Introduction to the Modern Calculus of Variations

MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom F.Rindler@warwick.ac.uk http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result. Only then do I specialize to the case of integral functionals on Sobolev spaces. • I do not attempt to trace every result to its original source as many results have evolved over generations of mathematicians. Some pointers to further literature are given in the last section of every chapter. Most of what I present here is not original, I learned it from my teachers and col- leagues, both through formal lectures and through informal discussions. Published works that influenced this course were the lecture notes on microstructure by S. Muller¨ [73], the encyclopedic work on the Calculus of Variations by B. Dacorogna [25], the book on Young measures by P. Pedregal [81], Giusti’s more regularity theory-focused introduction to the Calculus of Variations [44], as well as lecture notes on several related courses by J. Ball, J. Kristensen, A. Mielke. Further texts on the Calculus of Variations are the elementary introductions by B. van Brunt [96] and B. Dacorogna [26], the more classical two-part trea- tise [39, 40] by M. Giaquinta & S. Hildebrandt, as well as the comprehensive treatment of Comment... ii PREFACE the geometric measure theory aspects of the Calculus of Variations by Giaquinta, Modica & Soucekˇ [41, 42]. In particular, I would like to thank Alexander Mielke and Jan Kristensen, who are re- sponsible for my choice of research area. Through their generosity and enthusiasm in shar- ing their knowledge they have provided me with the foundation for my study and research. Furthermore, I am grateful to Richard Gratwick and the participants of the MA4G6 course for many helpful comments, corrections, and remarks. These lecture notes are a living document and I would appreciate comments, corrections and remarks – however small. They can be sent back to me via e-mail at F.Rindler@warwick.ac.uk or anonymously via http://tinyurl.com/MA4G6-2015 or by scanning the following QR-Code: Moreover, every page contains its individual QR-code for immediate feedback. I encourage all readers to make use of them. F. R. January 2015 Comment... Contents Preface i Contents iii 1 Introduction 1 1.1 The brachistochrone problem ......................... 2 1.2 Electrostatics ................................. 4 1.3 Stationary states in quantum mechanics ................... 5 1.4 Hyperelasticity ................................ 6 1.5 Optimal saving and consumption ....................... 8 1.6 Sailing against the wind ............................ 9 2 Convexity 11 2.1 The Direct Method .............................. 12 2.2 Functionals with convex integrands ...................... 14 2.3 Integrands with u-dependence ........................ 18 2.4 The Euler–Lagrange equation ......................... 19 2.5 Regularity of minimizers ........................... 24 2.6 Lavrentiev gap phenomenon ......................... 31 2.7 Invariances and the Noether theorem ..................... 33 2.8 Side constraints ................................ 37 2.9 Notes and bibliographical remarks ...................... 41 3 Quasiconvexity 43 3.1 Quasiconvexity ................................ 44 3.2 Null-Lagrangians ............................... 51 3.3 Young measures ................................ 54 3.4 Gradient Young measures ........................... 62 3.5 Gradient Young measures and rigidity .................... 66 3.6 Lower semicontinuity ............................. 68 3.7 Integrands with u-dependence ........................ 73 3.8 Regularity of minimizers ........................... 74 3.9 Notes and bibliographical remarks ...................... 75 Comment... iv CONTENTS 4 Polyconvexity 77 4.1 Polyconvexity ................................. 78 4.2 Existence Theorems .............................. 79 4.3 Global injectivity ............................... 85 4.4 Notes and bibliographical remarks ...................... 86 5 Relaxation 87 5.1 Quasiconvex envelopes ............................ 88 5.2 Relaxation of integral functionals ....................... 91 5.3 Young measure relaxation ........................... 95 5.4 Characterization of gradient Young measures . 101 5.5 Notes and bibliographical remarks . 105 A Prerequisites 107 A.1 General facts and notation . 107 A.2 Measure theory ................................109 A.3 Functional analysis ..............................111 A.4 Sobolev and other function spaces spaces . 112 Bibliography 117 Index 125 Comment... Chapter 1 Introduction Physical modeling is a delicate business – trying to balance the validity of the resulting model, that is, its agreement with nature and its predictive capabilities, with the feasibility of its mathematical treatment is often highly non-straightforward. In this quest to formulate a useful picture of an aspect of the world, it turns out on surprisingly many occasions that the formulation of the model becomes clearer, more compact, or more descriptive, if one introduces some form of a variational principle, that is, if one can define a quantity, such as energy or entropy, which obeys a minimization, maximization or saddle-point law. How “fundamental” such a scalar quantity is perceived depends on the situation: For example, in classical mechanics, one calls forces conservative if they can be expressed as the derivative of an “energy potential” along a curve. It turns out that many, if not most, forces in physics are conservative, which seems to imply that the concept of energy is fundamental. Furthermore, Einstein’s famous formula E = mc2 expresses the equivalence of mass and energy, positioning energy as the “most” fundamental quantity, with mass becom- ing “condensed” energy. On the other hand, the other ubiquitous scalar physical quantity, the entropy, has a more “artificial” flavor as a measure of missing information in a model, as is now well understood in Statistical Mechanics, where entropy becomes a derived quantity. We leave it to the field of Philosophy of Science to understand the nature of variational concepts. In the words of William Thomson, the Lord Kelvin1: The fact that mathematics does such a good job of describing the Universe is a mystery that we don’t understand. And a debt that we will probably never be able to repay. Our approach to variational quantities here is a pragmatic one: We see them as providing structure to a problem and to enable different mathematical, so-called variational, methods. For instance, in elasticity theory it is usually unrealistic that a system will tend to a global minimizer by itself, but this does not mean that – as an approximation – such a principle cannot be useful in practice. On a fundamental level, there probably is no minimization as such in physics, only quantum-mechanical interactions (or whatever lies “below” quantum 1Quotes prove nothing, however. Thomson also said “X-rays will prove to be a hoax”. Comment... 2 CHAPTER 1. INTRODUCTION mechanics). However, if we wait long enough, the inherent noise in a realistic system will move our system around and it will settle with a high probability in a low-energy state. In this course, we will focus solely on minimization problems for integral functionals of the form (W ⊂ Rd open and bounded) Z f (x;u(x);∇u(x)) dx ! min; u: W ! Rm; W possibly under conditions on the boundary values of u and further side constraints. These problems form the original core of the Calculus of Variations and are as relevant today as they have always been. The systematic understanding of these integral functionals starts in Euler’s and Bernoulli’s times in the late 1600s and the early 1700s, and their study was boosted into the modern age by Hilbert’s 19th, 20th, 23rd problems, formulated in 1900 [47]. The excitement has never stopped since and new and important discoveries are made every year. We start this course by looking at a parade of examples, which we treat at varying levels of detail. All these problems will be investigated further along the course once we have developed the necessary

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