Symmetrization And Applications (Series in Analysis)

\

Symmetrization & Applications \t

(acav

Symmetnzation & Applications

SERIES IN ANALYSIS Series Editor: Professor Roderick Wong City University of Hong Kong, Hong Kong, China

Published Vol. 1

Wavelet Analysis edited by Ding-Xuan Zhou

Vol. 2

Differential Equations and Asymptotic Theory in Mathematical Physics edited by Hua Chen and Roderick S.C. Wong

Vol. 3

Symmetrization and Applications by S Kesavan

Symmettizatiort & Applications

S Kesavan The Institute of Mathematical Sciences, India

l | p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING

• SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

SYMMETRIZATION AND APPLICATIONS Series in Analysis — Vol. 3 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-733-X

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface

The study of isoperimetric inequalities goes back to antiquity. The solution of Dido's problem states that, of all plane domains of given perimeter, the circular disc, and the disc alone, maximizes the enclosed area. Equivalently, of all domains of given area, the disc minimizes the perimeter. This is expressed via the classical isoperimetric inequality L2

> 4TTA,

where L stands for the perimeter and A for the enclosed area of a domain in the plane. Equality is attained only for the disc. In three dimensional space, if S is the surface area of a body and V its volume, then S 3 > 367ry2 and equality is attained only for the sphere. Thus, of all bodies of given volume, the sphere, and the sphere alone, has least surface area. (Why are soap bubbles spherical in shape? The soap bubble will attain a position of equilibrium when the potential energy due to surface tension is minimal. This energy is proportional to the surface area of the bubble. Hence, for a given volume of air blown to form a bubble, it will take the spherical shape which has the least surface area.) These two inequalities can be generalized to all space dimensions. In a broader sense, an isoperimetric problem tries to optimize a given domain dependent functional keeping some geometric parameter of the domain (like its volume) fixed. The study of such problems started with the conjecture of Saint Venant in 1856 regarding the optimal shape of the cross section of a prism in order to maximize its torsional rigidity. This was finally settled by Polya in 1948. In 1894, Lord Rayleigh, in his treatise on the theory of sound, made conjectures regarding the vibrations of cerV

VI

Symmetrization and Applications

tain elastic bodies. He conjectured that, of all fixed membranes of given area, the circular membrane has the lowest fundamental frequency of vibration. This was proved by Faber and Krahn towards the end of the first quarter of the twentieth century. He also conjectured that of all vibrating thin clamped plates of given area, the circular plate has the lowest fundamental frequency. This conjecture resisted solution for nearly a century and was finally settled by Nadirashvili in 1992. While the Faber - Krahn result can be extended to all dimensions, the conjecture regarding plates remains unsolved in dimensions greater than three even today. Poincare conjectured that of all bodies with given volume, the ball alone has the least electrostatic capacity and this was proved by Szego in 1930. The study of isoperimetric inequalities involves a fascinating interplay of analysis, geometry and the theory of partial differential equations. Several conjectures have been made and while many have been resolved, a large number still remain open. One of the first comprehensive treatises on this subject is the book by Polya and Szego (1951). It has since been complemented by several review articles and also by books such as those by Bandle (1980) and Mossino (1984). Since then, several new results have been proved, and a few conjectures have been resolved, especially those concerning eigenvalues of elliptic partial differential operators. As seen from the examples cited above, Nature often seems to choose the perfect symmetric form, viz. spherical symmetry, when optimizing various characteristics of bodies. One of the principal tools in the study of isoperimetric problems, especially when spherical symmetry is involved, is Schwarz symmetrization, which is also known as the spherically symmetric and decreasing rearrangement of functions. The aim of this book is to give an introduction to the theory of Schwarz symmetrization and study some of its applications. Other equally important types of symmetrization (for example, Steiner symmetrization) are not treated here. The first chapter introduces the notion of Schwarz symmetrization and proves several of its properties. The principal result of the second chapter is the famous inequality of Polya and Szego regarding Dirichlet integrals. Since this proof depends on the classical isoperimetric inequality and the coarea formula, these are also studied. The relationship between the classical isoperimetric inequality and Sobolev's inequality is also discussed. Usually, in their full generality, the proofs of the isoperimetric inequality, the co-area formula and the Polya - Szego theorem involve heavy doses of geometric measure theory. Here, a simple and elementary proof of the

Preface

vn

isoperimetric inequality (that I learnt from a lecture by X. Cabre) has been presented in the case of smooth domains. Simple versions of the co-area formula that are enough to prove the Polya - Szego theorem are proved using techniques mainly drawn from the study of partial differential equations. The third chapter centers around Talenti's theorem which compares the solution of a second order elliptic boundary value problem with that of a 'symmetrized problem'. Several applications of this result are studied. The fourth chapter looks at various isoperimetric inequalities involving eigenvalues of elliptic operators. The fifth (and last) chapter deals with isoperimetric inequalities involving positive solutions of some non-linear problems and uses them to obtain the radial symmetry of such solutions when the domain is a ball. Effort has been made to keep the exposition as simple and as selfcontained as possible. Occasionally, in order that we do not get mired in technical details and thus lose the main thread of the argument, some proofs, which are long and which involve completely different techniques, have been omitted and the 'interested reader' is given appropriate references. But such instances are few and far between. A knowledge of the existence theory of weak solutions of elliptic partial differential equations in Sobolev spaces is, however, assumed. Apart from this and a general mathematical maturity at the graduate level, there are no other prerequisites. The text is peppered with several exercises. While the bibliography is fairly extensive, it is, as is to be expected, far from exhaustive. Further references to a vast and rich literature can be found in the works cited here. At the end of each of the last three chapters, bibliographic comments indicate results and directions not treated in the text. I have taught most of the material covered in this book, as a short introductory course, to graduate students at the Universita degli Studi di Roma, La Sapienza, Rome, Italy. I wish to take this opportunity to thank the university for its hospitality. Portions of this book were also taught at instructional conferences in India organized by the Aligarh Muslim University, Aligarh, and the Tata Institute of Fundamental Research (TIFR) at its Bangalore Centre (Indian Institute of Science Campus). I thank the organizers of these conferences for giving me the opportunity to deliver those lectures. I would like to thank the Institute of Mathematical Sciences, Chennai for the excellent facilities it provided which greatly eased the task of bringing out this volume. In particular, I wish to record my appreciation

viii

Symmetrization and Applications

of the cooperation extended by the members of our library staff (and, of course, their counterparts in the libraries of the TIFR Centre, Bangalore and the TIFR, Mumbai) in tracing and procuring copies of research papers in very old journals. I also thank Dr. M. Rajesh, who read portions of the manuscript and made several helpful suggestions. I learnt much about symmetrization from the works of, and through discussions with, several experts. It is a pleasure to express my thanks to Professor G. Talenti (especially for inviting me to the conference at Cortona, which turned out to be a valuable experience, and for all that he has done for me), to Professors G. TYombetti and V. Ferone (and all their colleagues) for very enjoyable and useful visits to Naples, to Professor Jacqueline Mossino for her warm hospitality at Orsay and to Professor Filomena Pacella for giving me the opportunity to teach this course in Rome and for fruitful collaboration (some of our joint work appears in this volume). I am grateful to my family for the constant support given to me. I am also very grateful to Professor P. G. Ciarlet who has been a constant source of inspiration and encouragement. I also wish to thank Professor R. Wong, the editor of the 'Series in Analysis', and the staff of World Scientific for their cooperation in bringing out this volume. Finally, I would like to dedicate this book to the memory of Professor Jacques-Louis Lions, who was one of my teachers and to whom I owe more than I can possibly express. S. Kesavan

Notations • MN denotes the N ~ dimensional Euclidean space. • If x,y £ RN, then x.y is the usual scalar product in R ^ , i.e. if x = (xi,...,xN), and y = (i/i,...,yjv), then x.y = J2iLixiViN • Ifxe R , then M = ( E ^ ! AfnN • If E C R , then \E\ denotes its (N - dimensional) Lebesgue measure. • If E C RN, then £* denotes the ball with centre at the origin such that \E*\ = \E\. • If E C R N , then |-E|jv_i denotes a suitable (iV — 1) - dimensional measure (cf. Section 2.1). • £ ( # ; r) denotes the open ball in MN with centre at x and radius r. Br will also be used to denote the ball with centre at the origin and radius r. • LOW denotes the volume of the unit ball in MN (cf. Section 1.3). • If Q C R ^ is a bounded domain, then d£l denotes its boundary. Let u : Q C RN -> K. • u + = max{u, 0} denotes the positive part of u. u" — — min{u,0} denotes the negative part of u. • Vu denotes the gradient of u. • Au denotes the Laplacian of u, i.e.,

Au

\—\ a u

= Eas =

. ,„ .

dlv v

(^

• u# (resp. u#) denotes the unidimensional decreasing (resp. increasing) rearrangement of u (cf. Section 1.1 (resp. Section 1.4)). • u* (resp. u*) denotes the spherically symmetric and decreasing (resp. increasing) rearrangement of u (cf. Section 1.3 (resp. Section 1.4)). Function Spaces Let Vt C RN be an open set. • C(fl) denotes the space of continuous functions on U. ix

x

Symmetrization

and

Applications

• Ck(Q.) denotes the space of functions which are k times continuously differentiable in Q and whose derivatives upto this order possess continuous extensions to Q. • V(Q) denotes the space of C°° functions with compact support in

n. Let 1 < p < oo. • The norm in the usual Lebesgue space L P (H) is denoted by ||-||p,n• If m is a positive integer, then Wm'p(Cl) is the usual Sobolev space of order m of functions in LP(U) all of whose (distribution) derivatives upto order m are also in LP(U). • The norm in this space is denoted by ||.|| m , p ,n• The closure of £>(H) in this space is denoted by W™,P(Q). • When p = 2, the above spaces are denoted by Hm(Q) and H™^) respectively.

Contents

Preface

v

Notations

ix

1.

Symmetrization 1.1 The Decreasing Rearrangement 1.2 Some Rearrangement Inequalities 1.3 Schwarz Symmetrization 1.4 Variations on the Theme

1 1 8 13 15

2.

Some Classical Inequalities 2.1 The Isoperimetric Inequality 2.2 The Co-area Formula 2.3 The Polya - Szego Theorem 2.4 Sobolev's Inequality

19 19 28 35 43

3.

Comparison Theorems 3.1 Talenti's Theorem 3.2 The Equality Case 3.3 Sobolev Imbeddings 3.4 The Obstacle Problem 3.5 Electrostatic Capacity 3.6 The Saint Venant Problem 3.7 Comments

47 47 53 63 66 70 73 80 xi

xii

Symmetrization

and

Applications

4.

Eigenvalue Problems 4.1 The Faber - Krahn Inequality 4.2 The Szego - Weinberger Inequality 4.3 Chiti's Theorem 4.4 The Payne - Polya - Weinberger Conjecture 4.5 Rayleigh's Conjecture for Clamped Plates 4.6 The Buckling Problem 4.7 Comments

83 83 89 95 98 105 116 122

5.

Nonlinear Problems 5.1 Payne - Rayner Type Inequalities 5.2 A System of Semilinear Equations 5.3 Comments

125 125 132 139

Bibliography

141

Index

147

Chapter 1

Symmetrization

1.1

The Decreasing Rearrangement

Schwarz symmetrization is a particular kind of rearrangement of functions defined on a domain Q C M.N. Given a real valued function on such a domain, we construct an associated function, on the ball centered at the origin and of the same measure as fi, assuming the same range of values and having special properties. In particular, we wish that this new function be radial and radially decreasing. In order to define this, we first construct the unidimensional decreasing rearrangement of the given function, which we now proceed to do. First of all, we need some notation. Given a (Lebesgue) measurable subset E C M.N, we denote its N - dimensional (Lebesgue) measure by \E\. Let fi C R ^ be a bounded measurable set. Let u : Q, —• R be a measurable function. For t 6 R, the level set {u > t} is defined as {u > t} = {x £ £1 | u(x) > £}. The sets {u < t}, {u > £}, {u = t} and so on are defined by analogy. Then the distribution function of u is given by

This function is a monotonically decreasing function of t and for t > ess. sup(u), we have fiu(t) = 0, while for t < ess. inf('u), we have (iu(t) = \Q\. Thus the range of fxu is the interval [0, |H|]. Definition 1.1.1 Let Q C R ^ be bounded and let u : Cl —» R be a measurable function. Then the (unidimensional) decreasing rearrange1

2

Symmetrization

and

Applications

m e n t of u, denoted u # , is defined on [0, |fi|] by u # (0) = ess. sup(u) 1 u*(s) = inf{« | /i u (t) < s}, s > 0. J

(1.1.1)

R e m a r k 1.1.1 Essentially, vft is just the inverse function of the distribution function fiu of u. However, since fiu(t) is just monotonically decreasing, it can have jump discontinuities. If t is a point of discontinuity, then the above definition fixes the value of u * in the interval [fiu(t+), fiu(t—)] as tM R e m a r k 1.1.2 In classical texts (cf. [Hardy, Littlewood, and Polya (1952)] or [Polya and Szego (1951)]), the distribution function is usually defined as fj,u(t) = |{|«| > t}\. Consequently, the definition of the rearrangement would correspond to, in our notation, that of \u\#. Throughout this text, we will, however, deal with the 'rearrangement with sign' i.e. our definition of the rearrangement will be based on the distribution function which takes the sign of the function into account. • E x a m p l e 1.1.1 [Kawohl (1986)]

0

0.5

Let H = ( - 2 , 2) C K. Define u : ft -> R as follows.

u(y)

2 + y, - 2 < y < - l l,-lN is bounded. Then u& is a non-increasing and left-continuous function. Proof: (i) Let si < s2. Then \{u > t}\ < s\ implies that \{u > t}\ < s2. Thus, {t \ i*u{t) < si} C {t

\fiu(t)<s2}.

Thus, by definition, it follows that u^(s\) > u*(s2). (ii) Let s G (0, \Q\). By definition of u#, given e > 0, there exists a t such that u#(s) < t < u#(s) + e and fiu(t) < s. Choose h > 0 such that /iu(£) < s — h < s. Then, for all 0 < h' < /i, we have (iu(t) < s — h' < s and so u#(s) < u&(s ~~ hf) < t < u*(s) + e. This proves that vft is left-continuous. • Proposition 1.1.2 The mapping u \—• u# is non-decreasing, i.e. if u < v, where u and v are real valued functions on fi, then u^ < v&.

4

Symmetrization and Applications

Proof: Since {u > t} C {v > t}, we have that {t\ \{v>t}\<s}

C {* |

\{u>t}\<s}

and the result follows from the definition.

•

Definition 1.1.2 Two real valued functions (with possibly different domains of definition) are said to be equimeasurable if they have the same distribution function. Equimeasurable functions are said to be rearrangements of each other. We now show that u# is indeed a rearrangement of u in this sense. Proposition 1.1.3 The functions u : Q —> R and u* : [0, |fi|] —> R are equimeasurable, i.e., for all t, \{u>t}\

= \{u*>t}\.

(1.1.2)

Proof: If vft(s) > t, then, by definition, it follows that \{u > t}\ > s. Thus, {5 | u*{s) >t)

C {s\ \{u > t}\ > s}.

Since u# is non-increasing, we have \{u#>t}\

-

sup{s | u#(s) >t}

< |{u>t}|.

(1.1.3)

On the other hand, let |{u* > t}| — s. By the left-continuity and the non-increasing nature of u # , it follows that u#(s) = t. Then, by definition, \{u>t}\ <s. Thus,

|{«>«}| < |{ti # >OI-

(1-1-4)

Applying (1.1.3) and (1.1.4) for t + h instead of t, we get \{u#>t

+ h}\ < \{u>t

+ h}\ < | { w # > t + ft}|.

Passing to the limit as h j 0, we get

\{u*>t}\ which proves (1.1.2).

^ l{«>*}l < K« # >*}l •

Symmetrization Corollary 1.1.1

5

With the preceding notations, we have \{u>t}\ \{u>t}\

= =

\{u*>t}\. \{u*>t}\.

|{« 0 -XF t (x) i / t < 0 .

TTien i-oo

/

b(t,x)dt.

(1.2.4)

-OO

Proof: If u(z) > 0, pu{x)

+oo

/ If u(a;) < 0, then

b(t,x)dt

— /

-oo +oo

/

dt =

u(x).

di =

u(x).

^0 /*0

b{t,x)dt

=

(x)

-

«/u(:

-oo

Lemma 1.2.2 Let / , g : fi —> R wt/i # integrable over fl. Let a < f < b < -f oo with a G R. T/ien / f(x)g(x)dx

7n

= a

g{x)dx + /

Vn

I /

Ja \J{f>t}

s(z)cfo ) dt.

(1.2. 5)

J

Proof: Assume that a > 0 (the other case can be similarly treated). Setting Et = {/ > £}, we have, by the preceding lemma, /(a) =

/ Jo

XEt(x)dt

Symmetrization

11

Thus, by Fubini's theorem, •6

/ f(x)g(x)dx Jo. which gives

=

/ g(x) / XEt(x)dtdx Ja Jo

/ f{x)g{x)dx Jo

"=

= 1 1 g(x)dxdt-\JO JO

g{x)\Et{x)dxdt Jo Jo,

I I g{x)dxdt Ja JEt

from which (1.2.5) follows immediately.

•

Exercise 1.2.1 In the above proposition, if b G R and if —oo < a < f