Calculus of Variations (Cambridge Studies in Advanced Mathematics)

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 64 EDITORIAL BOARD D.J.H. GARLING, W. FULTON, K. RIBET, T. TOM DIECK, P. WALTERS

CALCULUS OF VARIATIONS

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W . M . L . Holcombe Algebraic automata theory K . Petersen Ergodic theory P.T. Johnstone Stone spaces W,H. Schikhof Ultrametric calculus J.-P. Kahane Some random series of functions, 2nd edition H. Cohn Introduction to the construction of class fields J . Lambek & P.J. Scott Introduction to higher-order categorical logic H. Matsumura Commutative ring theory C . B . Thomas Characteristic classes and the cohomology of finite groups M. Aschbaeher Finite group theory J . L . Alperin Local representation theory P. Koosis The logarithmic integral I A. Pietsch Eigenvalues and S-numbers S.J. Patterson An introduction to the theory of the Riemann zeta-function H.J. Baues Algebraic homotopy V.S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups W. Dicks & M. Dunwoody Groups acting on graphs L . J . Corwin & F.P. Greenleaf Representations of nilpotent Lie groups and their applications R. Pritsch & R. Piccinini Cellular structures in topology H Klingen Introductory lectures on Siegel modular forms P. Koosis The logarithmic integral II M.J. Collins Representations and characters of finite groups H. Kunita Stochastic flows and stochastic differential equations P. Wojtaszczyk Banach spaces for analysts J . E . Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis A. Prohlich & M.J. Taylor Algebraic number theory K . Goebel & W . A . Kirk Topics in metric fixed point theory J . F . Humphreys Reflection groups and Coxeter groups D.J. Benson Representations and cohomology I D.J. Benson Representations and cohomology II C . Allday & V . Puppe Cohomological methods in transformation groups C . Soule et al Lectures on Arakelov geometry A. Ambrosetti & G . Prodi A primer of nonlinear analysis J . Palis & F . Takens Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations M. Auslander, I. Reiten & S. Smalo Representation theory of Artin algebras Y . Meyer Wavelets and operators C . Weibel An introduction to homological algebra W. Bruns & J . Herzog Cohen-Macaulay rings V . Snaith Explicit Brauer induction G . Laumon Cohomology of Drinfeld modular varieties I E . B . Davies Spectral theory and differential operators J . Diestel, H. Jarchow & A. Tonge Absolutely summing operators P. Mattila Geometry of sets and measures in Euclidean spaces R. Pinsky Positive harmonic functions and diffusion G . Tenenbaum Introduction to analytic and probabilistic number theory C . Peskine An algebraic introduction to complex projective geometry I Y . Meyer & R. Coifman Wavelets and operators II R. Stanley Enumerative combinatories I. Porteous Clifford algebras and the classical groups M. Audin Spinning tops V . Jurdjevic Geometric control theory H. Voelklein Groups as Galois groups J . Le Potier Lectures on vector bundles D. Bump Automorphic forms G. Laumon Cohomology of Drinfeld modular varieties II P. Taylor Practical foundations of mathematics M. Brodmann & R. Sharp Local cohomology J . Jost & X . Li-Jost Calculus of variations

Calculus of Variations Jiirgen Jost and Xianqing Li-Jost Max-Planck-Institute

for Mathematics Leipzig

in the

CAMBRIDGE U N I V E R S I T Y PRESS

Sciences,

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in Publication

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data

Jost, Jurgen, 1956Calculus of variations / Jurgen Jost and Xianqing Li-Jost. p. cm. Includes index. I S B N 0 521 64203 5 (he.) 1. Calculus of variations. I . Li-Jost, Xianqing, 1956I I . Title. QA315.J67 1999 515'.64-dc21 98-38618 C I P I S B N 0 521 64203 5 hardback

Transferred to digital printing 2003

Dedicated to Stefan Hildebrandt

Contents

Preface and summary Remarks on notation

page x xv

P a r t one: One-dimensional variational problems

1

1

T h e classical theory

3

1.1

The Euler-Lagrange equations. Examples

3

1.2

The idea of the direct methods and some regularity results The second variation. Jacobi fields Free boundary conditions Symmetries and the theorem of E. Noether

10 18 24 26

2

A geometric example: geodesic curves

32

2.1

The length and energy of curves

32

2.2

Fields of geodesic curves

43

2.3

The existence of geodesies

51

3

Saddle point constructions

62

3.1

A finite dimensional example

62

3.2

The construction of Lyusternik-Schnirelman

67

4

T h e theory of H a m i l t o n and J a c o b i

79

4.1

The canonical equations

79

4.2

The Hamilton-Jacobi equation

81

4.3

Geodesies

87

1.3 1.4 1.5

4.4

Fields of extremals

89

4.5

Hilbert's invariant integral and Jacobi's theorem

92

4.6

Canonical transformations

95 vii

viii 5 5.1 5.2 5.3

Contents D y n a m i c optimization Discrete control problems Continuous control problems The Pontryagin maximum principle

104 104 106 109

P a r t two: Multiple integrals in the calculus of variations

115

1 1.1 1.2

Lebesgue measure and integration theory The Lebesgue measure and the Lebesgue integral Convergence theorems

117 117 122

2 2.1

B a n a c h spaces Definition and basic properties of Banach and Hilbert spaces Dual spaces and weak convergence Linear operators between Banach spaces Calculus i n Banach spaces

125

2.2 2.3 2.4 3 3.1 3.2 3.3 3.4

4 4.1 4.2 4.3 4.4 4.5 5 5.1 5.2

6 6.1

p

125 132 144 150

L and Sobolev spaces L spaces Approximation of LP functions by smooth functions (mollification) Sobolev spaces Rellich's theorem and the Poincare and Sobolev inequalities

159 159

T h e direct methods in the calculus of variations Description of the problem and its solution Lower semicontinuity The existence of minimizers for convex variational problems Convex functional on Hilbert spaces and MoreauYosida approximation The Euler-Lagrange equations and regularity questions

183 183 184

Nonconvex functionals. Relaxation Nonlower semicontinuous functionals and relaxation Representation of relaxed functionals via convex envelopes

205 205

T-convergence The definition of T-convergence

225 225

p

166 171 175

187 190 195

213

Contents

ix

6.2 6.3

Homogenization T h i n insulating layers

231 235

7

BV-functionals and T-convergence: the example of M o d i c a and M o r t o l a The space BV{Q) The example of Modica-Mortola

241 241 248

7.1 7.2

Appendix A Appendix B 8 8.1 8.2 8.3

9 9.1 9.2 9.3 Index

The coarea formula The distance function from smooth hypersurfaces

Bifurcation theory Bifurcation problems in the calculus of variations The functional analytic approach to bifurcation theory The existence of catenoids as an example of a bifurca­ tion process T h e Palais—Smale condition and unstable critical points of variational problems The Palais-Smale condition The mountain pass theorem Topological indices and critical points

257 262 266 266 270 282

291 291 301 306 319

Preface and summary

The calculus of variations is concerned with the construction of optimal shapes, states, or processes where the optimality criterion is given in the form of an integral involving an unknown function. The task of the calculus of variations then is to demonstrate the existence and to deduce the properties of some function that realizes the optimal value for this integral. Such variational problems occur in many-fold applications, in particular in physics, engineering, and economics, and the variational integral may represent some action, energy, or cost functional. The cal­ culus of variations also has deep and important connections with other fields of mathematics. For instance, in geometrically defined classes of objects, a variational principle often permits the selection of a unique optimal representative, and the properties of this representative can fre­ quently be used to much advantage to deduce additional information about its class. For these reasons, the calculus of variations is a rich and ample mathematical subject, and a good impression of this diversity can be obtained by reading the beautiful book by S. Hildebrandt and A. Tromba, The Parsimonious Universe, Springer, 1996. In this textbook, we have attempted to present some of the many faces of the calculus of variations, and a brief summary may be useful before putting the contents into a broader perspective. A t the same time, we shall also describe the logical connections between the various chapters, in order to facilitate reading for readers with a specific aim. The book is divided into two parts. The first part treats variational problems for functions of one independent variable; the second, problems for functions of several variables. The distinction between these two parts, however, is also that the first treats the more elementary and more classical aspects of the subject, while the second is concerned w i t h some more difficult topics and uses somewhat more abstract reasoning. I n this second part, x

Preface and summary

xi

also some examples are presented in detail that occurred in recent ap­ plications of the calculus of variations. This second part leads the reader to some topics and questions of current research in the calculus of vari­ ations. The first chapter of Part I is of a somewhat introductory nature and attempts to develop some intuition for the properties of solutions of vari­ ational problems. I n the basic Section 1.1, we derive the Euler-Lagrange equations that any smooth solution of a variational problem has to sat­ isfy. The topics of the other sections of that chapter contain some reg­ ularity questions and an outline of the so-called direct methods of the calculus of variations (a subject that will be taken up in much more de­ tail in Chapter 4 of Part I I ) , Jacobi's theory of the second variation and stability of solutions, and Noether's theorem that deduces conservation laws from invariance properties of variational integrals. A l l those results will not be directly applied in subsequent chapters, but should rather serve as a motivation. I n any case, basically all the chapters of Part I can be read independently, after the reader has gone through Section 1.1. In Chapter 2, we treat one of the most important variational prob­ lems, namely that of geodesies, i.e. of finding (locally) shortest curves under smooth geometric constraints. Geodesies are of fundamental im­ portance in Riemannian geometry and several physical applications. We shall make use of the geometric nature of this problem and develop some elementary geometric constructions, to deduce the existence not only of length-minimizing curves, but also of curves that furnish unstable criti­ cal points of the length functional. I n Chapter 3, we present some more abstract aspects of such so-called saddle point constructions. A t this point, however, we can only treat problems that allow the reduction to a finite dimensional situation. A deeper treatment needs additional tools and therefore has to wait until Chapter 9 of Part I I . Geodesies will only occur once more in the remainder, namely as an example in Section 4.3. Chapter 4 is concerned with one of the classical highlights of the cal­ culus of variations, the theory of Hamilton and Jacobi. This theory is of particular importance in mechanics. Presently, its global aspects are resurging in connection w i t h symplectic geometry, one of the most active fields of present mathematical research. Chapter 5 is a brief introduction to dynamic optimization and control theory The canonical equations of Hamilton and Jacobi of Section 4.1 briefly reoccur as an example of the Pontryagin maximum principle at the end of Section 5.3. As mentioned, Part I I is of a less elementary nature. We therefore need

xii

Preface and summary

to develop some general theory first. I n Chapter 1 of that part, Lebesgue integration theory is summarized (without proofs) for the convenience of the reader. While in Part I , the Riemann integral entirely suffices (with the exception of some places in Section 1.2), the function spaces that are basic for Part I I , namely the LP and Sobolev spaces, are es­ sentially based on Lebesgue's notion of the integral. I n Chapter 2, we develop some results from functional analysis about Banach and Hilbert spaces that will be applied in Chapter 3 for deriving the fundamen­ tal properties of the L and Sobolev spaces. (In fact, as the tools from functional analysis needed in subsequent chapters are of a quite varied nature, Chapter 2 can also serve as a brief introduction into the field of functional analysis itself.) These chapters serve the purpose of making the book self-contained, and for most readers the best strategy might be to start with Chapter 4, or at most with Chapter 3, and look up the results of the previous chapters only when they are applied. Chapter 4 is fundamental. I t is concerned with the existence of minimizers of vari­ ational integrals under appropriate convexity and lower semicontinuity assumptions. We treat both the standard method based on weak com­ pactness and a more abstract method for minimizing convex functionals that does not need the concept of weak convergence. Chapters 5-7 essen­ tially discuss situations where those assumptions are no longer satisfied. Chapter 5 deals with the method of relaxation, while Chapters 6 and 7 present the important concept of T-convergence for minimizing func­ tionals that can be represented only in an indirect manner as limits of other functionals. Such problems occur in many applications, including homogenization and phase transitions, and several such examples are treated in detail. Chapter 8 discusses bifurcation theory. We first dis­ cuss the variational aspects (Jacobi fields), taking up the constructions of Sections 1.1 and 1.3 of Part I , then develop a general functional an­ alytic framework for analyzing bifurcation phenomena and then treat the example of minimal surfaces of revolution (catenoids) in the light of that framework. Chapter 8 is independent of Chapters 4-7, and of a more elementary nature than those. The key tool is the implicit function theorem in Banach spaces, proved in Section 2.4. The last Chapter 9 re­ turns to the topic of the existence of non-miminizing, unstable critical points of variational integrals. While such solutions usually cannot be observed in physical applications because of their unstable nature, they are of considerable mathematical interest, for example in the context of Riemannian geometry. Chapter 9 is independent of Chapters 4-8. p

Preface and summary

xiii

The present book is self-contained, with very few exceptions. Prere­ quisites are only the calculus of one and several variables. Although, as indicated, there are important connections between the calculus of variations and geometry, the present book is of an analytic nature and does not explore those connections. One such connection con­ cerns the global aspects of the space of solutions of one-dimensional vari­ ational problems and their trajectories that started w i t h the qualitative investigations of Poincare and is for example represented in V . I . Arnold, Mathematical Methods of Classical Mechanics, G T M 60, Springer, New York, 2nd edition, 1987. Here, geometric methods are used to study variational problems. I n the opposite direction, variational methods can often be used to solve geometric problems. This is the topic of geometric analysis; we refer the interested reader to J. Jost, Riemannian Geome­ try and Geometric Analysis, Springer, Berlin, 2nd edition, 1998, and the references contained therein. There is one important omission in this textbook. Namely, the reg­ ularity theory for solutions of variational problems is not treated, w i t h the exception of the one-dimensional case in Section 1.2 of Part I , and the simplest example of the multi-dimensional theory, namely harmonic functions (plus an easy generalization) in Section 4.5 of Part I I . There­ fore, the solutions of the variational problems that are discussed usually only are obtained in some Sobolev space. We think that a detailed treat­ ment of regularity theory more properly belongs to the realm of partial differential equations, and therefore we have to refer the reader to text­ books and monographs on partial differential equations, for example D. Gilbarg and N . Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2nd edition, 1983, or J. Jost, Partielle Differentialgleichungen, Springer, Berlin, 1998. In any case, the present textbook cannot cover all the many diverse aspects of the calculus of variations. For readers who are interested in a more extensive treatment, we strongly recommend M . Giaquinta and St. Hildebrandt, Calculus of Variations, several volumes, Springer, Berlin, 1996 ff., as well as E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. I l l and I V , Springer, New York, 1984 ff. (a second edition of Vol. I V appeared in 1995). Additional references are given in the course of the text. Since the present book, however, is neither a research monograph nor an account of the historical development of the calculus of variations, references to individual contributions are usually not given. We just list our sources, and refer the interested readers as well as the contributing mathematicans to those for references to the original contributions.

xiv

Preface and summary

The authors thank Felicia Bernatzki, Ralf Muno, Xiao-Wei Peng, Marianna Rolf, and Wilderich Tuschmann for their help in proofreading and checking the contents and various corrections, and Michael Knebel and Micaela Krieger for their competent typing. The present authors owe much of their education in the calculus of variations to their teacher, Stefan Hildebrandt. In particular, the pre­ sentation of the material of Chapters 1 and 4 in Part I is influenced by his lectures that the authors attended as students. For example, the regularity arguments in Section 1.2 are taken directly from his lectures. For these reasons, and for his generous support of the authors over many years, and for his profound contributions to the subject, in particular to geometric variational problems, the authors dedicate this book to him.

Remarks on notation

A dot

d

always denotes the Euclidean scalar product in R , i.e. if d

x = (x\...,x )

d

,y =

d

(y\...,y )eR ,

then d x%

x -y —

%

y

x%

%

— y

(Einstein summation convention)

,

2=1

and 2

|x| =

X • X.

For a function u(t), we write

In Part I , the independent variable is usually called t, because in many physical applications, it is interpreted as the time parameter. Here, the dependent variables are mostly called u(t) or x(t). I n Part I I , the inde­ pendent variables are denoted by x = ( x , . . . , x ) , conforming to estab­ lished conventions. We use the standard notation 1

d

k

c (n) for the space of A;-times continuously differentiable functions on some open set Q C M , for k = 0 (continuous functions), 1,2,..., oo (infinitely often differentiable functions). For vector valued functions, w i t h values in M , we write d

d

k

d

C (fl,R ) xv

xvi

Remarks on notation

for the corresponding spaces. Co°°(ft) denotes the space of functions of class C°° on ft that vanish identically outside some compact subset K C ft (where K may depend on the function, of course). Occasionally, we also use the notation fc