Chipot m (ed ) handbook of differential equations vol 4

This page intentionally left blank... equa-The topics covered by this volume include rearrangements techniques and tions, Liouville-type theorems, similarity solutions of degenerate boun

This page intentionally left blank

Institute of Mathematics, University of Zürich, Zürich, Switzerland

Paris•San Diego•San Francisco•Singapore•Sydney•Tokyo

North-Holland is an imprint of Elsevier

Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands

The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK

First edition 2007

Copyright © 2007 Elsevier B.V All rights reserved

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or

by any means electronic, mechanical, photocopying, recording or otherwise without the prior writtenpermission of the publisher

Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in ford, UK: phone (+44) (0) 1865 843830; fax (+44) (0)1865 853333; email: permissions@elsevier.com.Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/

Ox-locate/permissions, and selecting Obtaining permission to use Elsevier material

Notice

No responsibility is assumed by the publisher for any injury and/or damage to persons or property as

a matter of products liability, negligence or otherwise, or from any use or operation of any methods,products, instructions or ideas contained in the material herein Because of rapid advances in the med-ical sciences, in particular, independent verification of diagnoses and drug dosages should be made

Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

ISBN-13: 978-0-444-53036-3

Set ISBN: 0 444 51743-x

For information on all North-Holland publications

visit our web site at books.elsevier.com

Printed and bound in The Netherlands

07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

This handbook is the volume IV in a series devoted to stationary partial differential tions As the preceding volumes, it is a collection of self contained, state-of-the-art surveyswritten by well-known experts in the field

equa-The topics covered by this volume include rearrangements techniques and tions, Liouville-type theorems, similarity solutions of degenerate boundary layer equations,monotonicity and compactness methods for nonlinear variational inequalities, stationaryNavier–Stokes flow in two dimensional channels, the investigation of singular phenom-ena in nonlinear elliptic problems It includes also a very complete study of the maximumprinciples for elliptic partial differential equations I hope that these surveys will be use-ful for both beginners and experts and help to the diffusion of these recent deep results inmathematical science

applica-I would like to thank all the contributors for their elegant articles applica-I also thank ArjenSevenster and Andy Deelen at Elsevier for the excellent editing work of this volume

M Chipot

v

This page intentionally left blank

List of Contributors

Brock, F., Departamento de Matemáticas, Universidad de Chile, Casilla 653, Chile (Ch 1) Farina, A., LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de

Mathématiques et d’Informatique, 33, rue Saint-Leu, 80039 Amiens, France (Ch 2)

Guedda, M., LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de

Mathématiques et d’Informatique, 33, rue Saint-Leu 80039 Amiens, France (Ch 3)

Kenmochi, N., Department of Mathematics, Chiba University, L-33 Yayoi-cho,

Inage-Ku 263, Chiba, 263-8222 Japan (Ch 4)

Morimoto, H., Department of Mathematics, Meiji University, 1-1-1 Higashi-mita,

Tanaka-ku, Kanagawa, Kawasaki, 214 8571, Japan (Ch 5)

Pucci, P., Dipartimento di Matematica e Informatica, Università degli Studi di Perugia,

Via Vanvitelli 1, Perugia, Italy (Ch 6)

R˘adulescu, V.D., Department of Mathematics, University of Craiova, 200585 Craiova,

Ro-mania and Institute of Mathematics of the RoRo-manian Academy, P.O Box 1-764, 014700 Bucharest, Romania (Ch 7)

Serrin, J., Department of Mathematics, University of Minnesota, Minneapolis, MN, USA

(Ch 6)

vii

This page intentionally left blank

P Pucci and J Serrin

7 Singular Phenomena in Nonlinear Elliptic Problems: From Blow-Up Boundary

V.D R˘adulescu

ix

This page intentionally left blank

Contents of Volume I

1 Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via

C Bandle and W Reichel

G.P Galdi

W.-M Ni

4 On Some Basic Aspects of the Relationship between the Calculus of Variations

This page intentionally left blank

Contents of Volume II

T Bartsch, Z.-Q Wang and M Willem

G Rozenblum and M Melgaard

S Solimini

xiii

This page intentionally left blank

Contents of Volume III

1 Elliptic Equations with Anisotropic Nonlinearity and Nonstandard Growth

S Antontsev and S Shmarev

A Braides

M del Pino and M Musso

J Hernández and F.J Mancebo

This page intentionally left blank

Rearrangements and Applications to Symmetry

Problems in PDE

F Brock

Departamento de Matemáticas, Universidad de Chile, Casilla 653, Chile

E-mail: fbrock@uchile.cl

In honor of the 65th birthday of Albert Baernstein II

Contents

1 Introduction 3

2 Basic definitions 6

3 Rearrangements 6

3.1 General properties 7

3.2 Two-point rearrangement 15

3.3 Symmetrizations 19

4 Inequalities for symmetrizations 24

5 Symmetry results 37

5.1 Uniformly elliptic case 37

5.2 Degenerate elliptic case 42

6 Other symmetry results 51

List of notations 54

Acknowledgement 55

References 55

Abstract

This survey is meant as a general introduction to the theory of rearrangements on RN

We give proofs of all the basic integral inequalities for symmetrization, and a number of ap-plications to symmetry problems in PDE A particular rôle plays the method of two-point rearrangement

HANDBOOK OF DIFFERENTIAL EQUATIONS

Stationary Partial Differential Equations, volume 4

Edited by M Chipot

© 2007 Elsevier B.V All rights reserved

1

This page intentionally left blank

Let v denote the Schwarz symmetrization of v (for a definition see Section 3.3) sume that Ω= RN , or Ω = B R (R > 0), F = F (v) and F is continuous, and K contains

As-only nonnegative functions and has the property that, if v ∈ K, then also v ∈ K Then

However, if the global minimizer is not unique, then the question arises whether there still

holds equality in (1.2) if v = v Unfortunately, this cannot be excluded in general, as the

following simple example shows (see [38])

EXAMPLE 1.1 Let F ≡ 0 Then there are nonnegative smooth functions v with

com-pact support which are not radially symmetric and satisfy J (v) = J (v) Their superlevel

sets{v > c}, c > 0, are nested balls which, however, might be nonconcentric, and the set {∇v = 0} has nonempty interior, that is the graph of v has “plateaus”.

Physically relevant are not only the global minima but also the local minima and critical

points of (P) To show symmetry properties of these functions, the above argument fails,

because in general the Schwarz symmetrization v is not close to v Even though one

expects symmetric solutions in many cases, there are again exceptions Here is anothertypical example

EXAMPLE 1.2 (Semilinear problem for the p-Laplacian) Let B be a ball inRN with

center 0, f ∈ C(R+0) , p > 1, and let u ∈ C2(B)satisfy

− p u≡ −∇|∇u| p−2∇u= f (u), u > 0 in B,

However, if p > 2 or if f is not smooth, then the conclusion (1.5) holds only under some

additional assumptions Below we give a short (but not complete) list of sufficient criteriafor (1.5)

(i) p = 2 and f = f1+ f2, where f1is smooth and f2is increasing, [64];

(ii) p = N and f (v) > 0 for v > 0, [75] (see also [87] for the case p = N = 2);

(iii) f ∈ C1(R+0)and∇u vanishes only at 0, [17];

(iv) f ∈ C1(R+0) and 1 < p 2, [50] and [103]

The proofs for (i), (iii) and (iv) use the so-called Moving Plane Method (MPM) which

turned out to be a very powerful technique in proving symmetry results for positive tions of elliptic and parabolic problems in symmetric domains during the last two decades.Important references for uniformly elliptic equations are [102,64,65,24–26,82,83,80,81],and [42] There have also been obtained symmetry results for some degenerate elliptic

solu-equations including in particular the case of the p-Laplacian operator for p ∈ (1, 2) during

the last 5 years, see [49–51,67,52,53,103], and [54]

The (MPM) exploits comparison principles for elliptic equations while using the variance of the equation with respect to reflections We emphasize that the nonnegativityassumption on the solution is needed to make the method work Furthermore, if the differ-

in-ential operator of the problem degenerates and/or the nonlinearity f in (1.3) is not smooth,

then the (MPM) is often applicable only under additional assumptions on the solution This

concerns, for instance, the p-Laplacian operator for p > 2 (compare the cases (iii) and (iv)

above)

Notice that the result (ii) was proved by combining an isoperimetric inequality and a

Pohozaev-type identity However this method is not applicable if p = N.

One can construct radially symmetric solutions of (1.3) for which the second condition

in (1.5) fails if either p > 2 and f is smooth, or if p ∈ (1, 2] and f is Hölder continuous

(see [67]) Moreover, if p = 2 and f is only continuous and changes sign, then we cannot

hope that the solution of (1.3) is radially symmetric Below we give examples of solutions

(see [33]) Note that similar examples can also be found in the recent paper [103]

“plateau” at height 1 while the other two are congruent to each other with their “feet” lying

on the plateau After a short computation we see that u is a solution of (1.4) with B = B6

If p = 2 and s > 2 then we have f ∈ C∞( [0, 2] \ {1}) ∩ C1−(2/s) ( [0, 2]) The difference

quotient of f is not bounded below near u = 1, i.e f /∈ C1( [0, 2]) In contrast, if p > 2

and s > p/(p − 2), then we have f ∈ C1( [0, 2]).

On the other hand, the functions in the above examples are distinguished by some “local”symmetry which can be described as follows:

(LS) Every connected component of {x ∈ B: u(x) > 0, ∇u = 0} is an annulus A of the

form B R2(z) \ B R1(z) (R2> R1 0, z ∈ B), u is radially symmetric in A, that is

u(x) = v(|x − z|) and ∂v/∂r < 0 for R2> r > R1.

Our aim is to obtain those weak – and also other – symmetries for stationary solutions

of problems like (P), using rearrangement arguments Our approach is closely related to

the corresponding variational problems of the differential equations But in contrast to the(MPM), we can deal with a large class of degenerate operators, and also with nonsmoothnonlinearities in the equations Furthermore, we may sometimes drop the nonnegativity

assumption provided that the solution is a minimizer of the corresponding variational

prob-lem

We would like to refer the interested reader to the nice survey of H Brezis [29], where

a lot of challenging open symmetry problems in PDE can be found

We now outline the content of our work In Section 3.1 we investigate a general class

of rearrangements onRN There is no doubt that slight modifications of these results hold

as well in other situations In Section 3.2 we study the two-point rearrangement This very

6 F Brock

simple type of rearrangement will then be helpful in Sections 3.3 and 4 to show manyproperties of the Schwarz, Steiner and cap symmetrizations, including all the basic in-equalities that compare an integral of some given functions with the same integral of theirsymmetrizations In Section 5 we obtain symmetry properties for minimizers of ellipticvariational problems, where we combine the two-point rearrangement with two importanttools in PDE: the Principle of Unique continuation (Section 5.1) and the Strong MaximumPrinciple (Section 5.2) These results are contained in two recent articles of the author,[35,36]) Finally we briefly report on other symmetry results in Section 6, in particularthose which have been obtained using the method of continuous Steiner symmetrization(see [30,32,33]

2 Basic definitions

We will assume N ∈ N, N  2 throughout our work We write x = (x1, , xN ) = (x1, x ),

y = (y1, , yN) = (y1, y ) , , for points inRN , and x · y = N

i=1xi yi,|x| =√x · x for

the Euclidean scalar product and norm, respectively

of allL-measurable – measurable in short – sets of R N By · p,M we denote the usual

norm in the space L p (M) (M ∈ M), and we write · p:= · p,RN (1 p  +∞) If Ω

is an open set inRN , and if u ∈ L∞(Ω) we define the modulus of continuity ω u,Ω by

ω u,Ω (t ):= sup u(x) − u(y): x, y ∈ Ω, |x − y| < t

(Here and in the following sup (inf) means ess sup (ess inf).) We also write ω u,RN = ω u

Notice that if u ∈ C(Ω) then u is equicontinuous on Ω iff lim t 0ω u,Ω (t )= 0

general-ized partial derivatives ∂u/∂x i ∈ L p (Ω), i = 1, , N, and we denote by W 1,p

0 (Ω)the

u : Ω→ R+0 by zero outside Ω, so that W01,p (Ω) ⊂ W 1,p (RN )in that sense (see [1])

By C00,1 (Ω) we denote the space of Lipschitzean functions with compact support in Ω.

For any of the above function spaces, let the lower index “+” indicate the corresponding

subset of nonnegative functions, e.g L p+(RN ), W01,p+(Ω), C00,1+(Ω), .

Finally, a function G :R+0 → R+0 is called a Young function if G is continuous and

3 Rearrangements

Since the times of Steiner [106], Hardy, Littlewood and Polya [68] rearrangements havebeen used to prove isoperimetric inequalities in mathematical physics The monographs[68,96,45,39,44,93,20,72,69,85] and the surveys [101,5,76,59,18,112,113,37,84] providemany results and further references

3.1 General properties

In this section we introduce a general concept for rearrangements which is appropriate forall the symmetrizations onRN, and we prove general properties

We often treat measurable sets only in a.e sense, that is we identify a set M with its

equivalence class given by all measurable sets MwithL N (M  M) = 0 If M1, M2∈ M,

we will then write

Some rearrangements satisfy, in addition to (3.3)–(3.5),

if M1, M2∈ M, M1⊂ M2, thenL N (M2\ M1) = L N (T M2\ T M1) (3.6)

REMARK3.1

(1) The notion of rearrangement is reserved for measurable sets If a rearrangement

has certain smoothing properties, as for instance the symmetrizations and the point rearrangement (see Sections 3.2 and 3.3), then one can introduce pointwise

two-representatives for the set T M when M is open or compact.

(3.4) and (3.6) imply in particular, that T is continuous from the outside, that is, if

{M n } is a nonincreasing sequence in M, then

8 F Brock

usu-ally not distinguish between u and its equivalence class given by all measurable functions which differ from u on a nullset only We say that u belongs to S – the set of ‘symmetriz- able functions’ – if

Obviously, μ u is a nonincreasing, right-continuous function with μ u (λ) = +∞ ∀λ < inf u,

μ u (λ) = 0 ∀λ > sup u, and μ u (λ) < ∞ ∀λ ∈ (inf u, ∞).

We will say that two functions u, v ∈ S are equidistributed, u ∼ v, if μ u (λ) = μ v (λ)

∀λ ∈ R.

The distribution function of μ u – that is, the right-continuous inverse of μ u– is called

the symmetric decreasing rearrangement of u and is denoted by u It is easy to see that u

is a nonincreasing, right-continuous function onR+0 with u ( 0) = sup u, lim s→+∞u (s)=

Furthermore, if u ∈ S, or if T satisfies (3.6), then we obtain from (3.15) and (3.16) that

i.e u is the superposition of the characteristic functions of its level sets Notice that

the integrals in (3.22) are à la Bochner, i.e., if 0= t k

Notice that the second integral in (3.22) and (3.23) is zero when u 0.

(3) If u ∈ L1(RN )then we have that



RN

u dx=

 ∞0

immedi-LEMMA 3.1 Let T be a rearrangement and u ∈ L1(RN ) Furthermore, let either u 0,

or let T satisfy property (3.6) Then

The next result seems to be well known as well, but I could not find a reference

non-decreasing function Furthermore, assume that either u ∈ S or T satisfies (3.6) Then

lim

s t ϕ(s) = λ1 λ  λ2= lim

s t ϕ(s),

and obviously λ1 ϕ(t)  λ2 We consider three cases:

(i) λ = λ2 Setting t = max{s  t: ϕ(σ ) = λ2 ∀σ ∈ [t, s]}, we have then, for every

(ii) ϕ(t)  λ < λ2 Then we have for every s∈ R,

s > t ⇐⇒ ϕ(s) > λ,

so that (3.27) holds with t replaced by t

(iii) λ1 λ < ϕ(t) Then we have for every s ∈ R,

The following important theorem can be seen as a form of Cavalieri’s principle (see

e.g [72])

THEOREM 3.1 Let T be a rearrangement, f : R → R continuous or nondecreasing,

u:RN → R measurable, and f (u) ∈ L1(RN ) Furthermore, assume either that u ∈ S,

or that T satisfies (3.6) Then f (T u) ∈ L1(RN ) and

PROOF If f is nondecreasing, then (3.28) follows from Lemmata 3.1 and 3.2.

Next let f be continuous For every λ > 0, the set G := {t ∈ R: f (t) > λ} is open.

Hence there is representation G=∞i=1I i , with mutually disjoint intervals I i = (a i , b i ),

i = 1, 2, Since f (u) ∈ L1(RN ), and using (3.15)–(3.17), we then deduce,

+∞ > L N f (u)  λ= L N f (T u)  λ ∀λ < 0. (3.30)

Theorem 3.2 below has many applications, too It has been shown by Crowe, bloom and Zweibel [48] for the Schwarz symmetrization of nonnegative functions How-ever, it has been observed in [34] and in [120] that their proof carries over to arbitraryrearrangements without difficulties

Rosen-12 F Brock

THEOREM3.2 Let F ∈ C(R2) , F (0, 0) = 0, and

F (A, B) − F (a, B) − F (A, b) + F (a, b)  0

∀a, b, A, B ∈ R with a  A, b  B, (3.31)

and let T be a rearrangement Furthermore, let u, v measurable such that F (u, 0), F (0, v),

F (u, v) ∈ L1(R N ) Finally, let either

 s0

Choosing s = u(x) and t = v(x) in (3.34) and then integrating we find

RN F (T u, T v) dx Now one obtains (3.32) from these

representations and from the inequalities below, which follow from (3.6), (3.15) and (3.17),

Choosing F (s, t) := −G(|s − t|) with a Young function G in (3.32) we obtain

COROLLARY 3.1 (Nonexpansivity of the rearrangement) Let G be a Young function,

T a rearrangement , and let u, v be measurable such that G( |u|), G(|v|), G(|u − v|) ∈

L1(RN ) Furthermore, assume that one of the conditions (i) or (ii) from Theorem 3.2 is

Furthermore, (3.32) with F (u, v) = uv gives the following inequality which is attributed

to Hardy and Littlewood, see [68]

COROLLARY 3.2 Let u, v ∈ L2(RN ) , and let T be a rearrangement Furthermore, let

either u and v be nonnegative, or T satisfies (3.6) Then

14 F Brock

Rearrangements are nonexpansive in L∞(RN ), too

COROLLARY 3.3 Let u, v ∈ L∞(RN ) , and let T be a rearrangement Furthermore, let

either u ∈ S or T satisfies (3.6) Then

T u − T v∞ u − v∞. (3.40)

monotonicity (3.18) and (3.26) this means that

T v − C = T (v − C)  T u  T (v + C) = T v + C, a.e on R N,

REMARK 3.4 For applications it is useful to define rearrangements of functions which

{−∞}, with c  inf M u , we extend u ontoRNby setting

u(x) := c if x ∈ R N \ M, (3.41)

and define T u by (3.14) (Notice that in fact T u does not depend on the particular choice

of c Notice also that if L N (M) < ∞, and if c > −∞ then this implies u, T u ∈ S.) Since

let T be a rearrangement If f : R → R is continuous or nondecreasing and f (u) ∈ L1(M) then

3.2 Two-point rearrangement

We now study the two-point rearrangement This simple rearrangement can be used toprovide simple proofs of many geometric and functional inequalities which are related tosymmetrizations (see [68,123,22,57,58,19,18] and [37])

of the two open halfspaces into whichRN is subdivided by Σ Let σ H denote reflection

in Σ = ∂H , that is if H = {y: y · ν < λ}, (ν exterior unit normal to ∂H , λ ∈ R), and if

x∈ RN then σ H x := x + 2(λ − x · ν)ν Furthermore, if u : R N→ R is measurable, then

we define its reflexion σ H uby

σ H u(x) := u(σ H x), x∈ RN ,

and its two-point rearrangement T H u (with respect to H ) by

T H u(x):=



max{u(x); u(σ H x) } if x ∈ H,

min{u(x); u(σ H x) } if x ∈ R N \ H. (3.46)

rearrange-ment of its characteristic function, i.e

REMARK3.5 In the case that u is continuous and M is open or closed, equations (3.46) and (3.47) have to be understood in the pointwise sense It then follows that if u is contin-

uous onRN then so does T H u , and if M is an open or closed subset ofRN , then T H Misopen, respectively closed, too The following, more precise statement was proved in [19]

u ∈ C(Ω) ∩ L∞(Ω) , or u ∈ C(Ω) ∩ L∞(Ω) , then so does u H and

PROOF For any t > 0 and ε ∈ (0, ω u H ,Ω (t )) , we find x, y ∈ Ω such that |x − y|  t and

|u H (x) − u H (y) |  ω u H ,Ω (t ) − ε Now consider two cases.

(i) Let x, y ∈ H ∩ Ω or x, y ∈ Ω \ H Then one verifies easily the inequality

max u(x) − u(y); u(x H ) − u(y H )

 max u H (x) − u H (y);u H (x H ) − u H (y H ).

The following lemma shows that the two-point rearrangement depends continuously onits defining halfspace, see [37].

n→∞σH n x = σ H x, uniformly in compact subsets ofRN

This leads to (3.51) in case that u is continuous with compact support.

In the general case let ε > 0 We choose a continuous function v with compact support

such thatu − v p < ε/ 3, and then n0large enough such that

and (3.51) follows.

(2) Let the sequence{H n } as in case (2) and let u  0 If u has bounded support then

there is some n0∈ N such that

uH n = u ∀n  n0.

We now state a convolution-type inequality (see [19] and [18])

Young function, and let u, v be measurable functions onRN Then

PROOF An elementary calculation shows that∀x, y ∈ H ,

Gu(x H ) − v(y) + Gu(x) − v(y H )

|x − y H|

+ Gu H (x H ) − v H (y H )w

|x H − y H|.

It is easy to see that if u and v are nonnegative then we may replace the function G(s, t)

by ( −st) in the above proof.

for some p ∈ [1, +∞], then u H ∈ W 1,p (Ω) More precisely, if H = {x: x · ν < λ}

(ν: exterior unit normal to ∂H , and λ ∈ R), and if ∂u/∂ν denotes the directional derivative

ν · ∇u and ∇ ν⊥u the gradient w r.t the (N − 1)-dimensional subspace orthogonal to ν

(that is,∇ν⊥u = ∇u − (ν · ∇u)ν), then ∇u p = ∇u H p,∂u/∂ν p = ∂u H /∂ν p , and

∇ν⊥u p= ∇ν⊥uH p Finally, if u ∈ W 1,∞(Ω) , and if G is a function satisfying

u(x) − (u(x) − v(x))+for all x ∈ H ∩ Ω, we have that u H , w ∈ W 1,1

COROLLARY 3.5 Let Ω⊂ RN be an open set, let H be a halfspace and u ∈ W 1,p

PROOF Extending u by zero outside Ω we find that u H ∈ W 1,p

then we choose a sequence {u n } ⊂ C 0,1

0 +(Ω) which converges to u in W01,p (Ω) Since

(u n ) H → u H in L p (Ω H ) , and since the functions (u n ) H are equibounded in W01,p (Ω H ),

we find a function v ∈ W 1,p

0 (Ω H ) and a subsequence (u n ) H which converges to v weakly

in W 1,p (Ω H ) This means that for every ϕ ∈ C∞

(1) Let M ∈ M If L N (M) < ∞, then let M the ball B RwithL N (B R ) = L N (M), and

ifL N (M) = ∞ then let M = R N Correspondingly, for any function u ∈ S we introduce

u by formula (3.1) with T u = u and T {u > λ} = {u > λ} An equivalent definition is

u(x) := u 

κ N |x| N

(Here κ N = L N (B1).)

The objects M and u are called the Schwarz symmetrizations of M and u,

and radially nonincreasing’, that is, u depends on the radial distance |x| only, and is

non-increasing in|x| The superlevel sets {u > λ} are balls centered at zero, and they have the

same measure as{u > λ}, λ ∈ R Notice that (3.57) implies that two functions u, v ∈ S are

20 F Brock

(Here{u(·, x ) > λ } is a short-hand for {x1: u(x1, x ) > λ }.) The function u∗is even in the

variable x1and nonincreasing in x1for x1 0, and

L1 u( ·, x ) > λ

= L1 u∗( ·, x ) > λ

∀λ ∈ R and for a.e x ∈ RN−1

(3.60)the set on the right-hand side of (3.60) being an interval centered at zero Hence the Steinersymmetrization is a rearrangement, too

(3) Let P := (1, 0, , 0) – the ‘north pole’ If M ∈ M then there is for a.e r > 0

CM:= x∈ RN

: x ∈ CM(r), r > 0 (3.61)

define its cap symmetrization Cu by

The mapping C is a rearrangement which satisfies (3.6) The superlevel sets {Cu > λ} ∩

∂B r are spherical caps centered at P and have the same measure as {u > λ} ∩ ∂B r (r > 0,

λ ∈ R) Hence Cu depends only on the radial distance r = |x| and on the geographical

latitude θ1:= arccos(x1/ |x|), and is nonincreasing in θ1∈ [0, π].

(4) Now we introduce pointwise representatives of the symmetrization of open andclosed sets and of continuous functions

If M is an open set, then we agree that the definitions of M and M∗given above are

taken in pointwise sense If M is closed, then let M the closed ball B Rhaving the same

measure as M if L N (M) < ∞, and M = R NifL N (M)= ∞, and let

is lower, respectively upper semicontinuous in x , if M is open, respectively closed, so that

M∗given by the above definition is open, respectively closed, too.

Furthermore, if M is open, respectively closed, then let CM(r) the spherical cap

B ρ ∩ ∂B r , respectively B ρ ∩ ∂B r, such thatL N−1(M ∩ ∂B r ) = L N−1(B

ρ ∩ ∂B r ), when

r > 0 We also let CM(0) = {0} iff 0 ∈ M in both cases The pointwise representative CM

is then given by formula (3.61) Notice that the function

g(r, λ) := L N−1

{u > λ} ∩ ∂B , λ ∈ R, r  0, (3.65)

is lower, respectively upper semicontinuous, if M is open, respectively closed, so that CM

is then open, respectively closed, too

Finally, if u is a continuous function, then the superlevel sets {u > λ} (λ ∈ R), are open.

pointwise representatives for{u > λ} (λ ∈ R).

REMARK 3.6 It can be seen from simple examples that the Schwarz and Steiner metrizations do not satisfy property (3.6) This can lead to pathologies if one applies for-mula (3.14) to measurable functions which are not inS:

sym-For instance, if u(x) = sin |x| and v(x) = w(x ) sin x1, for some positive measurable

function w, then (3.1) gives u(x) ≡ 1 and v∗(x) = w(x ) , so that the negative parts of u

and v do not show up at their symmetrizations.

We now introduce the following notation: LetH be the set of all affine halfspaces of R N,and

(1) It is easy to see that, if u ∈ S, then (u H ) = u ∀H ∈ H and (u H )∗= u∗∀H ∈ H∗.

(2) The two-point rearrangement can be used in order to identify symmetric situations

u ∈ S, u = u∗, then u = u H ∀H ∈ H∗

0 Finally, if u :RN→ R is measurable and

u = Cu, then u = u H ∀H ∈ CH P

A more interesting result is the following

LEMMA3.8 Let u ∈ L p (RN ) for some p ∈ [1, ∞).

(1) If L N ( {u > 0)}) > 0, and if for every H ∈ H∗ we have either u = u H or σ H u=

uH , then u ∈ S+, and there is a point ξ = (λ0, 0, , 0)∈ RN (λ0∈ R), such that

u( · − ξ) = u∗( ·) In particular, if u = u H ∀H ∈ H∗

0then u ∈ S+and u = u∗.

(2) If L N ( {u > 0}) > 0, and if for every H ∈ H we have either u = u H or σ H u = u H,

then there is a point ξ∈ RN such that u ∈ S+and u( · − ξ) = u(·) In particular, if

u = u H ∀H ∈ H0then u ∈ S+and u = u.

(3) If for every H ∈ CH we have either u = u H or σ H u = u H then there is a rotation

ρ about zero such that u(ρ ·) = Cu(·) In particular, if u = u H ∀H ∈ CH P then

u = Cu.

22 F Brock

PROOF We only prove the first parts of the assertions (1)–(3) and leave the second parts

to the reader

(1) For simplicity, we write H λ = {x: x1> λ }, u H λ = u λ and σ H λ u = σ λu (λ∈ R) Since

L N ( {u > 0}), and since u is decaying at infinity, we have that u = u λ when λ is small enough and σ λ u = u λ when λ is large enough By the continuity of the mapping λ → u λ (Lemma 3.5) we find an intermediate value λ0∈ R such that u = σ λ0u = u λ0 Assume that

there is another value λ

0= λ0with the same property Then u(x1, x ) = u(2λ0− x1, x )=

u( 2λ

0− x1, x ) = u(x1+ 2(λ

0− λ0), x ) ∀(x1, x )∈ RN , that is, u is periodic in x1withperiod 2|λ

0−λ0| But this is not possible since u ∈ L p (RN ) Hence we have that u = u λfor

λ  λ0and σ λu = u λ for λ  λ0, which means that u is nonincreasing in x1for x1 λ0

Since u decays at infinity this in particular implies that u 0, and the first part of (1)

follows

(2) Proceeding as in part (1) of the proof for any of the x i -directions, we find that u ∈ S+,

and that there is point ξ = (ξ1, , ξN )∈ RN such that the function v( ·) := u(· − ξ) is even

in the variables x i and nonincreasing in x i for x i  0 (i = 1, , N).

continuity this means that if e a unit vector, and H = {x: x · e > 0}, then one has v = σ H v

Hence v depends on the radial distance |x| only, and is nonincreasing in |x|, and the first

part of (3) follows

(3) Fix N mutually orthogonal unit vectors e i (i = 1, , N), and let e1= (1, 0, , 0)

and e2 = (0, 1, 0, , 0) Introducing polar coordinates x1 = r cos ϕ, x2 = r sin ϕ

(ϕ ∈ [−π, π], r  0), the function w(ϕ, r, x ) := u(x) (x = (x3, , xN) ), is (2π periodic in ϕ From the assumption it follows that, if λ ∈ [−π, π], then either w(ϕ, ·) 

)-w( 2λ − ϕ, ·) ∀ϕ ∈ [λ, λ + π] or w(ϕ, ·)  w(2λ − ϕ, ·) ∀ϕ ∈ [λ, λ + π], where “·” stands

for (r, x ) We claim that there is a value λ∗∈ [−π, π] such that w(ϕ, ·) = w(2λ∗− ϕ, ·)

∀ϕ ∈ [−π, π] and such that w is nonincreasing in ϕ for ϕ ∈ [λ∗, λ∗+ π].

First observe that by periodicity and continuity there exists a value λ0∈ [0, π] such that

w(ϕ, ·) = w(2λ0− ϕ, ·) ∀ϕ ∈ [−π, π] Assume that there is a sequence of values {λ n} with

the same symmetry property, and such that λ n = λ0∀n ∈ N and lim n→∞λ n = λ0 Then

we have that w(ϕ, ·) = w(2(λ0− λ n) + ϕ, ·) ∀n ∈ N and ∀ϕ ∈ [−π, π], which means that

w(ϕ, ·) = w(0, ·) ∀ϕ ∈ [−π, π], that is w is independent of ϕ.

Next assume that a sequence with the above property does not exist By periodicity

and continuity, we then find two numbers λ i ∈ [−π, π] with λ2∈ (λ1, π + λ1] such that

w(ϕ, ·) = w(2λ i − ϕ, ·) ∀ϕ ∈ [−π, π] (i = 1, 2), and such that w(ϕ, ·)  w(2λ − ϕ, ·)

∀ϕ ∈ [λ, λ + π] and ∀λ ∈ [λ1, λ2] Assume that λ2< λ1+ π Then we have that w(ϕ, ·) 

w( 2(λ1− λ) + ϕ, ·) = w(2(λ2− λ) + ϕ, ·) ∀ϕ ∈ [λ2, λ1+ π] and ∀λ ∈ [λ1, λ2] This again

implies that w is independent of ϕ It remains to consider the case that λ2= λ1+ π It is

then easy to see that w is nonincreasing in ϕ for ϕ ∈ [λ1, λ1+ π].

We have thus proved the claim in any of the above cases

Let l1 = (cos λ∗, sin λ∗, 0, , 0) Notice that l

1 ∈ span{e1, e2} It follows that if

l ∈ span{e1, e2} and H = {x: x · l > 0} such that l1∈ H , then u = u H Analogously

we find a vector l2∈ span{l1, e3} such that the following holds: If l ∈ span{e1, e2, e3} and

H = {x: x · l > 0} such that l ∈ H , then u = u

Continuing in this manner we find unit vectors l k ∈ span{e1, , e k+1} (k = 1, ,

N ư 1), such that the following holds: If l ∈ span{e1, , e k+1} and H = {x: x · l > 0}

such that l k ∈ H , then u = u H (k = 1, , N ư 1).

By continuity and periodicity this also implies that if H ∈ CH such that l Nư1∈ ∂H ,

then u = σ H u

Finally, let ρ a rotation of the coordinate system about the origin such that ρl Nư1= P =

( 1, 0, , 0), and introduce a function v by v(ρ ·) := u(·) Then the above properties imply

that v depends on the variables |x| and θ1:= arccos(x1/ |x|) only and is nonincreasing in

θ1∈ [0, π] This means that v = Cv, and the first part of (3) follows. 

The next separation property will be crucial in the proof of inequalities between theSobolev norm of a function and its symmetrization It has been proved in [19] for the capsymmetrization and in [18] for the Schwarz and Steiner symmetrization

LEMMA3.9 Let u ∈ L p (RN ) for some p ∈ [1, ∞).

(1) If u  0, and if u = u then there is an H ∈ H0such that

An integration of (3.69) over H then leads to u H ưu p  uưu p Therefore to prove

(3.66) it suffices to show that for a suitable choice of H the inequality (3.69) becomes strict

on a subset of H of positive measure.

Since u = u, we find some number c > 0 such that L N ( {u > c}{u > c}) > 0.

Let x1 and x2 density points of the sets {u > c} \ {u > c} and {u > c} \ {u > c},

respectively We choose a halfspace H such that x1= σ H x2and x2∈ H (Note that from

u(x1)  c < u(x2)it follows that 0∈ H ) Hence there is a subset K of H of positive

measure containing x2such that

But this means that the inequality (3.69) becomes strict on the set K.