction theorem) there exists ν, the (internal) normal to ∂ ∗ A defined H 1 almost everywhere on ∂ ∗ A. The structure theorem above, in particular allows to define perimeter functionals as integrals on the reduced boundary as in the following definition. Note that the usual perimeter is obtained by taking the constant 1 as ϕ. Definition 3.4 (Perimeter Functionals) Let X = {A : A of finite perimeter}. A functional F : X → R is a (homogeneous) perimeter functional if it is given by F (A) =

3 Note

∂∗A

ϕ(ν(x)) dH 1(x),

(3.2)

η

that if (i) holds, in order to prove (ii) it suffices to show that for all η > 0 there exists uε η converging to A such that F (A) ≥ lim supε→0 Eε (uε ) − η. Indeed by a diagonal argument then we obtain a sequence uε converging to A that satisfies F (A) ≥ lim supε→0 Eε (uε ), which is a recovery sequence by (i). 4 The Hausdorff measure H 1 can be thought as the extension of the notion of length of a curve to arbitrary sets. Even though sets of finite perimeter may be very irregular, in our context it is sufficient to think of them as having a piecewise-regular boundary with finite length. In a d-dimensional setting the one-dimensional Hausdorff measure must be replaced by the d − 1dimensional Hausdorff measure.

34

3 Discrete-to-Continuum Limits of Planar Lattice Energies

where ν(x) is the internal normal to ∂ ∗ A at x and ϕ : S 1 → R is a continuous function.5 Since -limits are lower semicontinuous with respect to the convergence in which they are computed, we may restrict our analysis to lower-semicontinuous functionals, characterized in the following remark. Remark 3.1 (Lower-Semicontinuity Conditions) The functional F in (3.2) is lower semicontinuous with respect to the L1loc convergence if and only if ϕ defines a norm; i.e., ϕ ≥ 0 and (the one-homogeneous extension of) ϕ is convex; that is, the function z z → ϕ(z) = zϕ( z ) is convex. ♦ Remark 3.2 (Representation of Perimeter Functionals in Terms of Sublevel Sets) After extending ϕ : S 1 → R by one-homogeneity, a perimeter functional F given by (3.2) is characterized by the set Bϕ = {z ∈ R2 : ϕ(z) ≤ 1}, which is a convex set if F is lower semicontinuous. Identifying F with Bϕ is a handy way to describe our functionals in a pictorial way. • Example I (Euclidean perimeter). If ϕ(ν) ≡ 1, then F (A) = Per (A) and the 1-homogeneous extension of ϕ is ϕ(z) = z. We have that Bϕ is the unit (euclidean) ball centered in 0, that is Bϕ = B1 (0). • Example II (square-crystalline perimeter). If ϕ(ν) = ν1 , then the 1homogeneous extension is ϕ(z) = z1 and the set Bϕ is the square with vertices in ±(0, 1) and ±(1, 0). ♦ We introduce the following notion of a Wulff shape, which can be defined (up to a dilation) as a set minimizing a perimeter with given area, or conversely, maximizing the area with given perimeter (Dido’s problem). We prefer the second one as it is more convenient in the description of homogeneous perimeter functionals, which are completely described by their Wulff shape (which, in a sense, is a dual definition of the corresponding Bϕ ). The knowledge of Wulff shapes will also be used in the following to obtain a class of sets on which the description of the variational evolution is particularly relevant. Definition 3.5 (Wulff Shape of ϕ) Let F be defined as in (3.2). A Wulff shape of ϕ is a (convex) set Wϕ maximizing6 max{|A| : F (A) ≤ 1}.

define the Borel measure H 1 ∂ ∗ A by H 1 ∂ ∗ A(B) = H 1 (B ∩ ∂ ∗ A). The integral in (3.2) is performed with respect to that measure. 6 Equivalently, we may define Wulff shapes as W = B/F (B), where B is a solution of min{F (B) : ϕ  |B| ≥ 1}. Conversely, minimizers of this last problem are given by Wϕ / |Wϕ |. The equivalence of the two problems is obtained by using the fact that H 1 is positively one-homogeneous. 5 We

3.2 Limits of Homogeneous Energies in a Square Lattice

35

If ϕ is even; that is, ϕ(ν) = ϕ(−ν), then we can choose uniquely Wϕ by centering it at 0. We can characterize the Wulff shape of a norm ϕ as the unit ball Bϕ ∗ of the dual norm ϕ ∗ divided by the factor F (Bϕ ∗ ); that is,7 Wϕ =

  Bϕ ∗ with Bϕ ∗ = w ∈ R2 : max v, w ≤ 1 . F (Bϕ ∗ ) ϕ(v)=1

Referring to the examples of perimeter functionals in Remark 3.2 we have the following characterization of Wulff shapes. • Example I. If F (A) = Per(A), the problem characterizing Wϕ is the classical Dido’s problem, and the resulting Wulff shape for the corresponding ϕ(z) = z is given by the Euclidean ball of perimeter 1; i.e., Wϕ = B1/2π (0). 1 1 2 • Example II. If ϕ(z) = αz1 , the Wulff shape is the square [− 8α , 8α ] . Remark 3.3 (Inhomogeneous Perimeters) We have considered homogeneous perimeter functionals; more in general, we may consider inhomogeneous perimeter functionals of the form ϕ(x, ν) dH 1. F (A) = ∂∗A

For these, necessary conditions for the lower semicontinuity are more complicated to state, since we have to take into account the behaviour of ϕ on 1-rectifiable sets, but if ϕ is lower semicontinuous again convexity in ν is a sufficient condition. ♦

3.2 Limits of Homogeneous Energies in a Square Lattice We now turn to the description of the -limit of lattice energies in the case of nonnegative interactions (ferromagnetic case). The case when also negative interactions are taken into account is more complex and largely open, but some cases will be taken into account separately in the following.8 In this section we describe some easy examples of -convergence of ferromagnetic lattice systems; i.e., we consider energies Eε (u) =



ε aij |ui − uj |,

(3.3)

i,j ∈Z2

7 This

geometric characterization of the Wulff shape can be found for instance in [7, Th. 10.6].

8 We refer to the paper [2] for examples showing a variety of limit behaviours for non-ferromagnetic

systems, and also for systems with coefficient aij not with uniform range, but with a controlled decay in terms of i − j . Some systems with long-range interactions and not controlled decay can be treated by coarse graining [5].

36

3 Discrete-to-Continuum Limits of Planar Lattice Energies

Fig. 3.1 The set Aεu corresponding to a discrete 2 function u ∈ {0, 1}εZ

where u : εZ2 → {0, 1} and ui = u(εi). We will suppose that the coefficients aij be non-negative and uniformly of finite range (i.e., there exists R > 0 such that aij = 0 if j − i > R). Remark 3.4 (Coerciveness by Nearest-Neighbour Interactions) We recall that in the simplest case of nearest-neighbour interactions, when  aij =

1

if i − j  = 1

0

otherwise,

(3.4)

the value Eε (u) equals the perimeter of Aεu (see Fig. 3.1). This implies that for general energies Eε in (3.3) with coefficients satisfying aij ≥ c0 > 0 for all i, j such that i − j  = 1,

(3.5)

if {uε } is a sequence with equi-bounded energy Eε (uε ), the corresponding Aεuε have equi-bounded perimeter, so that they are precompact thanks to Theorem 3.1. Hence, we may suppose uε → A, and only describe the asymptotic behaviour of Eε when A is a set of finite perimeter. ♦ We have noticed that homogeneous perimeter functionals are described by their Wulff shapes. We now note that if aij are translation-invariant (or homogeneous) and coercive, then from the -convergence we can deduce an approximate description of Wulff shapes, as follows. Theorem 3.3 Let aij depend only on j − i, have finite range, and let aij > 0 if j − i = 1. If Eε -converge to a continuum energy F of the form (3.2) according to Definition 3.3, then there exists a family {uε } of minimizers of   min Eε (u) : #{i ∈ Z2 : ui = 1} ≥ ε−2 ,

3.2 Limits of Homogeneous Energies in a Square Lattice

37

which converges to a minimizer of min{F (A) : |A| ≥ 1} in the sense of Definition 3.2. Proof Note first that by the finite-range condition, by testing the minimum problems above with u the characteristic function of a unit cube, we obtain that the minimum values above are equibounded. Let uε be a minimizer of the problem above (which exists since Eε takes values on a set composed of isolated points). Consider the sets Aεuε in (3.1). Since our energies are invariant by translation then we can suppose that 0 ∈ Aεuε . Note that the εR-neighbourhood of Aεuε is connected; otherwise we may translate a connected component not containing 0 towards the origin until it it does not interact with another component, in this way decreasing the energy, which is a contradiction. By the condition aij > 0 if j − i = 1 we obtain that Aεuε are sets of equi-bounded perimeter. Since they are connected and contain the origin they are also contained in a fixed ball, so that they are precompact. The -convergence of Eε to F then ensures the convergence of minimizers, once we observe that the constraint #{i ∈ Z2 : ui = 1} ≥ ε−2 translates into |A| ≥ 1 for the limit sets.  

3.2.1 The Prototype: Homogeneous Nearest Neighbours We consider aij given by (3.4), for which the energy Eε is Eε (u) =



ε|ui − uj |

(3.6)

i,j

where i, j is a shorthand for nearest-neighbour interactions; that is, the sum is performed over indices i, j ∈ L such that i − j  = 1, with the convention that pairs of indices are accounted for only once. We have noted (see Remark 3.4) that Eε (u) equals the perimeter of Aεu . However, since the normal to ∂Aεu may only take the values ±e1 and ±e2 , in order to have a better lower bound it is convenient to view Eε (u) as the value of an anisotropic perimeter functional with a convex integrand ϕ. The only conditions to require on ϕ are that ϕ(±e1 ) = ϕ(±e2 ) = 1. The largest such ϕ is the 1-norm. Hence, it is convenient to write Eε (u) = ν1 dH 1 . ∂Aεu

We now show that this choice of ϕ is indeed sharp; i.e., - lim Eε (A) = F (A) := ε→0

∂∗A

ν1 dH 1 .

38

3 Discrete-to-Continuum Limits of Planar Lattice Energies

We have to show that the two conditions of Definition 3.3 are verified. (i) (lim inf inequality) Let uε → A; i.e., Aεu converges to A. By the lower semicontinuity of F (Remark 3.1) we have lim inf Eε (uε ) = lim inf F (Aεu ) ≥ F (A). ε→0

ε→0

(ii) (recovery sequence) Let A be a polyhedral set. Setting uε = χA in εZ2 , we obtain lim Eε (uε ) = F (A)

ε→0

(it suffices to note that for each segment of the boundary, the approximation by discretization gives exactly the sum of the lengths of the components of the segment). For a general A, we note that the polyhedral sets are dense in energy for F ; which means that there exist polyhedra Pj converging to A such that F (Pj ) → F (A). We then construct recovery sequences by discretization as above for each such Pj and conclude by using a diagonal argument. Remark 3.5 Note that if we scale our lattice by a factor λ > 0; i.e., we consider the same energies Eε , but defined on v : ελZ2 → {0, 1}, then Eε (u) equals λ1 Per Aελ v , according to definition (3.1), and hence the limit is scaled by the factor λ1 . ♦

3.2.2 Next-to-Nearest Neighbour Interactions We consider the functionals given by Eε (u) =



ε|ui − uj |

i,j

where i, j

denotes the family of nearest neighbours and next-to-nearest neighbours in Z2 , the latter corresponding to ‘cross links’ on the square lattice; √ that is, the sum is performed over indices i, j ∈ L such that i − j  ≤ 2, with the convention that pairs of indices are accounted for only once. The network of interactions is represented in Fig. 3.2, where, in the picture on the right-hand side, we have highlighted the sites interacting with a given site. The energy corresponding to next-to-nearest-neighbour interactions can √ be viewed as the superposition of two lattice energies on lattices of spacing 2ε. 2 2 Hence, we may introduce Eε1 , Eε,odd and Eε,even as nearest-neighbour interaction energies in the three lattices in Fig. 3.3, respectively, and write 2 2 Eε (u) = Eε1 (u) + Eε,odd (u) + Eε,even (u).

(3.7)

3.2 Limits of Homogeneous Energies in a Square Lattice

39

Fig. 3.2 Scheme of next-to-nearest neighbour interactions

Fig. 3.3 Decomposition of next-to-nearest-neighbour interactions into three families of nearestneighbour interactions

√ 2 2 Note that the lattices corresponding to Eε,odd and Eε,even are scaled by 2 and rotated of π/4 with respect to the initial lattice Z2 . Setting d1 = e1√+e2 and d2 = e1√ −e2 , 2

2

and noting that the 1-norm of ν in the reference frame rotated by π/4 is 1 | ν, d1 | + | ν, d2 | = √ (|ν1 + ν2 | + |ν1 − ν2 |) 2 1 1 = √ max{|ν1 |, |ν2 |} = √ ν∞ , 2 2

recalling Remark 3.5 with λ = 2 Eε,even is 1 √ 2

√

2 2, we have that the -limit of both Eε,odd and



(| ν, d1 | + | ν, d2 |) dH = 1

∂∗A

∂∗A

ν∞ dH 1 .

(3.8)

40

3 Discrete-to-Continuum Limits of Planar Lattice Energies

We can now proceed to the computation of the -limit on sets of finite perimeter. (i) (lim inf inequality) If uε → A note that also the interpolations of uε in the 2 2 lattices corresponding to Eε,odd and Eε,even converge to the same A (for the details, see the general case, Proposition A.1).9 Hence, we can separately estimate the contribution of each term as in Sect. 3.2.1, regarding them as 2 nearest-neighbour interaction energies. Since the contributions of Eε,odd and 2 Eε,even coincide, we finally obtain 2 2 (uε ) + lim inf Eε,even (uε ) lim inf Eε (uε ) ≥ Eε1 (uε ) + lim inf Eε,odd ε→0

ε→0

≥

∂∗A



ν1 dH 1 + 2

∂∗A

ε→0

ν∞ dH 1.

(ii) (recovery sequence) Exactly as in Sect. 3.2.1, given a polyhedral set A we choose the recovery sequence uε = χA , and for a general A, we use the density and a diagonal argument. Note that in general the -limit of a sum of energies is not the sum of the -limits since the constructions of recovery sequences for the different terms may be incompatible. In this ‘fortunate case’, however, the choice of the recovery sequence for nearest-neighbour interactions in Sect. 3.2.1 is independent of the lattice, so that the same sequence is simultaneously a recovery sequence for the three lattices (for more details, again see Proposition A.1). This shows that - lim Eε (A) = F (A) := ε→0

∂∗A

(ν1 + 2ν∞ ) dH 1.

We can compute the set Bϕ and the Wulff shape Wϕ for ϕ(z) = z1 + 2z∞ , obtaining that Bϕ is the equilateral octagon with vertices (± 13 , 0), (0, ± 13 ) (± 14 , ± 14 ). The corresponding Wulff shape is the octagon with vertices (±1, ±3), 1 (±3, ±1) scaled by the factor 56 (see Fig. 3.4). To check this, note that for the computation of the Wulff shape we have to maximize the linear function tx + sy over the pairs (t, s) ∈ ∂Bϕ ; the maximum is then attained at one of the vertices of ∂Bϕ . By symmetry, it is sufficient to consider the set {(x, y) : x ≥ y > 0} and determine the set where max{ 13 x, 14 (x + y)} = 1. Hence, we obtain x ≥ 3y, so that x = 3 and 3y ≥ x ≥ y > 0, which gives x + y = 4. The perimeter F of the resulting octagon is 56, which gives the scaling factor.

9 This

can be seen noting that the limit of such interpolations correspond to the weak limit of Aεuε intersected with the odd or even ε-checkerboard, using the strong convergence of Aεuε and the weak convergence of the latter.

3.2 Limits of Homogeneous Energies in a Square Lattice

41

Fig. 3.4 The sets Bϕ and Wϕ for ϕ(z) = z1 + 2z∞

3.2.3 Directional Nearest-Neighbour Interactions We now consider a nearest-neighbour system where horizontal and vertical interactions have different strengths. With fixed 0 < α < β < +∞, define aij =

 α β

if i − j = ±e1 if i − j = ±e2 ,

and the nearest-neighbour energies Eε (u) =



εaij |ui − uj |.

(3.9)

i,j

Note that this is a homogeneous energy: i.e., the value of aij depends only on j − i, with horizontal bonds (corresponding to vertical interfaces, in turn corresponding to a horizontal normal) of strength α, and vertical bonds (corresponding to horizontal interfaces, in turn corresponding to a vertical normal) of strength β (see the left-hand side picture in Fig. 3.5). We have that - lim Eε (A) = F (A) := (α|ν1 | + β|ν2 |) dH 1. (3.10) ε→0

∂∗A

The proof is exactly the same as that of the -limit in Sect. 3.2.1 with ϕ(z) = α|z1 |+ β|z2 |. 1 The Wulff shape is a (coordinate) rectangle with length of the vertical side 4α 1 and of the horizontal side 4β (see the right-hand side picture in Fig. 3.5).

42

3 Discrete-to-Continuum Limits of Planar Lattice Energies

Fig. 3.5 Directional energy interactions and the corresponding Wulff shape

3.2.4 General Form of the Limits of Homogeneous Ferromagnetic Energies The limit procedure highlighted in the previous examples can be used to prove a general homogenization theorem when (with a slight abuse of notation) aij = aj −i for some {ak }. Before stating the corresponding -limit result, we note a couple of technical facts. Remark 3.6 If in the computation of the -limit we can limit our analysis to uε → A such that the perimeters of Aεuε are equibounded, then we may treat also degenerate coefficients. In particular in Sect. 3.2.3 we may take α = 0 and obtain  β ∂ ∗ A |ν2 | dH 1 as a limit on sets of finite perimeter.10 ♦ Remark 3.7 If uε : εZ2 → {0, 1} converge to A according to Definition 3.2, then the restrictions of uε on any square sub-lattice εL of εZ2 (in particular for the sublattices defined in Sect. 3.2.2) still converge to the same A if the definition of Aεuε is given with the periodicity cube11 of L in place of Q. We refer to Sect. 3.4 for details in a general lattice. ♦ Theorem 3.4 Let aij = aj −i = ai−j ≥ 0 describe a system of interactions with finite range and let ae1 ae2 > 0. Then the functionals12    Eε (u) = εaij |ui − uj | = εak |ui − ui+k | (3.11) i,j ∈Z2

i∈Z2 k∈Z2

check this, in the liminf inequality we may add and subtract ηPer(Aεuε ), and use the result of Sect. 3.2.3 and the arbitrariness of η > 0, while the construction of recovery sequences remains unchanged. 11 This can be seen using the strong convergence of u and the weak convergence of the ε (interpolations of the) characteristic functions of εL. 12 Note that, contrary to the case of previous examples, in the definition of E below pairs of indices ε are accounted for twice. 10 To

3.2 Limits of Homogeneous Energies in a Square Lattice

43

-converge to F (A) :=

∂∗A

ϕ(ν) dH 1, where 

ϕ(ν) =

ak | ν, k |

(3.12)

k∈Z2

is a crystalline energy density; i.e., the corresponding Wulff shape is a (symmetric) convex polyhedron. Proof Recall that by the coerciveness of the energies we can compute the lim inf inequality on sequences uε with Aεuε with equibounded perimeters (see Remark 3.4). For any k = (k1 , k2 ) ∈ Z2 let k ⊥ = (k2 , −k1 ) and let Qk = [0, 1)k × [0, 1)k ⊥ be the half-open square with two sides k and k ⊥ , and for all  ∈ Qk consider 

Eεk, (u) =

εak |ui − ui+k |,

i∈Z2 +

so that Eεk (u) :=



εak |ui − ui+k | =

i∈Z2



Eεk, (u).

∈Qk

The functionals Eεk, can be interpreted as nearest-neighbour energies on a translation of the square lattice generated by k and k ⊥ and the coefficient in direction k ⊥ equal to 0. By Remarks 3.6 and 3.5 their -limit is 1 k



 k   ak  ν,  dH 1. ∗ k ∂ A

Note that #Qk ∩ Z2 = k2 , so that if uε → A then recalling Remark 3.7 lim inf Eε (uε ) ≥ ε→0

  k∈Z2 ∈Qk

≥

  k∈Z2 ∈Qk

=



k∈Z2

∂∗A

lim inf Eεk, (uε ) ε→0

1 k2

∂∗A

ak | ν, k | dH 1

ak | ν, k | dH 1 .

The proof is concluded after remarking that we may take the same recovery sequence for any k and  as in Sect. 3.2.2.

44

3 Discrete-to-Continuum Limits of Planar Lattice Energies

We finally note that Bϕ is a symmetric convex polyhedron since the sum in k is finite, which implies that such is also Wϕ .  

3.3 Limits of Inhomogeneous Energies in a Square Lattice In this section we consider inhomogeneous interactions; i.e., such that aij does not depend on j − i only. While homogeneous energies can be directly seen as a discrete decomposition of a crystalline perimeter, inhomogeneous energies require the construction of optimal discrete interfaces highlighting a ‘homogenization process’ in the discrete-to-continuum procedure.

3.3.1 Layered Interactions A slightly more complex arrangement of interactions than that of Sect. 3.2.3 is with ⎧ ⎪ ⎪ ⎪α1 ⎪ ⎨β 1 aij = ⎪α1 ⎪ ⎪ ⎪ ⎩ β2

if j − i = e1 and j1 even if j − i = e1 and j1 odd if j − i = e2 and j2 even if j − i = e2 and j2 odd,

and aij = aj i if j − i ∈ {−e1 , −e2 }. In this case horizontal bonds are of alternating strength α1 and β1 , and vertical bonds of alternating strength α2 and β2 . If we let γ1 =

α1 + β1 , 2

γ2 =

α2 + β2 , 2

then for the nearest-neighbour energies (3.9) we still can describe the -limit as - lim Eε (A) = F (A) := ε→0

∂∗A

(γ1 |ν1 | + γ2 |ν2 |) dH 1 .

(3.13)

To check this, note that an interface with average normal ν crosses at least as many α1 and β1 -bonds with proportion ν1 and at least as many α2 and β2 -bonds with proportion ν2 , so that we have the liminf inequality in (3.13), while a recovery sequence for a polyhedral set is still the same as in the previous constructions above. Again, the Wulff shape is a rectangle. As a particular case, we may take α1 = α2 = α,

β1 = β2 = β.

3.3 Limits of Inhomogeneous Energies in a Square Lattice

45

Fig. 3.6 An interface in a layered structure

This situation is pictured in Fig. 3.6, where aij =

 α

on lighter bonds

β

on darker bonds,

and an interface is pictured, highlighting zones with strength α (dashed lines) and β (solid lines). In this case we get - lim Eε (A) = ε→0

α+β 2

∂∗A

ν1 dH 1 ; i.e., ϕ(z) =

α+β z1 , 2

and the corresponding Wulff shape Wϕ is the (coordinate) square with side length 1 2(α+β) . Note that these interactions can be seen as a ‘rearrangement’ of the interaction in the example in Sect. 3.2.3, but give a different -limit.

3.3.2 Alternating Nearest Neighbours (‘Hard Inclusions’) We now construct a second example of inhomogeneous interactions, in which again we have an equal number of α-bonds and β-bonds. Differently from the previous cases, now recovery sequences will have to satisfy some (very simple) optimality conditions; namely, they will ‘avoid’ the stronger connections. Fix 0 < α < β < +∞ and define ⎧ ⎪ ⎪ ⎨α aij = α ⎪ ⎪ ⎩β

if i − j  = 1, i1 = j1 and max{i2 , j2 } is even if i − j  = 1, i2 = j2 and max{i1 , j1 } is even otherwise.

46

3 Discrete-to-Continuum Limits of Planar Lattice Energies

Fig. 3.7 A path avoiding ‘hard inclusions’

and consider the inhomogeneous nearest-neighbour interaction (3.9). A pictorial representation of the interactions is given in Fig. 3.7 where aij =

 α

on lighter bonds

β

on darker bonds.

We now prove that - lim Eε (A) = F (A) := α ε→0

∂∗A

ν1 dH 1 ;

i.e., the limit integrand is ϕ(z) = αz1 , independent of β. Indeed, the liminf inequality trivially holds by the computation in Sect. 3.2.1 since α < β. Conversely, for a target polyhedral set, note that we can take interfaces avoiding the (stronger) β-connections (discretizing the set A on a grid of double lattice spacing). As for homogeneous nearest-neighbour interactions the 1 corresponding Wulff shape Wϕ is the (coordinate) square with side length 4α .

3.3.3 Homogenization and Design of Networks We note that in the last example above the optimal interface, however simple, is not trivial and follows a path of least energy. This is a first example of homogenization. Even though we will not use it in the following, we state a general result for the homogenization of periodic lattice energies. In order to avoid some technical assumptions, we simply deal with the case of systems with a finite range of interaction. This condition also ensures that the limit perimeter is crystalline, which implies that such perimeters are a ‘natural’ framework for the approximation of general lattice systems.

3.3 Limits of Inhomogeneous Energies in a Square Lattice

47

Theorem 3.5 (Homogenized Crystalline Perimeters13 ) Let aij be of finite range, satisfy (3.5), and be periodic; that is, there exists a period T ∈ Z such that ai+T ek j +T ek = aij for any i, j ∈ Z2 and k = 1, 2. Let Eε be defined by Eε (u) =



εaij |ui − uj |.

i,j ∈Z2

Then there exists a convex and even ϕ : R2 → [0, +∞), positively homogeneous of degree one, such that - lim Eε (A) = ε→0

∂∗A

ϕ(ν) dH 1.

(3.14)

Moreover, ϕ is crystalline; that is, the Wulff shape Wϕ is a polygon (symmetric with respect to the origin). In general, the function ϕ is given by an asymptotic minimal-interface formula, which can also be interpreted as a minimal-path formula in the case of nearestneighbour interactions only. A particular case is when aij depend only on i − j (or, equivalently, the period T = 1), which is the case treated in Theorem 3.4. An application of homogenization theory is the design of optimal structures. An example in the case of design of networks is considering mixtures of two types of bonds (α and β-bonds). As we have seen above, we may have cases when these bonds appear in the same proportion, but the -limits have different forms, depending on the arrangement of the connections. A first step towards optimal design is the description of homogenized energies with a given proportion of such bonds. We only give an example of such a description when the proportions are equal, which shows a nice way of using the representation of functionals F through their Wulff shapes.14 Theorem 3.6 (Design of Periodic Networks) The closure of the set of all (symmetric polyhedral) Wulff shapes of functionals F obtained as the homogenization of nearest-neighbour interaction energies Eε (u) =



ε aij |ui − uj |,

i,j

13 This is a ‘classical’ type of homogenization result, proved by characterizing optimal interfaces through homogenization formulas, for which we refer to [4]. A proof of the fact that ϕ is crystalline is due to Chambolle and Kreutz [6]. 14 We give a statement only for the case of equal proportion of α and β-bonds. A similar statement hold for a given proportion of α-bonds in (0, 1), with a different equation of the curve in (3.15). The interested reader can find an elementary proof of this result in [3].

48

3 Discrete-to-Continuum Limits of Planar Lattice Energies

Fig. 3.8 A possible limit of Wulff shapes satisfying a design constraint

where {aij } is a periodic family with aij ∈ {α, β} and such that #{aij = α} = #{aij = β} (in the period) is the family W(α, β) of all convex sets, symmetric with respect to the origin and intersecting the four regions of the plane x-y delimited by the curve 1 1 + = 8(α + β), |x| |y| and the coordinate square with side length

1 4α

(3.15)

(the grey zones in Fig. 3.8).

This theorem states that for any Wuff shape W ∈ W(α, β) and for all η > 0 we can find a period T , T -periodic coefficients {aij } with an equal number of α and β-bonds (in the period) such that the (polyhedral) Wulff shape of the corresponding homogenized ϕ satisfies |W Wϕ | ≤ η. Note that the largest Wulff shape in 1 W(α, β) is the coordinate square with side length 4α (obtained in Sect. 3.3.2 with ‘hard inclusions’), while the smallest square Wulff shape in W(α, β) is obtained by taking x = y > 0 in (3.15), which gives the coordinate square with side length 1 2(α+β) (obtained in Sect. 3.2.3 by a layered structure).

3.4 Limits in General Planar Lattices by Reduction to the Square Lattice Many of the results obtained for the square lattice can be extended to arbitrary (planar) lattices. Now we consider the case of a discrete lattice L ⊂ R2 spanned over Z by two independent vectors, for which we can refer to the results in Z2 by a linear change of variable.

3.4 Limits in General Planar Lattices by Reductionto the Square Lattice

49

The convergence of a sequence uε ∈ {0, 1}εL is defined in analogy with Definition 3.2. Given a basis {w1 , w2 } of the lattice L, we denote by K the dual cell given by K = {x ∈ R2 : x = tw1 + sw2 , t, s ∈ [0, 1]} −

w1 + w2 2

and we identify each uε ∈ {0, 1}εL with the set Kuεε =



(εK + εi).

i∈L uεi =1

We say that uε → A, where A is a set of finite perimeter, if |Kuεε A| → 0 as ε → 0. Note that this notion of convergence does not depend on the choice of the basis. Let Eε be defined in {0, 1}εL by Eε (u) =



εaij |ui − uj |,

(3.16)

i,j ∈L

where aij satisfy the corresponding hypotheses to those in Sect. 3.2.4. Then the limit of Eε is of the form (3.14) with ϕ(ν) =

cL  a0i | ν, i |. 2

(3.17)

i∈L

and cL = | det(w1 , w2 )|−1 (note that this coefficient depends on L and not on the choice of the basis). The proof of this result is given in Appendix A.1 (Proposition A.1). Example 3.1 (Nearest-Neighbour Interactions on the Triangular Lattice) As an example we now consider a nearest-neighbour system on a triangular lattice, which corresponds to an  ‘asymmetric’ next-to-nearest  neighbour system on the square lattice. Let vn = cos((n−1) π3 ), sin((n−1) π3 ) for n = 1, 2, 3, and L = Zv1 +Zv2 ; the coefficients aij are given by  aij =

1

if i − j ∈ {±v1 , ±v2 , ±v3 }

0

otherwise.

50

3 Discrete-to-Continuum Limits of Planar Lattice Energies

Fig. 3.9 Interactions, Bϕ and Wϕ for the triangular lattice

Fig. 3.10 Interactions, Bϕ and Wϕ in Z2 for an asymmetric system on the square lattice corresponding to the triangular lattice (analog of Fig. 3.9)

Since cL =

√2 , 3

by (3.17) we get 2  ϕ(ν) = √ | ν, vn |. 3 n=1 3

The set Bϕ and the Wulff shape Wϕ are represented in Fig. 3.9. Note that the change of variable carries the triangular lattice in a square lattice with asymmetric next-to-nearest neighbour interactions for which (3.12) gives ϕ(ν) = |ν1 | + |ν2 | + |ν1 + ν2 | = ν1 + |ν1 + ν2 |. The ball Bϕ and the Wulff shape Wϕ are represented in Fig. 3.10.

♦

References 1. A. Braides, Approximation of Free-Discontinuity Problems. Lecture Notes in Mathematics, vol. 1694 (Springer, Berlin, 1998) 2. A. Braides, M. Cicalese, Interfaces, modulated phases and textures in lattice systems. Arch. Ration. Mech. Anal. 223, 977–1017 (2017)

References

51

3. A. Braides, L. Kreutz, Design of lattice surface energies. Calc. Var. Partial Diff. Equ. 57, 1485– 1520 (2018) 4. A. Braides, A. Piatnitski, Homogenization of surface and length energies for spin systems. J. Funct. Anal. 264, 1296–1328 (2013) 5. A. Braides, M. Solci, Compactness by coarse-graining in long-range lattice systems. Adv. Nonlin. Studies 20(4), 783–794 (2020) 6. A. Chambolle, L. Kreutz. Personal communication (2020) 7. F. Morgan, Riemannian Geometry. A Beginner’s Guide (AK Peters, CRC Press, Boca Raton, 1998)

Chapter 4

Evolution of Planar Lattices

In this chapter we describe some evolutions in the plane using the method of minimizing movements along families of ferromagnetic energies as those studied in the previous chapter. As shown in Theorem 3.5 (and also Theorem 3.4) the discreteto-continuum -limits of such energies are crystalline perimeters. In Sect. 4.1 we first describe motion by square crystalline curvature, which is obtained as a minimizing movement for the square perimeter (flat flow) using a dissipation D introduced by Almgren and Taylor.1 This evolution justifies the definition of discrete dissipations Dε introduced in Sect. 2.2.1 and provides a natural environment for a general evolution of lattices whose energies are asymptotically described by a square perimeter in the extreme regime described in Theorem 2.1(a). In the final and most important section of this chapter we will examine a number of energies which have the square perimeter as a -limit but whose minimizing movements are influenced in different ways by the local arrangements of interactions, showing how local minimization may influence the evolution through homogenization and pinning effects.

4.1 Flat Flows We now describe two geometric evolutions obtained as minimizing movements from crystalline perimeters which will be comparison ‘extreme’ evolutions for evolutions on lattices.

1 Motion by crystalline curvature is an active and exciting research topic, which requires complex notions of evolving sets. We refer the interested reader e.g. to [6, 7]. In the two-dimensional case the analysis often reduces to the description of a finite number of sides of a polygon, which makes instead this evolution easier to describe as a system of ODEs.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Braides, M. Solci, Geometric Flows on Planar Lattices, Pathways in Mathematics, https://doi.org/10.1007/978-3-030-69917-8_4

53

54

4 Evolution of Planar Lattices

4.1.1 Flat Flow for the Square Perimeter In this section we analyze an evolution obtained taking as energy the 1-crystalline perimeter; i.e., the anisotropic perimeter given by Per1 (A) =

∂∗A

ν1 ds

on sets of finite perimeter, where ν is the (inward) normal to ∂ ∗ A and ν1 = |ν1 | + |ν2 |. Remark 4.1 (Comparison with ‘Coordinate Rectangles’) We note that if (a piece of) the boundary of the set A can be parametrized by a curve γ = (γ1 , γ2 ) with γ1 and γ2 both monotone functions, then

ν1 ds = γ

γ 

ν1 ds =

ν1 ds, γ

where  γ and γ are the coordinate curves (i.e. curves following two segments parallel to the axes) with the same endpoints as γ (pictured in Fig. 4.1). We then deduce that Per1 (A) ≤ Per1 (R), where R is the least rectangle with sides parallel to the coordinate axes containing A up to negligible sets. ♦ We consider the minimizing-movement scheme for Per1 with the dissipation D given by D(A, B) = AB

dist∞ (x, ∂B) dx

for B smooth enough. The choice of dist∞ instead of the Euclidean distance in R2 is due to duality arguments. We already noted that choosing a different distance will only result in a positive constant (mobility factor). The resulting evolution is called Fig. 4.1 Three curves with the same L1 -length

4.1 Flat Flows

55

a flat flow, which may be described as a motion by crystalline curvature.2 In the next sections we briefly describe the features of motion by crystalline curvature that are relevant to our analysis.

4.1.2 Motion of a Rectangle In order to explain how motion by crystalline curvature is defined, we consider the example where the initial datum A0 is a rectangle. We fix τ > 0 and describe the orbit Aj = Aτj obtained by minimizing A → Per1 (A) +

1 D(A, Aj −1 ) τ

(4.1)

iteratively. Remark 4.2 (Minimizers Are Rectangles) We first note that the minimizer A1 is a rectangle included in A0 with the same center. Indeed, this is a consequence of the fact that, for test sets A, we have • A is contained in A0 . Indeed if A ⊂ A0 both the perimeter and the dissipation decrease by taking A ∩ A0 in the place of A; • (each connected component of) A is a coordinate rectangle. Indeed by Remark 4.1 replacing each connected component by the corresponding minimal coordinate rectangle we decrease the dissipation and we do not increase the 1perimeter. This argument can be repeated if (the closure of) two such rectangles intersect; • A (has a unique connected component which) is centered at the center of A0 . Indeed, if (a connected component of) A is a coordinate rectangle with a different center, there exists a vertical (or horizontal) translation reducing the distance between the centers such that the translated set does not intersect any other connected component, and has a strictly lower dissipation, as pictured in Fig. 4.2. This also shows that A can only have one connected component. By induction we deduce that minimizers Aj of (4.1) are rectangles centered in the centre of A0 and decreasing with respect to inclusion. ♦ We denote by Lj h and Lj v the lengths of the horizontal and vertical edges of Aj , respectively. We now describe L1h and L1v in terms of L0h and L0v .

2 Almgren and Taylor in [1] show that it coincides with the motion by crystalline curvature as previously defined by Taylor in [10].

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4 Evolution of Planar Lattices

Fig. 4.2 A translation decreasing the dissipation

Fig. 4.3 The initial datum A0 and the minimizer A1

Remark 4.3 We remark that the contribution of the corners (highlighted in gray in Fig. 4.3) to the dissipation is negligible. We first note that Lh =√L0h − L1h = o(1)τ →0 , and the same for Lv , from which it follows that Lh ≤ τ and Lv ≤ √ τ for τ small enough. Indeed, 1 D(A1 , A0 ) τ  L 2 1  L 2 1 h v ≥ (L0v − Lv ) √ + (L0h − Lh ) √ . 4 4 τ τ

2(Lh + Lv ) = Per1 (A0 ) − Per1 (A1 ) ≥

Hence, the contribution of the corners is O(τ 3/2)τ →0 , and it is negligible. The same argument can be applied to the description of Aj for any index j , so that at every step the contribution of the corners can be neglected in the computation of the side lengths of Aj . ♦

4.1 Flat Flows

57

Remark 4.4 (Characterization of Aj ) Up to the negligible terms corresponding to the contributions in the corners (see the previous remark), we have to minimize 2(L1h + L1v ) +

 1 L1h (L0v − L1v )2 + L1v (L0h − L1h )2 4τ

with respect to L1h and L1v . By neglecting terms of order higher than τ in the expansion of the solution, we get the two equations 4 L1h − L0h =− τ L1v 4 L1v − L0v =− . τ L1v Finally, we have the equations Lj h − L(j −1)h 4 =− τ Lj v Lj v − L(j −1)v 4 =− τ Lj h

(4.2)

by applying the same argument to each j in the iterative minimization procedure. ♦ Remark 4.5 (Passage to the Limit) Following Sect. 2.1.2, we define the timediscrete lengths given by Lτh (t) = Lt /τ h and Lτv (t) = Lt /τ v . Since the sets Aj are decreasing, so are the time-discrete lengths, and there exist limit functions denoted by Lh (t) and Lv (t), respectively, that are the horizontal and vertical lengths of a limit time-depending rectangle (the minimizing movement). Then, the equations describing Lj h and Lj v can be written as Lτh (t + τ ) − Lτh (t) 4 =− τ τ Lv (t + τ ) 4 Lτv (t + τ ) − Lτv (t) =− τ . τ Lh (t + τ ) Passing to the limit as τ → 0, we get the system of ODE ⎧ 4 ⎪ ⎨(Lh ) = − Lv 4 ⎪ ⎩(Lv ) = − Lh describing the motion of the limit rectangle, as long as Lh and Lv are positive.

(4.3)

♦

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4 Evolution of Planar Lattices

Definition 4.1 (Crystalline Curvature) The 1-crystalline curvature of a side of a rectangle with length L is given by κ = L2 . We can finally reread Eqs. (4.3) as the property that each side with length L moves inward with velocity v equal to its crystalline curvature; that is, v = −κ

(motion by 1-crystalline curvature)

(4.4)

after noting that the derivative of the length of each side is twice the velocity of the corresponding orthogonal side. In (4.4) we have adopted the notation that the velocity is negative if the side moves inwards.

4.1.3 Motion of a General Set In this section we describe motion by 1-crystalline curvature for general sets. The proof that the flat flow for the 1-perimeter for general sets is characterized as a motion by 1-crystalline curvature is rater complex, with several technical points beyond the scope of these notes, a delicate one being that a connected set remains connected during the evolution. The main point in this section is highlighting that flat flow is determined by the motion of rectangles. We first consider the case of A a coordinate polyrectangle; i.e., a finite union of coordinate rectangles. In this case the boundary sides of A can be of three types: convex (i.e., A is convex in a neighbourhood of the side), concave (i.e., the complement of A is convex in a neighbourhood of the side), or neutral. In the three cases, we may define the crystalline curvature as ⎧ 2 ⎪ ⎪ ⎨L κ = − L2 ⎪ ⎪ ⎩0

if the side is convex if the side is concave

(4.5)

if the side is neutral.

If the initial datum A0 is a coordinate polyrectangle then, arguing locally as in the case of a rectangle, we may show that each element of an orbit Aτj is still a polyrectangle, and the same type of computations as in the previous section can be carried on, to conclude that the minimizing movement is a time-depending polyrectangle the motion of whose sides are still described by (4.4). This means that convex sides move inwards, concave sides move outwards with velocity equal to 2/L (L being the corresponding side length), while neutral sides do not move. In Fig. 4.4 we picture a polyrectangle with the corresponding velocities of the sides. This description of the motion is valid until ‘extinction’ or until a time t0 when two sides collapse into one, in which case the motion must be ‘updated’ considering the set at time t0 as initial datum.

4.1 Flat Flows

59

Fig. 4.4 A polyrectangle and its curvature-driven velocities

Fig. 4.5 A generic set and its curvature-driven velocities

Fig. 4.6 Motion from a ball as initial datum (sets at different times)

The 1-crystalline curvature for a (quite) general set A can be understood as a limit case of polyrectangles, so that at a point where the normal does not point in a coordinate direction we have 1-crystalline curvature equal to 0. A side with normal pointing in a coordinate direction and side length L has curvature ±2/L with the sign corresponding to a convex/concave geometry. This includes also the degenerate case with L = 0 and infinite curvature. A pictorial description is given in Fig. 4.5. A minimizing movement from a general set will consist in a motion of the sides pointing in the coordinate directions (and possibly degenerating to a point), while the rest of the boundary of the set does not move. For instance, if A0 is a ball, then we have an immediate creation of four sides, which then move inwards until they touch creating a square, and then the set moves as described in the previous section (see Fig. 4.6).

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4 Evolution of Planar Lattices

4.1.4 An Example with Varying Initial Data In the rest of this chapter we will compute minimizing movements along families of lattice energies parameterized by a lattice dimension ε. By the definition of these energies, the initial data will belong to domains varying with ε. In our problems those initial data will be chosen as discretization of a given regular set, and the dependence of the process on ε will be all determined by the details of the energy. We note however that in general, the dependence on the initial data can greatly influence the evolution, even in the case of a single energy. An example is given in Remark 2.2, where we may chose initial data that are local minima (starting from which the evolution is trivial) converging to an initial datum from which the evolution is not trivial. We now give a similar example in the case of the 1-crystalline perimeter with varying initial data Aτ0 converging to some A0 and show that in this case the resulting minimizing movement may depend on the speed of convergence of Aτ0 to A0 . Note that this may be seen as an example of minimizing movements along a sequence of energies, where the dependence on a further parameter ε is indeed only in the initial data (which can be seen as Aε0 rather than Aτ0 ). Fixed L > 0 and dτ > 0 such that lim dτ = 0, for i = 1, . . . , 4 we define the τ →0

points Ciτ =

L 2

+ dτ

 0 −1i−1 1 . 1 1 0

The initial datum is given by Aτ0 =

4   i=1

−

 L L 2 , + Ciτ ; 2 2

that is, Aτ0 is the union of four unit coordinate squares as pictured in Fig. 4.7. Fig. 4.7 The initial datum Aτ0

4.1 Flat Flows

61

Now we study the evolution problem for Aτ0 . The same arguments as in Sect. 4.1.2 allow to deduce that a minimizer of Per1 (A) +

1 D(A, Aτ0 ) τ

(4.6)

is either the union of four disjoint squares or a coordinate square symmetric with respect to the origin. In detail, we have one of the following situations. Case 1 The minimizer Aτ1 is given by 4  

−

i=1

2  L L + δ, − δ + Ciτ 2 2

(4.7)

for some√δ ∈ (0, L) (see Fig. 4.8). Note that, as in Remark 4.3, we can assume δ = O( τ )τ →0 . In this case, neglecting the contribution of the corners and the higher-order terms as in Remark 4.3, we have that the minimum is achieved for δ = 2τ L + o(τ )τ →0 , and Per1 (Aτ1 ) +

 1 2τ  D(Aτ1 , Aτ0 ) = 16 L − + o(τ )τ →0 . τ L

(4.8)

Case 2 The minimizer Aτ1 is a coordinate square symmetric with respect to the origin and included in [−L − dτ , L + dτ ]2 , that is Aτ1 = [−L − dτ + , L + dτ − ]2 for some  ∈ (0, L + dτ ) (see Fig. 4.9). In this case, the minimum is achieved for  = τ + o(τ )τ →0 , where τ =

2τ +dτ2 2L .

We now show that if dτ is small enough with respect to τ , then the minimizer is a square (Case 2), while if dτ is large then we are in Case 1. We first note that √ dτ = o( τ )τ →0 ⇒ a minimizer of (4.6) is a square.

Fig. 4.8 A disconnected minimizer and the detail of the first quadrant

(4.9)

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4 Evolution of Planar Lattices

Fig. 4.9 A connected minimizer and the detail of the first quadrant

Indeed, let A be the union of four coordinate squares each one included in one of the initial squares; that is, A=

4   i=1

−

2  L L + δ, − δ + Ciτ 2 2

for some δ ∈ (0, L). Setting !2 A = − L − dτ + δ, L + dτ − δ , we obtain 1 1 D(A , Aτ0 ) − Per1 (A) − D(A, Aτ0 ) τ τ  (L − δ)dτ2 + dτ3  . ≤ 4 − 2L + 6δ + 2dτ + τ √ Recalling that if A is a minimizer √ then δ = O( τ )τ →0 (see Remark 4.3), this inequality shows that if dτ = o( τ )τ →0 , then A cannot be a minimizer. Indeed, we get Per1 (A ) +

Per1 (A ) +

1 1 D(A , Aτ0 ) − Per1 (A) − D(A, Aτ0 ) ≤ −8L + o(1)τ →0 , τ τ

which proves that the square A is more convenient than A, giving a contradiction. We now prove that √ √ dτ = c τ with c ∈ (0, 2) ⇒ a minimizer of (4.6) is a square.

(4.10)

4.1 Flat Flows

63

√ To see this, we assume that dτ = c τ √ for some c > 0 and prove that a minimizer of (4.6) is a square if and only if c < 2. We suppose that the minimizer A is a 2 square of the form [−L − dτ + , L + dτ − ]2 . Since τ = 2+c 2L τ we obtain Per1 (A ) +

√ √ 1 D(A , Aτ0 ) = 4(2 + c2 )L + 8c τ + o( τ )τ →0 . τ

(4.11)

Now, we compare this energy with the energy of a set A of the form (4.7) with δ = 2τ L + o(τ )τ →0 . Recalling (4.8) we obtain that 1 1 D(A, Aτ0 ) − Per1 (A ) − D(A , Aτ0 ) τ τ √ √ = 4(2 − c2 )L − 8c τ + o( τ )τ →0 .

Per1 (A) +

Hence, for τ small enough, the minimizer Aτ1 is the union of four squares given by √ (4.7) if and only if c ≥ 2. The same argument shows that more in general dτ ≥

√

2τ ⇒ a minimizer of (4.6) is the union of four squares.

(4.12)

In particular, we note that if dτ is large enough then the minimizing movement A(t) is such that A(0) is disconnected, while the limit of the initial data Aτ0 as τ → 0 is a square.

4.1.5 Flat Flow for an ‘Octagonal’ Perimeter The minimizing-movement scheme can be repeated for general perimeters and dissipations. An important case is that of Eulidean perimeter and dissipation, which have been proved to give motion by mean curvature in the seminal work by Almgren, Taylor and Wang. In the case of crystalline perimeters, the final evolution can be described similarly to the case of the 1-perimeter, in terms of a motion by a crystalline curvature which is not zero only on sides parallel to the sides of the Wulff shape. In particular, the role of rectangles for the 1-crystalline perimeter is played by polyhedra whose sides are parallel to the sides of the Wulff shape (Wulfftype sets). The motion by crystalline curvature is then defined taking into account a mobility factor, which depends on the interplay between perimeter and dissipation. We consider in detail an example for the perimeter which has been shown to be obtained as a limit of next-to-nearest neighbour lattice energies (Sect. 3.2.2). In this case the 1-crystalline perimeter is substituted by the functional given by F (A) =

∂∗A

(ν1 + 2ν∞ ) dH 1.

(4.13)

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4 Evolution of Planar Lattices

In order to highlight the dependence of the mobility on the dissipation, and for an interesting comparison with the lattice case in the following, we consider the minimizing-movement scheme for F with a general dissipation given by Dp (A, B) =

dp (x, ∂B) dx,

(4.14)

AB

where B is smooth enough and dp (x, C) = inf{x − yp : y ∈ C}. Since the Wulff shape Wϕ with ϕ(ν) = ν1 + 2ν∞ is a scaling of the octagon with vertices (±1, ±3), (±3, ±1) (see Sect. 3.2.2), we consider as an initial datum A0 given by an octagonal Wulff-type set; that is, an octagon with sides oriented in the same directions as the sides of the corresponding Wulff shape, that is parallel to e1 , e2 , e1 + e2 and e1 − e2 (possibly with one or more sides missing). We denote this family of sets by W. We denote by 0k , k = 1, . . . , 8 (mod. 8) the sides of A0 , labeled clockwise as in Fig. 4.10, so that if k is odd then the side 0k is parallel to one of the coordinate axes; L0k will be the length of 0k . Remark 4.6 (Octagonal Shape of the Minimizers) Following the argument used in the case of the crystalline perimeter with ϕ(z) = z1 , we can show that a minimizer of F (A) +

1 Dp (A, A0 ) τ

(4.15)

is again given by an octagon belonging to W and included in A0 . Indeed, let A1 = Aτ1 be a minimizer of (4.15); note that it is a convex set included in A0 . In particular, its boundary is a Lipschitz closed curve. Now, if by contradiction A1 ∈ W we can construct an octagon A∗ ∈ W such that A1 ⊆ A∗ ⊆ A0 and F (A∗ ) +

Fig. 4.10 Evolution of a Wulff-type set for ϕ(ν) = ν1 + 2ν∞

1 1 Dp (A∗ , A0 ) < F (A1 ) + Dp (A1 , A0 ). τ τ

4.1 Flat Flows

65

Fig. 4.11 A ‘Wulff-type envelope’

Fig. 4.12 Comparison with a ‘Wulff-corner’

Referring to Fig. 4.11, since A1 is convex we can view A∗ as the intersection of all the closed half-planes containing A1 and defined by straight lines in the directions e1 , e2 , e1 + e2 and e1 − e2 . Since A1 is strictly included in A∗ , then Dp (A∗ , A0 ) < Dp (A1 , A0 ). In order to show that the perimeter F does not increase, we consider the case in which A∗ is not degenerate; the other case can be obtained with a slight adaptation of the argument. Note that if we consider two (adjacent) sides of A∗ with a common vertex V , then for any two points P and Q belonging to these sides we have ϕ(P − Q) = ϕ(ν) dH 1 γ∗

= min



 ϕ(ν) dH 1 : γ Lipschitz path joining P and Q γ

where γ ∗ is the union of the segments P V and QV (see Fig. 4.12).

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4 Evolution of Planar Lattices

Since any side 0k of A∗ contains (at least) a point of ∂A1 , there exist eight points Pk ∈ 0k ∩ ∂A1 . Since ∂A1 is a Lipschitz closed curve, by the previous remark we get F (A∗ ) =

8 

ϕ(Pk+1 − Pk ) ≤ F (A1 ).

k=1

Hence, A1 ∈ W.

♦

Now, we consider a minimizer A1 of (4.15) and set δk = δk,p as the Euclidean distance between the side 0k of A0 and the corresponding side 1k of A1 . Note that, even though we omit the dependence on p in the minimizing set A1 , in the conclusions of this section we will √ highlight it in the computation of δk,p . As in Remark 4.3, we can show that δk ≤ τ for any k and the dissipation is given by Dp (A1 , A0 ) =

 k odd

√ 1 δk2 δ2 2  L0k + 2 p L0k k + o(τ )τ →0 , 2 2 2

(4.16)

k even

where the remainder is a sum of terms of the form τ1 P3 (δ1 , . . . , δ8 ) where P3 is a homogeneous polynomial of degree 3 with coefficients depending only on the lengths L0k . The length L1k of the side 1k is given by L1k = L0k + 2δk −

√ 2(δk−1 + δk+1 )

(4.17)

(where we recall that δ0 = δ8 and δ9 = δ1 ); hence, 

F (A1 ) = 3

 √  √  L1k + 2 2 L1k = F (A0 ) − 2 δk − 2 2 δk .

k odd

k even

i odd

k even

Then, in order to determine δk we have to minimize −2

 k odd

√  1 1  δ2 δ2  1  − δk − 2 2 δk + L0k k + 2 p 2 L0k k τ 2 2 k even

k odd

k even

(up to a negligible term). Recalling the form of the remainder in the dissipation which allows to neglect the resulting terms in the expansions, we obtain, up to higher order terms ⎧ 2τ ⎪ ⎨ if k odd L δk = δk,p = 2τ0k 1 1− ⎪ ⎩ 2 p if k even. L0k

4.1 Flat Flows

67

By proceeding iteratively, we may conclude that each set Aτj of the discrete orbit belongs to W and is decreasing with j . We may then pass to the limit obtaining a time-continuous evolution A(t) ∈ W for all t, whose sides k (t) with length Lk (t) move inwards with a velocity (depending on p) given by

vk,p

⎧ 2 ⎪ ⎨− Lk = 1 2 ⎪ ⎩− 21− p Lk

if k odd (4.18) if k even

as long as Lk > 0. Note that, upon suitably defining the crystalline curvature κ of a side  this equation is of the form v = −mp (ν)κ, where mp (ν) is a mobility factor depending on the orientation of the normal ν to . Remark 4.7 (Some Effects of the Mobility) We now examine some effects due to the dependence on p by examining the system of ODE satisfied by Lk . By (4.17) we obtain L1k − L0k τ

⎧ 4 5 1 1 1  2−p ⎪ − 2 + ⎪ ⎨ L0k L0(k−1) L0(k+1) = 3− p1  1 ⎪ 3 2 1  ⎪ ⎩ − 22 + L0k L0(k−1) L0(k+1)

if k odd if k even.

so that, applying the same argument to each step in the iterative minimization procedure and passing to the limit as τ → 0 as in Remark 4.5, we get the system of ODE ⎧ 4 5 1 1  1 ⎪ 2−p ⎪ if k odd − 2 + ⎨L Lk−1 Lk+1 k  (Lk ) = 3− 1  1 ⎪ 3 1  2 p ⎪ ⎩ if k even. − 22 + Lk Lk−1 Lk+1 where Lk = Lk (t) is the length of the k-th side of the limit octagon at the time t. Now, we consider the case when the initial datum is such that the odd (and the even) sides have the same length Lo (0) (and Le (0), respectively). The system is ⎧ 7 1 − ⎪ 4 22 p ⎪  ⎪ ⎨(Lo ) = − Lo Le 5 3− p1 ⎪ ⎪ 2 22 ⎪ ⎩(Le ) = − Le Lo

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4 Evolution of Planar Lattices

where Lo and Le denote the length of the odd and even sides, respectively. For any p ∈ [1, +∞] there exists a proportion αp given by  2(1 + √2α )  1 p = log2 √ p αp (αp + 2) such that, if Le (0) = αp Lo (0), then the solution of the system of ODE satisfies Le (t) = αp Lo (t) and it is given by " Lo (t) = 3

Note that αp < 2 2

− p1

L2o (0) −

8 23 − p1 (2 − αp )t. αp

for any p. We also observe that αp is strictly increasing √ √

with respect to p, with α1 = 1 and α∞ = 2+2 10 . In Fig. 4.13 two flux lines are pictured in a diagram where vector field (Lo , Le ) is qualitatively represented. We note that there is a particular p which maintains Wulff shapes. If A0 is a (scaled) Wulff shape of ϕ; that is, Le (0) =

√ 2Lo (0),

then the condition 3 1 = log2 p 2 Fig. 4.13 Qualitative description of the (normalized) vector field (Lo , Le )

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

69

for the value p in the dissipation Dp guarantees that the minimizing movement be a (scaled) Wulff shape for any t. ♦

4.2 Discrete-to-Continuum Geometric Evolution on the Square Lattice We now finally turn to the core of these notes; i.e., to the description of the effects of the heterogeneities in lattice energies on the corresponding minimizing movements. We follow the discrete analog of the definition of flat flow in a square-lattice environment. Let Eε be a lattice energy defined by Eε (u) =



εaij |ui − uj |.

i,j ∈Z2 p

With fixed p the dissipation Dε will be given by p

Dε (u, v) =



ε3 dp (I ; {j : vj = 0})

i:ui =0,vi =1

+



ε3 dp (I ; {j : vj = 1}),

(4.19)

i:ui =1,vi =0

where dp is the discrete Lp -distance. Recalling that functions u, v : εZ2 → {0, 1} are identified with the sets A = Aεu and B = Aεv as in (3.1), respectively, then the p dissipations Dp (A, B) := Dε (u, v) can be interpreted as dissipations between sets ‘discretizing’ the ones defined in (4.14) for the continuum problems. In view of the results of Sect. 4.1.1, in a square-lattice environment the case p = ∞ (i.e., the dissipation as in (2.8)) is particularly relevant, so that we simply write Dε (u, v) = Dε∞ (u, v)

(4.20)

to shorten the notation. Following Definition 2.3, given a sufficiently regular A0 in R2 for each ε > 0 we define uε0 as the characteristic function of A0 ∩ εZ2 (note that some care should be taken when we have ε-depending initial data, but we will always take sufficiently regular A0 , thus avoiding the issues highlighted in Sect. 4.1.4), and, for each positive τ,ε ε τ , we define recursively uτ,ε 0 = u0 and uk+1 as a minimizer of u → Eε (u) +

1 p D (u, uτ,ε k ). τ ε

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4 Evolution of Planar Lattices

We then define the piecewise-constant set-valued functions Aτ,ε (t) = Aεuτ,ε , t/τ 

and finally a minimizing movement along Eε as any limit A of Aτj ,εj as j → +∞. Scope of this part of the notes is to describe some relevant features of such a minimizing movement through some prototypical examples. In all cases we will have a geometric motion with initial datum A(0) = A0 . Before entering in the details of the single examples, we make some observations. Remark 4.8 (Critical Scales) In all the examples below, the critical scales (i.e., those ε-τ regimes such that we are not in the extreme cases described by Theorem 2.1) are with τ ∼ ε; i.e., (i) if τ > ε, then any minimizing movement along Eε is a minimizing movement of the -limit F . This fact will be obtained as a consequence of our computation, but likely holds for a wide class of lattice energies. Upon the extraction of subsequences, we may always suppose that there exists γ such that τε → γ . The cases γ = 0 and γ = +∞ are the extreme cases (i) and (ii) above. From now on we will then consider the case when τ = γ ε for some γ > 0,

(4.21)

which is not restrictive, up to taking into account the presence of a small error in computations the proofs. ♦ Remark 4.9 (Restriction to the Study of Wulff-Type Evolutions) In order to describe the interesting issues in the sequel it will be sufficient to deal with simple initial data; namely, Wulff-type shapes. We will not be concerned in the description of evolutions starting from general initial data, which often can be dealt with following the arguments in Sect. 4.1.3.3 ♦ Remark 4.10 (Description Through Degenerate ODEs) Minimizing movements will be described by (systems of) ODEs of the form y  = f (y), where y is the vector of all the length of the sides of a Wulff-type shape. The function f is in general discontinuous in a discrete set; as a consequence, solutions to this equation must be understood in a generalized sense. In the relevant study cases, it is sufficient to understand the equation as solved by y = y(t) for almost all values of t. Otherwise, the equation can be generalized using differential inclusions. We will not describe

3 The extension from rectangles to general initial data for ferromagnetic energies in the square lattice is explained in detail in [4].

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

71

Fig. 4.14 Evolution of a discrete rectangle

the evolution in all exceptional cases, which may give non-uniqueness phenomena, but only treat the model case in Examples 4.14 and 4.15, the other cases being treated similarly. ♦

4.2.1 A Model Case: Nearest-Neighbour Homogeneous Energies We consider the nearest-neighbour energies Eε (u) =



ε|ui − uj |,

i,j

whose continuum limit is the 1-crystalline perimeter (Sect. 3.2.1).4 In analogy with Sect. 4.1.2, we consider the minimizing-movement scheme along Eε with the dissipation Dε in (4.20). Following Remark 4.9 we limit our analysis to considering a coordinate rectangle R0 as initial datum, that is uε0 = χR0 on εZ2 . Note that the initial datum Aεuε actually 0 depends on ε, being a rectangle with side length a multiple of ε, but we still let R0 denote it, and u0 denote uε0 , with a slight abuse of notation. The length of the horizontal sides are denoted by L0h and of the vertical sides by L0v . We now describe discrete orbits, first noting that the minimizer u1 = uτ,ε 1 gives a set Aεu1 which is still a coordinate rectangle, denoted by R1 = R1τ,ε (with side lengths denoted by L1h for the horizontal sides and L1v for the vertical sides, and L1h , L1v ∈ εN), as pictured in Fig. 4.14. Indeed, we may follow the argument of

4 The arguments of this section can be repeated for nearest-neighbour energies on other lattices taking into account Sect. 3.4. For the triangular lattice an explicit computation is contained in [9].

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4 Evolution of Planar Lattices

Sect. 4.1.2: we first note that a minimizer is included in R0 ; then, we observe that if we consider a (discrete) connected subset A of R0 , we can choose the smallest (discrete) coordinate rectangle including A, which has a lower perimeter energy and a lower dissipation term. Hence, the minimizer is a union of coordinate rectangles included in R0 , and we have to show that it has only one connected component. This can be done by contradiction, showing that if we have two connected components, we can translate them towards the center of R0 until their boundaries intersect, so that the perimeter energy and the dissipation terms both decrease, since the resulting dissipated area is closer to the boundary of R0 . In the case of more than two connected components, this argument can be used recursively with some more technical details. We note that the rectangle R1 is centered at the same point as R0 . The problem of the description of the evolution of the rectangle R0 is then reduced to the computation of the distance between the boundaries of R0 and R1 ; that is, of the (integer) values Nh =

L0h − L1h L0v − L1v and Nv = . 2ε 2ε

Since R1 ⊂ R0 , the dissipation is given simply by 

Dε (u0 , u1 ) =

ε3 d∞ (i; {j : εj ∈ R1 }).

εi∈R0 \R1

Remark 4.11 (Negligible Contribution of the Corners) We√show that the distance between the boundaries of R0 and R1 is of order lower than ε. We assume (without loss of generality) that Nh ≥ Nv . Since Eε (u1 ) +

1 Dε (u1 , u0 ) ≤ Eε (u0 ), τ

(4.22)

the bound on the contributions of the corners in the dissipation gives the estimate ε2  Nh (Nh + 1)(2Nh + 1) Nh (Nh + 1)  + ≤ 2(L0h + L0v ), 2γ 6 2 so that Nh3 ε2 ≤ C (and the same holds for Nv ); hence, again by (4.22) we have  ε (L0v − 2εNv )Nh (Nh + 1) + (L0h − 2εNh )Nv (Nv + 1) ≤ 4ε(Nh + Nv ) γ

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

73

for ε small enough. Recalling that Nh3 ε2 ≤ C, this implies  1  Nh = O √ . ε ε→0

(4.23)

From these estimates, we deduce that the contribution of the dissipation in the √ ♦ corners is O( ε)ε→0 . Since by Remark 4.11 the contribution of the corners is negligible as ε → 0, we obtain that, up to a higher order term, Dε (u0 , u1 ) ∼ 2ε2 L0v

Nh 

i + 2ε2 L0h

i=1

Nv 

i

i=1

= ε2 L0v (Nh + 1)Nh + ε2 L0h (Nv + 1)Nh . Hence, we have to minimize   L − L  L − L  1 0h 1h 0h 1h +1 2(L1h + L1v ) + L0v τ 2 2   L − L  L − L 0v 1v 0v 1v +1 +L0h 2 2 with the restriction L0h − L1h , L0v − L1v ∈ 2εN. We compute L1h , which corresponds to minimizing the function (up to a constant) Nh → −4εNh +

  ε2 L0v Nh (Nh + 1)L0v = ε − 4Nh + Nh (Nh + 1) τ γ

for Nh ∈ N. Since the minimum of the parabola P (x) = −(4 − obtained for x=

2γ 1 − , L 2

then the (discrete) minimum of (4.24) is achieved for the value Nh ∈

 2γ L0v

−

 1  2γ 1 , +1 − 2 L0v 2

L γ )x

(4.24) +

L 2 γx

is

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4 Evolution of Planar Lattices

Fig. 4.15 Minimization of the discrete parabola and computation of Nhmin

minimizing the distance from

2γ L0v

− 12 , that is Nhmin =

2γ  L0v

(see Fig. 4.15). Note that for L2γ0v ∈ N we have two possible values of Nhmin . This may give some non-uniqueness phenomena for the limit flow. Anyhow, these situations can be considered as exceptional and will be briefly analyzed below. The computation for L1v being the same, we obtain that the minimizers satisfy 2ε L1h − L0h 2 2γ  , = − Nhmin = − τ τ γ L0v L1v − L0v 2ε 2 2γ  = − Nvmin = − . τ τ γ L0h Iterating the procedure and passing to the limit as τ → 0 we obtain that the minimizing movement starting from R0 is then a rectangle R(t) with R(0) = R0 , with side lengths Lh (t) and Lv (t), and we (formally) get the system of ODEs ⎧ 2 2γ  ⎪ ⎨(Lh ) = − γ Lv  2 2γ ⎪  ⎩(Lv ) = − . γ Lh As anticipated in Remark 4.10, note the right-hand sides in the equations are discontinuous, and locally the velocity of a side is constant and depends on the inverse of the length of the other side. Then, a side of R(t) with length L(t) moves (inward) with velocity v=−

1 2γ  γ L

(4.25)

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75

since the velocity of a side equals a half of the ‘variation’ of the length of the orthogonal side. Note that Eq. (4.25) may be written in terms of the 1-crystalline curvature κ as 1 v = − γ κ γ and can be extended to more general sets, as v=−

$ sign(κ) # |κ|γ . γ

where κ is defined in (4.5). The extension of the motion to more general initial data can be then performed as in Sect. 4.1.3. Remark 4.12 (Quantized Motion of ‘Small’ Sets) If L < 2γ , then the sides move with (piecewise-constant) velocity an integer multiple of γ1 , as pictured in Fig. 4.16. The velocity changes when

2γ L

∈ N, that is L=

2γ , n ∈ N. n

Hence, we have • unique motion (since the discontinuities of the right hand side of the evolution equation are negligible); • extinction in finite time. ♦ Remark 4.13 (Pinning) From (4.25), in particular we deduce that if a side has length L such that L > 2γ , then it does not move. The value 2γ is called the Fig. 4.16 Evolution of the length of a ‘small’ side

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4 Evolution of Planar Lattices

Fig. 4.17 Partial pinning if Lh > 2γ = Lv

pinning threshold for L, over which the motion is trivial. We may have • total pinning of large rectangles: if the starting rectangle R0 is ‘large’; that is, both Lh and Lv are greater than 2γ ; • partial pinning: if (for instance) Lh > 2γ > Lv In this case there is motion but not ‘immediate detachment’ from the boundary of the initial set, as pictured in Fig. 4.17. In particular, only a weak comparison principle holds between sets at different times; • pinning after an initial motion (for non-rectangular initial data) when the length 2γ is reached at a positive time. This occurs for example if the initial datum is a large ball, whose evolution follows the one pictured in Fig. 4.6 until the coordinate sides reach the critical length, after which the minimizing movement is constant. ♦ Remark 4.14 (Possible Non Uniqueness) If R0 = 2γ Q, Q being the square [− 12 , 12 ]2 , so that the length of the side is L = 2γ , we have as possible evolutions: • a pinned solution (corresponding to choosing the minimizers N min = 0 at all steps), • a (unique) motion of contracting squares (corresponding to choosing the first minimizers N min = 1, after which the motion is unique), or even a motion with initial pinning until some time t0 , after which we have a motion of contracting squares (corresponding to choosing first the minimizers N min = 0, and then N min = 1). Note that in order to have two minimizers N min for the discrete orbits, we have also to choose particular values of ε (which are not explicit since we have neglected some higher-order term in the computation of N min ). ♦ Remark 4.15 (‘Exceptional’ Cases of Initial Data) We consider the case where the initial datum is a rectangle with vertical side length Lv = 2γ n for some integer n, and horizontal side length Lh strictly greater than 2γ . In this case, in the minimization we have a discrete parabola with two (integer) minimizers, and the velocity can be different for the two vertical sides, as pictured in Fig. 4.18. Indeed, the only restriction for the velocity v of each vertical side is that |v| ∈

n n+1 1 [n, n + 1] that is ≤ |v| ≤ . γ γ γ

Here, not even a weak comparison principle holds.

♦

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77

Fig. 4.18 An example of non uniqueness and non-symmetric evolution

Remark 4.16 (Recovery of the Extreme Motions: Interpolation) We check a posteriori the validity of the statement in Remark 4.8, and that minimizing movements with γ > 0 are an interpolation between the extreme cases. For any γ = τε , we denote by Aγ (t) the minimizing movements for Eε starting from the fixed rectangle R0 . Then • for γ small enough, Aγ (t) = R0 for any t, hence lim Aγ (t) = A0 (t)

γ →0

where A0 (t) ≡ R0 for any t; • we have lim Aγ (t) = A∞ (t)

γ →+∞

where A∞ (t) is the flat flow for the 1-crystalline perimeter. Moreover, this is true also considering τ, ε → 0 with τε → 0 and τε → +∞ respectively. We may then regards γ as an interpolation parameter for minimizing movements between the two extreme cases. ♦

4.2.2 Next-to-Nearest-Neighbour Homogeneous Energies We analyze the evolution for energies given by Eε (u) =



ε|ui − uj |,

i,j

where i, j

denotes the family of nearest neighbours and next-to-nearest neighp bours in Z2 , as in Sect. 3.2.2, with respect to the dissipation Dε given by (4.19) with p ∈ [1, +∞]. Also in this case we consider the evolution at the critical scale τ = γ ε.

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4 Evolution of Planar Lattices

Fig. 4.19 A discretization of a Wulff-type set

The -limit of Eε is the ‘octagonal’ perimeter F in (4.13) obtained in Sect. 3.2.2, and whose flat flow has been studied in Sect. 4.1.5. Following Remark 4.9 we only consider Wulff-type shapes as initial data. The set of such polygons is denoted by W and it is given by octagons whose sides are parallel to e1 , e2 , e1 +e2 and e1 −e2 , also including the degenerate cases when one or more sides vanish (see Sect. 4.1.5 and in particular Remark 4.6). In order to apply the results of Sect. 4.1.5, for each u ∈ 2 {0, 1}εZ we introduce a corresponding octagon Aˆ εu ∈ W defined as the smallest element of W containing Aεu . Now, we choose as initial datum u0 = uε0 such that, setting Aˆ 0 = Aˆ εu0 , we have that u0 coincides with the characteristic function of the internal part of Aˆ 0 in εZ2 . 2 We also set A0 = Aεu0 (see Fig. 4.19). We say that u ∈ {0, 1}εZ is octagonal if it satisfies such a property; that is, u = χint(Aˆ ε )∩εZ2 . u

With these definitions, for an octagonal uε a computation gives Eε (uε ) =

∂ Aˆ εuε

(νε 1 + 2νε ∞ ) dH 1 − 4ε,

where νε is the normal to ∂ Aˆ εuε . Following the notation of the continuum case (see Sect. 4.1.5), the Euclidean lengths of the sides 0k of Aˆ 0 are denoted by L0k , in such a way that k odd corresponds to sides oriented as the Cartesian axes. Now we consider discrete orbits; that is, we face the problem of minimizing iteratively Eε (u) +

1 p ε D (u , u). τ ε 0

(4.26)

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79

A slight adaptation of the argument in Remark 4.6 shows that a minimizer u1 = uτ,ε 1 is octagonal. We define (omitting the dependence on τ and ε) A1 = Aεu1 and Aˆ 1 = Aˆ εu1 and denote the side lengths of Aˆ 1 by L1k , correspondingly to the ones of Aˆ 0 . Since A1 ⊆ A0 , the dissipation is given by 

Dεp (u0 , u1 ) =

ε3 dp (i; {j : εj ∈ A1 }).

εi∈A0 \A1

As in the continuum case, δk denotes the Euclidean distance between the side 0k of Aˆ 0 and the corresponding side 1k of Aˆ 1 (see Fig. 4.20). We define hk ∈ N by ⎧ δk ⎪ ⎨ ε hk = √ ⎪ ⎩ 2δk ε

if k odd (4.27) if k even;

that is, the integer which gives the number of discrete steps in the directions orthogonal to the coordinate axes (see again Fig. 4.20). As for the dissipation, we have Dεp (u0 , u1 ) ∼



ε2 L0k

k odd

+

hk 

i

i=1

hk % hk % % %   L0k   % i − 1 i + 1 % % i i % , ε2 √ % % , % % + p 2 2 2 2 p 2 i=1 k even i=2



i odd

i even

up to a term due to the negligible contribution of the corners given by ε3 P3 (h1 , . . . , h8 ), with P3 a homogeneous polynomial of degree 3 and hk = Fig. 4.20 Local motion of a Wulff-type set

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4 Evolution of Planar Lattices

Fig. 4.21 Pictorial representation of the computation of the dissipation

O( √1ε )ε→0 as in the nearest-neighbour case (see Remark 4.11). The computation p

of Dε for the even sides is pictured in Fig. 4.21. Recalling (4.17), we get (up to the negligible term 4ε) Eε (u1 ) = 3



8  √  L1k + 2 2 L1k = Eε (u0 ) − 2ε hk . k even

k odd

k=1

Hence, the lengths of the sides L1k are determined by computing

hmin k,p

⎧  ⎪ ⎪ ⎨arg min − 2h +  = ⎪ ⎪ arg min − 2h + ⎩

 L0k 2 (h + h) : h ∈ N 2γ  L0k √ φp (h) : h ∈ N 2γ

if k odd (4.28) if k even,

where the function φp is given by ⎧ h  h ⎪   i − 1 p  i + 1 p  p1 1  i ⎪ ⎪ p ⎪ + +2 ⎨ 2 2 2 i=2 φp (h) = i=1 i odd i even ⎪ ⎪ ⎪ 1 1 h + 1 ⎪ ⎩ (h2 + h) + 4 2 2

if p ∈ [1, +∞) if p = +∞.

Note that the complex form of φp is due to the inhomogeneity of the minimal distance between slanted sides; i.e., for k even (see Fig. 4.22), while the dependence is homogeneous in the coordinate directions. Finally, we obtain for any p that hmin k,p =

2γ  L0k

if k is odd,

while the computation in the case k even is more complex and depends on p. We consider more in detail the cases p = 1 and p = ∞, for which a complete analysis is simpler since the sums in the definition of φp (h) can be explicitly written as polynomials in h.

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

81

Fig. 4.22 Different discrete sets of ‘equal dissipation’ in terms of the distance from 0

• If p = 1, we have to minimize in N the parabola L0k h ∈ N → −2h + √ h(h + 1). 2 2γ It follows that

hmin k,1

⎧ 2γ  ⎪ ⎨ if k is odd 0k = L  2γ √ ⎪ ⎩ 2 if k is even. L0k

• If p = ∞ we have to minimize L0k L0k h + 1  h ∈ N → −2h + √ (h2 + h) + √ 2 4 2γ 2 2γ obtaining

hmin k,∞

⎧ 2γ  ⎪ ⎨ if k is odd 0k = L  √ 2γ ⎪ ⎩ 2 2 if k is even. L0k

Proceeding iteratively, we then conclude that the velocity of the limit octagon in the two cases is given by

vk,1

⎧ 1 2γ  ⎪ if k is odd ⎨− γ Lk =  √ 1 2γ ⎪ ⎩− 2 if k is even, γ Lk

vk,∞

⎧ 1 2γ  ⎪ if k is odd ⎨− γ Lk =  √ 1 2γ ⎪ ⎩− 2 2 if k is even, γ Lk

respectively (Lk denoting the side length of the k-th side).

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4 Evolution of Planar Lattices

In the general case p ∈ (1, +∞), we can still give an estimate on hmin k,p . Indeed, we note that, for k even, φp satisfies 1

2p

−2

1

(h2 + h) ≥ φp (h) ≥ 2 p

−2

(h2 − h),

and this gives " 2γ 23 − p1 2 − L0k

" 2γ 32 − p1 2γ 23 − p1 2 ≤ hmin 2 + k,p ≤ L0k L0k

2γ 32 − p1 2 . L0k

(4.29)

Hence, the minimizing movement starting from A0 is an octagon A(t) with A(0) = A0 and lengths of the sides Lk (t), moving (inward) with velocity

vk,p = lim

min δk,p

τ →0

τ

⎧ 1 2γ  ⎪ ⎪ k odd ⎨− γ L k  = 1 2γ 32 − p1 ⎪ ⎪ 2 + p (γ , Lk ) k even, ⎩− √ 2γ Lk

min is given by where δk,p

min δk,p

⎧ min ⎪ ⎨εhk,p = εhmin k,p ⎪ ⎩ √ 2

if k odd if k even

as in (4.27), hmin k,p is defined in (4.28) and " |p (γ , Lk )| ≤

2γ 23 − p1 2 . Lk

Even though their form is not explicit, we can check that the velocities vk,p give the ‘extreme regimes’ passing to the limit as γ → +∞. We recover for any p the corresponding flat flow in the continuum case, that is

∞ vk,p

⎧ 2 ⎪ ⎨− L k = 1 2 ⎪ ⎩− 21− p Lk

k odd k even

as in (4.18). Remark 4.17 (A Discrete Mobility) Summarizing the observations above, the discrete nature of the interactions, and of the dissipation in particular, has influenced the form of the mobility, whose dependence on p is much more complex than in

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

83

the continuum case due to the geometry of the set of lattice points in which the dissipation is computed at the ε-scale. ♦

4.2.3 Evolutions Avoiding Hard Inclusions We consider the evolution of the inhomogeneous lattice energy Eε (u) =



εaij |ui − uj |,

(4.30)

i,j

with nearest-neighbour connections given by

aij =

⎧ ⎪ ⎪ ⎨α α

⎪ ⎪ ⎩β

if i1 = j1 and max{i2 , j2 } is even if i2 = j2 and max{i1 , j1 } is even otherwise,

where 0 < α < β < +∞ (see Sect. 3.3.2 and Fig. 4.23). The dissipation is given by (4.19) as in the homogeneous case.  As we noticed in Sect. 3.3.2, the -limit F (A) = α ∂ ∗ A ν1 dH 1 is the same as in the homogeneous case (in particular it is the 1-crystalline perimeter if α = 1); hence, the extreme cases for the minimizing movements along Eε are the same. As in the previous examples, we analyze the motion at the critical scale τε = γ . In the sequel, when it is possible without ambiguity we omit the dependences on τ and ε. We will show that the minimizing movements along Eε are simply obtained considering sets whose boundary ‘avoids β-connections’; hence, it can be seen as a minimizing movement for energies on a lattice with double spacing 2εZ2 . Also in this case, we can consider the evolution starting from a coordinate rectangle A0 = Aεu0 . We say that a coordinate rectangle R = Aεu is an α-type rectangle if the sides of R only intersect α bonds and analogously we define β-type rectangles. Fig. 4.23 A layered lattice with ‘hard inclusions’

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4 Evolution of Planar Lattices

Fig. 4.24 The α-rectangularisation of a set A

We show that if A0 is a coordinate rectangle then it evolves in α-type rectangles. Indeed, as in the evolution of homogeneous energies, a minimizer of Eε (u) +

1 Dε (u, u0 ) τ

(4.31)

2

is a function u ∈ {0, 1}εZ such that Aεu is a connected set included in A0 . Let 2 A = Aεu for some u ∈ {0, 1}εZ be a connected subset of A0 and define its αrectangularisation as the union of A and the maximal α-type rectangle R ⊂ A0 2 such that each side of R intersects A (see Fig. 4.24). We let uR ∈ {0, 1}εZ denote the function such that R = AεuR . We have Eε (max{u, uR }) ≤ Eε (u), and since A ⊂ A ∪ R ⊂ A0 we also get Dε (max{u, uR }, u0 ) ≤ Dε (u, u0 ); therefore, a minimizer of (4.31) is identified with the union of an α-type rectangle R and some ‘β-protrusions’, that is subsets of Rβ \ R, where Rβ is the minimal β-rectangle including R. We note that, as in the homogeneous √ case, the distance between the boundaries of A0 and R is of order less than ε (see Remark 4.11). Indeed, we denote the distance between the vertical (respectively, horizontal) sides of A0 and R by εNh (respectively, εNv ), and assume (without loss of generality) that Nh ≥ Nv . If R  = AεuR is the minimal α-type rectangle including R and whose sides do not intersect R, then Eε (uR ) +

1 Dε (uR  , u0 ) ≤ Eε (u0 ) ≤ β Per(A0 ). τ

(4.32)

As in Remark 4.11, a computation allows to show that the bound on the contributions of the corners in the dissipation gives the estimate Nh3 ε2 ≤ C; hence, by (4.32) we have that for ε small enough  ε  Lv (Nh − 2)(Nh − 1) + Lh (Nv − 2)(Nv − 1) − εα(Nh + Nv ) 8γ ≤

β −α (Lh + Lv ), 2

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

85

Fig. 4.25 A side of R and the elimination of a β-protrusion

where Lh and Lv denote the lengths of the horizontal and vertical sides of A0 , respectively. This implies  1  Nh = O √ ε ε→0

(4.33)

and clearly the same for Nv . Now, we prove that for ε small enough each minimizer of (4.31) is an α-type rectangle (with the usual identification of a discrete function with its characteristic set). Indeed, we consider a side of R as pictured in Fig. 4.25, where we have a βprotrusion B of length εM. If we modify the set A ∪ R by removing the protrusion B, we have that Eε (u) ˜ ≤ Eε (max{u, uR }) + εM(α − β) where u˜ is the function identified with the modified set (A ∪ R) \ B, and ˜ u0 ) = Dε (max{u, uR }, u0 ) + ε3 MN, Dε (u, where εN is the distance between the corresponding sides of A0 and R. Therefore, recalling (4.33), Eε (u) ˜ +

1 1 Dε (u, ˜ u0 ) ≤ Eε (max{u, uR }) + Dε (max{u, uR }, u0 ) τ τ √  C ε +εM (α − β) + . γ

Since α < β, for ε small enough we obtain that the minimizing set A1 does not contain β-protrusions; hence, it is an α-type rectangle. Then, the motion of the rectangles ‘lives’ in a lattice of size 2ε. As in formula (4.24) for the homogeneous case, this corresponds to minimizing   Lv Nh (Nh + 1) Nh → ε − 4αNh + γ

(4.34)

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4 Evolution of Planar Lattices

(again neglecting the contribution of the corners) for Nh ∈ 2Z. The same holds for Nv . The even integer minimizing the discrete parabola is then given by Nhmin = 2

1  2γ α 2

Lv

γ α 1 1 1 =2 , − )+ + 2 2 Lv 4

so that the limit velocity of each side (with length L) is v=−

2 γα 1 + , γ L 4

v=−

2 γ κα 1  + γ 2 4

or

in terms of the 1-crystalline curvature κ. Note the difference with respect to the homogeneous case. The discontinuity set of the right-hand side gives the lengths 4γ where the velocity changes, that is L = 4n−1 , n ∈ N. In particular, the pinning threshold is 4γ 3 , which is lower than that for only nearest-neighbour interactions (Remark 4.13).

Remark 4.18 If we change the geometry of the interactions taking one layer of αconnections each k layers (see Fig. 4.26 for k = 4); namely, with nearest-neighbour connections given, for vertical connections (i.e., for i and j with i1 = j1 ), by aij =

 α β

if max{i2 , j2 } = 0 (mod k) otherwise,

and analogously for horizontal connections, then the motion of the rectangles takes place in a lattice of size kε, with k ∈ N. We leave to the reader the modifications to the proof, which involve similar geometric arguments on the shape of competing Fig. 4.26 A layered lattice with period 4 and 3 × 3 ‘hard inclusions’

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

87

sets.5 As in (4.34) we can compute the optimal Nh by minimizing the parabola Nh → −4αNh +

Lv Nh (Nh + 1) γ

in kZ. We get Nhk,min = k

1  2γ α k

Lv

2γ α k − 1  1 1 =k , − )+ + 2 2 kLv 2k

so that the limit velocity of each side (with length L) is vk = −

k 2γ α k − 1  + . γ kL 2k

Note that for k = 2 we recover the case just analyzed, and for k = 1 (and α = 1) the velocity in the homogeneous case (4.25). ♦ Note that this example gives a different interpolation between (the same) A0 and for each k.

A∞

4.2.4 Asymmetric Motion We may consider a variation of the layered media examined in the previous section, taking a different number of layers in horizontal and vertical direction. Namely, in (4.30) we take, for vertical connections (i.e., for i and j with i1 = j1 ), aij given by  aij =

α

if max{i2, j2 } = 0 (modulo kv )

β

otherwise,

and, for horizontal connections, (i.e., for i and j with i2 = j2 ), by  aij =

α

if max{i1, j1 } = 0 (modulo kh )

β

otherwise,

with possibly  kv = kh (see Fig. 4.27). Note that the -limit of Eε is still given by F (A) = α ∂ ∗ A ν1 dH 1 . The same arguments as in the previous section can be followed, noting that if the initial datum is a rectangle we may consider orbits composed of rectangles avoiding β-connections, and that the computation of the velocity of each side is independent

5 The

more complex geometric constructions of this case can be found in [3].

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4 Evolution of Planar Lattices

Fig. 4.27 A layered lattice with asymmetric layers

from that of the other sides, so that it can be reduced to the one in Remark 4.18. We conclude that the limit minimizing movement is a rectangle with vertical and horizontal sides moving with velocity vv = −

kv − 1  kv 2γ α , + γ kv L 2kv

vh = −

kh − 1  kh 2γ α . + γ kh L 2kh

respectively. In particular, if e.g. kv = 1 and kh = 2, and α = 1 we can ‘mix’ the cases studied in the previous two sections obtaining (in terms of the 1-crystalline curvature κ of each side) vv = −

1# $ γκ , γ

vh = −

1 2 κ γ + . γ 2 4

Note that the overall velocity has the form v = −f (ν, κ) depending both on the orientation of each side and on their 1-crystalline curvature.

4.2.5 Homogenized Motion The previous sections introduce the very interesting question of the determination of the minimizing movements when the distribution of aij is geometrically complex. This seems to be a non-trivial problem even if aij ∈ {α, β} and we only have nearestneighbour interactions. We pursue further this issue when we still have a layered

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

89

Fig. 4.28 A layered lattice with more consecutive α-layers

lattice. As in Sect. 4.2.3 we suppose that aij are invariant by rotation of π/2, so that we can just define vertical interaction, as aij =

 α β

if 1 ≤ max{i1 , j1 } ≤ kα (mod k) otherwise,

(4.35)

with 2 ≤ kα < k. We still have a k-periodic lattice with alternating α and β layers, but, contrary to the case in Remark 4.18 (corresponding to kα = 1) we have more consecutive α-layers, giving more freedom and complexity to the discrete orbits (Fig. 4.28). We consider a rectangular initial datum and fix τ and ε. Referring to Sect. 4.2.3 for details in the proofs, we remark that • each element of a discrete orbit is an α-rectangle. The proof is the same as in Sect. 4.2.3 if k − kα = 1 (i.e., in each direction we have isolated single layers of β-connections), or otherwise follows the slightly more complex geometrical arguments referred to in Remark 4.18; • the location of the next rectangle is again obtained by solving minimum problems as in (4.34), the difference now being that, if M0 ∈ Z is such that a vertical size is located at x0 = εM0 at the initial step, then (up to the usual approximations that are negligible as τ → 0) the position of the same vertical size is given by εMj where Mj = Mj −1 + Nhj and Nhj solves problem (4.34) among Nh with 1 ≤ Mj −1 + Nhj ≤ kα modulo k. Note that if kα = 1 then, up to a fixed translation, this restriction reduces to Nhj ∈ kZ, and we are back to Remark 4.18. The analog description holds for horizontal sides. In this second step, the orbit depends on the initial point M0 . However, this orbit ‘stabilizes’ after a finite number of steps (at most kα ) and the final velocity of the side after τ → 0 can be characterized independently of M0 . In order to illustrate this fact, we picture the behaviour of discrete orbits in the simplest case k = 3 and kα = 2. Example 4.1 In Fig. 4.29 we give a pictorial representation of discrete orbits in dependence of the value Nmin of problem (4.34) in Z. We consider the case k = 3

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4 Evolution of Planar Lattices

Fig. 4.29 Representation of periodic orbits in dependence of Nmin

and kα = 2; the circles represent α-connections and crosses the ‘forbidden’ βconnections. The arrows represent the displacements in the orbit; arrows with a dashed line indicate a possible displacement of Nmin that cannot be achieved since it would end up in a β-connection. If Nmin = 0, then the problem is pinned from the start. If Nmin = 1, we may have a starting motion, but if the value of the parabola in (4.34) is lower for N = 0 than N = 2, then the motion is pinned at the second step. Conversely, if the value of the parabola in (4.36) is lower for N = 2 than N = 0 we have alternating motion with N = 1 and N = 2. The same holds if Nmin = 2 but the value of the parabola in (4.34) is lower for N = 1 than N = 3. If Nmin = 2 but the value of the parabola in (4.36) is lower for N = 3 than N = 1 then the periodic orbits have constantly N = 3. The same clearly holds if Nmin = 3, and for Nmin = 4 if the value is lower for N = 3 than N = 5, and so on. ♦ Going back to the general case, we note that, after at most kα steps, the values Mj are periodic modulo k; i.e., there exists q ∈ N (and by the structure of the problem q ≤ kα ) such that Mj +q = Mj modulo k for all j ≥ kα . Moreover, since orbits with initial data M0 < M0 satisfy Mj < Mj for all j we conclude that the velocity of the periodic orbits is independent of the initial M0 and we may directly consider Mj periodic modulo k as in the following definition.6 Definition 4.2 (Homogenized Velocity for Layered Media) Let α be fixed, let k, kα be given by (4.35). The homogenized velocity for given ξ is defined as fhom (ξ ) =

Mj +q − Mj q

6 For a simple example in the continuum where a general proof of the existence of homogenized velocity is given we refer to [2].

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

91

(independent on j and q), where q ∈ N and Mj is any sequence in Z with Mj +q = Mj modulo k for all j , Mj = Mj −1 + Nj and Nj minimizing the problem N → −2αξ N + N(N + 1)

(4.36)

among all N with 1 ≤ Mj −1 + N ≤ kα modulo k. Example 4.2 Referring to Fig. 4.29, the resulting value of fhom is 0 for case 1, it is 3/2 for cases 2 and 3, and it is 3 for cases 4, 5 and 6. ♦ Proceeding as in the previous sections we conclude that the minimizing movement along Eε starting from a rectangle is a family of rectangles whose sides have velocity v=−

1 fhom (γ κ), γ

where κ is the 1-crystalline curvature of the side. Note that fhom (κ) ∈ kQ and with q above not greater than kα , so that the left-hand side is a discontinuous function, and the meaning to the solution of this system of ODE must be understood as holding for all but a discrete set of times, as usual.

4.2.6 Motions with an Oscillating Forcing Term We consider the energies given by Eε (u) =



ε|ui − uj | +

i,j



ε 2 b i ui ,

i

where i = (i1 , i2 ) ∈ Z2 and bi =

 1 −1

if i1 even if i1 odd,

where the nearest-neighbour ferromagnetic energy is perturbed by a forcing term with a layered structure in the vertical direction. The forcing term continuously converges to 0, hence the -limit is the 1-crystalline perimeter. The dissipation Dε is given by (4.19), and we analyze the motion at the critical scale τε = γ . Note that the forcing term alone does not favor Wulff-type shapes, so that the description of

92

4 Evolution of Planar Lattices

discrete orbits is delicate even for rectangular initial data. The fact that also in this case the elements of the orbits are rectangles is proved in detail below.7 In the following, we say that a column [ε(i1 − 12 ), ε(i1 + 12 )] × R is negative if i1 is odd and positive if i1 is even. In the following, the initial datum is a coordinate rectangle R0 , with the usual identification with the corresponding discrete function u0 . The following proposition holds. Proposition 4.1 (Rectangular Shape of Minimizers) A minimizer of Eε (u) +

1 Dε (u, u0 ) τ

corresponds to a coordinate rectangle included in R0 and symmetric with respect to the horizontal axis of R0 . Proof We first note that each connected component of a minimizer intersects R0 . Indeed, for any row of a connected component not intersecting R0 we consider the difference between the number of ‘negative’ and ‘positive’ cells; if this number is m > 0, then the contribution of the perimeter is at least 2mε, while the contribution of the forcing part is −mε2 . Hence, since the dissipation is positive, the elimination of the connected component gives a lower energy for ε small enough. We assume for simplicity that the length of the vertical side of R0 is an even multiple of ε (otherwise, we have to slightly adapt the proof). Hence, it is not restrictive to fix R0 = [0, εN0 ] × [−εM0 , εM0 ] with N0 , M0 ∈ N. We divide the proof in some steps. Property 1 A minimizer is the union of connected columns with a vertical symmetry with respect to the horizontal axis of R0 . Indeed, given a (bounded) set A corresponding to some u, for any  ∈ Z we define h+  =

  1  A ∩ [ε( − 1), ε] × [0, +∞)  2 ε

and correspondingly h−  , and we modify each column of A by substituting it with + [ε( − 1), ε] × [−εh−  , εh ] as pictured in Fig. 4.30.

7 The content of this section is completely original. The interested reader may compare the results with those of [5, 8], where oscillating terms are considered in the continuum. An interesting feature there is that elements of an orbit starting from a rectangle are not rectangles, but contain some indentations that are absorbed during the motion. Conversely, in those papers the resulting homogenized motions are inhomogeneous but not as influenced by microscopic oscillations as those in this section.

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

93

Fig. 4.30 Pictorial description of Property 1

Fig. 4.31 Reduction to a rectangle with ‘protrusions’

Upon a reflection with respect to the central horizontal axis of R0 , it is not − restrictive to assume that h+  = h = h . If A does not satisfy Property 1, the set obtained in this way has a strictly lower energy. Property 2 A minimizer is a subset of R0 . Assume that A satisfies Property 1 and denote by H the largest integer such that A∩{(x, y) : x ≥ ε(N0 +H )} = ∅. Now, we suppose that A∩{(x, y) : x > εN0 } = ∅ (that is, H ≥ 1) and prove that A cannot be a minimizer. Setting K = N0 + H , we consider the pair of columns corresponding to K and K − 1. If we remove from A the set [ε(K −2), εK]×[−2ε min{hK−1 , hK }, 2ε min{hK−1 , hK }] (with the same notation of the previous step), each term of the energy decreases or remains unchanged (even in the extreme case H = 1). If after this operation the column indexed by K is not empty then it consists of a pair of connected components and by an isoperimetric estimate we can remove it for ε small enough, obtaining a set with a strictly lower energy (see Fig. 4.31a). Otherwise, we repeat the translation toward the central axis of R0 for the column in position K − 1 as pictured in Fig. 4.31b. Note that the sequence of these two operations gives a strictly lower energy, hence A cannot be a minimizer. The same argument can be applied to the left-hand side of R0 . Now, we show that if A ∩ {(x, y) : y > εM0 } = ∅ then A cannot be a minimizer. Let C be a connected component of A ∩ {(x, y) : y > εM0 }. We denote by εc1 , . . . , εcm the heights of the columns of C. If the sum of the heights of the positive columns (denoted by εh+ ) is greater than or equal to the one of the negative columns (denoted by εh− ) we can remove C from A obtaining a set such that each

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4 Evolution of Planar Lattices

Fig. 4.32 Elimination of protrusions

component of the energy decreases (in particular, the dissipation strictly decreases); otherwise, we note that Per(A) − Per(A \ C) ≤ ε

m 

 |ci − ci−1 | + c1 + cm ,

i=2

while the term

 i

ε2 bi ui increases by ε2 (h− − h+ ). Since m 

|ci − ci−1 | + c1 + cm ≥ h− − h+ ,

i=2

we conclude that by removing C the energy is strictly reduced (see Fig. 4.32), and A cannot be a minimizer. Property 3 Each connected component of a minimizer is a coordinate rectangle. We note that if we add to a subset of R0 an ε-square at a distance εd from the boundary of R0 and belonging to a positive column, then the balance of the energy for the dissipation and the forcing term is given by −

ε2 d + ε2 . γ

(4.37)

Hence, if the distance from the boundary is greater than εγ and the global perimeter does not increase, we can always add squares. Conversely, if the distance is lower than εγ and the perimeter does not increase, we can remove the squares in the positive columns. We define Rγ = R0 ∩

 i:dist∞ (εi,∂R0 )≥εγ

  1 1  ; εi + ε − , 2 2

note that Rγ is strictly included in R0 if and only if γ ≥ 1. As in the case of connected components outside R0 , we observe that it is always convenient to remove a connected component included in R0 \ Rγ .

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

95

Fig. 4.33 Reduction to ‘almost’ rectangles

Now, let A be a connected minimizer satisfying the properties of the previous steps. If we substitute A ∩ Rγ with the smallest coordinate rectangle including it, the energy decreases (strictly, if A ∩ Rγ is not a rectangle); hence, if γ < 1, the minimizer A is a rectangle. If γ ≥ 1, by the previous remark we can assume that A is given by the γ + (connected) union of a rectangle RA = [εn− h , εnh ] × [−εnv , εnv ] ⊂ Rγ with a set outside Rγ , as pictured in Fig. 4.33. We show that also in this case A is a rectangle. Let RA = [εNh− , εNh+ ] × [−εNv , εNv ] be the smallest coordinate rectangle including A. Assume now that Nv > N0 − γ  (hence nv = N0 − γ ), the other case being simpler. We prove that any positive column of A (with the possible exception of the first and the last ones) has the same height of the two adjacent columns. Indeed, if it is higher, since we are outside Rγ and the perimeter strictly decreases we can remove the exceeding part. Then, we substitute the union of the three columns with the smallest rectangle including them, and each component of the energy does not increase (see Fig. 4.33). We note that if Nh+ > M0 − γ  or Nh− < γ , then we can assume that the corresponding columns are in a negative position (otherwise, we can remove it and strictly decrease the energy). Hence, in this case the previous argument gives a coordinate rectangle with energy lower than A (strictly lower if A is not a rectangle), concluding the proof. Otherwise, if A is not a rectangle and either Nh− ≥ γ  or Nh+ ≤ M0 − γ  the iterated rectangularization of the union of groups of three adjacent columns gives a set of the form R\ where R is a rectangle (R = RA up to at most the first or the last column) and  is the union of two or four ‘corner rectangles’ with horizontal length ε (see Fig. 4.33). If the corners are in a negative position, we can add them to the set and the energy strictly decreases, while if they correspond to a positive contribution of the forcing term we have to compare two possibilities: • remove the entire corresponding negative column, which gives a variation of the energy Erem = −2ε +

ε2  dist∞ (i, 0 ) − ε2 k, γ i∈A

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4 Evolution of Planar Lattices

where εh is the height of the corner and εk the height of the part of the column in Rγ , 0 is the scaled boundary of R0 and the sum is taken over the integer points such that εi belongs to the removed column; • add the corner and a further positive column with the same height, which gives a variation of the energy Eadd = 2ε −

ε2  dist∞ (i, 0 ) − ε2 k γ i∈B

where the sum is taken over the points of the added set. Since the difference between the distances of corresponding points from the boundary is at most 1 and h > 0, by summing the two variations we get Erem + Eadd
12 ) If γ > 12 , the optimal N is even at any step up to (at most) the first step; in particular, if the first column of R0 is in a negative position, this property holds at any subsequent step; if it is positive, it becomes negative at the first step and then remains negative. Hence for the velocity of the side we obtain v=−

2 γ 1 + (even motion) γ L 4

(as in the case of the hard inclusions with α = 1).

♦ 1 2)

1 2,

Remark 4.20 (Even and Odd Motion for γ < If γ < we have two possibilities for each step of the motion of the side. If the distance of xmin from the set of the odd positive integers is less than 12 − γ , then the optimal N is odd at any step, and the resulting velocity of the inward motion of the side is given by v=−

1 1 2  γ − + (odd motion) . γ L 4 2

Otherwise, up to (at most) the first step, the optimal N is even at any step; in particular, if the first column of R0 is in a negative position, this property holds at any subsequent step; if it is positive, then it remains positive when the distance of

4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice

99

xmin from the set of the non-negative even integers is less than 12 − γ , otherwise it becomes negative at the first step and then remains negative. We then have v=−

1 2 γ + (even motion) . γ L 4 ♦

as the resulting velocity. Remark 4.21 (Pinning and Motion) • If γ > 12 , the pinning threshold is given by L=

4γ . 3

If L < 4γ 3 , then the velocity of the side changes when L = Recalling Remark 4.19, we have an even motion with v=−

2n γ

if

4γ 4n−1 ,

n ∈ N.

4γ 4γ

1 2

(a) and γ
4γ . Hence, • if 4γ ≤ 1 any i ∈ (2Z)2 ∩ R0 belongs to Au1 ; • if 4γ > 1, the isolated squares in Au1 are ‘dissipated’ in the subsequent step of the iterative minimization procedure, since the distance from the boundary of Au1 is now 1, which is lower than 4γ . Note that if the distance between R0 and R1 is large enough (greater than 4γ ), then it is always convenient to have a mushy layer. Since the mushy layer is composed by the weak islands in R0 \ R1 at a distance larger than 4γ from the boundary of R0 , in order to describe the evolution it is sufficient to compute the distance of each side of R1 from the corresponding side of R0 . Up to the (negligible) contribution of the corners, we can consider separately the motion of each side, as in Sect. 4.2.1. The velocity of each side is determined by the characterization of the discrete orbits, and depends on whether the elements of the discrete orbits are simply rectangles or have a mushy layer. We then compute the solutions of the two corresponding minimum problems and the corresponding discrete velocity. Optimizing the choice of the latter and passing to the limit will give a description of the actual velocity. • Minimizers without mushy layers. The set Au1 does not contain isolated sites if L0h − L1h < 4γ and 2ε

L0v − L1v < 4γ 2ε

(5.1)

where L0h and L1h denote the lengths of the horizontal sides of R0 and R1 , respectively, and correspondingly L0v and L1v for the vertical sides; in this case the computation of L1h is the same as in the case of the hard inclusions described in Sect. 4.2.3; that is, L0h − L1h 4 γ 1 = + and τ γ L0v 4

4 γ 1 L0v − L1v = + . τ γ L0h 4

5.1 High-Contrast Ferromagnetic Media: Mushy Layers

107

Fig. 5.3 Motion of a side with a mushy layer

• Minimizers with mushy layers. If the length of the sides of R0 is small enough, then the minimizer Au1 contains isolated sites. In this case, we compute the motion of a side of R0 with length L. We denote the distance between this side and the corresponding side of R1 by 2εN; N0 is the minimal odd integer greater than 4γ , so that the layer without weak islands has a width of 2εN0 (see Fig. 5.3). Then, up to constants and to negligible (higher-order) contributions due to the corners, the optimal N is determined by minimizing the sum of the following contributions: – (variation of) the perimeter of the bulk part: −4εN; – dissipation of the bulk part: 2N L ε2  L j = ε (2N 2 + N) γ ε γ j =1

– (variation of the) total energy due to the weak islands: N     L L  2 ε2 4ε − (2j + 1) = ε C + 2LN − nN2 , 2ε γ 2γ

j =N0

where C is a constant not depending on N. Hence (up to a constant) we have to minimize in N ∈ N the discrete parabola N →

3L 2 L N − N(2 − L − ) 4γ 2γ

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5 Perspectives: Evolutions with Microstructure

obtaining Nmin =

4γ 3L

−

2γ 1 1  4γ 2γ 1 − + = − + . 3 3 2 3L 3 6

Optimizing the choice of the minimizer and passing to the limit, we have that each side moves (inward) with velocity v=−

 γ 2 1  4γ 2γ 1  max + , − + , γ L 4 3L 3 6

(5.2)

where L denotes its length, or, in terms of the 1-crystalline curvature v = − max

 2 γ κ γ

2

+

1  2 2γ κ 2γ 1  , − + . 4 γ 3 3 6

(5.3)

Remark 5.1 (Extreme Cases) We denote by Aγ (t) the minimizing movement along Eε with γ = τε . • As γ → 0, we have the convergence Aγ (t) → R0 (pinning). • As γ → +∞, Aγ (t) converges to the motion of a rectangle whose sides move with velocity ⎧  8 4 ⎪ ⎨− − 3L 3 v= 2 ⎪ ⎩− L

1 2 1 if L ≥ , 2

if L
4 if κ < 4.

Note that in this last case, for large curvatures we have a motion by crystalline curvature with a forcing term. In particular, the minimizing movement along Eε (corresponding to τ >> ε) is not the minimizing movement of the -limit, whose velocity is given by v = −κ; hence, the ‘mushy layer’ accelerates the motion. ♦ Remark 5.2 (General Issue Raised by This Example) In this example we can obtain minimizing movements along a sequence with τ ∼ ε, but the commutation with the minimizing movement of the limit does not hold at the extreme scales. The reason why the claim of Theorem 2.1 is not verified is that the sequence Eε is not equicoercive in the space of sets of finite perimeter (that is, with respect to the L1 convergence); hence, the hypotheses of Theorem 2.1 are not satisfied. This is a case of ‘wrong’ computation: since we may not apply compactness results for sets of

5.2 Some Evolutions for Antiferromagnetic Systems

109

finite perimeter, the -limit should be computed with respect to some other topology weaker than the L1 -topology in which the energies should be equicoercive (e.g., the weak∗ L∞ -topology, or some topology which may keep trace of the perimeter of the mushy layers). ♦

5.2 Some Evolutions for Antiferromagnetic Systems Antiferromagnetic systems are governed by energies of the form E(u) = −



aij |ui − uj |,

(5.4)

i,j

with aij ≥ 0, or, rather, E(u) =



aij (1 − |ui − uj |),

(5.5)

i,j

so as to have positive integrands. Their minimization favors states u with alternating 0 and 1. For arbitrary lattices and coefficients aij the arrangements of minimizers can be very complex or almost arbitrary due to frustration issues (i.e., it is not possible to have |ui − uj | = 1 for all pairs of indices with aij = 0). Notably, this occurs even for nearest-neighbour interactions in the triangular lattice. In the square lattice we may treat some antiferromagnetic cases by introducing some new variables. The nearest-neighbour antiferromagnetic energy can be reduced to a ferromagnetic energy by setting vi = ui if i = (i1 , i2 ) and i1 + i2 is even, and vi = 1 − ui if i1 + i2 is odd, so that for nearest neighbours we have 1 − |ui − uj | = 1 − |vi + vj − 1| = |vi − vj |, and  i,j

aij (1 − |ui − uj |) =



aij |vi − vj |.

i,j

Note that the two uniform states v = 0 and v = 1 correspond to the even checkerboard (ui = 1 if i1 + i2 is even, and ui = 0 if i1 + i2 is odd) and to the odd checkerboard, respectively. Hence the boundary of the set Av = A1v corresponding to v can be interpreted as an interface between the checkerboard with different parity. This change of variables allows to describe the discrete-to-continuum limit on scaled lattices by computing the -limit in terms of the variable v, but it is ‘incompatible’ with our minimizing-movement scheme since it highly affects dissipation terms. However, it highlights the fact that motions governed by antiferromagnetic energies will describe an evolution of microstructure. We now describe two such

110

5 Perspectives: Evolutions with Microstructure

evolutions obtained by minimizing-movement schemes in the square lattice without entering in the details of the proofs, which are technical.

5.2.1 Nearest-Neighbour Antiferromagnetic Interactions: Nucleation We consider as an initial datum a single point (the origin) of Z2 , and describe the evolution starting from this point (nucleation) obtained by a minimizing-movement scheme for the antiferromagnetic energy2 Eε (u) = −



ε2 |ui − uj |.

i,j

Note that the energy Eε is scaled differently than in the surface-scaling case. This is motivated by scaling arguments for the perimeter of checkerboards. The energy Eε itself is not bounded from below, but its sum with the dissipation is, so that we may easily obtain the existence of discrete orbits. Note that the initial datum, the function u0 defined by (u0 )i = 1 only if i = 0, has energy −4ε2 . p We study the minimizing-movement scheme along Eε with dissipation Dε and τ,ε initial datum u0 , and with the constraint of increasing sets; namely, uk = uk satisfy Aεuk ⊂ Aεuk+1 .

(5.6)

It can be proved that the relevant regime for such a minimizing scheme is when τ = γ ε. The first u1 is then obtained by solving    1  min − |vi − vj | + ip , γ i,j

(5.7)

i:vi =1

among v : Z2 → {0, 1} with v0 = 1, and setting u1 (εi) = vi . In general this is not an easy problem, due to the asymmetries of the p-dissipation already encountered in Sect. 4.1.5. It can be proved that, except for a discrete set of γ , the minimizer v of (5.7) is unique and it is a part of an (even) checkerboard. Once this is proved, then it is easy to characterize such v: for each i such that vi = 1 we have a contribution −4 due to the first term (the perimeter of the square centered in i) plus the corresponding dissipation. The total contribution of energy and dissipation

2 The

content of this section is a partial account of the results in [4].

5.2 Some Evolutions for Antiferromagnetic Systems

111

Fig. 5.4 Example of the first and second step of the nucleation of a point with p = 2

should be negative, which gives −4+

1 ip ≤ 0; that is, ip ≤ 4γ . γ

(5.8)

Note that this does not characterize uniquely the minimizer if γ is such that ip = 4γ for some i in the even checkerboard, for which we indeed have more than one minimizer. After removing those exceptional cases, for which we refer to Remark 5.3 below, (5.8) gives that the minimizer corresponds to the part of the even checkerboard composed of squares with centers inside the ball of centre 0 and radius p 4γ for the p-norm, which we call the nucleus of the evolution, and denote by Nγ (Fig. 5.4). The minimization process can be then iterated (again, this requires a geometrically very complex proof), and the orbit can be completely described in terms of γ γ p Nγ : the function uk = uτ,ε k is described by uk (εi) = 1 if and only if there exist p i1 , . . . , ik ∈ Nγ such that i = i1 + · · · + ik . This means that the k-th set of the orbit ε,γ ε,γ Ak = Aεuγ is given recursively by A0 = ε(− 12 , 12 )2 and k

ε,γ

Ak γ

ε,γ

= Ak−1 + εNγp . ε,γ

As usual, we denote Aτ (t) = At /γ ε (recall that τ = γ ε). In order to give a continuum description passing to the limit as ε → 0, we remark that the γ characteristic functions of the sets Aτ (t) weakly∗ converge (in L∞ (R2 )) to 12 the characteristic function of a set Aγ (t) increasing with t. Finally (under some p technical assumption on Nγ ) the characterization above gives that Aγ (t) = p

p

t p P , γ γ

(5.9)

where Pγ is the convex envelope of Nγ . Hence, the evolution is given by a linear expansion of a set obtained as the convex envelope of the nucleus (Fig. 5.5).

112

5 Perspectives: Evolutions with Microstructure

Fig. 5.5 Nucleation from a point and the sets Pγ2 for different choices of γ

The shape of the nucleus depends on p and γ , except for p = ∞, in which case Pγ∞ is always a coordinate square3 and for p = 1, in which case Pγ1 is always a square rotated by π/4 with respect to the coordinate directions. Note that if 4γ < 1 then the nucleus is actually (− 12 , 12 )2 , corresponding to the point 0, so that Aγ (t) = {0} at all times, and the motion is pinned. Conversely, if p p γ → +∞ then the scaled sets γ1 Pγ converge to the ball B4 of centre 0 and radius 4 for the p-norm. This gives that the extreme cases when τ 0. In this energy we have split interactions into nearest neighbour interactions and second-neighbour interactions. We have used the notation √ i, j 2 to mean that the sum is performed over indices i, j ∈ Z2 such that i − j  = 2, with the convention that pairs of indices are accounted for only once.4

4 The evolution of these energies has been studied in [3], to which we refer for the very complex details of the proofs. There the energy is defined on functions u taking values in {−1, +1} and has the form   ui uj + c2 ui uj , E(u) = c1 i,j

but the analysis is the same.

i,j 2

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5 Perspectives: Evolutions with Microstructure

In order to understand the behaviour of such energies, for every square of connections we may consider the combined interactions corresponding to the four sides of strength −c1 (with a factor 1/2 since they are accounted for twice) and the two cross links of strength −c2 ; i.e., if we denote by i, j, , k the four nodes in a counterclockwise order, we consider the value −

c1 (|ui − uj | + |uj − u | + |u − uk | + |uk − ui |) − c2 (|ui − u | + |uj − uk |). 2

This expression is minimized (up to rotations) by ui = uj = 1, u = uk = 0 (stripe structure) if 2c2 > c1

(5.11)

or by ui = u = 1, uj = uk = 0 (checkerboard structure) otherwise. We consider the case (5.11), which is richer than the checkerboard case (which in turn may be treated similarly to that of nearest neighbours). Indeed in this case ground states minimizing the energy at each square of connections are of four types (vertical and horizontal stripes, with two different possible parities each). We use the vectors ±e1 as a parameter to label ground states with vertical stripes, and ±e2 to label ground states with horizontal stripes. This is an arbitrary choice, but reminds the orientation of the stripes (orthogonal to the corresponding vector) and the dependence on the parity. We regroup the interactions in the energy so as to avoid +∞ − ∞ indeterminate forms. The scaled energies are then of the form Eε (u) = −

   c1 (|ui − uj | + |uj − u | + |u − uk | + |uk − ui |) ε 2 squares i,j,,k

 +c2 (|ui − u | + |uj − uk |) − c1 − 2c2 ,

with the sum performed over all squares of interactions in εZ2 with i, j, , k parameterizing the four sites of the square. Subtracting the normalization constant c1 + 2c2 makes the energy of ground states 0. The -limit can be then described as an energy defined on u : R2 → {±e1 , ±e2 }, which is not a perimeter energy anymore but is an energy on partitions. This limit energy is again crystalline, in that if it restricted to functions taking only two values, it can be written as a crystalline perimeter. Geometric evolution problems for partitions are an almost completely unexplored fields for anisotropic perimeters and in particular for crystalline energies.5 We may nevertheless study some evolution problems when the limit initial datum corresponds to a function taking only two values in {±e1 , ±e2 } and the correspond-

5 We

refer for example to [5] and [6].

5.2 Some Evolutions for Antiferromagnetic Systems

115

ing characteristic functions are Wulff-type set for the limit energy (when restricted to these two values). The interesting case is when these two values are ±e1 (or ±e2 ), parameterizing stripes of the same orientation, for which the Wulff shape is an hexagon with horizontal or 45-degree sides (for stripes of different orientation; e.g, if the limit initial datum takes only the values e1 and e2 , the Wulff shape is a square and the situation is similar to that of a ferromagnetic energy). A discrete Wulff-type set for two types of vertical stripes is pictured in Fig. 5.7. Note that the transition for the horizontal interfaces is achieved producing a single layer of squares which are not of ground-state type, while for the slanted sides the minimal interface is obtained with a double spacing in the horizontal direction. We consider as a starting set A0 a Wulff-type set as in Fig. 5.7 parameterized with e1 on a bounded region (and −e1 on an unbounded region) in the parameterization above, and u0 = uε0 the corresponding function defined on εZ2 with value 1 on black circles and 0 on white circles. We define iteratively uk = uε,τ k as minimizing   1 min Eε (u) + Dε1 (u, uk−1 ) . τ The dissipation should be properly defined in terms of the parameters ±e1 , so that in the following it is understood that it involves the 1-distance from the set where the parameter of u (defined on squares) is −e1 , and not from the set where u itself (defined on the nodes) is 1. We can describe the discrete orbits as follows. • each uk corresponds either to a Wulff-type set as in Fig. 5.7, or to a Wulff-type set with a corner defect as in Fig. 5.8. The appearance of interactions which are not minimal for the interfacial energy (the black square in Fig. 5.8) is due to a combined effect of energy and dissipation: a defect may appear if a minimizer without defect requires too much dissipation; • the squares where the parameter describing vertical stripes is e1 (i.e., such that uk equals to the stripes parameterized by e1 on the square) decrease with k, the squares where the parameter describing vertical stripes is −e1 increase with k. This means that the Wulff-type sets are ‘contracting’; Fig. 5.7 A discrete Wulff-type set for the NNN antiferromagnetic energy

116

5 Perspectives: Evolutions with Microstructure

Fig. 5.8 A discrete Wulff-type with a corner defect

• the rate of contraction of the slanted sides can be computed by separately considering the possibility of minimizer either not to have defects, or to toggle between defected and non-defected Wulff-type sets. These two cases occur in dependence not only of the length of the corresponding side, but also of the neighboring slanted side. The rate of contraction of the horizontal side only depend on their length. We omit the rather lengthy computations. The critical regime can be again proved to be described by τ = γ ε. In this case γ we define as Ak the union of the ε-squares where the parameter of uk is e1 , and γ γ γ Aτ (t) = At /τ . The sets Aτ (t) converge to sets Aγ (t). This family is composed of Wulff-type hexagons, whose horizontal sides move with velocity 1 v = − αγ κ, γ where we set κ = 2/L, L being the side length, and α is a constant depending on c2 . The slanted sides move with a velocity depending on the curvature κ of the side and on the curvature κ  of the neighboring slanted size, with a velocity 1 v = − √ fhom (βγ κ  , βγ κ), 2γ where β is a constant depending on c1 and c2 and the non-local homogenized velocity fhom : [0, +∞) × [0, +∞) → N is described in Fig. 5.9: fhom takes integer values which are the same in horizontal stripes. The horizontal stripes are themselves subdivided in smaller regions, corresponding alternately to regimes where the homogenized velocity is achieved at the discrete level with alternating defected and non-defected sets, or with only non-defected sets. The regions where fhom equals the constant n are bounded by the graphs of some functions vn and vn+1 , whose computation we do not include. It is close in spirit to that for ferromagnetic systems with an oscillating forcing term, in the sense that

5.2 Some Evolutions for Antiferromagnetic Systems

117

Fig. 5.9 The curves y = vn (x) defined in (5.12) and (5.13)

also there we had a choice between two types of minimum problems giving different forms of the energy. The functions vn have a form as follows. • For n ≥ 1 odd and m ≥ 0 we have  nx    ⎧ c1 nx ⎪ min , n 1 + , ⎪ ⎪ m 2(c2 − c1 ) 2x − (m + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if m ≤ x ≤ m + 1, m even, ⎨ vn (x) =  nx  ⎪ c1  nx  ⎪ ⎪ ⎪ max , , n 1 − ⎪ ⎪ 2x − m 2c2 m + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ if m ≤ x ≤ m + 1, m odd.

(5.12)

• If n = 0 then v0 (x) ≡ 0; if n ≥ 2 is even and m ≥ 0 then

vn (x) =

 nx  ⎧ c1  nx  ⎪ max , n 1 − , ⎪ ⎪ 2x − m 2c2 m + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if m ≤ x ≤ m + 1, m even, ⎨  nx    ⎪ c1 nx ⎪ ⎪ ⎪ min , n 1 + , ⎪ ⎪ m 2(c2 − c1 ) 2x − (m + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ if m ≤ x ≤ m + 1, m odd.

(5.13)

118

5 Perspectives: Evolutions with Microstructure

5.3 More Conclusions In this chapter we have briefly examined three examples, which also in the static case lead us towards new and interesting types of energies involving the formation and description of microstructure. In Sect. 5.1 the lack of equicoerciveness in the space of sets of finite perimeter of a family of ferromagnetic energies forces the emergence of microstructure producing mushy layers in discrete orbits. While we are able to describe minimizing movements along these energies, the characteristics of the problem suggest that a finer description of the effects of the concentration of the perimeter measure on the boundary of sets might be given, providing a different -limit, which would allow for the characterization of extreme regimes as its minimizing movements. The model studied in Sect. 5.2.1 introduces energies which favor the formation of bulk microstructure, for which the limit motion is obtained as a motion of sets in which bulk and surface interactions mix at the lattice level in a way that has no direct continuum counterpart. We have only examined the case of nucleation from a point, which gives a linearly expanding ‘crystal’ inside which the perimeter is ‘maximized’ at a discrete level. The third model in Sect. 5.2.2, finally, possesses four ‘striped’ microscopical ground states, and its behaviors can be described with an energy on partitions into four sets of finite perimeter with crystalline interfacial energies. We have examined the problem of the motion of a single crystal of one of the minimizing striped states within a matrix of the same kind of stripes but with different parity. A new phenomenon has emerged, consisting in discrete orbits favoured by the appearance of corner defects (a kind of boundary dislocation), which make the final motion depending on the combined effect of neighboring sides of a crystal. The general case when more than two microscopical ground states are present is completely open, also by the absence of results on anisotropic or crystalline motion of ‘polycrystals’ on the continuum that could be used as a comparison in the place of motion by crystalline curvature.

References 1. A. Braides, G. Scilla, Nucleation and backward motion of discrete interfaces. C.R. Acad. Sci. Paris. Ser. I 351, 803–806 (2013) 2. A. Braides, M. Solci, Motion of discrete interfaces through mushy layers. J. Nonlinear Sci. 26, 1031–1053 (2016) 3. A. Braides, M. Cicalese, N.K. Yip, Crystalline motion of interfaces between patterns. J. Stat. Phys. 165(2), 274–319 (2016) 4. A. Braides, G. Scilla, A. Tribuzio, Nucleation and growth of lattice crystals. Preprint (2020) 5. D. Kinderlehrer, C. Liu, Evolution of grain boundaries. Math. Models Methods Appl. Sci. 11, 713–729 (2001) 6. C. Mantegazza, M. Novaga, V.M. Tortorelli, Motion by curvature of planar networks. Ann. Scuola Norm. Sup. Pisa 3, 235–324 (2004)

Appendix A

-Limits in General Lattices

We prove a -convergence result for the sequence of energies Eε defined in {0, 1}εL by Eε (u) =



εaij |ui − uj |,

(A.1)

i,j ∈L

where L is a discrete lattice spanned over Z by two independent vectors w1 and w2 . We assume that aij is of finite range, non-negative, symmetric and such that aij depends only on i − j ; that is, the period of the interactions is 1. Moreover, we assume that the set σ ({aij }) = {i ∈ L : a0i = 0} is a system of generators for L; that is, {aij } satisfies an equicoercivity condition on nearest-neighbours. Note that, up to changing the basis of L, we can assume that w1 , w2 ∈ σ ({aij }). The finite set σ ({aij }) and the corresponding values a0i completely determine the interaction. We set cL = | det(w1 , w2 )|−1 , and note that it is an invariant of the lattice (i.e., it does not depend on the choice of the basis). In these hypotheses, we have the following result. Proposition A.1 Let Eε be given by (A.1), with aij of finite range, non-negative, symmetric, depending only on i − j and such that spanZ σ ({aij }) = L. Then the -limit of Eε is of the form (3.14); that is, - lim Eε (A) = ε→0

∂∗A

ϕ(ν) dH 1 ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Braides, M. Solci, Geometric Flows on Planar Lattices, Pathways in Mathematics, https://doi.org/10.1007/978-3-030-69917-8

(A.2)

119

120

A -Limits in General Lattices

and we can give an explicit formula for ϕ; that is, ϕ(ν) =

cL  a0i | ν, i |. 2

(A.3)

i∈L

Proof We subdivide the proof in some steps. Decomposition Let 2N + 4 = #σ ({aij }). In order to decompose the interactions (and the functional), we write σ ({aij }) = {±w1 , ±w2 , ±v 1 , . . . , ±v N } and define the (possibly empty) sets σ1 = {v r : {w1 ; v r } are independent}

and

σ2 = {v r : v r ∈ span(w1 ) ∩ L}.

We consider the following pairs of independent vectors in σ ({aij }): • (v1s , v2s ) with v1s = w1 and v2s ∈ σ1 , s = 1, . . . , N1 = #σ1 ; for these pairs, we define aijs by setting aijs = 0 if i − j ∈ {±w1 , ±v2s } and aijs =

a0w1 if i − j ∈ {±w1 }, aijs = a0v2s if i − j ∈ {±v2s }; N1 + 1

• (v1s , v2s ) with v2s = w2 and v1s ∈ σ2 , s = N1 +1, . . . , N; for these pairs, we define aijs by setting aijs = 0 if i − j ∈ {±w2 , ±v1s } and aijs =

a0w2 if i − j ∈ {±w2 }, aijs = a0v1s if i − j ∈ {±v1s }; N − N1 + 1

• (v1N+1 , v2N+1 ) = (w1 , w2 ); we set aijs = 0 if i − j ∈ {±w1 , ±w2 } and aijN+1 =

a0w1 a0w2 if i − j ∈ {±w1 }, aijN+1 = if i − j ∈ {±w2 }. N1 + 1 N − N1 + 1

 s s s s In this way, we get the decomposition aij = N+1 s=1 aij . Setting R = Zv1 + Zv2 s s s and observing that L is the union of cL | det(v1 , v2 )| copies of (translations) of R , we can write Eε (uε ) =

N+1  s=1

Eεs (uε ) :=

N+1  s=1

cL | det(v1s , v2s )|



εaijs |uεi − uεj |.

(A.4)

i,j ∈Rs

Passage to εZ2 and -Convergence For any s we define the linear map Ms : Rs → Z2 such that Ms (v1s ) = e1 and Ms (v2s ) = e2 (see Fig. A.1); note that det(MN+1 ) = cL . Moreover, we define the dual cell C s of the lattice Rs by setting C s =

A -Limits in General Lattices

121

Fig. A.1 The lattice Rs , its subdivision into cells and their image in Z2

(Ms )−1 (Q) and the set Cεs = of the lattice L.

&

i∈L (εC

s

+ εi). Note that C N+1 is the dual cell

We define a family of nearest-neighbours functionals defined on the lattice εZ2 by setting Gsε (v) =

 cL s εaˆ hk |vh − vk |, | det(Ms )|

s = as where aˆ hk

Ms−1 (h)Ms−1 (k)

- lim Gsε (B) = ε→0

. Recalling Sect. 3.2.3,

cL | det(Ms )|

∂ ∗B

  s s | ν, e2 | dH 1 . aˆ 0e1 | ν, e1 | + aˆ 0e 2

Now, let uε ∈ {0, 1}εL be such that Eε (uε ) ≤ C; then we can assume uε → A; 2 setting uˆ ε,s = uε ◦ (Ms )−1 ∈ {0, 1}εZ , by (A.4) we have Eε (u ) = ε

N+1 

Gsε (uˆ ε,s ).

(A.5)

s=1

For any s < N + 1 we define the set sε = convergences

&

i∈Rs (εC

N+1 + εi);

thanks to the weak

χsε  | det(Ms )|(cL )−1 and χMs (sε )  | det(Ms )|(cL )−1 in L2 (R2 ),

122

A -Limits in General Lattices

we have uˆ ε,s → Ms (A). By a change of variable, we get for any s

 ∂ ∗Ms (A)

 s s aˆ 0e | ν, e1 | + aˆ 0e | ν, e2 | dH 1 1 2

=

 s  s s s 1 | det(Ms )| a0v s | ν, v1 | + a0v s | ν, v2 | dH ;

∂ ∗A

1

2

hence, N+1 

- lim Gsε (Ms (A)) =

s=1

ε→0

N+1 

cL

s=1

=

∂ ∗A

cL 2

  s s s 1 a0v s | ν, v1s | + a0v s | ν, v2 | dH

∂ ∗A

1



2

a0v | ν, v | dH 1 ,

v∈σ ({aij })

where the coefficient 1/2 comes from the fact that if v ∈ σ ({aij }) also −v does, with the same coefficient. Now we can give the lower bound for Eε (uε ), as lim inf Eε (uε ) ≥ ε→0

N+1  s=1

lim inf Gsε (uˆ ε,s ) ≥ ε→0

∂ ∗A

cL 2



a0v | ν, v | dH 1 .

v∈σ ({aij })

As for the upper bound, we consider a polygonal set A and choose uε = χA∩εL . Since uˆ ε,s = χMs (A)∩εZ2 → Ms (A), then for any s uε gives a corresponding recovery sequence for Gsε , so that lim sup Eε (uε ) = lim sup ε→0

N+1 

ε→0

≤

∂ ∗A

cL 2

Gsε (uˆ ε,s ) =

s=1



N+1  s=1

lim Gsε (uˆ ε,s )

ε→0

a0v | ν, v | dH 1 .

v∈σ ({aij })

The proof for A of finite perimeter follows by density, using a diagonal argument.   Note that using Proposition A.1 we may in particular recover the form of energy densities for long-range interaction systems in the square lattice. For example, we consider L = Z2 and the coefficients aij as in Sect. 3.2.2. Hence σ ({aij }) = {±(1, 0), ±(0, 1), ±(1, 1), ±(−1, 1)}

A -Limits in General Lattices

123

and by (A.3) we obtain ϕ(ν) = |ν1 | + |ν1 + ν2 | + |ν2 | + | − ν1 + ν2 | = ν1 + 2ν∞ . Similarly, adding ±(2, 0) e ±(0, 2) we may include neighbours at distance 2 obtaining ϕ(ν) = 3ν1 + 2ν∞ .

Appendix B

A Non-trivial Example with Trivial Minimizing Movements

We give the proof of Proposition 2.1, which shows that condition (H) is not necessary to have that any minimizing movement along a sequence Eε at any scale τ is a minimizing movement for the (limit) functional E. We recall the definition of the functionals E and Eε , which do not satisfy condition (H). Fixed X = [0, +∞) with the distance d(u, v) = |u − v|, we set E(u) = −u and Eε (u) = −u1+ε and consider the dissipation given by D(u, v) =

1 (u − v)2 . 2

The initial datum for E is u0 = 0, and the same for any Eε . We will compute the minimizing movements for Eε at any fixed ε, showing that their limit coincides with the minimizing movement of E, and then we prove that it coincides with the minimizing movement along the sequence Eε at any scale τε by using the result of Theorem 2.1(a) for small time scales and some monotonicity properties. 1. Minimizing movement for E and Eε . The minimizing movement for E is u(t) = t. As for Eε , the following proposition holds, where we denote by uε the minimizing movement (instead of uε as used everywhere else in the notes) to avoid ambiguity with the power uε . The notation for the orbits remains unchanged. Proposition B.1 Let Eε = −u1+ε . Then the only minimizing movement with initial datum uε0 = 0 and dissipation D is   1 1−ε . uε (t) = (1 − ε2 )t

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Braides, M. Solci, Geometric Flows on Planar Lattices, Pathways in Mathematics, https://doi.org/10.1007/978-3-030-69917-8

125

126

B A Non-trivial Example with Trivial Minimizing Movements

The proof of Proposition B.1 is based on the following estimate on the orbits, showing that the function 0 is not a minimizing movement for Eε . ε Lemma B.1 Let {uτ,ε k }k be the orbit of Eε with initial datum u0 = 0. Then, for any τ > 0 an for any k ∈ N the following estimate holds

  1 1−ε 2 ≥ (1 − ε )kτ . uτ,ε k

(B.1)

Proof (of Lemma B.1) We introduce the function ϕ τ,ε : [0, +∞) → R defined by ϕ τ,ε (u) = u − τ (1 + ε)uε

(B.2)

(see Fig. B.1); the orbit {uτ,ε k } is then (implicitly) defined by the equation τ,ε ϕ τ,ε (uτ,ε k ) = uk−1 .

(B.3)

We prove estimate (B.1) by induction on k. If k = 1, the estimate holds since 1

1−ε . uτ,ε 1 = (τ (1 + ε))

Now we assume that the claim is true for k − 1; that is,   1 1−ε 2 uτ,ε , k−1 ≥ (1 − ε )(k − 1)τ

Fig. B.1 The function ϕ τ,ε and the construction of the orbit uτ,ε k

(B.4)

B A Non-trivial Example with Trivial Minimizing Movements

127

and deduce that (B.1) holds for k. The function ϕ τ,ε defined in (B.2) is strictly τ,ε increasing in the interval where it is positive; that is, in [uτ,ε 1 , +∞). Since uk > τ,ε u1 , the orbit is then given by τ,ε τ,ε uτ,ε k = ψ (uk−1 ),

where ψ τ,ε is the inverse of the restriction of ϕ τ,ε to [uτ,ε 1 , +∞). Since (the restriction of) ϕ τ,ε is strictly increasing, we deduce that (B.1) is satisfied for k if we show that τ,ε uτ,ε k−1 ≥ ϕ



(1 − ε2 )kτ



1 1−ε

 .

(B.5)

1

Now we consider the function u → u 1−ε defined in [0, +∞); since ε ∈ (0, 1), it is convex, so that for any k we have 1

1

(k − 1) 1−ε ≥ k 1−ε −

ε 1 k 1−ε . 1−ε

 1  1−ε we get Multiplying this inequality by (1 − ε2 )τ   1   1   ε 1−ε 1−ε 1−ε (1 − ε2 )(k − 1)τ ≥ (1 − ε2 )kτ − (1 + ε)τ (1 − ε2 )kτ = ϕ τ,ε

  1  1−ε (1 − ε2 )kτ .

We apply the induction hypothesis (B.4) and deduce that (B.5) holds, concluding the proof of (B.1).   Proof (of Proposition B.1) By applying Lemma B.1 we obtain the estimate 

t   1 1 1−ε uτ,ε (t) ≥ (1 − ε2 ) τ ≥ ((1 − ε2 )t) 1−ε ; τ hence, if uε is a minimizing movement for Eε we deduce 1

uε (t) = lim uτj ,ε (t) ≥ ((1 − ε2 )t) 1−ε > 0 τj →0

for t ∈ (0, +∞).

(B.6)

Now, we compute the minimizing movement. Since uτ,ε (t) ≥ 0, we can write −Eε (uτ,ε (t)) =

uτ,ε (t) − uτ,ε (t − τ ) τ

128

B A Non-trivial Example with Trivial Minimizing Movements

in [τ, +∞). Fixed 0 < δ < T , for any φ ∈ Cc∞ (δ, T ) we obtain for τ ≤ δ − δ

T

Eε (uτ,ε (t))φ(t) dt



T

=

δ T

=

uτ,ε (t) − uτ,ε (t − τ ) φ(t) dt τ uτ,ε (t)

δ

φ(t) − φ(t + τ ) dt. τ

Passing to the limit as τ → 0, we get

T

− δ

Eε (uε (t))φ(t) dt

=

T





T

uε (t)(−φ (t)) dt =

δ

(uε ) (t)φ(t) dt

δ

so that the equation −Eε (uε ) = (uε ) is satisfied in the sense of distributions in (0, +∞) since δ and T are arbitrary. By (B.6), every minimizing movement is strictly positive for t > 0; hence, by solving the equation with initial datum uε (0) = 0 it follows that the (unique) minimizing movement is   1 1−ε uε (t) = (1 − ε2 )t in [0, +∞).

(B.7)  

2. Minimizing movements along the sequence Since the limit of the sequence uε as ε → 0 is the minimizing movement of E, by Theorem 2.1(a) we have in particular that there exists a time-scale τ (ε) such that τ (ε) → 0 as ε → 0 and lim uτε ,ε (t) = u(t) = t

ε→0

(B.8)

(uniformly) for any scale τε ≤ τ (ε). It is not restrictive to assume that the sequence τ (ε) is strictly increasing. Now, we show that (B.8) holds for any time-scale. We start by proving that also in this case the minimizing movements along the sequence vanish only in 0. By contradiction, suppose that there exist εj → 0 and τj → 0 both decreasing as j → +∞ (this is not restrictive, up to subsequences) such that for some t > 0 uτj ,εj (t) → 0 as j → +∞. Hence, up to a finite number of j , τj > τ (εj ). By induction, thanks to the monotonicity of τ (ε) and of τj , we can construct a sequence of integers jk such

B A Non-trivial Example with Trivial Minimizing Movements

129

that, for any k, jk+1 > jk , jk > k and τjk ≤ τ (εk ). This implies lim uτjk ,εk (t) = u(t) = t.

k→+∞

Now, in order to ‘reparameterize’ the sequence and conclude the argument, we need to show that the orbits uτ,ε (t) are non-increasing with respect to ε at any fixed τ . Lemma B.2 There exists c > 0 such that if ε < ε < c, then for any τ and k  1 τ,ε  τ,ε  ⇒ uτ,ε ∈ 0, uτ,ε k , uk k > uk . e Proof (of Lemma B.2) Solving the equation for the first step of the orbit, we obtain   1 1−ε uτ,ε = τ (1 + ε) ; 1 hence there exists c > 0 such that   1 1 1−ε uτ,ε = τ (1 + ε) ≤ 1 e for any τ, ε < c. The sequence {uτ,ε k }k increases to +∞ for any fixed ε, τ > 0. Now, we show the claim by induction. Let ϕ τ,ε be the function introduced in (B.2). With fixed u and τ , the function ε → ϕ τ,ε (u) is strictly increasing with 1 respect to ε if u < e− 1+ε . 1 Since e− 1+ε > 1e , for any τ and u ≤ 1e 

ε < ε ⇒ ϕ τ,ε (u) < ϕ τ,ε (u) (see Fig. B.2). Recalling that ϕ τ,ε (u) is strictly increasing (with respect to u) in τ,ε [uτ,ε 1 , 1/e], we consider the inverse of this restriction, again denoted by ψ , which τ,ε is strictly increasing with respect to u; moreover ε → ψ (u) is strictly decreasing with respect to ε, so that 

ε < ε ⇒ ψ τ,ε (u) > ψ τ,ε (u). The first step of the induction is then true, since 



τ,ε τ,ε (0) = uτ,ε uτ,ε 1 = ψ (0) > ψ 1 . 

τ,ε If we assume that uτ,ε k−1 > uk−1 , we obtain the claim for k; that is, 







τ,ε τ,ε τ,ε τ,ε τ,ε τ,ε uτ,ε (uτ,ε k−1 ) = uk , k = ψ (uk−1 ) > ψ (uk−1 ) > ψ

130

B A Non-trivial Example with Trivial Minimizing Movements

Fig. B.2 Monotonicity of ϕ τ,ε with respect to ε

where we used the monotonicity of ψ τ,ε (·) and the induction hypothesis for the first inequality, and the monotonicity with respect to ε in the second. Hence, the claim holds for any k.   Now, we go back to the proof of the fact that a minimizing movement along Eε vanishes only in 0, recalling that we supposed by contradiction that uτj ,εj (t) → 0 as j → +∞ for some t > 0. If t ≤

1 2e

and k large enough, we have that uτjk ,εk (t) ≤

1 1 , uτjk ,εjk (t) ≤ . e e

Since jk > k and εj is monotone, then εjk < εk and we can apply Lemma B.2 obtaining t = lim uτjk ,εk (t) ≤ lim uτjk ,εjk (t) = 0, k→+∞

k→+∞

1 which gives a contradiction if t ≤ 2e . Hence, any minimizing movement along the sequence Eε with initial datum 0 vanishes only in 0. Finally, we have to prove that any minimizing movement along Eε with initial datum uε0 = 0 is the minimizing movement of E.

B A Non-trivial Example with Trivial Minimizing Movements

131

Let uτε ,ε any orbit converging to a minimizing movement w. As above, we can write −Eε (uτε ,ε (t)) =

uτε ,ε (t) − uτε ,ε (t − τε ) τε

in [τε , +∞). Fixed 0 < δ < T arbitrary, taking φ ∈ C0∞ (δ, T ) we get − δ

T

Eε (uτε ,ε (t))φ(t) dt



T

= δ

uτε ,ε (t)

φ(t) − φ(t + τε ) dt τε

for τε ≤ δ. The minimizing movement w is continuous and strictly positive in [δ, T ], hence uτε ,ε takes values in a compact set included in (0, +∞). Since Eε uniformly converges to 1 in the compact subsets of (0, +∞), and uτε ,ε uniformly converges to w, by taking the limit as ε → 0 we get that w = 1 in the sense of distributions in [δ, T ]. Since the choice of δ and T is arbitrary, it follows that w(t) = t = u(t) in [0, +∞), concluding the proof of Proposition 2.1.

Index

C Corner defect, 115 Critical scale, 70 Crystalline curvature, 5, 58 motion by –, 5, 55, 58, 75 motion by – with a forcing term, 108 Curve of maximal slope, 24 D De Giorgi variational interpolant, 24 Descending slope, 23 Dissipation, 8 scaled –, 16, 69 E Energy on partitions, 114 Evolution of microstructure, 109 F Flat flow, 12, 54 Forcing term, 91 Frustration, 5, 109 G Gamma-convergence, 19 – on lattices, 33 Gradient flow, 4, 22 H Homogenization, 14, 46 Homogenized evolution, 4

Homogenized velocity, 90 non-local –, 116 shallow-well –, 100

I Implicit Euler scheme, 9 Ising system, 2

L Lattice energies, 15 anti-ferromagnetic –, 5, 110 ferromagnetic –, 3, 35 Local minima, 26

M Metric derivative, 23 Minimal-interface formula, 47 Minimizing movement, 13 – along a sequence, 18 – along lattice energies, 70 Mobility factor, 63, 67 discrete –, 82 Mushy layer, 104

N Nearest neighbours, 36 Normal to ∂ ∗ A, 33 Nucleation, 110

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Braides, M. Solci, Geometric Flows on Planar Lattices, Pathways in Mathematics, https://doi.org/10.1007/978-3-030-69917-8

133

134 P Perimeter, 31 crystalline –, 16, 34, 47, 54 – functionals, 33 set of finite –, 32 Pinning, 75 – threshold, 75, 99

Q Quantized velocity, 75

R Reduced boundary, 33

Index S Slope-cone property, 26 Spin functions, 2 convergence of – to a set, 32 Surface scaling, 15

T Time-discrete orbit, 8 Triangular lattice, 49 W Weak upper gradient, 23 Wulff shape, 34, 36, 43 Wulff-type set, 63, 70