Calculus of Variations (Cambridge Studies in Advanced Mathematics 64)

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

for Mathematics Leipzig

in the

CAMBRIDGE UNIVERSITY PRESS

Sciences,

P U B L I S H E D BY THE PRESS S Y N D I C A T E OF THE U N I V E R S I T Y OF C A M B R I D G E

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data

Jost, Jiirgen, 1956Calculus of variations / Jiirgen Jost and Xianqing Li-Jost. p. cm. Includes index. ISBN 0 521 64203 5 (he.) 1. Calculus of variations. I. Li-Jost, Xianqing, 1956II. Title. QA315.J67 1999 515'.64-dc21 98-38618 CIP ISBN 0 521 64203 5 hardback

Transferred to digital printing 2 0 0 3

Dedicated to Stefan Hildebrandt

Contents

Preface and summary Remarks on notation

page x xv

Part one: One-dimensional variational problems

1 3 3

1.3 1.4 1.5

The classical theory The Euler-Lagrange equations. Examples 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 2.1 2.2 2.3

A geometric example: geodesic curves The length and energy of curves Fields of geodesic curves The existence of geodesies

32 32 43 51

3 3.1 3.2

Saddle point constructions A finite dimensional example The construction of Lyusternik-Schnirelman

62 62 67

4 4.1 4.2 4.3 4.4 4.5 4.6

The theory of Hamilton and Jacobi The canonical equations The Hamilton-Jacobi equation Geodesies Fields of extremals Hilbert's invariant integral and Jacobi's theorem Canonical transformations

79 79 81 87 89 92 95

1 1.1 1.2

vn

Vlll

5 5.1 5.2 5.3

Contents Dynamic optimization Discrete control problems Continuous control problems The Pontryagin maximum principle

104 104 106 109

Part 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

Banach spaces Definition and basic properties of Banach and Hilbert spaces Dual spaces and weak convergence Linear operators between Banach spaces Calculus in Banach spaces

125

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

159 159

The 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 functionals 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

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

125 132 144 150

166 171 175

187 190 195

213

Contents

DC

6.2 6.3

Homogenization Thin insulating layers

231 235

7

BV-functionals and T-convergence: the example of Modica and Mortola The space BV(ft) The example of Modica-Mortola

241 241 248

Appendix A The coarea formula Appendix B The distance function from smooth hypersurfaces

257 262

8 8.1 8.2 8.3

266 266 270

7.1 7.2

9 9.1 9.2 9.3 Index

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 bifurcation process The Palais—Smale condition and unstable critical points of variational problems The Palais-Smale condition The mountain pass theorem Topological indices and critical points

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 calculus 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 frequently 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. At 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 with some more difficult topics and uses somewhat more abstract reasoning. In this second part, x

Preface and summary

XI

also some examples are presented in detail that occurred in recent applications of the calculus of variations. This second part leads the reader to some topics and questions of current research in the calculus of variations. The first chapter of Part I is of a somewhat introductory nature and attempts to develop some intuition for the properties of solutions of variational problems. In the basic Section 1.1, we derive the Euler-Lagrange equations that any smooth solution of a variational problem has to satisfy. The topics of the other sections of that chapter contain some regularity 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 detail in Chapter 4 of Part II), 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. All those results will not be directly applied in subsequent chapters, but should rather serve as a motivation. In 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 problems, namely that of geodesies, i.e. of finding (locally) shortest curves under smooth geometric constraints. Geodesies are of fundamental importance 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 critical points of the length functional. In Chapter 3, we present some more abstract aspects of such so-called saddle point constructions. At 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 II. 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 calculus of variations, the theory of Hamilton and Jacobi. This theory is of particular importance in mechanics. Presently, its global aspects are resurging in connection with 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 II is of a less elementary nature. We therefore need

xii

Preface and summary

to develop some general theory first. In 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 II, namely the LP and Sobolev spaces, are essentially based on Lebesgue's notion of the integral. In Chapter 2, we develop some results from functional analysis about Banach and Hilbert spaces that will be applied in Chapter 3 for deriving the fundamental properties of the Lp 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. It is concerned with the existence of minimizers of variational integrals under appropriate convexity and lower semicontinuity assumptions. We treat both the standard method based on weak compactness and a more abstract method for minimizing convex functionals that does not need the concept of weak convergence. Chapters 5-7 essentially 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 functionals 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 discuss the variational aspects (Jacobi fields), taking up the constructions of Sections 1.1 and 1.3 of Part I, then develop a general functional analytic 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 returns 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.

Preface and summary

xin

The present book is self-contained, with very few exceptions. Prerequisites 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 concerns the global aspects of the space of solutions of one-dimensional variational problems and their trajectories that started with the qualitative investigations of Poincare and is for example represented in V.I. Arnold, Mathematical Methods of Classical Mechanics, GTM 60, Springer, New York, 2nd edition, 1987. Here, geometric methods are used to study variational problems. In 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 Geometry and Geometric Analysis, Springer, Berlin, 2nd edition, 1998, and the references contained therein. There is one important omission in this textbook. Namely, the regularity theory for solutions of variational problems is not treated, with 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 II. Therefore, the solutions of the variational problems that are discussed usually only are obtained in some Sobolev space. We think that a detailed treatment of regularity theory more properly belongs to the realm of partial differential equations, and therefore we have to refer the reader to textbooks 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. Ill and IV, Springer, New York, 1984 ff. (a second edition of Vol. IV 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 presentation 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 '•' always denotes the Euclidean scalar product in Wd, i.e. if

x = (x\...y),y

=

(y\...y)eB*,

then d

x • y — 2_\ xlyl = %lyl

(Einstein summation convention)

,

2=1

and

=xx.

|x| For a function it(£), we write

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

ck{rt) for the space of fc-times continuously differentiate functions on some open set fi C K d , for k = 0 (continuous functions), 1, 2 , . . . , oo (infinitely often differentiate functions). For vector valued functions, with values in Md, we write k d

c (n,m ) XV

XVI

Remarks on notation

for the corresponding spaces. c0°°(fi) 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). Occassionally, we also use the notation

c 0 fc (n) for Ck functions on fi that again vanish outside some compact subset

Ken. Finally, we use the notation

to indicate that the expression on the left of this symbol is defined by the expression on the right of it.

Part one One-dimensional variational problems

1 The classical theory

1.1 The Euler-Lagrange equations. Examples The classical calculus of variations consists in minimizing expressions of the form I(u) = I Ja

F(t,u(t),u(t))dt,

where F : [a, 6] x Rd x Rd —> E is given. One seeks a function u : [a, 6] —> Rd minimizing J. More generally, one is also interested in other critical points of J. Usually, u has to satisfy some constraints, the most common one being a Dirichlet boundary condition u(a) = u\ u(b) = U2>

Also, one needs to specify a class of admissible functions among which one seeks a minimizing u. For example, one might want to take the class of continuously difFerentiable or piecewise continuously difFerentiable functions. Let us consider some examples of such variational problems: (1) We want to minimize the arc-length of the graph of a function u : [a, 6] —• 1R, i.e. the length of the curve (t,u(t)) C K2 among all graphs with prescribed boundary values u(a),u(b). This leads to the variational problem b

y/l -f u(t)2dt -+ min. / Of course, one knows and easily proves that the solution is the straight line between u(a) and u(b), i.e. satisfies il(t) = 0. 3

4

The classical theory (2) Historically, the calculus of variations started with the so-called brachystochrone problem that was posed by Johann Bernoulli. Here, one wants to connect two points (to,yo) and (t\,y\) in R 2 by such a curve that a particle obeying Newton's law of gravitation and moving without friction travels the distance between those points in the fastest possible way. After falling the height t/, the particle has speed (2gy)z where g is the gravitational acceleration. The time the particle needs to traverse the path y = u(t) then is

«=xV

'=>

.

/

^

(3) A generalization of (1) and (2) is rb y/l + u(t) 2

I{u) = f

Ja

l(t,u(t))

dt,

where 7 : [a,fe] x R - > 1 is a given positive function. This variational problem also arises from Fermat's principle..That principle says that a light ray chooses the path that needs the shortest time to be traversed among all possible paths. If the speed of light in a given medium is y(t,u(t)), we obtain the preceding variational problem. If one seeks a minimum of a smooth function / : fi -+ E

(fi open in Md),

one knows that at a minimizing point ZQ € fi, one necessarily has

Df(z0) = 0, where Df is the derivative of / . The first variation of / actually has to vanish at any stationary point, not only at minimizers. In order to distinguish a minimizer from other critical points, one has the additional necessary condition that the Hessian D2f(z0) is positive semidefinite and (at least for a local minimizer) the sufficient condition that it is positive definite. In the present case, however, we do not have a function / of finitely many independent real variables, but a functional Z o n a class of functions. Nevertheless, we expect that a first derivative of J — something still to be defined — needs to vanish at a minimizer, and moreover that a suitably defined second derivative is positive (semi)definite.

1.1 The Euler-Lagrange equations. Examples

5

In order to investigate this more closely, we assume that F is of class C 1 and that we have a minimizer or, more generally, a critical point of / that also is C1. We also assume prescribed Dirichlet boundary conditions u(a) = ui, u(b) = U2- In other words, we assume that u minimizes / in the class of all functions of class C 1 satisfying the prescribed boundary condition. We then have for any 77 G CQ ([a, &],Rd)f and any s G E I(u + sri) > I(u). Now I(u + sr)) = I F(t,u(t) + sr){t),u{t) + sr}{t))dt. Ja Since F , it, and 77 are assumed to be of class C 1 , we may differentiate the preceding expression w.r.t. s and obtain at s = 0 ^ ( « + *»)|.-o

(1-1-1)

= f {Fu(t,u(t),u(t))-r){t)+Fp(t,u(t),u(t))-r,(t)}dt, Ja where Fu is the vector of partial derivatives of F w.r.t. the components of u, and Fp the one w.r.t. the components of u(t). We now keep 77 fixed and let s vary. We are thus just in the situation of a real valued / ( s ) , s G R, (/(s) = I(u + S77)), and the condition /'(0) = 0 translates into

0 = / {Fu(t,u(t),ii(t)) Ja

- r](t) + Fp(t,u(t),ii(t))

- fj(t)}dt,

(1.1.2)

and this actually then has to hold for all rj £ CQ. We now assume that F and u are even of class C2. Equation (1.1.2) may then be integrated by parts. Noting that we do not get a boundary term since 77(a) = 0 = 77(6), we thus obtain

0=

/ { ( ^ (*>"(*)»«(*)) - 1 (FP(*'«(*)'«(*)))) •'»(*)}* (i-L3>

for all 7] G Co([a, 6],R d ). In order to proceed, we need the so-called fundamental lemma of the calculus of variations: f This means that rj is continuously differentiable as a function on [a, b) with values in Rd and that there exist a < a\ < b\ < b with rj(x) = 0 if x is not contained in [