An Introduction to Maximum Principles and Symmetry in Elliptic Problems (Cambridge Tracts in Mathematics)

CAMBRIDGE TRACTS IN MATHEMATICS General Editors B. BOLLOBAS, F. KIRWAN, P. SARNAK, C.T.C. WALL 128 An Introduction t...

Author: L. E. Fraenkel

24 downloads 579 Views 3MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

CAMBRIDGE TRACTS IN MATHEMATICS General Editors

B. BOLLOBAS, F. KIRWAN, P. SARNAK, C.T.C. WALL

128

An Introduction to Maximum Principles and Symmetry in Elliptic Problems

An Introduction to Maximum Principles and Symmetry in Elliptic Problems L.E. Fraenkel School of Mathematical Sciences University of Bath

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi

Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521461955

© Cambridge University Press 2000

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2000

A catalogue record for this publication is available from the British Library ISBN 978-0-521-46195-5 hardback Transferred to digital printing 2008

Contents

page vii

Preface 0

Some Notation, Terminology and Basic Calculus

1

Introduction A glimpse of objectives What are maximum principles? On reflection in hyperplanes What is symmetry? Exercises

1.1

1.2 1.3

1.4 1.5

2 2.1

2.2 2.3 2.4 2.5 2.6 2.7 2.8

1

17 17 19

24 27 32

Some Maximum Principles for Elliptic Equations 39 39 Linear elliptic operators of order two The weak maximum principle 41 The boundary-point lemma and the strong maximum principle 50 A maximum principle for thin sets S2 56 Steps towards Phragmen-Lindelof theory 61 Comparison functions of Siegel type 72 Some Phragmen-Lindelof theory for subharmonic functions 77 Exercises

84

3

Symmetry for a Non-linear Poisson Equation in a Symmetric

3.1

Set n The simplest case

87

87 93

3.2 3.3

A discontinuous non-linearity f Exercises

101

4

Symmetry for the Non-linear Poisson Equation in IRN Statement of the main result Four lemmas about reflection of v Proof of Theorem 4.2 and a corollary

106 106

4.1

4.2 4.3

v

111

120

Contents

vi

4.4 4.5

Application to some Newtonian potentials

5

Monotonicity of Positive Solutions in a Bounded Set Sl Prospectus On the geometry of caps and reflected caps Monotonicity in fl A little topology Exercises

5.1

5.2 5.3 5.4 5.5

Exercises

Appendix A. On the Newtonian Potential A.1 Point sources in R3 A.2 The Newtonian potential: first steps A.3 Continuity of the force field Vu A.4 Multipoles and the far field A.5 Second derivatives of u at points in G A.6

Exercises

Appendix B. Rudimentary Facts about Harmonic Functions and the Poisson Equation B.1 Real-analytic functions B.2 Smoothness and mean-value properties of harmonic functions B.3 The Kelvin transformation B.4 On the Dirichlet and Neumann problems B.5 The solution of the Dirichlet problem for a ball B.6

Exercises

122 133 141

141

142 153 159 162 167 167 174 194 199 203 213 221 221

224 232 235 248 262

Appendix C. Construction of the Primary Function of Siegel Type

270

Appendix D. On the Divergence Theorem and Related Matters D.1 A first divergence theorem D.2 Extension to some sets with edges and vertices D.3 Interior approximations to the boundary 00

279 279 285 293 300

D.4

Exercises

Appendix E. The Edge-Point Lemma E.1 Preliminaries E.2 Bluntness and ellipticity under co-ordinate transformations E.3 Two stages of the edge-point lemma

305 305 309 311

Notes on Sources

324

References

332

Index

337

Preface

During the academic year 1987-8 a group of young mathematicians at the University of Bath prepared (for the first time) a pamphlet, Master of Science in nonlinear mathematics, that contained the following entry. PG14 Symmetry and the Maximum Principle The maximum principle for elliptic operators will be proved from first principles and developed to the extent where the work on symmetry of positive solutions of semi-linear elliptic problems of Gidas, Ni, Nirenberg may be proved.

Naturally, the pamphlet did not state how this goal was to be reached in twenty lectures to students who could not be assumed to have any experience whatever of partial differential equations. Nor were detailed suggestions issued to me when, in the autumn of 1988, I joined the University of Bath and was ordered to give these lectures. What the authors of the pamphlet did do, however, was to attend the lectures themselves, to ask awkward questions, to imbue the course PG14 with their own youthful verve, and to appeal to my vanity by suggesting that I prepare something like the present book. This explanation should indicate that the word Introduction in the title of the book is no gloss. I offer genuine apologies to B. Gidas, W.-M. Ni and L. Nirenberg for the extent to which I have used their paper Symmetry and related properties via the maximum principle (1979), to H. Berestycki and L. Nirenberg for my use of the easiest part of On the method of moving planes and the sliding method (1991), and to D. Gilbarg

and N.S. Trudinger for the extent to which I have lifted theorems from their book Elliptic partial differential equations of the second order (1983).

But I hope that it will be clear from the following pages that I have not merely copied, and that my aim has been to prepare beginners (and vii

viii

Preface

plodders like myself) for the more advanced and abbreviated treatment in those sources. Theorems of `Gidas, Ni and Nirenberg type' have their roots in work of Alexandrov (see H. Hopf 1956, p.147) and of Serrin (1971); such theorems establish monotonicity properties and, where the situation allows it, some

type of symmetry for solutions of certain elliptic equations that may be slightly non-linear or highly non-linear. The symmetry may be invariance under reflection in a particular hyperplane or may be spherical symmetry. By now there is a profusion of such theorems. This book proceeds to simple, but perhaps basic, results of this kind, mainly for the equation Au + f (u) = 0. Chapters 3 and 5 concern positive solutions in a bounded set SZ c RN, with u(x) = 0 on all or part of the boundary an. In Chapter 3 the set S2 has either what will be called Steiner symmetry or spherical symmetry; once this has been specified, there is no need for worry about the form of 752. In Chapter 5 no symmetry of SZ is required; then worry about 752 is a necessary preliminary to the monotonicity results that are established there for solutions u. In both cases, our main task is patient analysis of the growth of u(x) as the point x is moved inwards from the boundary an. Chapter 4 deals with solutions of Du + f (u) = 0 on the whole space RN; it precedes Chapter 5 because there is no boundary an to cause grief. But neither is there a boundary condition to initiate the analysis; instead, one demands that u(x) have what will be called admissible asymptotic behaviour as jxi -+ co. Ultimately it emerges that there is a centre q of symmetry, that u depends only on Ix - qI and that u decreases as Ix - qI increases.

The book is organized as follows. Chapter 0 contains the terminology

and apparatus from which later chapters proceed. Appendices A and B are intended to be a leisurely, elementary but serious introduction to methods and results that are basic for elliptic equations. This requires a naive but flexible divergence theorem, which I have tried to provide in Appendix D. Some (by no means all) results in Appendices A and B are used to build comparison functions and examples elsewhere in the book. Because the treatment is to be elementary, there is no whisper of distributions, weak derivatives and Sobolev spaces, but test functions and weak solutions play a central part. For the beginner, Appendices

A, B and D may be at least as important as the rest of the book. This material is outside the main text only in deference to the reader who knows everything about the Newtonian potential but knows nothing about proofs of symmetry by means of the maximum principle.

Preface

ix

Chapters 1 to 5 proceed from modest beginnings to results like Theorem 4.2 (described above) with only two digressions. The first is a sketch of symmetry, in §1.4, that is short but more general than later chapters

demand. The second digression, in §§2.5 to 2.7, concerns PhragmenLindelof theory for subharmonic functions. (The meaning of this phrase is explained at the beginning of §2.5.) These sections are included to show maximum principles at work on questions quite different from symmetry

and of importance in their own right. However, §§2.5 to 2.7 are not needed for Chapters 3 to 5.

Appendices C and E contain proofs that would interrupt the story unduly if they were placed in the main text. The subject of Appendix C is construction of the first of three comparison functions of a type introduced by D. Siegel (1988); these are the main tools of the PhragmenLindelof theory in §2.7. The edge-point lemma in Appendix E is a version of a maximum principle, previously called the boundary-point lemma at

a corner, developed by Serrin (1971) and by Gidas, Ni and Nirenberg (1979). The lemma plays an important part in those papers, but has been circumvented in much recent work. It is not used in this book, but would be needed if Chapter 5 were taken further at our level. Moreover, it seems likely that, as a natural step beyond the boundary-point lemma, the edge-point lemma will find applications quite different from those for which it was devised. The mnemonics C for construction, D for divergence and E for edgepoint may be helpful where reference is made to the last three appendices.

Acknowledgements I am heavily indebted to a reviewer, of an early and incomplete draft of the book, who criticized it authoritatively and helpfully. The description of symmetry in §1.4 represents my attempt to meet one of his wishes. Points of detail throughout the text have been improved by his suggestions. However, I have rejected some of the reviewer's advice (for example, that Appendix A be moved into the main text), and I have failed to close the gap (which the reviewer deplored) between general theory and particular applications. I am much indebted, also, to H. Brezis and to E.N. Dancer for tuition, at different times, on recent work in symmetry by way of the maximum

principle; to B. Buffoni for suggesting various improvements, one of which has simplified greatly the proof of Theorem 2.11; to G.R. Burton for instruction in the material of §1.4, for suggesting the phrase Steiner symmetry (Definition 3.2) and for contributing Exercise D. 15; to G. Keady for the essential idea in Exercise 5.30; to W. Reichel for various references

Preface

x

and for attempts to bring me up to date; to G.C. Smith for correction of blunders in Chapter 0 and §1.4; and to J.F. Toland, who not only led me to Exercises 3.18 and 3.19, but also showed extraordinary patience with my questions as to whether I should do this or do that, and made valuable comments on first drafts of several sections. Three conventions Square brackets [...] enclose explanations that will be obvious to some readers and, strictly speaking, are redundant. However, it may be that not all remarks in square brackets offend knowledgeable readers.

The abbreviated form if (of if and only if) is used in definitions, despite its vulgarity, because repeated use of ìf and only if' causes undue length, while plain ìf' is dangerous. The symbol marks the end of a proof, of a definition, of an exercise

or of any statement that is not italicized and requires separation from the general commentary.

0


Preamble Let fl denote, now and henceforth, an open, non-empty subset of the real, N-dimensional, Euclidean space RN, and let S1 denote its closure. The critical steps in this book will be of the following kind. An inequality

w(x) < 0 for all x E S2, which holds by hypothesis or for obvious reasons, will often be sharpened, by means of more nebulous properties of w and strenuous analysis, to

w(x) < 0 for all x E S2.

This improvement may seem too small to be interesting or useful, but the ultimate result of many steps of this type can be astonishing. To prove such fine distinctions, we need careful language, precise notation and some classification of functions and of boundaries OK2 according to their smoothness properties. It is appropriate, although unexciting, to begin with a list of such items. However, even the conscientious reader is advised merely to glance at the list now, referring to it more carefully later, as the need arises. Moreover, if this preliminary material should suggest that the text demands fluency in Lebesgue integration and sophisticated analysis, then that impression is false. Inspection of Chapter 1 and Appendix A, for example, should show that the reader is assumed to know only fragments of undergraduate material in calculus, linear algebra and analysis. However, a primitive grasp of inequalities between integrals is required for the basic material

in Appendices A and B; the reader with no experience of the Holder and Minkowski inequalities for integrals, and of the Lebesgue spaces Lp(S2), should perhaps give special attention to sections (xii) to (xiv) of this preliminary chapter. 1

2

0 Some Notation, Terminology and Basic Calculus

(i) Basic usage Some symbols in this list are not basic, but are included to prevent confusion with basic ones. P or Q : at least one of the statements P and Q (possibly both). Iff: if and only if. Greater than, positive, increasing, ... have the strict meaning (strict inequality: a > b), unless inverted commas are added to the word. Thus a real-valued function f defined on the real line is decreasing if t > s . f (t) < f (s); it is non-increasing (alternatively, `decreasing') if t > s f (t) < f (s). x, y E A: abbreviation for x E A and y E A.

A c B if x E A x E B (possibly A= B; we shall not use g). In words, A is a subset of B; it is a proper subset iff A c B and A

B.

A\B:= {x I xEA and A- B := { a- b a E A and b E B} when A and B are subsets of a I

linear space (or vector space). A x B := { (a, b) a E A and b E B }, the Cartesian product of the sets A and B. I

0: the empty set. N { 1,2,3 ....}, the set of positive integers or natural numbers. No {0} U N = {0, 1, 2,3,...1, the set of non-negative integers. 7L {... , -2, -1, 0,1, 2, 3, ... }, the set of all integers. R: the set of real numbers, either as a field or as an inner-product space; for all s, t E R, the inner product is st and the norm is IsI. IR# := R U {-oo} U {oo}, the extended real-number system. The rules postulated for the symbols -oo and oo (such as t E IR t+ (-oo) = -oo) are listed by Apostol (1974, p.14), by Rudin (1976, p.11) and in many other texts. C: the set of complex numbers. Re, Im: respectively the real part, imaginary part of a complex number or complex-valued function. (a, b): either an ordered pair (as in the definitions of A x B and IR2), or the open interval { t c= R I a < t a. In words, t tends to a from above, or from the right.

t j b if t, b E R, t < b and t - b. In words, t tends to b from below, or from the left.

For any function f having values in R, fixed] {f

f +(x) := max1x

(x), 0} defines the non-negative (or `positive') part

f + of the function f.

f -(x) := minx

fixed] {f

(x), 0} defines the non-positive (or `negative') part

f- off.

The Landau symbols O(.) [òh' or `big oh'] and o(.) [`little oh'] will not be used with great generality, so that the following examples of their meaning should suffice. For functions f, g from a subset of RN to R m,

f (x) = g(x) + 0(r2 log r) as r := Ix) -+ oo if there exist positive numbers C1 and C2 such that r > C1

11(x) - g(x)I < C2r2log r.

6


f (x) = g(x) + o(r2 log r) as r := IxI -> oo if If (x) - g(x)I/r2 log r -+ 0.

f (x) = g(x) + o(1) as x - c if If (x) - g(x) I - 0. f (x) = g(x) + O(Ix - cl-1) as x -> c if there exist positive numbers 6 and C such that 0 < Ix - cl < 6 If (x) - g(x)l < C/Ix - cI.

_ J 1 ift>0, 0

f x (t)

if t < 0

the characteristic function of (0, co) c ]8, is the

Heaviside function.

(vi) Partial derivatives Consider points x E fl [where f2 is an open non-empty subset of R'N] and a function f : SZ

R.

ej := (8jj)N 1 = (0,..., 0, 1, 0,..., 0), where 1 is the jth entry. a j f (x) = a f (x1, ... , XN)/axe := limt..o (f (x + tef) - f (x)) It whenever the limit exists.

Here a; means differentiation with respect to the jth argument, whatever that argument may be, and a; f is a function in its own right: (aif)(Y + z) = of (x) axl

x=Y+z

V = grad := (01, ... , ON), the gradient operator.

A = V V = div grad := a, +... + 02v, the Laplace operator. f x, f y, f ...:= a f/ax, a f /a y, a f /ar, ... respectively, when, for example, points of R2 are denoted by x = (x, y) = (r cos 0, r sin 0). (vii) Sets of continuous, and of continuously differentiable, functions Let A be a non-empty subset of RN ; let fl have its standard meaning.

C(A) := { f : A --* R I f is continuous on A }. f = (f 1, ... , fm) : A -> RM I each fj is continuous on C(A, RM) A }.

C [a, b], C [a, b)..... abbreviations for C([a, b] ), C([a, b)).... when N = 1.

C'(S2) :_ If : S -> R I f and all partial derivatives a j f, aia j f, ... of order < m are continuous in n I. Cm(S2) := { f : fi -> R I f c C(S2) f1 Cm(S2) and all partial derivatives of order < m have extensions continuous on fi }. It is necessary to introduce extensions; if fl c R2 is the triangular open set in Figure 0.1, and f is defined only on SZ, then (al f)(a), (a2 f)(b) and (a2 f)(c) cannot be calculated even as the right-hand limit or left-hand limit of the relevant difference quotient, because this quotient is unknown.


7

c

Fig. 0.1.

supp cp

A2

DO

((E

CC"

(Q) Fig. 0.2.

Cm(52)

cp E Cm(52) supp T is compact, supp cp c 52 }. A basic theorem states that a subset of RN is compact if and only if I

it is bounded and closed relative to ]IS N. If (P E Cm(52) and RN, then dist(supp cp, 752) > 0 [because supp (p is compact, 752 is closed, and the two are disjoint], as is shown in Figure 0.2.

52

Co(.) := C(.) and Q.) := Cc(.). nmo C`"(.) and Cc(K2) := nmo Cm(52). (viii) Boundaries of class Ck Let N E f2,3,4 .3,4,. ..} and k E No. The boundary 852 [of an open non-empty set SZ c RN] is of class Ck if (a) 852 = 852;

(b) for each point p E 752, there exist a set U = U(p) open in RN and

8


Fig. 0.3.

containing p, `local' co-ordinates y' := (Y1, ... , yN-1) and YN, with y = 0 at x = p, and a function It = h(., p) such that OQ n U has a representation YN = h(y'),

Y' E G,

h E C'(G),

(0.1)

where G = G(p) is open in RN-1 and convex (Figure 0.3).

The co-ordinate transformation is y = A(p)(x - p), where A(p) is an orthogonal N x N matrix depending on p, and x1, ... , XN are the co-ordinates in terms of which SZ is defined.

Note that condition (a) rules out sets with isolated boundary points, such as 9(0,1) \ {0}, and sets òn both sides of their boundary', such as

][8N\{x I xNO}. If Ofl is [at least] of class C1, then the outward unit normal n(x) _ A(p)-'v(y'), and the element dS(x) = da(y') of surface area, are defined as follows at points x E M n U(p). Let the yN-axis point into 0; then

v(Y) = da(Y) :=

dy

wN(Y')I

((a,h)(Y'),... , (ON-1h)(Y'), -1) { (01 h)(Y')2 + ... + (ON-1 h)(Y')2 + 111/2'

(0.2)

= { (a,h)(Y )2 + ... + (ON-1h)(Y )2 + 1 }1/2 dy', (0.3)

where the components vj of v are relative to the y/-axes, and

dy' := dy, dY2... dyN_1.

(ix) Integrals For a function f : A -+ R, where A c RN, integrable


9

will mean Lebesgue-integrable, or, in symbols, that f E Li(A); the word measurable, applied to a subset of RN or to a function, will mean Lebesgue-measurable. The notation

(dx := dxl dx2... dxN)

f f := JA f (x) dx A

(0.4)

A

is often convenient.

In contrast to many texts, we do not admit integrals equal to ±oo. Therefore the statements (a) the integral fA f exists, (b) f is integrable over A, (c) f E LI(A), all have the same meaning. Each implies that

-oo R I u is measurable, fn lute exists }, (0. 10)

Ilullp := (fn IuIP) 'P for all u E Lp.

J

1

The semi-normed linear space Lp(S2) is the pair (Lp, II . Ilp); here p E [1, co).

An explanation is in order. The function 11p is not a norm because the condition II u II p = 0 implies only that u = 0 a.e. in 52, not that u is II

the zero function [which equals zero everywhere]. However, II . IIp satisfies (0.8) for all u, v E LP and all a E R; that it satisfies the triangle inequality will be shown in (xiii). Therefore, if addition and scalar multiplication of elements of LP are defined by

(u+v)(x) := u(x)+v(x) and (au)(x) := au(x) for all x E S2 and a E R, then LP becomes a linear space over R. (c) The semi-normed linear space L,,(n) is the pair (L,,, II . I.), where

L,,

u : S2 -> JR Ilull

I

u is measurable, ess supxEn lu(x)I < co },

:= esssupxEn lu(x)l for all u r= L. (0.11)

The explanation is much as before. The reason for the notation is that, if n has finite volume (IfII < co) and u E L,,(S2), then IIullp IuII,, as p

oo.

(d) Caution In (xv) we shall meet normed linear spaces .p(12) and


12

Fig. 0.4.

',,(S2) that resemble LP(S2) and respectively. Even the most precise and respected sources (Rudin 1970, pp. 65-66; Weir 1973, pp. 165-166) blur the distinctions between LP(S2) and 2 ,(S2) and between Lo,,(S2) and

written Ilu function u.

I

We shall follow this practice: for example, IIuIIP (also LP(s2)II) will be called, quite incorrectly, the LP norm of the

(xiii) The Holder and Minkowski inequalities for integrals

The statement 1 1. If p, q E [1, oo] and 1/p + 1/q = 1, then q is the Holder conjugate of p, and p is the Holder conjugate of q. The pairs (1, oo),(4, 5),(2,3) and (2,2) are examples of such conjugates. By uv we mean here the pointwise product: (uv)(x) := u(x)v(x). Young's inequality For real numbers a, b E [0, oo), P

9

p

q

1+1=1, 1 0; the other cases follow readily once the inequality has been proved for this case. Since Fubini's theorem allows us to change the order of integration, we may imitate the proof of (0.14). In fact, P

S2

fy

dx = U Iu(x,y)I a(y) dy } U(x)P-1 dx

Iu(x,y)I a(y) dy I

S2

Y

)))

J tf lu(x,y)l U(x)P-1 dx))) a(y) dy Y

2. Therefore it might seem likely that, by perturbing aG, one could construct a pair of nearly spherical bodies for which the potential is exactly constant on the boundary when Icl is sufficiently large. The theorem shows that this cannot be done. Although Theorem 1.1 establishes the shape of a set, we shall prove it by means of results for solutions on RN of a non-linear elliptic equation

of order two. A corresponding problem for the same equation on a bounded set 0 (open as always) is that of finding a function u E C2(S2) such that

Au+f(u)=0 in f2'

(1.3)

ulan=0,

} where f : R - R is a given function in C' (R), say. Conclusions about the set 92 are again an important part of the results of Gidas, Ni and Nirenberg, but the theory also establishes properties of u when it is given

1.2 What are maximum principles?

19

that S is a ball. The next theorem is valuable because, for the general problem (1.3), existence of many solutions (or of none) tends to be the rule rather than the exception.

Theorem 1.2 Suppose that in (1.3) the set fl is a ball, say f) = R(O,a) c RN, and that u is a positive solution: u(x) > 0 for all x E 91. Then u is spherically symmetric: it depends only on r := lxJ, and du/dr
t. (iv) minXEA f (x) :_ - maxXEA{-f (x)}, infXEA f (x) := - SUpXEA{-f (x)}.

According to this definition, a maximum is a supremum that is attained at some point. To say that a maximum is attained (as is often said in the literature) states nothing beyond our definition; to say that a maximum

is attained at a specified point, or in a specified subset, may be useful information; to say that a supremum is attained is always cause for rejoicing, because then the function has a maximum. [For example, the functions from R into 11 defined by f(x) = x,

g(x) =

J0 e-X

ifx 0,

have no maximum.]

Item (iv) of Definition 1.3 is somewhat indirect, but is useful for passing from a theorem about maxima to one about minima; no doubt the reader can supply an equivalent statement that is more direct. Notation We shall often use abbreviations like in place of maxXEAf(x),

(1.5a)

f 0 on [a, b], then the following hold.

(i) Either u(x) < max{u(a), u(b)} for all x E (a, b), or u is constant on [a, b]

1 Introduction

22

(ii) If u(b) >- u(a), then either the outward derivative u'(b) > 0, or u is constant on [a, b]. (iii) If u(a) > u(b), then the outward derivative -u(a) > 0.

Proof Integration by parts gives fX

u(x) = u(a) +

u'(t) dt

u(a) + u'(a) (x - a) + f u"(t) (x - t) dt,

a < x < b.

(1.6)

To eliminate the ùnknown' u(a) in favour of the `data' u(a), u(b) and u" 0, set x = b; then

-

b

u'(a) =

b

1

{ u(b) - u(a) - I (b - t) u"(t) dt } .

a l Substituting (1.7) into (1.6), and tidying the result, one obtains

(1.7)

JJJJ

b

u(x) = v(x) - J. G(x, t) u"(t) dt

J.

(a < x < b),

(1.8)

where v is the solution of v" = 0 given by

v(x) := u(a) +

u(b) - u(a)

b-a

(x - a),

and, on the square [a, b] x [a, b] (Figure 1.2), G(x, t)

(b-x)(t-a)/(b-a) ifx> t, (x - a)(b - t)/(b - a) if x < t.

Thus G > 0 in (a, b) x (a, b) and G = 0 on the boundary of the square. Since also u" >- 0 on [a, b], it follows from (1.8) that u(x) < v(x)

(a< x< b),

(1.9)

with strict inequality for x E (a, b) if u"(xo) > 0 at some point xo E [a, b], because then u" > 0 in an interval of positive length, by the continuity of u".

(i) To prove assertion (i) of the Remark, we consider three cases. (1) If u(a) u(b), then u(x) < v(x) < max{u(a), u(b)} for x E (a, b). (2) If u(a) = u(b) and u"(xo) > 0 at some xo E [a, b], then, by the remark following (1.9), u(x) < v(x) = u(a) for x E (a, b).

(3) If u(a) = u(b) and u" = 0 on [a, b], then u(x) = v(x) _ u(a) for x E [a, b].

1.2 What are maximum principles?

23

t

b

a

_L

I

a

b

x

Fig. 1.2.

(ii) Differentiating (1.8) and setting x = b, or else forming the analogue of (1.7) for u'(b), we obtain b

u'(b)

=

b

1

a

{u(b) - u(a) + I (t - a) u"(t) dt}

0

JJJ

when u(b) > u(a), with equality only for case (3). (iii) If u(a) > u(b), then u'(a) < 0 by (1.7).

The next observation is almost trivial because it is the implication of a strict differential inequality. Nevertheless, a slight variant of this remark will play a part in the proof of Theorem 2.5.

Remark 1.6 If u E C2 (92) and Du > 0 in Sl, then u cannot have a local (let alone a global) maximum at a point of 52. Proof Assume (for contradiction) that u has a local maximum at q E fl. Then for each j E {1, ... , N} we have (a3u)(q) = 0 and (8?u)(q) < 0. Hence (Au)(q) < 0.

1 Introduction

24

k

T1, (k)

Fig. 1.3.

1.3 On reflection in hyperplanes

With this method, as with maximum principles, the proof of the pudding will be in the eating. However, we can take a first step in this section, and look ahead a little. Figure 1.3 illustrates the objects in the next definition; the notation will be shortened when we come to use it in earnest.

Definition 1.7 For each unit vector k E R' and each number u E R, define a hyperplane by z Tµ(k):_ {BERN I k=p}

(Iki=1).

1.3 On reflection in hyperplanes

25

The reflection in T,,(k) of any point x E RN is xµ'k

and the reflection in Tµ(k) of any function p : RN -> R is defined by (pµ,k(x) :_ cp(xµ,k)

for all x E RN.

The reader should verify that (x" )P k = x. There is no sign convention in the definition of hyperplanes, so that T-,,(-k) = T,(k). If (p is defined only on a proper subset A of RN, then we define spa only on {x E RN xP,k E Al, which is the reflection in I

Tµ(k) of A.

Lemma 1.8 If v : RN -> R has the property that v(x°°k) = v(x) for each unit vector k and all x E RN, then v is spherically symmetric (depends only on lxl).

Proof We need prove only that, for any two points y and z satisfying IYI = Izl and y * z, we have v(y) = v(z). Choose k = (z - y)/lz - yl. If z = y°k, as Figure 1.4 suggests, then v(z) = v(Y°'k) = v(Y),

as desired. To check that z = y°,`, we have yOk

y°'`

y z-y

=

- YI

z

-yIz

1z-y1'

z y - z = y-z+2(Y-z)y Iz-yI2

Iz - yI2

=0

{ (Izi2

- 2z y + IYI2) + 2(y z - IYI2) }

because IYI = Izl

For a preliminary sketch of the method of reflection in hyperplanes, we return to Theorem 1.2 and consider a function u such that ,Lu + f (u) = 0 in 52 = 4(0, a),

uIan =0,

uIa > 0,

u E C2(S2).

Our aim is to prove the hypothesis of Lemma 1.8, adapted to the ball 0: that u(x°") = u(x) for each unit vector k and all x E S2. To this end,

1 Introduction

26

Fig. 1.4.

we fix the unit vector k and contemplate all positions of the hyperplane Tµ(k) from y = a to it = 0 (Figure 1.5). Let

Z(µ,k) := {z E1

I

p}.

The procedure is to prove that (for fixed but arbitrary k) w(z,µ) := u(z) - u (zµ.k) < 0

if z E Z(p,k)

(*)

whenever it E (0, a). Once this has been established, a continuity argument for p 10 shows that w(z, 0) < 0 if z E Z (0, k); in other words, that u(x) 0 and x E S2.

But the same result for the direction -k implies that u (x°k) < u(x)

if x k > 0 and x E S2.

These two inequalities yield the desired result that u (x °°k) = u(x) for all

x E 0 [since, if x k = 0, we have xO,k = x, while points x with x k < 0 are points y°" with y k > 0].

That du / dr < 0 for 0 < r < a is an easy consequence of (*) for 0 < p < a and of a maximum principle called the boundary-point lemma for balls.

The proof of (*) for all It E (0, a) proceeds in two stages. The first is for small, positive values a - p and begins with application to w of a maximum principle for thin sets; this is possible because w satisfies an

elliptic equation of order two and suitable boundary conditions. The

1.4 What is symmetry?

27

k

Z (µ, k)

T.M:

=,u

Fig. 1.5.

second stage is a demonstration that, once (*) holds in an interval (µo, a),

it can fail only as y 10. For equations less simple, or functions f significantly less smooth, than those in Theorem 1.2, the second stage requires tenacity and use of a substantial tool-kit. This sketch of the method for the case of a ball serves also for any bounded set S2 that has appropriate symmetry under reflection in a hyperplane. But when the equation Au + f (u) = 0 holds in the whole space R', and no boundary condition is available, one must first locate a possible centre of symmetry by means of information about u(x) for IxI --> oo, and then pursue an analogue of (*) in which Z(p,k) is a half-space. Again there are two stages, but in the first stage one now labours to organize implications of the asymptotic behaviour of u(x) for

1xI -+ oo.

1.4 What is symmetry? A small answer to this large question will be sufficient for our purpose, because the symmetries that will be established in Chapters 3 to 5 are

1 Introduction

28

simple enough to require no reference to any general theory. Nevertheless, some indication of a wider viewpoint seems in order. By an isometry of RN we mean a function f : RN --> RN such that

for all x,y E RN.

If(x)-f(y)I = Ix - yl

(1.10)

x= y], but does not demand a priori that f be surjective [that the range This definition ensures that f is injective [one-to-one: f (x) = f (y)

f(RN) = RN], nor that f be of any particular form. However, (1.10) has strong consequences, as follows.

Theorem 1.9 If f is an isometry of RN, then it is affine [is a linear function plus a constant] and bijective [both injective and surjective]. In fact,

f (x) = f (O) + Tx

for all x E R",

equivalently, N

fi(x) = f i(0) +

for each i E { 1, ... , N} and all x E RN,

Ti.jx j J=1

where T = (Ti,j) is an orthogonal N x N matrix independent of x. The proof is outlined in the hint to Exercise 1.22. Let A be a subset of RN that contains N + 1 affinely independent points a°, al, ... , aN ; this means that the N vectors al -a0,. .. , aN - a° are linearly independent. A symmetry transformation of A is an isometry f of RN that maps A onto itself: f (A) = A. For example, if A c R2 is a regular pentagon centred at the origin, or a good specimen of carissa grandfora centred at the origin (Figure 1.6), then rotation through 72° is a symmetry transformation; if anti-clockwise, this rotation is the map 0 H 0 + 2ir/5 for points (r cos 0, r sin 0), or, more fully,

a(x) .-

sc I

in a

cos x I [

X2

where a = 2n/5 = 72°. If the regular pentagon has a vertex on the x1-axis, then reflection in the xl-axis is a symmetry transformation of the pentagon, but not of carissa grandora because each petal is asymmetric; this transformation is (X 1, x2) -* (X 1, -x2), or 0 H -0 for points (r cos 0, r sin 0), or

b(x)

[

01 0-1

1

1

X2 X,


29

Fig. 1.6.

For a subset A of RN that contains N + 1 affinely independent points, the set of all symmetry transformations of A forms a group if we define

the group operation o to be composition [so that g o f is defined by (g o f)(x) := g(f(x)) for all x E RN] and define the unit element to be the identity map I [such that Ix := x for all x E RN]. This group is the symmetry group of A; we denote it by G(A). The symmetry group of carissa grandilora (centred at the origin) is 2

3

4

where a is as in (1.11) with a = 2n/5, the symbol an means composition aoao

o a with n factors a [so that a3 is the map 0 H 6 + 6n/5 for

points (r cos 0, r sin 0)], and it is understood that a5 = I. The symmetry group of the regular pentagon (centred at the origin and having a vertex on the x1-axis) is a

a

where a is as before, b is as in (1.12), ba" := b o a", and it is understood that b2 = I. To extend these definitions of 1'5 and A5, we set

a := 2n/n,

n E N,

in (1.11). Then

r" := {I,a,...,an-1}

(1.13)

1 Introduction

30

is the symmetry group of a flower with n identical petals that are placed about the origin at equal angular intervals but lack individual symmetry. Also,

An := {I,a,...,an-1, b,ba,...,ban-1 }

(1.14)

is the symmetry group, if n >_ 3, of a regular polygon centred at the origin, having n sides and with a vertex on the xl -axis. The group Fn is a copy of the more abstract group usually written Cn and called the cyclic group of order n, the order of a group begin the number of its elements. Similarly, An is a copy of the dihedral group of order 2n, often written Dn. [In Dn, the elements a and b are arbitary objects for which a group operation and a unit element can be defined.] Let A be an isometric image of A; that is, there is an isometry qp of RN such that cp(A) = A. [For example, A might be the pentagon in Figure 1.6, and A a translated and rotated version of it] Then to each symmetry

transformation g of A there corresponds a symmetry transformation k := cp o g o (p-1 of A; conversely, to each g E G(A) there corresponds g := cp-1 o g o (p E G(A). In other words, the map g 1--* cp o g o cp-1

from G(A) onto G(A)

(1.15)

is a group isomorphism; we regard G(A) and G(A) as essentially the same and say that they are equal up to isomorphism.

The symmetry group G(A) indicates the symmetry of a set A. If G(A) c G(B) and G(A) # G(B), then A is less symmetric than B; indeed,

A is less symmetric than each isometric image B of B, because G(B) and G(B) are equal up to isomorphism. If G(A) has many elements, then A has a large measure of symmetry; if G(A) = {I }, then A has one symmetry transformation but no symmetry in the everyday sense of the word. The group G(RN) contains all isometries of RN and is therefore the largest symmetry group for given N. The symmetry group G(B) of a ball B := R(0, p) in RN is almost as large: it contains all those isometries f of RN for which f (0) = 0. The irregular polyhedron N

P :=

x E RN

I 1: jxj < 1,

each xj > 0

N > 2,

j=1

has symmetry group G(P) = {I }. We turn now to the symmetry of functions. Again let A be a subset of RN that contains N + 1 affinely independent points; an invariance transformation of a function u := A -+ RM is a symmetry transformation f of


31

A that leaves u unchanged : u o f = u. The invariance transformations of u form a group if the group operation is composition and the unit element is the identity map I ; this group is the invariance group of the function u and we denote it by H(u). (Some authors use the phrases symmetry of u and symmetry group of u for what are called here invariance transformation of u and invariance group of u.) Here are two examples. 1.

If v : JR -> JR2 is defined by v(t) := (a cos t, fl sin t) for t E JR, the

numbers a and /3 being positive constants, and if f (x) := x + 27r for x E JR, then the substitution t = f(x) shows that (v o f)(x) = v(x) for all x E JR, so that f is an invariance transformation of v. The invariance group of this function v is H(v) := {f(. , k)

I

where f (x, k) := x + 2kir for all x E R.

k e Z},

Note that H(v) is quite different from the symmetry group G(v(J)) of the range of v; when a f3, we have G(v(J)) = A2, a copy of the dihedral group of order 4. 2. The function W : JR2 \ {0} -+ R defined by fi(x)

-

2 = -sine 20

2 xlx 2

(xi +X22)

for all x = (r cos O, r sin 0) ER 2\ {0}

r

is the stream function of a certain quadrupole at the origin, according to (A.55) and Exercises A.19 and A.21. The level sets of W are displayed in Figure A.10. We contemplate both V and the set

V:={xER2\{0}

IW(x)I>1}.

This set has the symmetry group G(V) = A4. [Recall that, up to isomorphism, A4 is the symmetry group of any square in JR2] The function 1P has a smaller invariance group, isomorphic to A2; in fact, H(yp) = {I, a, b, ba},

where a is the rotation 0 H 0 + it, while b is the reflection 0 - f 7r/2 - 0. Four symmetry transformations of V are absent from H(ip) because they change sin 20 to - sin 20. It seems desirable to link results like those in Chapters 3 to 5 to the theory of symmetry that has been sketched here, but so far there has been little work in this direction. We note one difficulty, and one encouraging feature, for any attempt to build a bridge between the two topics. For the sake of definiteness, consider a positive solution u of (1.3). The difficulty is that the invariance group of this function u depends not only on the symmetry group of 92 but also on other properties of 12,

1 Introduction

32

such as its topological type, as well as on details of the function f in (1.3). In particular, if the ball B := .4(0,a) in Theorem 1.2 is replaced by an annulus or spherical shell A centred at the origin, then these two sets have the same symmetry group: G(A) = G(B), but the first result in Theorem 1.2, that u(x) depends only on lxi, does not extend from KI = B to 0 = A. (Exercise 5.30 shows that (1.3), with 52 = A, admits positive solutions u which do not depend only on jxl.) The encouraging feature is the fundamental role of reflection in both theories. Let us rewrite a part of Definition 1.7 as

R(x) := x + 2(µ - x k)k

for all x E RN;

(1.16)

then the reflection operator R is an isometry of R' that sends x to its reflection in the hyperplane called T,(k) in Definition 1.7 and Figure 1.3. In order to cast (1.16) into the form of an isometry displayed in Theorem 1.9, we write it as

R(x) = 2µk + Sx,

where Si3 := bid - 2kikj

(iki = 1);

(1.17)

one checks without difficulty that the matrix S is orthogonal. It should be clear from §1.3 that the operator R is basic in Chapters 3 to 5; maximum principles and reflection in hyperplanes are the principal tools in those

chapters. It has not yet been stressed in this section that reflection is basic to symmetry groups, but this is implied by the following theorem.

Theorem 1.10 If f is an isometry of RN, then it can be written as a composition product

f =RooRlo."oRm,

m- 0 to be specified, or of class C00. [Refer to Chapter 0, (viii).] The letters a, b, c denote positive constants.

1.5 Exercises

33

(i) The elliptic disk D x E R2 xi/a2 + x2/b2 < 1 }. (ii) The elliptic cone K x E R3 x1 > 0, x22/b2 + x3/c2 < xi }. (iii) The double cone G x E R3 X1 E R \ {0}, x2 + x3 < xi }. (iv) The hyperboloidal set H x E R3 x; > a2, x2/b2 + x3/c2
0ifxx`sinifx>0 . x

Exercise 1.12 Given that in Theorem 1.1 the constant value of u I OG is /3, where /3 < 0 if N < 2, and /3 > 0 if N > 3, find the radius of the ball G in terms of A,# and N. [Refer to (A.23) in Appendix A] Exercise 1.13 The factorial function (.) ! is defined on IR \ {-1, -2, -3,.. .} by

(x +

x!=I e_ttxdt if x>-1, 0

The second equation is implied by the first for x > -1, and defines x! for non-integer x < -1. The factorial function is related to the gamma function F by x ! = I'(x + 1). Given that (_j) ! = 7r1/2, prove that (sin t)" dt= o

Znl/2 12n- 2 I!/

\

\

I

Zn I!

for n E {0,1,2,...},

\\\

111

by showing that, if An denotes either expression, then AO = x/2, Al = 1 and An/An-2 = (n - 1)/n for n E f2,3,4 .3,4,.. .}.

Exercise 1.14 Use the result of Exercise 1.13 and induction over N to calculate the volume of a ball in RN :

ON(0, P)l := f ...

f

N12

7r dx1 ... dxN = (N/2)! PN.

Infer that the surface area of the bounding sphere is ,N/2 10-4N(0,01 = (N/2)! N PN 1

1 Introduction

34

Exercise 1.15 The function

RN -> R is defined by

- -1r2

V() W=

if r

1

lxi < 1,

if r1.

0

Prove that W E CC°(RN).

[It may be helpful to consider first all derivatives of

g(t) := exp -1

1

0< t < 1,

t

and ultimately to use multi-index notation (Definition A.3) for derivatives of ip.]

Exercise 1.16 Let f E C(O). Prove that, if fn cpf >- 0 whenever T E C,00(fl)

and cp >- 0, then f >_ 0 in n; if fn (pf < 0 whenever (p E C,1 (Q) and cp >- 0, then f < 0 in S2; if fn cpf = 0 whenever cp E C,°(Q) and cp > 0, then f = 0 in f). [The abbreviation (1.5b) has been used. The function 1p in Exercise (x 1.15 has a useful variant defined by 1(x) = 1p c) for any fixed 8 c E RN and 5>0.] Exercise 1.17 Prove the one-dimensional case (N = 1) of Theorem 1.1.

[Any open set in R is a countable union of disjoint open intervals (Apostol 1974, p.51). It may be helpful to consider the case of finitely many intervals before proving the full result.] Exercise 1.18 (i) Show that, when we operate on C2 functions depending only on r := lxi [say u(x) =u(r)], r-N+1 d

rN_1 d

dr

dr

if r > 0.

(ii) For N >- 3, let s := (x2 + + x2)1/2 and define 9 E [0, n] by x1 = r cos 6, s = r sin 0. Show that, when we operate on C2 functions depending only on x1 and s,

°=

a

Ox1)2

+

a s- N+2 as (SN_2

(s

as)

r-N+1 ar (r"_'_) + r-2(sin

B)-N+2

> o) {(sinOy\T_2

a6

a6 } (s > 0).

1.5 Exercises

35

(iii) Write x = (r cos 0, r sin 0) for points of R2. By adapting the final result of (ii), or otherwise, show that

__la (ra-) ao r ar ar + r2 Call 1

(N = 2, r > 0).

Exercise 1.19 The Bessel function Jv, v > 0, is defined on [0, oo) by

Jv( t) _

J!

(v

(t)v+2i, +.j)!

t > 0;

2

satisfies

{tat (tdt)+I-

t>0;

c2}JV(t)=0,

and has denumerably many positive zeros, say fv,p, ordered according to 0 < /3v,1 < $v,2 < . [See, for example, Burkill 1975, p.83; or Simmons 1972, p.121; or Whittaker & Watson 1927, p.367.] Referring to Exercise 1.18, (iii), show that the function u,,,p defined by u,,,p(x) = cJ (/3,,,p a) cos(nO + K),

nEN0, pEN,

0 < r < a,

x=0 if n=0,

c E R \ {0},

KER if n> 1,

is in C0°(B), where B = R(0, a) c R2, and satisfies Lu + (/3n,p/a)2u = 0

in B,

ul BB

= 0.

0 for n >_ 1, it is to be understood that u,,,p(x) = 0 for n

[Since

1 and x = 0. However, to establish the C00 property at x = 0, one must use co-ordinates xl and x2.] Exercise 1.20 Writing x = (r cos 0, r sin 0 cos cp, r sin 0 sin (p), 0 < 0 < it, for points of R3, show that for N = 3, and r sin 0 > 0,

r

2

ar

Cr

a) + sin 0 as (sin0) + (sin 0)2 (7)

[It is not necessary to make six lengthy applications of the chain rule; see, for example, Kellogg 1929, p.183; or Sobolev 1964, p.393; or Spiegel 1959, p.151.]

1 Introduction

36

Exercise 1.21 Let n E No and m E No. The Legendre function P,m is defined on [-1,1] by Po(t) n

satisfies ( {

= NO _

2nn1

Pn(t) =

(1 - t2)m/2 (

(_)

+n(n+1)-

(-)(t2_l)n, (Tt

d)

Pn(t),

m2

-1 < t < 1;

}Pn(t)=0-1 0 and all ki(x) = W(x)/ x E RN, so that kp > 0 on RN and

kp E C,(RN), suppkp c M0, p),

fkp = RN

J A0,p)

kp = 1.

We call kp a smoothing kernel, or averaging kernel, of smoothing radius p.

Now let f : RN - R be locally integrable, that is, integrable on every compact subset of ]ESN. Then we form f p(x) :

J

kp(x - y) f (y) dy =

.f kp(z) f (x - z) dz,

X E RN ,

RN

RN

and call fp a regularization of f, or a mean function of f. Prove the following. (i) fp E C°°(RN).

(ii) If f E C(RN), then fp(x) - f(x) as p -> 0, uniformly on each compact subset of RN. (iii) If f E C1(RN), then smoothing and differentiation commute:

a;(fp) = (a;f)p

for each j E {1,...,N},

so that 8j f p(x) - 8jf (x) as p - 0, uniformly on each compact subset of RN.

[For (i), multi-indices (Definition A.3) may be useful: so may a glance at the proof of Theorem A.5. For (ii), observe that

f (x) - f p(x) _ .f0,p) kp(z){ f (x) - f (x - z) } dz. gs(

1 Introduction

38

For (iii), we have (8/axj)kp(x - y) = -(8/8yj)kp(x - y) and integration by parts.] Exercise 1.24 Continuity of translation for functions in Lp(RN), 1 0

as

0,

Ih1

where h E RN and f ( -h) is the function with values f (x - h). [There is a sequence ((pn) in C,(RN) such that If - con II -> 0 as n --> 00. The symbol Q.) refers to continuous functions of compact support; see Chapter 0, (iv) and (vii).] Exercise 1.25 Averaging functions in Lp(RN), 1 0 and all x E RN. Given p E [1, co) and f E Lp(RN), define a mean function, as before, by

f

fp(x) :_, Nap(z)f(x-z) dz,

x E RN.

0 asp

Writing 11.11 := II . I Lp(]RN) II, prove that If - fp II

0.

[Observe that the substitution z = py yields IIf-fP11

= 0, there is a number M = M(s, f) such that 2IIf

IIfIvl>M Iai(Y)I dy < 's;

and that for IyI < M continuity of translation is relevant.]

2

Some Maximum Principles for Elliptic Equations

2.1 Linear elliptic operators of order two As always, S2 denotes an open non-empty subset of RN.

Definition 2.1 (i) The operator L, defined by N

N

E aij(x) 8iOj +

Lu(x)

bj(x) 8j + c(x)

U(x)

(2.1)

j=1

i,j=1

whenever u E C2(S2) and x E S2, is a linear partial differential operator, of order two. Here

a = (aij) : f - RN2,

b = (bj)

:

--+ RN,

c:

--+R

are given measurable functions. The N x N matrix a is symmetric : a ji(x) = ai j(x) for all i, j e {1,. .. , N} and all x e 0. [This involves no loss of generality because ajOju = 8i3ju.]

(ii) We say that L is elliptic at x E S2 if there is a number ).(x) > 0 such that N

E aij(x)

for all

E RN;

(2.2)

i,j=1

that L is elliptic in n if it is elliptic at every x E S2; and that L is uniformly elliptic in 0 if there is a constant AO > 0 such that 2(x) > 20 for all x E S2. The best (largest) values 2(x) and 20 are, respectively, the pointwise and uniform moduli of ellipticity of L.

Here are three examples to which we can apply the definition with almost no calculation. 39

2 Some Maximum Principles for Elliptic Equations

40

1. If L = .+ lower order terms, then aij(x) = 8 [the Kronecker delta, Chapter 0, (v)], so that L is uniformly elliptic in every n, with 20 = 1. 2. Let X 1 ,--- , NN-1

operators ai +

be space variables, while xN denotes time. Then the

+ 0N_1 - aN of the wave equation, and a; +

+

a2_1 - ON of the heat equation, are not elliptic: choose l;i = SNi in (2.2).

3. The Tricomi operator a; + x102 is elliptic in the half-plane { x E xl > 0 } but not uniformly so; the pointwise modulus of ellipticity

1182

I

is x1

if0<x11.

A(x)

Exercise 2.2 Given that a(x) is symmetric and satisfies (2.2), prove the following.

(a) Ellipticity is invariant under rotation of co-ordinate axes. That is, if R is a constant, orthogonal N x N matrix, y := Rx and h(y) := Ra(x)R-1, then a`j(x)

a

a

a

a

axi axe = E hpq(Y) a- aYq

and

> hpq(Rx)rlprlq >- 2(x)I, I2 for all 1 E RN. p,q

(b) The pointwise modulus of ellipticity is the smallest eigenvalue of a(x).

(c) Let g(x) be a non-positive N x N matrix; we write g(x) < 0, meaning that g(x)l; < 0 for all E RN. Then trace(a(x) g(x))

aij(x) gj,(x) < 0. i,J

(Here the rule for matrix multiplication is summation over adjacent subscripts:

gij(x) (.i

g(x) J

i gt.i(x) cj,

g(x) C i=1

i,1

so that row and column vectors need not be distinguished in such expressions.)

2.2 The weak maximum principle

41

2.2 The weak maximum principle Definition 2.3 The operators to be considered in this section and the next two are N

N

Eaij(x) aiaj + Ebj(x) aj,

Lo

j=1

i,j=1

L

:=

Lo + c(x), N

with c(x) < 0 for all x E 0, N

E aij aiaj + E bj aj + c, with c < 0;

L1

i,j=1

j=1

in L1 all coefficients ai j, b j and c are constants. Thus Lo is the particular L with c = 0 (the zero function), while L1 is the particular L with constant coefficients. All three are uniformly elliptic : for all x E S2 and E RN, N

E aij(x)

AO = const. > 0.

(2.3)

i,j=1

All coefficients are bounded and measurable: in Lo and L, for all i and 3,

supxEn Jaij(x)I < oo,

supXEn Ibj(x)I < oo,

supXEO Ic(x)I < co.

Definition 2.4 We shall say that u is a C2-subsolution relative to L and 0

if u E C2(1)) and Lu > 0 in 0. (Here L may be replaced by Lo or L1.) We distinguish Lo from L because stronger conclusions are possible when c = 0, and L1 from L because a different kind of subsolution will be used for L1. However, in the following three versions of the weak maximum principle (which is not to be despised, relative to the strong maximum principle), hypothesis (a) is always the same; it ensures, as was noted in Exercise 1.4, that supn u = maxii u. Theorem 2.5 (the weak maximum principle for Lo). Suppose that (a) 0 is bounded, u E C(C2); (b) u is a C2-subsolution relative to Lo and 0. Then the supremum of u is attained on the boundary: max5 u = maxan u.


42

Proof (i) Define, for arbitrary e > 0 and for a constant K to be chosen presently, v(x) := u(x) + eeKx', x E S2.

Now, for all x E fl, L0(exx,) _ {au(x)K2+bt(x)K}exx,

>

(AoK2

>0

- {sup0 Ibtl}K) eKx'

if we choose K >

[in (2.3),

_ (K,0,...,0)]

1

supn Ibt I. TO

Hence Lov > 0 in n. (ii) Assume (for contradiction) that supn v is attained at xo E f2. Then (ajv)(xo) = 0 for all j e {1,...,N}, and the Hessian matrix H(xo) := ((ajajv)(xo)) < 0.

[Otherwise Z'H(xo) = a > 0, say, for some C E RN with ICI = 1, and the Taylor formula v(xo + h) = v(xo) + 0 + i E(0j0jv)(xo) h;hj + o (Ihl2)

i,j

leads to a contradiction, because we can choose h = Q4 with fi > 0 so small that v(xo + h) > v(xo).] The result of Exercise 2.2, (c), now shows that (Lov)(xo)

= Y. aij(xo) (ajatv)(xo) + 0 trace (a(xo) H(xo)) < 0,

which contradicts the result of step (i). (iii) Accordingly, for every e > 0 and all x E S2, u(x) < v(x) < maxan v < maxan u + eK1, where

Kt := maxxEa2 e Kxj

It follows that u(x) < maxan u for all x E S2.

[Otherwise u(xo) _ maxan u + 6 for some xo E S2 and some S > 0; we obtain a contradiction by choosing e = 6/2K,.] The weak maximum principle for L involves the non-negative part u+ of u [see Chapter 0, (v)] and states less than the theorem for Lo when


43

maxan u < 0. However, if maxan u >- 0, then maxi u = maxao u exactly as before, because in that case maxan u+ = maxan u, so that strict inequality in (2.4) is impossible.

Theorem 2.6 (the weak maximum principle for L). Suppose that (a) f2 is bounded, u e C(S2); (b) u is a C2-subsolution relative to L and SZ. Then

maxi u < maxaf, u+.

Proof Let n+ := { x E f2

u(x) > 0 }. This set is open in RN: if y E 92+, say u(y) = a > 0, then there is a number 6 > 0 such that both I

M(y, 6) c f2 [since f2 is open] and u(x) > a/2 whenever x E .1(y, 6) [since u is continuous], so that M(y, d) is in f2+. If 12+ is empty, then maxis u < 0 and the theorem is true. Suppose then that f2+ is not empty. The hypotheses Lou >- -c(x)u in

f2 and c(x) < 0 in f2 imply that Lou > 0 in f2+; by Theorem 2.5, the maximum of u over f2+ equals that over aft+; hence there is a point xo E 8f2+ such that u(xo) = maxjj+ u > 0.

If xo E K1 (Figure 2.1) we have a contradiction: by continuity, u > 0 in .4(xo, p) for some p > 0; on the other hand, -I(xo, p) contains points of f2 \ f2+, because xo E 8f2+, and u < 0 at such points. Therefore xo E aft.

Remark 2.7 If u is a C2-supersolution relative to L and Q, which means that u E C2 (Q) and Lu < 0 in f2, then -u is a C2-subsolution. If also condition (a) holds, then

maxi(-u) < maxan(-u)+, where

(-u)+(x) = max[x fixed] { -u(x), 0 } =

- min]X fixed] { u(x), 0 }

-u-(x), so that

maxi(-u) < maxan(-u-). Equivalently,

mind u > minan a-.

(2.5)

44


A2 Fig. 2.1.

If u is a C2-solution relative to L and fl, which means that u E C2(f2) and Lu = 0 in fl, and condition (a) holds, then (2.4) and (2.5) imply that

minan u < u(x) < maxan u+

for all x E S2.

(2.6)

Similarly, all our results for subsolutions have implications for supersolutions and solutions. Remark 2.8 The Dirichlet problem for L in a bounded set S2 is to find v such that

Lv = f in n, v lasz = g,

v E C(S2) n C2(12),

(2.7)

where f and g are given functions. This problem has at most one solution, because the difference u := Vt - v2 of two solutions satisfies Lu = 0 in fl,


45

u = 0 on M, and has the smoothness required for (2.6); therefore u = 0 on S2.

Remark 2.9 (i) The condition c < 0 in S2 (Definition 2.3) cannot be omitted from Theorem 2.6. Once again this is illustrated by eigenfunctions of the Laplace operator. For example, let S2 be the rectangle (0, a) x (0, f3) in R2, and let

u(x) = sin

m7rx1

a

sin

n 7Ex2,

(2.8)

m, n E N.

fl

Calculating L u, we see that

Au + cu = 0 in

2

92,

where c = (m) +IQI a

2

\ /

> 0,

and, in contrast to (2.4), maxi u = 1 while maxan u+ = 0. (ii) We cannot replace u+ by u in (2.4). [As was noted earlier, this would

give maxii u = maxan u] For, let S2 be the unit ball M(0,1) in RN, let L = A - 1, and let u(x) = -3N - Ix12 on S2. Then Au = -2N, so that

Lu(x) = Du(x) - u(x) = N +x12 > 0 in S2, and

maxi u = -3N > -3N - 1 = maxan u. The requirement in Definition 2.4 that subsolutions be in C2(S2) can cause embarrassment. For example, the Newtonian potential of constant

density in a bounded open set G is not twice differentiable at points of 8G; when 8G is unknown a priori and may be unpleasant, a need to consider second derivatives of the potential would be a source of difficulty. We now define subsolutions for which membership of C'(Q)

is ample smoothness. However, we do this only for the operator LI, because a proof of something like Theorem 2.11 for an operator with variable coefficients requires (I believe) considerably more machinery. Definition 2.10 We shall say (a) that u is a generalized subsolution relative to LI and S if u E C 1(S2) and N

N

i,1=1

i=1

{_aii()(3iu)+biiu+cu in

AI((P,u;fl) >_

0 whenever p E Cc(K2) and tp >_ 0;

46


(b) that u is a distributional subsolution relative to LI and S2 if u is locally integrable in n (integrable on each compact subset of S2) and N

N

u - E b;

at; in E i>;=1

Ala((P, u; n)

z

u + ccpu

j=1

0 whenever ( E C,°(L) and qp >- 0.

Then u is a generalized supersolution if -u is a generalized subsolution;

u is a generalized solution if it is both a generalized subsolution and a generalized supersolution (cf. Remark 2.7). Distributional supersolutions and distributional solutions are defined similarly.

Evidently the key to this definition is integration by parts: A,((p,u;S2) =

Jin

cpL1u if p E C,(1) and u E C2(S2);

(2.9)

AIAp,u;S2) = AI(cp,u;0) if qo E CC°(S2) and u E C'(S2).

(2.10)

Since a C2-subsolution u satisfies LIu > 0 in 92, we see from (2.9) that a C2-subsolution (relative to L1 and fl) is a generalized subsolution, and from (2.10) that a generalized subsolution is a distributional subsolution. On the other hand, a distributional subsolution is a generalized subsolution only

if it is also in C'(fl), and a generalized subsolution is a C2-subsolution only if it is also in C2(S2). [In this last case, we use (2.9) and Exercise 1.16 to deduce that L1u > 0 in Q.] Note that, in the following theorem, hypothesis (a) swamps the condition of local integrability demanded in Definition 2.10, (b). Theorem 2.11 (the weak maximum principle for LI). Suppose that (a) S2 is bounded, u E C(S2);

(b) u is a distributional subsolution relative to LI and 92. Then the previous conclusions hold:

if c = 0,

(2.11 a)

maxii u < maxan u+ if c < 0.

(2.11b)

maxjj u = maxan u

Proof (i) Let an arbitrary point theorem by showing that

r maxanu maxan u+

E 0 be given; we shall prove the

if c = 0, if c < 0.

(2.12a) (2.12b)


47

Adopting a standard trick, we choose the following test function 9 in the definition of distributional subsolution. (p(y) = kp(x - y) for all y E K2,

(2.13a)

where kp is a smoothing kernel as in Exercise 1.23; p and x are parameters satisfying

0 2p}, (2.13c)

x E G(p),

as is illustrated in Figure 2.2. This choice of cp is legitimate because kp(x - y) = 0 when Iy - xj >- p, so that supp kp(x -.) = 0, and certainly kp(x -.) is infinitely differentiable and non-negative in Q. (ii) Now let up(x) =

kp(x - y) u(y) dy, x E G(p),

(2.14)

J

where, equally well, the integral could be written as one over P'(x, p). Then up E C00(G(p)) by Exercise 1.23; the present boundary 30 plays no

part when x E G(p). The definition of distributional subsolution states that 0

0, kp(x - y) u(y) dy

vP(x) mx,P)

f = M+E,
0 and for all x E aG(p), if 3p < 5g; again

50


Fig. 2.3.

both sides of (2.16b) are zero. If M > 0, then maxan u+ = M, and (2.19) implies (2.16b) once more. 2.3 The boundary-point lemma and the strong maximum principle Lemma 2.12 (the boundary-point lemma for balls). Suppose that

(a)Bcz0isaball, uEC(B); (b) u is a C2-subsolution relative to Lo or L and B, or a distributional subsolution relative to Ll and B; (c) u(x) < u(p) for all x E B and some p E 3B, with u(p) >- 0 when the coefficient c is not the zero function. Let m be an outward unit vector at p (m n > 0 and Imi = 1, where n denotes the outward unit normal to aB at p). Then lim in ftlo

u(p) - u p - tm) > 0 ,

l

( 2 . 20)

which implies that

a m (P) :

= li mto

u(p) - p - tm) > 0

( 2 . 21 )

t

whenever this one-sided directional derivative exists.

Proof (i) As in Figure 2.3, let B =: R(q, p) and A := R(q, p) \ .4(q, 2 p); it will suffice to consider the annular set A. Also, let M := u(p) = SUPB u.

2.3 The boundary-point lemma and the strong maximum principle 51

If we can find a function v E C2(A) such that v(p) = 0,

(I)

u+v < M on A,

(III)

then we can prove (2.20) as follows. Let w := u + v. For 0 < t < 2p,

w(p) - w(p - tm) = M - w(p - tm) t

t

>0

[by (I) and (III)],

whence lim inftlo

u(p) - u(p - tm) t

= lim infrl0

> liminftl0 8m

{w(p) - w(p - tm)} - {v(p) - v(p - tm)} t

-v(p) + v(p - tm) t

(p) > 0

[by (II) ]

(ii) Consider the function defined on A by

v(x) := 6 (e Kr2 - e Kp2) ,

r := Ix - qI,

and shown in Figure 2.4; both positive constants S and K are still to be chosen.

Certainly v E C2(A); also (I) and (II) hold, since

0. Choose S = a; then u < M - a and v < a for r = 2p. Accordingly, maxaA (u + v) = M.

(iii) If we can choose K so that Lv >- 0 in A (hence so that L0v >- 0

in A, or Liv > 0 in A), then condition (III) will follow from one of


52

<S

.r

Fig. 2.4.

our three versions of the weak maximum principle, applied to u + v and A. The condition u(p) >- 0 when c # 0 banishes one difference in these versions. To deal with a distributional subsolution u, we add to the given condition,

Ald(cp, u; A) > 0 whenever (p e C°(A) and cp > 0, the condition Ald((p,v;A) >- 0 for the same gyp; this property of v will be implied by integration by parts [as in (2.9) and (2.10)] once we have Llv > 0 in A. Then, with u+v E C(A) and Ald(cp,u+v;A) >- 0, condition (III) will follow from Theorem 2.11. (iv) It remains to calculate Lv and choose K. Since 2

ai e -Kr

aiaj eKr2

= e -Kr' (-K)2(xj - q1), = eKr2 {4K2(x, - gi)(xi - qi) - 2KS,i}

we have

Lo(x) = exr2

4K2 E affi(x) (xi - gi)(x1 - qj) i,)

-2K Eaffi(x)-2K i

i

+ C(x) (eKr2 - eKP2 }

bi(x) (xi - qj)

2.3 The boundary-point lemma and the strong maximum principle 53 By the condition (2.3) of uniform ellipticity, I Lv(x)

>

4K2Aor2 - 2K supA

e-Kr2

I aiiI + I bl p I - SUPA ICI

i

>0

for xEA

if we choose K sufficiently large, because r2 > (2 p)2.

Note a change of direction in the statement of the next theorem: there is no mention of 52 or of 852.

Theorem 2.13 (the strong maximum principle). Suppose that (a) 12 is a region (open and connected, possibly unbounded); (b) u is a C2-subsolution relative to Lo or L and 52, or a generalized subsolution relative to Ll and 52; (c) supn u >- 0 when the coefficient c is not the zero function. Under these hypotheses, if supn u is attained at a point of 0, then u is constant in 52.

Proof Let M := supn u, and assume that this supremum is attained at z E Q. Define

F := {x E 52

I

u(x) = M),

G := {x E 52

I

U(X) < M} ;

then F is closed in the metric space 52, and not empty because z E F; the set G is open in the metric space Q. If G is empty, the theorem is true. Suppose then that there is a point xo E G. We shall obtain a contradiction by means of Lemma 2.12, first using the result that, because 52 is open and connected in RN, it is pathwise connected (Burkill & Burkill 1970, p.44; Cartan 1971, p.42). This implies existence of a continuous arc

y:={fi(t) I 0 0.

But since p E F, it is an interior maximum point of u E C'(0). Hence (Vu)(p) = 0 and we have our contradiction.

There are many boundary-point lemmas for elliptic operators and sets

other than balls, but Lemma 2.12 is probably the heart of the matter. Theorem 2.15 is a consequence of that lemma, seasoned by a touch of the strong maximum principle. First, we need a definition.

Definition 2.14 A set fl has the interior-ball property at a point p E as2 if there exists a ball B0 c ) such that p E aBo; it has the exterior-ball property at p if there exists a ball B1 c RN j j such that p E aB1.

Figure 2.6 shows two cases of the interior-ball property for 92, and therefore two cases of the exterior-ball property for RN \ S2. Note that a unit vector m at p, outward from an interior ball B0, need not be outward from S2.

Theorem 2.15 (a boundary-point theorem for fl). Suppose that (a) 0 is a region; (b) u is a C2-subsolution relative to Lo or L and S2, or a generalized subsolution relative to L, and S2;

2.3 The boundary-point lemma and the strong maximum principle 55

aS2

Fig. 2.6.

(c) there is a point p E 00 such that u E C(f) U {p}) and u(p) = supu u, with u(p) > 0 when the coefficient c is not the zero function; (d) ) has the interior-ball property at p. Let m be a unit vector at p, outward from an interior ball Bo at p. Then either liminftlo

u(p) - u(p - tm) > 0 t

(2.22)

(which implies that (au/am)(p) > 0 whenever this derivative exists), or u is constant in Q.

Proof Let xo be the centre of the ball Bo and let po :_ lp - xol, so that Bo = P4(xo, po). Now consider the smaller ball B := M(q, po) with z hence q := 1(p + xo). Since B c Bo U {p}, we have B c C1 U {p} and u E C(B).

If u(x) < u(p) for all x E B, then Lemma 2.12 implies (2.22). If u(z) = u(p) for some z E B, then u(z) = supu u and the strong maximum principle implies that u is constant in n.

Suppose that p is what may be called an edge point ; for example, f2 = (0,1)2 =R2 and p= (0, 0), or S = (0,1)3 c 1[83 and p = (1, 0, 0). Then S2 lacks the interior-ball property at p, but something can still be said, for a subsolution, about an outward derivative or difference quotient at p. This is the subject of Appendix E.

Remark 2.16 (on the condition c < 0 in S2). For a subsolution u, if supu u = 0 in an application of the weak or strong maximum principle, or if SUPB u = 0 in an application of the boundary-point lemma for a ball B, then the condition c < 0 in S2 (imposed in Definition 2.3) can be omitted.

56


Proof We use the decomposition c(x) = c+(x)+c (x) [defined in Chapter 0, (v)]. The foregoing theorems and lemma are valid for the operator Land bilinear form Ai defined by

L-u := Lu - c+(x)u (= Lou + c-(x)u Al (cp, u; S2) := Al (gyp, u; S2) - c+

Jn

cpu

(c+ = c > 0).

When supn u = 0, we can use L- in place of L because Lu 0 and u < 0 imply that L-u >- 0. Again, when supn u = 0, we can use Al in place of Al because A, (gyp, u; S2) >- 0, cp >_ 0 and u < 0 imply that Al (gyp, u; S2) >- 0.

2.4 A maximum principle for thin sets f

All our maximum principles so far have required that the coefficient c(x) < 0 for all x E S2, unless it happens to be known for a subsolution u that supn u = 0 (Remark 2.16), or for a supersolution v that infn v = 0. In this section we proceed to a weak maximum principle for thin sets f in which both c(x) and u(x) are unrestricted in sign a priori. By a thin set f) we mean one of specified diameter and small volume: 1921 < 6, where the positive number 6 depends only on diam S2 and on constants

independent of n. To derive this maximum principle, we need some form of the basic estimate for elliptic equations that is presented here as Theorem 2.18. This estimate, in turn, is a consequence of elementary

results in Appendix A for the Newtonian potential and of the weak maximum principle in Theorem 2.11. Given a bounded open subset G of RN, we define a modified Newtonian kernel K by

-ZIxj +1diamG 1

2 log

diam G Ixl

Ix11

KN

-2

ifN=1, ifN = 2,

(2.23)

ifN > 3,

where x 0 if N >- 2 and where KN is as in (A.18b) of Appendix A. This function differs from the Newtonian kernel K introduced by (A.18) only for N = 1 or 2, and then only by the addition of a constant which ensures that K(xo - x) >_ 0 whenever xo,x E G. The corresponding

2.4 A maximum principle for thin sets S)

57

modified Newtonian potential of a suitable density function g : G - R is defined by v(xo) :=

Jc

K(xo - x) g(x) dx,

xo E RN,

(2.24)

but here we restrict attention to field points xo E G.

Lemma 2.17 Let G be a bounded open subset of RN and let v be the modified Newtonian potential defined by (2.23) and (2.24). If g E Lp(G)

with 1 - 2, then v E C(G) and v is a distributional solution (Definition A.7) of -A v = g in G. Moreover, Iv(xo)I < I(N, p) (diam G)2-(N/P)

for all xo E G,

II g I Lp(G) II

(2.25)

where, with the notation 1/p + 1/q = 1 and ?N := I0-IN(0,1)1, 1/q

2F

(I'

1

for N > 3,

(log p F(N, p) =

1/q

q

1

t(2, p) _

f or 1_ 0

j=1

i,j=1

whenever (p E Cc(Q) and (p > 0.

(2.27)

Theorem 2.18 (a basic estimate for the operator L1). Suppose that (a) 0 is bounded, u E C(ST); (b) L1u + f >_ 0 in fl in the distributional sense, where f E Lp(S2) with

10

on

(2.32b)

by Lemma 2.17 and because k (z - l; ) > 0 and (f) (o >- 0. Let w

v; then (2.31a), (2.31b) and (2.32a), (2.32b) imply that

0 in G in the d.s.,

(L - k2) w >- k2v WIaG - -g(x)u- >- 0 in 12 in the d.s., to recall that ul an < 0, and to apply Theorem 2.18 with f = gu+; the choice p = N is admissible for all N E N. This yields maxi u < A II gu+ I LN(G) 11 < A II g I L.(n) II maxj u+ 15211/N.

(2.36)

If maxi u < 0, then (2.34) holds. If maxis u >- 0, then (2.36) and our bound for 11 g I L,,,(52)11 imply that

(maxi u+) { 1- 2A 11 Y

I

L.(K2) 11 InI IIN } I

01.

(b) In the rest of this chapter, the dimension N > 2. (c) The condition u c C(S2) will be relaxed at only one or two boundary points; typically to u E C(S2 \ {p}), where p is a specified point of M.

(d) Only the Laplace operator A will be considered. There is no disgrace in this restriction; good answers to our question are sensitive to details of the differential operator L, and each proof involves a comparison function tailored rather closely to the task in hand. To launch here into the more general theory initiated by Gilbarg (1952) and E. Hopf (1952a) would be a catastrophic attempt to run before we have learned to walk.


62

Definition 2.20 We shall say that u is subharmonic in C1 if it is a distribu-

tional subsolution relative to A and fl; that is, if u is locally integrable in 0 and

r, (L p)u >- 0

whenever

(p e Cc(f2)

and

(p >- 0.

Then u is superharmonic in S2 if -u is subharmonic there; u is harmonic in 52 if it is both subharmonic and superharmonic in S2. This definition of `harmonic' (which follows inevitably from the useful definition of `subharmonic' that we have adopted) scarcely does justice to harmonic functions. Theorems B.6 and B.10 show that, if u is harmonic

in 92 according to Definition 2.20, then, after re-definition on a set of

measure zero, not only is u a C2-solution of Au = 0, but also u is real-analytic in 92.

Our opening question can now be replaced by the following. If u is continuous on b and subharmonic in D, to what rate of growth, as JxJ -+ oo, must u(x) be restricted in order that supD u = supaD u? If u is continuous merely on S2 \ {p} and subharmonic in S2, to what rate of growth, as x -> p, must u(x) be restricted in order that supn u = supan\{P} u?

We begin by inspecting some simple and explicit harmonic functions

that vanish on the boundary of the half-space D or on a punctured boundary 8S2 \ {p}; these functions indicate rates of growth that are too large in the context of our questions. Examples 1. The harmonic polynomials Pl(x) = xN, p2(x) = 2x1xx, p3(x) = 3xixN - $N,...,

pm(x) = Im (xl + ixN)m,...

(2.37)

all vanish on 8D; if N > 3, there are many more such polynomials. But, apart from the zero function, no function springs to mind that

is continuous on D, is harmonic in D, vanishes on 8D and is o(r) as r := JxI -> oo. This is significant: the critical rate of growth for the result supD u = supaD u will turn out to be close to growth like r as r -+ co.

2. If we seek functions that are continuous on D \ {0}, tend to zero at infinity, are harmonic in D and vanish on OD \ {0}, then the prototype is ql(x) = xN /rN

,

x E RN \ {0},

r := IxI.

(2.38)

This is the potential of a particular dipole (§A.4); more precisely, the

2.5 Steps towards Phragmen-Lindelof theory

63

potential of a multipole of type (0, ... , 0, 1). Differentiating this repeatedly

in horizontal directions (with respect to xj, j < N - 1), we generate multipole potentials like

q2(x) = algl(x) = -NxlxNr N-2 02g1(x) = NXN{ (N+2)x2 -r2 }r-N-4, q3(x) =

(2.39)

which retain the properties listed before (2.38), but have a stronger singularity at the origin, relative to ql, and a more rapid decay at infinity. However, no non-trivial function springs to mind that has the properties listed before (2.38) and is o(r-N+1) as x -> 0 with x E D. Again this is significant: the critical rate of growth for the result supra u = supan\{P} u will turn out to be close to growth like Ix - p1-N+1 as x --> p, when 892 is smooth.

3. We now allow a singularity at the south pole p := (0, .... 0, -a) of the ball B := _4(0, a) in RN. The Poisson kernel (§B.5) gives an example of a function continuous on {p}, harmonic in B and equal to zero on 8B \ {p}: P(x, p) = const. (a2 - r2)Ix - pI-N, x E B \ {p}, r := 1x4.

(2.40)

Again the singularity at p is of dipole type, and again appropriate differentiation generates a stronger singularity at p, while conserving the value zero on 8B \ {p}. Thus the harmonic function

Q(x) = (xNa1 -x1aN)P(x,p) = const. xl(a2 - r2)Ix - pl-N-2,

x E W\ {p},

(2.41)

has a quadrupole singularity at p. Rather as in Example 2, no non-trivial function springs to mind that is continuous on B \ {p}, harmonic in B, equal to zero on 8B \ {p} and is o(Ix - pj-N+1) as x -+ p; this is significant in the same way as before. The Phragmen-Lindelof theory that follows must be distinguished from Phragmen-Lindelof theory for holomorphic functions (complex-analytic functions). In that theory one supposes that, for example, supaD l u + iv I is known, where D is the upper- half of the complex plane C; the analogous situation for us, when N = 2, is that only supaD Iul (or only supaD lvi) is known. In the case of holomorphic functions (Hille 1973, Chapter 18; Titchmarsh 1932, §§5.6-5.8) much more can be inferred because much more is given. Definition 2.21 Let D := { x E RN I XN > 0 }, N >- 2; let Da = Dn-4(0, a)


64

Fig. 2.7.

and l7a := D n 8-4(0, a) (Figure 2.7); denote the equator of R(0, a) by E := 8D n 0-4(0, a). A function with values V (x, a) will be called a comparison function of

the first kind if (a) for each a E (0, oo),

V(., a) E C(Da \ E) n C2(Da)

and

(AV)(.,a) = 0 in Da, V (. ,

a) > 0

on ra;

V(., a) >_ 0 on Da \ E, (2.42) (2.43) (2.44)

(b)inf{V(x,a) I xEfa}->ooas a function A : D --+ (0, oo) such that V (xo, a) < A(xo) whenever xo E D and a >_ 21xol. E

Functions having these properties will be displayed in due course; first,

we prove the lemma for which the definition has been designed. This lemma shows that, if a function u is continuous on D, is subharmonic in D and is smaller, in order of magnitude, than V (. , a(n)) on some sequence (raw) of hemispheres marching to infinity, then we retain the result SUPD U = supaD u.


65

The proof will show that the important case of the growth condition (2.45) is that in which the limit inferior equals zero. Lemma 2.22 Let V be a comparison function of the first kind. If u E C(D), if u is subharmonic in D and if

lim info-,, sup

u(x)

xEra }50,

V (x, a)

(2.45)

then

SUPD U = SupaD u.

Proof (i) We may suppose that supaD u < oo, otherwise the result is trivial. Let u := u - supaD u. Then supaD u = 0 and Ii also satisfies the growth condition (2.45), because the definition of u and hypothesis (b) imply that sup

Iu(x) - u(xI

V (x a)

xEra }-->0 asa --goo.

Let both xo E D and s > 0 be given; we shall prove the lemma by showing that u(xo) < s.

(ii) If lim info-ao sup { u(x)

I

X E IF,, } < 0,

then no comparison function is needed. For, there is a sequence (a(n)) tending to infinity for which the supremum of u(x) over ra(n) tends to a non-positive limit. We choose a(k) so large that xo E Da(k) and so large that u(x) < s on ra(k). Then the weak maximum principle, Theorem D.11, applied to u on Da(k) shows that u(xo) < c as desired. (iii) It remains to consider the following case: there is a number A >- 0 such that

sup {u(x)

I

X E ra } > 0 whenever a > A,

and

lim infa-,,, sup

u(x) V (x, a)

= 0.

X E ra T

Let

s(a) := sup

u(x)

I

I

V(x,a)

l x E ra } for a > A; JJ

(2.46)


66

XN

D

Fig. 2.8.

then s(a) > 0. We so choose a that, at the given point x0, s(a) V(xo,a) < 1E;

(2.47)

this can be done because (2.46) states that there is a sequence (a(n)) for which s(a(n)) -- 0 as n -+ oo and a(n) -> oo, while hypothesis (c) ensures that V (xo, a(n)) < ..(xo) whenever a(n) > 21xoI. With a now fixed at this value, define cp(x) Da,a

for x E Da \ E, {x E D I IxI < a, dist(x,E) > 6 > 0}

u(x) - s(a) V (x, a)

(see Figure 2.8). Choose 6 so small that xo E Da,g and so small that

dist(x,E) = 6 and x E

u(x) < ZE;

this choice is possible because uI E < 0 and u E C(D).

(iv) Finally, we apply the weak maximum principle (Theorem 2.11) to pp on Da,g. Certainly cp E C(Da,g), and qo is subharmonic in Da,g, because

u is subharmonic there and V(., a) is harmonic. The boundary values of q' are as follows. On BDa,b n 8D we have u(x) < 0 and V (x, a) >_ 0, hence ep(x) < 0. On the part of ODa,g distant 6 from E we have u(x) < e and V (x, a) >- 0, hence cp(x) < E. On

i

z


67

3Da,b fl I'a we have cp(x) < 0 by the definition of s(a):

x E Fa

T(x) = V (x, a) V (xx a) - supYEra V (y, a)

0. JJJ

Therefore the weak maximum principle implies that cp(x) < 1c on Da,b; it follows from (2.47) that u(xo) = cp(xo) + s(a)V(xo, a) < s,

as desired.

The next item is a naive application of Lemma 2.22, based on a simple comparison function and intended to make Definition 2.21 less mysterious. In this example, V is independent of a, and does not have a discontinuity on the equator E. The full force of Lemma 2.22 will emerge only in §2.7, after more elaborate comparison functions have been constructed. Example 2.23 Let D :_ { x E R2

I

x2 > 0 }. If u E C(D), if u is subhar-

monic in D and if, for some constant /3 E (0, 1),

lim info-,,, sup { a-#u(x)

I

x E Fa } oo), then

SUPD U = supoD u.

Proof Denote points of D by x = (r cos 0, r sin 0), 0 < 0 < it. We claim that the formula k := (1 - /3) 2,

V(x) := r' sin(fl0 + k),

xE

defines a comparison function of the first kind. For, referring to Definition 2.21, we observe that V E C(D) n C2(D); that V >_ 0 on D, with V > 0 on {0}, because

sin(/30 + k) > sink

for 0< O< ir;

and that AV = 0 in D because V(x) = Ime'kzp

(z := x1 +ix2 = re'°).

Thus V satisfies condition (a) of Definition 2.21; it satisfies (b) because V(x) > a' sink when x E Ta; for (c), we may choose 2(xo) := V(xo) or 2(xo) := ro.


68

G G a

F

p

F

Fig. 2.9.

If the growth condition (2.48) implies (2.45) for the present function V, then Lemma 2.22 implies the present result. Now,

sink
2, let p E 8G be given, let Ga := G\R(p, a), let ya := Gfla (p, a) and let F := 8GnOR(p, a).

Here a E (0, ao) and ao is a positive constant depending only on G. (Three possible configurations are shown in Figure 2.9; in the third, G :_ R(p, ao) \ {p} and F is empty.) A function with values W (x, a) will be called a comparison function of the second kind if (a) for each a E (0, ao), W(., a) E C(Ga \ F) f1 C2(Ga)

and

W(., a) >_ 0

(AW)(. , a) = 0 in Ga, W(., a) > 0

on ya;

on Ga \ F,

(2.49)

(2.50) (2.51)

2.5 Steps towards Phragmen-Lindelbf theory

69

(b) inf{ W(x,a) I X E ya } - oo as a - 0; (c) there is a function A : G -> (0, oo) such that W (xo, a) < A(xo) whenever xo E G, a < 1 Ixo - pI and a < ao. Lemma 2.25 Let G be as in Definition 2.24 and let W be a comparison function of the second kind. Let S2 be a bounded open subset of G such that p E A2 n 8G. If U E C(S2 \ {p}), if u is subharmonic in S and if

lim infa.o sup

u(x) 1 W (x, a)

x E S2 n 8g(p, a) } < 0,

(2.52)

then

supn U = supan\{P} U.

Proof The proof resembles that of Lemma 2.22, but to condense it ruthlessly would be a false economy. (i) We may suppose that supao\{p} u < oo, otherwise the result is trivial. Let u := u - supa0\{P} u. Then supan\{p}u = 0 and u also satisfies (2.52),

because u - u is a (finite) constant and by condition (b) in Definition 2.24.

Let both xo E S and s > 0 be given; we shall prove the lemma by showing that u(xo) < E. To this end, write

fl,, := S2 \ V(p, a) and as := S2 n 8R(p, a). (ii) If

lim infa.o sup {u(x) I x E as } 0 instead of large hemispheres Fa(n) with a(n) -* oo.

(iii) It remains to consider the following case: there is a number a > 0 such that sup { u(x) I X E as } > 0 whenever a < a, and

lim infa.o sup

u(x) W (x, a)

Let r

s(a) := sup {

u(x)

l W(x,a)

x E as } = 0.

l x E as } J

for a < a;

(2.53)


70

an

Fig. 2.10.

then s(a) > 0. Choose a to be such that, at the given point xo, s(a) W (xo, a) < 1 e;

(2.54)

this can be done because of (2.53) and hypothesis (c) in Definition 2.24. With a now fixed at this value, define ip(x) := i(x) - s(a) W (x, a)

for x E S2a \ F.

If 8S2 intersects F, define Qa,b

{ x E 0 I Ix - pI > a, dist(x, F) > S > 0 }

(Figure 2.10); choose 6 so small that xo E S2a,b and so small that

dist(x, F) = S and x E S2

u(x) < 18;

this choice is possible because ul ffinF 0 and u E C (S2 \ {p}). If 8S2 does not intersect F (in particular, if F is empty), define S2a,b := Sta. In either case, W(. , a) is continuous on S2a,b and u(x) 0, hence ip(x) < Zs. On


71

af2a,b n as we have ip(x) < 0 by the definition of s(a). The maximum principle ensures that p(x) < E on a,b ; it follows from (2.54) that z

u(xo) = p(xo) + s(a) W (xo, a) < s, as desired.

O

Like Example 2.23, our first application of Lemma 2.25 will involve only simple comparison functions. But the result is better than that of Example 2.23; it is not restricted to R2 and it is best possible in a certain sense (Exercise 2.44) when no smoothness is demanded of 8S2 at p.

Theorem 2.26 Let SZ be bounded in RN, N > 2, let p E % and write as := 92 n 8-4(p, a). Let b be so large that f2 c R(p, b). If u E C(92 \ {p}), if u is subharmonic in S2 and if lim infa.o sup S

log(b/a)

lim info-o sup { aN-2u(x)

I

x E as } < 0 when N=2,

(2.55)

x E as } < 0

(2.56)

when N >_ 3,

then

supn u = supan\{P} U.

Proof The comparison functions are potentials of point sources (multiples of Newtonian kernels), discussed at some length in Appendix A.

(i) For N = 2, we choose G := .4(p, b) \ {p} for the set in Definition 2.24 and define

W(x) := log

x

for x E 4(p,b) \ {p}.

b pl

Then Ga = l(p, b) \ R(p, a) and ya = 0.4(p, a); condition (a) of Definition 2.24 is satisfied. Since W(x) = log(b/a) when x E ya, condition (b) holds. For (c), we choose , (xo) := W(xo). Thus W is a comparison function of the second kind. The growth conditions (2.52) and (2.55) coincide for this function W ; therefore Lemma 2.25 implies the present result. (ii) For N >_ 3, we choose G := RN \ {p} for the set in Definition 2.24 and define W(x) := Ix _ pl-N+2

for x E RN \ {p}.

One checks without difficulty, very much as in (i), that this function W is a comparison function of the second kind. The growth conditions (2.52)

72


and (2.56) coincide for this W ; again Lemma 2.25 implies the present result.

2.6 Comparison functions of Siegel type

This section concerns functions g(. ; a), ge(. ; a) and $2(. ; a, b) with the property that ag(. ; a) is a useful comparison function of the first kind, while a N+1ge(. ; a) and a N+l$2(. ; a, b) are corresponding comparison functions of the second kind; ge(. ; a) and $2(. ; a, b) are defined on different domains. These functions will enable us to extend Example 2.23 to half-spaces in RN for all N > 2; to improve the rate of growth allowed in Example 2.23 from approximately o(rfl), where r := Ixl and # < 1, to approximately o(r2/xN) as r -* oo; and to improve the rate of growth allowed in Theorem 2.26 from approximately o(Ix - pl-N+2), for N >- 3

and x - p, to something slightly bigger than o(Ix - pl-N+i), provided that S has the exterior-ball property at p (Definition 2.14). The functions g, ge and 92 will be called of Siegel type because g for N = 2, displayed here in (2.59), was introduced into Phragmen-Lindelof theory by D. Siegel (1988). The construction of g for all N >- 2, from the Poisson integral formula for functions harmonic in a ball (§B.5), is the subject of Appendix C. The functions ge and $2 result from applications to g of the Kelvin transformation (§B.3). It will be convenient to use the signum function, defined by

-1 sgn t :=

0 1

if if if

t < 0, t = 0,

(2.57)

t>0.

Theorem 2.27 Let B := R(0, a) in RN, N > 2, and let E

x E aB

xN = 0 } denote the equator of B. (a) There exists a function g = g(. ; a), which we call the primary function of Siegel type, such that g E C(B \ E) fl C0O(B) and

Ag = 0

in B, g(x) = a/xN on aB \ E, g(x)I < const. IxNI/a if r := Ixl < a/2, where the constant depends only on N. Also, sgn g(x) = sgn xN on and g(x; a) depends only on x/a.

(2.58a) (2.58b) (2.58c)

E,


73

(b) For N = 2, let (x, y) E R2 and z = x + iy E C. Then, on B \ E c R2, g(x, y; a) = Im (a

_

a z

ay

a

a+z

ay

(a - x)2 + y2 +(a + x)2 + y2

.

(2.59)

Proof See Appendix C.

Corollary 2.28 The exterior function ge = ge(. ; a) of Siegel type is defined by

ge(x; a) :=

(a) r

N-2

a2

g I r2 x; a I

,

x E l[8N \ (B U E),

(2.60)

where again r := Ixi and B, E are as in Theorem 2.27. It follows that ge E C (RN \ {B U E}) n C°°(RN \ B) and that

Age = 0

in RN \ B, ge(x) = a/xN on 8B \ E, Ige(x)I < const.aN-lr-NIxNI if r > 2a,

(2.61a)

(2.61b) (2.61c)

where the constant depends only on N. Also, sgn ge(x) = sgn xN on RN \ (B U E), and ge(x; a) depends only on x/a.

Proof A calculation, done fully in Theorem B.15, shows that (a)N+2(Ag)

(_x) a2= 0

Age(x) _ if r > a and hence Ia2x/r2I = a2/r < a. The remaining properties of Se are immediate consequences of the definition (2.60) and the corresponding properties of g.

Inspection of Corollary 2.28 and Definition 2.24 shows that a N+lgei restricted to D \ (B U E) (where D is our usual half-space), is a comparison

function of the second kind, with G = D and p = 0 in the notation of Definition 2.24. The restriction to D \ (B U E) is needed in order that ge >- 0. Therefore aN+lge can be used in Lemma 2.25 for sets 92 that are on one side of a hyperplane containing the point p of 8S2 at which u may be discontinuous; this is illustrated in Figure 2.11. In particular, a-N+lge can be used for convex sets S2. Suppose now that S2 has merely the exterior-ball property (Definition 2.14) at the specified point p E 852. Then a suitable comparison function

a N+1g2(. ; a, b) is found by inversion relative to a sphere as follows.


74

XN

Fig. 2.11.

Choose co-ordinates so that p and an exterior ball Bq at p are given by

p = (0,...,0,2b), q = (0,...,0,b), Bq =-V(q,b)

(2.62)

for some b > 0, as shown in the right half of Figure 2.12. Define

V := {

E RN N < 2b I, VQ := V \ -4(p, a) with 0 < a < b;

we shall use ge(p - ; a) for l; E Va \ E, where E now denotes the equator of R(p, a). Under the transformation 4b2

equivalently

x=b

x

(r.=lx1>0), (2.63)

2

(p := I c I > 0),

which is inversion relative to the sphere 0-4(0,2b),


75

Fig. 2.12.


76

the punctured half-space V \ {0} has image G := RN \ Bq ; (2.64a) the ball R(p, a) has image B* := -4(p*, a* ), where

1

P*

1 - (a/2b)2 p'

1- (a/2b)2

a*

a

the equator E has image F := 8G n 8B*.

(2.64b) (2.64c)

Note that p is a fixed point of the map (2.63) and that, near p, this transformation is approximately reflection in the hyperplane V. Therefore 8B* is close to 7_4(p, a) for small radii a. A more precise statement is that 4b 2

Ix-p12 = 727 b ( xN -PN + 2b

zzN

PN - b

\

I .

(2.65)

Here the factor 4b2/r2 will be unimportant when $2 comes to be used in Theorem 2.36 [because 4b2/r2 -_+ 1 as x -> p], but we shall need Z(x) := XN - PN + I

x-

2

Zb

I

,

(2.66)

which is almost dist(x, OG) for points x near OG. In fact, a calculation shows that 1 z(x) < dist(x, 8G) z Iac

= 0, 1- 2

b -

0 and any a E (0, b). Then the two-ball function $2 = $2( ; a, b) of Siegel type is defined by

(2b)N $2 (x; a, b)

2

4b ge

(p-

x; a)

,

x E G \ (B* U F U {0})

////

and by $2(0)

(2.68)

limr1o 92(x) = 0. It follows that $2 E C (G \ {B* U F}) n

Cc0(G \ B *) and that

A92=0 in G \ B*, $2(x) = C 2b r / N-4 z(x) a

on G n 8B*,

(2.69a)

(2.69b)

2.7 Some Phragmen-Lindelof theory for subharmonic functions

if x E G \ 4(p', 2a'), where p' := then 1$2()I - 2Ix0I,

where the constant depends only on N. Therefore the present theorem is implied by Lemma 2.22. The supremum over iTa in (2.45) of that lemma can now be written as a maximum over ra because the function with values XN u(x)/a2 is continuous on D; the maximum cannot be negative, because of values for XN = 0.


78

Corollary 2.31 Let G be an unbounded open subset of the half-space D. If u E C(G), if u is subharmonic in G and if

lim infa, max

xN u(x) a2

x E G,

Ix! = a

< 0,

(2.71)

then

SUPG U = SUNG U.

Proof In the proof of Lemma 2.22 we replace D by G, keeping the same comparison function V of the first kind. Thus Da is replaced by GQ := G n -4(0, a) and Fa now means G n 8-4(0, a). We define s(a) and choose the radius a exactly as before. If 8G does not intersect the equator E, we need not remove a neighbourood of E from GQ. The shape of 8G is unimportant; what matters is that u(x) < 0 on 8G by the definition of u, and that cp(x) < 0 on I'a by the definition of s(a). After this extension of Lemma 2.22, the present corollary results from

the choice V = ag made in the proof of Theorem 2.30. The maximum in the growth condition (2.71) may be negative once more, because XN need not descend to zero when x E G and IxI = a. Remark 2.32 (i) If we add to Corollary 2.31 the hypotheses: G is connected and u c C1(G), then

u(x) < supaG u for all x E G,

unless u is constant on This is an immmediate consequence of the strong maximum principle (Theorem 2.13). (ii) Let G be as in Corollary 2.31. If u E C(G), if u is superharmonic in G and if

lim supa,,, min {

XN u(x) a2

xEG, Ixl=a }>_0,

(2.72)

then

infG u = infaG u.

This follows from Corollary 2.31 by an argument like that in Remark 2.7.

(iii) For a function u that is continuous on and harmonic in G (hence is in C00(G), by Theorem B.6), we wish to conclude that infaG u < u(x) < supaG u for all x E G

2.7 Some Phragmen-Lindelof theory for subharmonic functions

79

(with strict inequality for x E G if G is connected and u is not a constant). It may be worthwhile to retain both (2.71) and (2.72) as hypotheses, but the simpler condition r l liminfa, max { xNI (x) x E G, lxi = a 0 (2.73) a2

111

is sufficient.

When a subset of D is significantly narrower near infinity than is D itself, much larger rates of growth are permissible. Corollary 2.31 is then far from sharp. We illustrate this by two examples; observe that, just as Theorem 2.30 extends to unbounded open subsets of D, so Examples 2.33 and 2.34 extend to unbounded open subsets of S and of H, respectively. Example 2.33 Consider the sector S := { (r cos 0, r sin 0) E R2

I

r > 0, 0 < 0 < Q }

,

$ E (0,27r).

If u E C(S), if u is subharmonic in S and if lim infR-,, max

sin(7cd/R/l)Pu(x, y)

(x, y) E S,

l

I(x, y)I = R y = 0, )))

(2.74)

then

sups U = supas U.

Proof This statement is merely a transcription of Theorem 2.30, for N = 2, under the conformal map of S onto D. Write z = x + iy = re'B for points of S, and _ + irl = pe" for points of D; the appropriate mapping is equivalently p = r"/p,

t = nO/l3,

r >_ 0, 0 < 0 < p.

(2'75)

This is a homeomorphism of the closed sector S onto the closed halfplane D; it is also a C°° map, with C°° inverse, of S \ {0} onto D \ {0}. Let u(c, rl) := u(x(l;, il), under the mapping (2.75). Then u E C(D) because u E C(S). Also, a satisfies the growth condition (2.70) because u satisfies (2.74). We now show that u is subharmonic in D. y), rl(x, y)). Given 0 E CC°(D) satisfying 0, define 'p(x, y) := Then 'p c Cc°(S), cp 0 and coxx+cpyy

+ Pnn = I

dl; / dz 12'

di; drl =

dt' dz

2

dx dy,

80


f

I G

b

H

x

Fig. 2.13.

so that

AD

ipnn) u d d'1 = .f

s((Px + (pyy) u dx dy > 0. S

The result 5UPD u = supaD u now implies that sups u = supas u.

If we allow /3 = 2n in the definition of S, then the foregoing result remains true even though S = R2 and (2.75) is no longer a homeomorphism of S onto D. Indeed, the weaker condition u E C(D) can replace u E C(S), provided that (2.74) is replaced by

lim infR-, sup

sin(0/2) u(x,.y) R1/2

1

(x, Y) E S>

I(x, Y)I = R T = 0 (2.76)

for /3 = 2ir. The condition u E C(D) is weaker for /3 = 2ir in that it allows limiting values u(x, 0+) as y 10 and u(x, 0-) as y T 0 such that u(x, 0+) * u(x, 0-) for x > 0. When contemplating supas u, we must then regard the upper and lower sides of 8S as distinct.

Example 2.34 Consider the half-strip H C(H), if u is subharmonic in H and if

lim inf,- max

sin(ny/b) u(c, y) exp(ne/b)

(0, oo) x (0, b) in ]R2. If u E

0 S y _0, 0 2, and assume that S2 has the exterior-ball property at p E 8SI (Definition 2.14). Let R(q, b), with b = l p - q I, be an exterior ball at p such that 2q - p S2 (Figure 2.14). Let dB (x) := dist (x, R(q, b)). If u E C(SZ \ {p}), if u is subharmonic in n and if

lim infa.o max { aN-2dB(x) u(x)

I

x E S1 n 8-4(p*, a*) } < 0,

(2.79)

where

p* :=P+

2

1

- q), a* 2 1 ((a/2b)2 (p

1 - (a/2b)2'

(2.80)

then

sup0 u = supan\{P} U.

Proof (i) We make two changes in Definition 2.24 and Lemma 2.25. (Readers who distrust such tinkering with previous results may prefer to prove the theorem by means of Exercise 2.45.) First, the ball _V(p, a) is replaced by 9(p*, a* ), where p* and a* are as in (2.80). Second, the

condition a < 1Ixo - p1, which accompanies the inequality W(xo, a) < 2(xo) in (c) of Definition 2.24, is replaced by

a' aN+l J 1 - 0 (b) } for x E GnaB*,

l

and (2.69c) implies a bound W* (xo, a) < A(xo)

for xo E G and a' < z Ixo - p'I

The growth conditions

lim info-o sup

u(x) W* (x, a)

X E fl n aB* } < 0

(2.82)


84

and (2.79) are equivalent because our choice of co-ordinates and (2.67) imply that x 2b

=1+ o(a) \b

zx =1+ o(a) for xEGnOB*, \b

and

dB(x)

and because the function with values aN-2dB(x)u(x) is continuous on SZ \ {p}. Therefore the modified Lemma 2.25 implies the present theorem.

2.8 Exercises

Exercise 2.37 Suppose that 1(p, a) c 0 c P1(q, b) in RN, and that f := S2 -> R satisfies 0 < k < f(x) < l for all x E fl. Prove that, if it exists, the solution u EC(?!) n C2(S2) of the Dirichiet problem -Du = f in .0, u I an = 0 is bounded by

(a2-Ix-p12) µ}, Y(µ):={xERNIXP EZ(p)},

(3.1)

and call these a cap and reflected cap, respectively. Note that Y(11) c S2

if µ > 0. Also, M := sup{ xl

I

x E E2 }, so that Z(µ) is empty if and

only if it >- M. Theorem 3.3 Let 0 be bounded, connected and Steiner symmetric relative to the hyperplane To. Assume that u := S2 --> R has the following properties. (a)

u EC(ii) fl C2(52),

(b)

Au + f (u) = 0 in S2,

u > 0 in S2,

uI an

= 0. (3.2)

where f : [0, oo) -> R is locally Lipschitz continuous. Then

u(z) < u(z'") if µ E (0,M) and z E Z(µ),

(3.3)

81u(x) < 0 if x1 > 0 and x E S2.

(3.4)

Proof (i) Let w(., p) := u - uµ on Z (µ) for µ E [0, M). Our main task is to prove (3.3), that

z E Z(µ) r w(z, µ) = u(z) - u(zµ) < 0 whenever p E (0, M). We begin by combining the equations

Au(z) + f (u(z)) = 0 for z E Z(µ), Au(y) + f (u(y)) = 0 for y E Y(µ),

(*)

3.1 The simplest case

89

Fig. 3.1.

where p E [0, M). In the second of these, set y = zµ. Then u(y) = u(zµ) _

uµ(z); also 8/8y1 = -8/8z1 and 8/8yj = 8/8zj for j = 2,...,N; hence 'Luµ(z) + f (uµ(z)) = 0 for z E Z(p).

Subtracting this from the equation for u(z), and defining f(uU

y(,p)

10

-f

uµuu)

at points z where u(z) uµ(z), at points z where u(z) = uµ(z),

we obtain Aw + y(z, p)w = 0 for z E Z(p),

(3.6)

where w = w(z, p). Also, the identity Z(p) = 52 n { x E RN I xl > p j implies that 8Z(µ) c 852 U Tµ; therefore w(z, p) < 0

for z E 8Z (p),

(3.7)

with strict inequality when z E 8Z(p) \ T. and z-u E 52 [because then z E 852, so that u(z) = 0, while u(zµ) > 0], and with equality when z E 8Z(µ) n T. Of course, there is a third possibility: that w(z,p) = 0 because z and zµ both belong to 00 \ T.

90

3 Symmetry for a Non-linear Poisson Equation

(ii) The next step is to show that (*) holds for all sufficiently small, positive values M - µ. (a) First, we apply the maximum principle for thin sets (Theorem 2.19) to the set Z(µ), the operator A + y(., µ) and the function w(. , µ). In order to have data independent of p, we use diam Z (0) for the diameter in Theorem 2.19. Since U E C(S2), there is a number U > 0 such that 0 < u(x) 0

such that If (u) - f(u) I < Al u - u; accordingly Iy(z, µ)I < A for all z E Z (p) and all it E [0, M). Since w(. , p) is a C2-solution of (3.6), it is certainly a distributional solution. Therefore Theorem 2.19 implies existence of a number 6 > 0, independent of µ E [0, M), such that

IZ(µ)I <S=w(z,p) 0 there exist points p E OZo(y) \ Tµ such that pµ E S [see a hint to Exercise 3.13] ; then w(p, p) < 0, as was noted after (3.7). Consequently, IZ(,a)I < 6

w(z,p) < 0 for z E Z(µ).

(3.9)

(iii) Let (m, M) be the largest open interval of y in which (*) holds; assume (for contradiction) that m > 0.

(a) Fix Z E Z(m) and let y 1 m. Since u(z) < u(zµ) for µ E (m,zl), and since u(zµ) varies continuously for fixed z and varying p, we have w(z, m) < 0. This inequality can be sharpened to w(z, m) < 0 for z E Z(m),

(3.10)

by means of (3.6), (3.7) and the strong maximum principle; the argument is like that in (ii)(b).

(b) Let 6 be the small positive number, independent of it E [0, M), inferred from Theorem 2.19 and introduced before (3.8). Let F C Z(m) be a compact set satisfying I Z(m) \ FI
0,

(3.11)

by (3.10), by the continuity of w(. , m) and by the compactness of F.

(c) Now consider wE := w(., m - e) for 0 < e < so, and let GE Z (m - e) \ F. The number eo is to be so small that I GE I < 6, that m - e > 0 and that wE(z) < -Zc for z E F. This last is possible by the uniform continuity of u on 0. We apply the maximum principle for thin sets (Theorem 2.19) to the set GE, the operator A + y(., m - e) and the function wE; the data diamZ(0), Iy(z,µ)I < A and S used in (ii)(a) can be used again here. By (3.7) and because wE < -Zc on F we have wE < 0 on BGE; therefore Theorem 2.19 implies that wE < 0 on GE. Consequently WE < 0 in Z (m - e); then, by the strong maximum principle once more

[applied as in (ii)(b) to each component of Z(m - e)], wE(z) < 0 for z E Z (m - e). This contradicts the maximality of the interval (m, M) with m > 0, and thus completes the proof that (*) holds for all u E (0, M). (iv) That 01u < 0 in Z(0) will now be inferred from the boundary-point lemma for balls (Lemma 2.12). Given p E S2 with p1 > 0, we set p = p1

92


and observe that there is a ball B := M(p + pe1, p) in Z(p) [because S is open, hence contains a ball M(p, 2p)]. Now w(., p) is a C2-solution relative to A+ y(., p) and B; also, w(z, p) < 0 for z E B while w(p, p) = 0. It follows from Lemma 2.12 and Remark 2.16 that -(alw)(p, p) > 0. This proves that (alu)(p) < 0 because the identity

aiw(x, p) = as { u(x) -u(2µ - xl, x")

(al u) (x) + (alu) (2p - xl, x")

l1

(3.12)

shows that (alw) (p, p) = 2 (alu) (p) when pl = p. Corollary 3.4 Under the hypotheses of Theorem 3.3, u is an even function

of xl: u(-xl, x") = u(xl, x") for all x E S2.

(3.13)

Proof Fix z E Z(0) and let p j 0. Since u(z) < u(zµ) for p E (0,zl), and since u(zu) varies continuously for fixed z and varying p, we obtain u(z) < u(z°) for z E Z(0). Then, by continuity of u on S2,

u(xl, x") < u(-xi,x") for xl >- 0 and x E 1. But, by applying to points of Y (O) the various arguments that we have used for points of Z(0) (in other words, by moving T. from p = -M to p = 0 rather than from p = M to p = 0), we obtain

u(-xl, x") < u(xl, x") for xl >- 0 and x E S1 Together, these two inequalities imply (3.13).

Corollary 3.5 If in Theorem 3.3 the set S is a ball, say KI = R(0, a) c then u is spherically symmetric (depends only on r := IxI) and du/dr < 0

for 0 < r < a. Proof (i) Let k be a given unit vector (k E RN and Jkl = 1); let R(k) be an orthogonal N x N matrix such that the transformation z = R(k)x

yields zl = x k; let u(z) = u(x) under this transformation. Then u satisfies the hypotheses of Theorem 3.3, hence is an even function of zl by Corollary 3.4. In the notation of Definition 1.7 we have u(x°") = u(x) for all x E RN, provided that we set u(x) := 0 outside R(0, a). This holds for every unit vector k; therefore Lemma 1.8 establishes the spherical symmetry of u.

3.2 A discontinuous non-linearity f

93

(ii) That du/dr < 0 for 0 < r < a follows from (3.4); indeed, now that u is known to depend only on r, it suffices to have (3.4) on a single ray from the origin. 0


There are two reasons for relaxing the condition in Theorem 3.3 that f be Lipschitz continuous. First, functions f with simple discontinuities arise naturally in problems of Newtonian potentials, steady vortex flows and magnetohydrodynamics (§4.4 and Exercises 3.16 and 4.21 to 4.24). Second, although Theorem 3.3 and its corollaries serve well as introduction to a strategy of proof, they do not show how robust and flexible the method is. The elliptic equation (3.6) satisfied by w was perhaps the most important ingredient of the proof of Theorem 3.3; in this section we shall see that Lemmas 3.7 and 3.8, which are mere shadows of equation (3.6), are quite enough to prove the results of the previous section for a less pleasant function f. Notation We continue to use the notation described after Definition 3.2. In particular, it is implicit that the unit vector k = el. Theorem 3.6 Let SZ be bounded, connected and Steiner symmetric relative to the hyperplane To. Assume that u : S2 -+ R has the following properties. (a) u E C(n) n C1(KI), u > 0 in 11, ul an = 0. (b) For all cp E CC°(S2f),

J

(3.14)

{

where f has a decomposition f = f 1 + f2 such that f 1

: [0, oo) --+ R is locally Lipschitz continuous, while f2 : [0, oo) -+ IR is non-decreasing and is identically 0 on [0, K] for some lc > 0.

Then the previous conclusions hold:

u(z) < u(z") 81u(x) < 0

if µ E (0, M) and z E Z(µ), if

x1 > O and x

J.

(3.15) (3.16)

Remarks 1. Equivalent hypotheses on f. Consider the conditions: f = 91 + 92, where g1 [0, oo) -+ R is locally Lipschitz continuous, while :

$2 : [0, oo) -> R is non-decreasing and is uniformly Lipschitz continuous

on [0, K] for some x > 0. These conditions are not more general than


94

those in the theorem: let $3(t)

I

Sl

g2(t)

if 0 < t < K,

$2(ac)

if t > K,

and define f, := 91 + g3, f2 := 92 - 932. Upward jumps and downward jumps of f. The function f in Theorem 3.6 may 'jump upwards' (as its argument increases) in that the hypotheses allow simple discontinuities

f (c+) - f (c-) > 0,

where f (c+) := limtlc f (t), f (c-) := limtTC f (t),

(3.17)

and where c > K [with f (K) = &-)]. If f jumps downwards, then the result need not hold (Exercise 3.19). Indeed, if f is merely Holder continuous [see remarks preceding Definition A.10] at a point of an interval in which it decreases, then (3.15) and (3.16) may be false (Exercise 3.18).

3. Notation and terminology. Let d(x) := dist(x, OQ) for points x E S2. The condition uI an = 0 and the uniform continuity of u on '2 imply that

3h > 0 such that d(x) < h

u(x) < x;

(3.18)

we define

Zh() := Z(p) n j x E 0 I d(x) < h }.

(3.19)

The phrase in the generalized sense will be a slight extension of the terminology in Definition 2.10; its meaning is displayed in Lemma 3.7. As before, w(. , p)

:= u - uµ on Z(µ) for µ E [0, M).

4. Method. Since equation (3.6), which was basic to the proof of Theorem 3.3, is no longer available, we must make do with the substitutes in the following two lemmas.

Lemma 3.7 If it E [0, M), then Aw + y(z, µ)w >- 0 in Zh(µ) in the generalized sense,

(3.20a)

where w = w(z,,u) and y is defined as before but with f 1 replacing f . That is,

fZh()

dz>0 whenever cp E C,°(Zh(p)) and q > 0,

(3.20b)

3.2 A discontinuous non-linearity f Y(., µ)

f1(uu

=

- u1(up)

0

95

at points z where u(z) uµ(z), at points z where u(z) = u, (z). (3.20c)

Proof (i) In order to derive an equation for w that will be useful both for this lemma and for the next, we abbreviate Z (p), Y (y) to Z, Y and proceed from for all cp E C,(Z ),

f{_vW Vu + Wf (u) } dy = 0

for all W E CC°(Y ).

Given cp E C°(Z), we choose W = cpµ [so that suppp c Y] and set y = zµ. Then a/ayl = -a/azl and a/aye = a/az; for j = 2,...,N; also, W(y) = (pp(y) = (p(yµ) = (p(z) and u(y) = u(zµ) = u,,(z), whence

f{_VVu,+f(u)} dz = 0 for all

E C°°(Z).

Subtract this equation from the equation for u(z); then { -VT Vw+4p[f1(u)-fi(uµ)] } dz

J

(3.21)

z

(p [f2(u) - f2(uN,)] dz

for all cp E CC°(Z).

(ii) Now restrict attention to cp E C,°(Zh) with cp >- 0, where Zh = Zh(µ),

and integrate only over Zh in (3.21). The definition (3.20c) of y shows that, on the left-hand side of (3.21), f i(u) - fi(uµ) = A. , µ)w.

On the right-hand side, f2(u) = 0 because u(z) < K in Zh, by (3.19) and (3.18), while f2(uµ) >- 0 because f2 >- 0 on [0, oo). Thus

- IZ, ( [ f2(u) - f2(u) ] dz > 0 whenever 4P E C(Zh) and cp >_ 0, from which (3.20b) follows.

Lemma 3.8 If u E [0, oo) and w(z, µ) < 0 for all z E Z(µ), then Ow - Aw >- 0

in Z(µ) in the generalized sense;

(3.22)

96


here w = w(z, p) and A is a Lipschitz constant for f 1 on the interval [0, supra u]. Thus A Z 0 and A is independent of p.

Proof In (3.21) we have

fl(u)-fl(uµ) -

-f P{fl(u)-fl(uµ)+f2(u)-f2(uµ)} dz z

Jz

(3.23)

cpAw dz,

and this is (3.22).

Proof of Theorem 3.6 (i) As before, our main task is to prove that

z E Z(p)

w(z, p) = u(z) - u(zµ) < 0

(*)

whenever p E (0, M). Again we begin with small values of M - p, and proceed very much as in the proof of Theorem 3.3, step (ii). (a) In order to apply the maximum principle for thin sets (Theorem 2.19) to the set Zh(p), the operator A + y(z, p) and the function w(. , p), we choose data A and 6 independent of p E [0, M), as in the earlier proof. Of course, f, replaces f ; the constant A is as in Lemma 3.8, so that Iy(z,p)I 0. (a) Fixing Z E Z(m) and letting p 1 m, we obtain [precisely as in the proof of Theorem 3.3, step (iii)(a)] that w(z, m) < 0. Then Lemma 3.8 applies for p = m and shows that w(. , m) is a generalized subsolution

relative to A - A and to each component of Z(m). As before, each component of Z(m) has boundary points p at which w(p,m) < 0; the strong maximum principle (Theorem 2.13) shows that

w(z, m) < 0 for z E Z (m).

(3.26)

(b) Next, we prove that

71u 0 and w(z,p) < 0 in this ball because p = pi >- m. The boundary-point lemma for balls (Lemma 2.12), applied as in the proof of Theorem 3.3, step (iv) [but now for a generalized subsolution relative to A - A], shows that (01u)(p) = 11(01w)(p, p) < 0.

(c) Letting wE := w(.,m-s) for 0 < E < Eo, as before, we wish to obtain a contradiction by showing that wE < 0 in Z(m - e) if Eo is sufficiently small. We continue to imitate the proof of Theorem 3.3, but a further construction is required because Lemma 3.7 applies only to sets near 751, while the hypothesis w(., p) < 0 of Lemma 3.8 remains to be proved for p < m. The new step is first to observe that (3.27) extends to certain subsets of Z(m - p) for some p > 0, and then to introduce a cylindrical set

EE := (m - E, m + i] XS, T>0, S (--R N-1, EE c Z(m - E),

(3.28)

as well as a compact subset F of Z(m); see Figure 3.3. We can prove that wE(z) < 0 for z E EE by integrating 71u from z'-E to z, provided that {m} x S is a compact subset of f1 0 and that i and so are sufficiently small.

Let GE := Z (m - E) \ (EE U F). We shall prove in (iii) that EE and F can be so chosen that, for 0 < E 0,

we < 0 on EE,

GE cZh(m-E),

wE < 0 on F, IGEI 0, each component of Z(m - E) has boundary points p at which wE(p) < 0; consequently, the strong maximum principle (Theorem 2.13) shows that wE < 0 in Z(m-E). This contradicts the maximality of the interval (m, M) with m > 0. (iii) Here is a detailed description of EE and F that leads to conditions (I) to (IV).

(a) Let Sm := {m} x S, where S is a closed subset of RN-1 such that Sm c (Tm n 0) and such that, in the hyperplane Tm, all points between Sm and an are sufficiently near an; say x E (Tm n n) \ Sm

dist(x, T. n On) < 1h.

(3.29)


99

The existence of such a set Sm follows from the representation of Tm n 91

as a countable union of dyadic squares, that is, of (N - 1)-dimensional dyadic cubes (Exercise 3.14).

By (3.27) and the continuity of alu on the compact set Sm, there is a number c = c(S) > 0 such that 81u < -c on Sm. Moreover, 81u is uniformly continuous on each compact subset of 0. Therefore, there exists a cylindrical set [m - p, m + a] x S within fl, with p > 0 and a > 0, such that

81u < 0 on [m - p, m + a] XS.

(3.30)

In view of (3.27), the number a is restricted only by the need to have this set within fl. Now, for every z > 0,

m-Zx-m-2z. Hence in the definition (3.28) of EE it suffices for condition (I) that 0 < i < a and Zr < p, and then that 0 < so < Zr; for it follows that, when z E E, the line segment from zm-E to z is in the set [m-p, m+a] x S, so that wE < 0 on EE, by (3.30).

(b) Let E' := EE n Z(m); the sets E' and F are to be independent of E. The complement of E' in Z(m) \ Z(m + i) is, or can be made, small in two senses. First, (3.29) and the fact (made precise in Exercise 3.13, (iii)) that Tµ n f2 cannot widen as y increases through positive values, imply that

z E { Z(m) \ Z(m + z) } \ E' - d(z) < 1h.

(3.31)

Second, if i is sufficiently small, then

{Z (m) \ Z (M + i)} \ El < 16.

(3.32)

The number i is now chosen so that the condition (I) and (3.32) hold; S and i are fixed henceforth, but so may be reduced further.

(c) Condition (II) will follow, as in the earlier proof, from the compactness of F and the uniform continuity of u(zµ) for fixed z and varying p, provided that so is sufficiently small. We now choose F so that (III) and (IV) hold for sufficiently small so.

Let F denote, for n E No, the union of all (closed) dyadic cubes of edge-length 2-" (Exercise 3.14) that are subsets of the open set Z(m). Then N1/22-n whenever x E Z(m) \ F,,; also, I Z(m) \ FnI -* 0 dist(x, 8Z(m))
m + 2; by (3.33), a point of 7Z(m) that is nearest z must be in 0 rather than in Therefore d(z) < h, by (3.33). (e) To prove (IV), we observe that IGE \ Z(m)I < 16 if go is sufficiently small, and that

IGEnZ(m)I N1/22-" (where n E No), then x E Fn. (ii) Fn c Fn+1 and S2 = Un o Fn. (iii) 92 is a countable union of dyadic cubes. (iv) In \ FnI 0 as n co. Exercise 3.15 We seek all functions u E C(ST) n C 1(92), other than the zero

function, that are generalized solutions (Definition A.7) of

u"+).fH(u-1)=0 in 92:=(-1,1)c]R,

Ulan = 0,

where A, is a positive constant and fH is the Heaviside function (see Chapter 0, (v)). Prove the following.

(a) For A. > 4 there are exactly two non-trivial, generalized solutions; they involve the points 2 (1 +

a1 i a2

1 - 4/2)

respectively,

and are, for j = 1, 2,

uj(x) =

1+x

if-1 <x0

(Exercise 3.16),

JJJ

and calculate 2 a posteriori. Prove that each problem has exactly one non-trivial, generalized solution for prescribed q > 0. Exercise 3.18 Consider, for arbitrary N E N,

Lu+f(u)=0

inB :=-4 (0,r6-) cIIBN

(3.35)

0, U I OB =

where, for a constant p > 2, 4p(p-1)(1-t)1-(2/p)+p(N+2p-2)(1-t)1-(1/0

if

0 1 and r := lxl < 2,

1-(Zr2_2)p

if 2 R is non-decreasing. In case (C), f 1 and f2 need be defined and need have these properties only on [0, oo).

4 Symmetry for the Non-linear Poisson Equation in RN

108

Let

_ q:

1 Km

c

1

-c K

for cases (A) and (C),

v(x) := u(x + q). (4.2)

for case (B),

Then v is spherically symmetric (depends only on r) and dv/ dr < 0 for

r>0. The proof of Theorem 4.2 will take the form of five lemmas and a proposition. A first step is to verify that, under the transformation (4.2), the function v has every good property of u and has better asymptotic behaviour in that the second-order terms in the expansion of u are absent from the expansion of v. Lemma 4.3 The function v in (4.2) satisfies conditions (a) and (b) of Theorem 4.2, but with improved asymptotic behaviour in that one of the following conditions holds for r >- R + 2IgI _: R, . As before, K, m, S and y are positive constants, with S E (0, 1]. (A)

v(x) = -Krm + g(x),

where IOg(x)I =

O(r"i-2_6),

and, ifm-1-80as r->co. (B)

v(x) = K log r + g(x),

where IOg(x)I = 0 (r-2-S)

,

and g(x)-* 0as r-+oo. (C)

v(x) = Kr-' + g(x),

where IVg(x)I = O(r m-2-s),

and g(x)->0as r->oo. Proof (i) It is clear from (4.2) that v E C1(R") and that v > 0 on RN in case (C).

(ii) Here is a merely formal calculation showing the removal of terms (c x)r" in Definition 4.1. For case (A), -K(r2 + 2q x + IgI2)mJ2 + C (x + q)Ix +

qlni-2

2+ so that terms of order rm-1 are cancelled by the choice q = c/Km. For case (B),

KlogIx+gl

2]}

4.1 Statement of the main result

=x

log

Y

r

- (q x)r-2 +

109

>

so that terms of order r-1 are cancelled by the choice q = c/K. For case (C), qI-m

iclx +

= xr-' { 1 - m(q x)r-2 + ... },

so that terms of order r-ni-1 are cancelled by the choice q = c/xm. (iii) To show that v satisfies (4.1), we suppose that any V in Cc(RN) is given. Choose cp(y) = yp(y - q) in (4.1); then { -(V1V)(Y JRN

- q) (Vu)(y) + 1V (y - q) f (u(y)) } dy = 0,

and the substitution y = x + q yields (4.1) for W and v.

(iv) For a rigorous justification of the claims made for Vg and g, we define functions F, G and H by Ix + qIa =: ra + a(q 1

+

log Ix

=: log gI

c (x + q) I x + q I

a-2

x)ra-2 + F(x,

a),

1

r - (q - x)r-2 + G(x),

=: (c . x)ra-2 + H(x, a),

where a E R is an arbitrary constant. Then, with h still denoting the remainder function in Definition 4.1, g is given by one of the formulae

g(x) _ -iF(x, m) + H(x, m) + h(x + q), g(x) = iG(x) + H(x, 0) + h(x + q),

g(x) = rcF(x, -m) + H(x, -m) + h(x + q). Accordingly, to prove the claims made for Vg it is sufficient to show that, for each k E { 1, ... , N}, a F(x, a) axk

< const. ra-3,

aak G(x) < const. r-3, a H(x, a) axk

< const. ra-3,

where the constants may depend on a, q, N and k, but not on x. To this

110


end, let cp(t) := Ix + tq la, 0< t< 1; then the definition of F becomes

= (p(1) - (p(0) - cp'(0) = 1(1 - t) T "(t) dt

F(x, a)

0

N

1

f(1 - t)

2

jEqj7j lx+tql°

dt,

(4.5a)

j=1

0

where x and q are fixed and a; means, as always, differentiation with respect to the jth argument of the operand. Similarly, N

E qj8j

(1 - t)

G(x) _

2

dt,

(4.5b)

{ c (x + tq) lx + tq I a-2 } dt.

(4.5c)

log

(j=1

Ix + tql

N

H(x, a) =

E qjaj

J

1=1

It is sufficient to justify differentiation with respect to xk under the integral sign for 21gI < r < M and each M. We may suppose that IgI > 0 (otherwise, there is nothing to prove), and then the bounds

0
- 2, N 4µh:5 s2], v

2 and

C2) > 0 if sa > 2km µ

V(Z) > sm-2-6h 2xm sa

(b) 4µh > s2. This condition implies that t2 > 252 and hence that f

tm - sm = tm { 1 -

r

()/2}

lm

\t/

> tm

1_

> 2 tm.

J

To bound g(y) - g(z), we observe that m - 1 - 8 >- 0 (because m >- 2 and 8 < 1). Reducing S slightly if necessary, we may suppose that m - 1 - 8 > 0 whenever m >- 2; then g can be estimated as follows by integrating radially outwards from 8-4(0, R(0,Re); we write ro := Ixol and g(xo) = g

-

( ro xo l + f 1olro R

Let xo be any point outside dS(Txo) dT

dT,


114

where dg(Txo)

= Ixo - (Vg)(Txo)I _

if C3 is sufficiently large. Consequently, (g(Y) - g(z)I < 2C3t

and, for this case [(A) with m >_ 2, N >- 2 and 4µh > s2], v(Y) - v(z)

> tm-1-S (K t1+o

- 2C3)

2 tm-1-a J K (21/2s)1+a - 2C3

>0

J

23/2-6/2C3

if S1+6 >

C4-

K

(c) Collecting results, we define 1/a

C2

R(µ) := max R (2Kmmin{µ,1})

C1/(1+a) 4

'

(4.9)

and the lemma is proved for case (A) with m >_ 2 and N >- 2. (iii) Case (C) with N >- 2. By Lemma 4.3, v(Y) - v(z) = K(s m - t-') + g(y) - g(z),

log(x)I

C5r-m-2-1

0,

for r > R,.

Again we estimate separately for 41th < s2 and for 41th > s2.

(a) 4µh< s2. By strict convexity of the function T H T m/2, 0 < T < 00,

s m - tm =

(s2)-m/2

- (s2 +

4µh)-m/2

> 2 (s2 +

(252)-m/2-14µh = C6 s m-2µh.

2

Integration along F gives, essentially as before, Ig(Y) - g(z)I 2 and 4µh < s2],

v(Y) - v(z) > s m-2-S h (C6Kµs ° - CSn) > 0 if s S >

C5

C6 K/2

(b) 4µh > s2. Since now t2 > 2s2, s

-m

m/2

lm - t m =S m{1- (S) }>s-m1

=C7sm.

-(2)

Next, for any point xo outside R(0, R,) we obtain more easily than before dg(ixo)

Ig(xo)I =

C5ro m- 1-6

dT

m+l+S'

d2

1

so that 2C5

1 g(Y) - g(z)I - 2 and 4µh > s2], v(Y) - v(z) > s' 1-6 (C7Ks'+S

+

-

> 0

m + 1

ifs

1+S

2C5

Cg.

> C7K(m+ 1 + 6)

(c) Collecting results, we define

R(p) := max

Rn,

( C6K

1/S

C5

l} /J

Cg/(1+S)

(4.10)

and the lemma is proved for case (C) with N >- 2.

(iv) Modifications for N = 1. When the domain of v is R, we must modify certain estimates of Ig(y)-g(z)I because no path outside the ball (-R,,, connects y and z if y < -R,,. In other words, we have no substitute for the arc F when N = 1 and y < -R,,. However, if y >- R we can replace integration of Vg along F by integration along R from y to z. Also, integrals of Vg along radial lines from (R,/ro) xo to xo, or from xo to infinity, can be used as before. (For N = 1 and xO < -R0, the latter are from xo to -oo.) Since the arc F was used only for 41h < s2, we need change only estimates that are subject to both 41,&< s2,

equivalently,

t2 < 2s2,

and

y = p - h = -s < -R,,, equivalently, h = s + µ >- R + µ.


116

Then, for case (C) with N = 1, by inequalities in (iii) that remain valid and by h > s, m- t m> C6 S ri-2µh > C6 s--n'-lµ, S g(y) - g(z) I

0

if S6 >

C9 2K µ

Lemma 4.6 There is a number µ* > 1 such that

µ > µ* and y E Y(µ)

v(y') < v(Y)

(II)

Proof (i) Let R(µ) be as in Lemma 4.5, so that R(µ) = R(1) for µ >- 1, and define R1 := max{ 1, R(1) },

c1 := min rsR, v(x).

[Here r := lxi, as before] (ii) For cases (A) and (B) of Lemma 4.3, choose µ* to be such that r >- µ*

v(x) < c1 - 1;

this is possible because v(x) --> -oo as r - oo. Then µ* > RI; Figure 4.3 illustrates the situation.

If µ >- µ*, y E Y(µ) and lyl > R1, then also lyl > R(µ) [because µ* > 1 and R1 >- R(1)]; Lemma 4.5 ensures that v(yµ) < v(y). If µ >- µ*,

4.2 Four lemmas about reflection of v

117

Fig. 4.3.

y E Y(µ) and IyI < RI, then v(y) >- cl [by definition of cl] and Iyµ) > µ* [because yl < Rl < p*] so that v(yµ) < cl - 1 < v(Y).

(iii) For case (C) of Lemma 4.3, choose y* to be such that

r > µ*

v(x) < 1 cl ;

this is possible because now cl > 0 and v(x) -+ 0 as r -+ oo. The rest is as in (ii), with Zc1 replacing cl - 1. O

Although all four results of this section will be needed to prove Theorem 4.2, it is the next lemma that is the heart of the matter. Observe that the generalized Poisson equation (4.1) is used only in Lemma 4.7,

and that the whole theorem rests on our ability to sharpen < to < in Lemma 4.7.

Lemma 4.7 Assume that for some y > 0 we have (a) v(yµ) < v(y) for all y E Y(µ), (b) v(Yo) * v(Yo)for some yo E Y(µ)


118 Then

v(yµ) < v(y) for all y E Y(µ),

(III.a)

alv(x) < 0 for all x E Tµ.

(III.b)

and

Here (III.a) implies (III.b).

Proof Let Y := Y (µ) and Z := Z (p) for the given u, and let w := vµ - v [which means, according to Definition 1.7, that w(x) = v(xµ) - v(x) for all x E RN]. By hypothesis, w(y) < 0 for y E Y ; we shall prove strict inequality by means of the maximum principle. (i) A generalized Poisson equation for w. By (4.1) and Lemma 4.3, we have J N { -OW Vv + Wf (v) } = 0

for all W E C,°(RN).

(4.11)

Given cp E C,(Y), choose W = cpµ in this equation; then suppcpµ c Z and

-Ocpµ(z) . Vv(z) + cpµ(z) f (v(z)) } dz = 0.

Set z = yµ. Then y = zµ E Y, a/azi = -a/ay, and a/azj = a/ayj for j = 2,...,N; also, cpµ(z) = T(zµ) = cp(y) and v(z) = v(yµ) = vµ(y). Accordingly, jr { -V q (Y) - Vv (Y) + cp(Y) f (vµ(Y)) } dy = 0.

(4.12)

Now choose p = T in (4.11), and subtract the result from (4.12); then f { -Ocp Vw + (p [f (vµ) - f (v)] } = 0 for all

E C,00(Y).

(4.13)

(ii) Application of the strong maximum principle.

In order to control f, we consider balls in Y rather than the whole of Y. Let D :_ M(c, p),

c E Y,

p := min{ 1, dist(c, Tµ) },

so that D c Y. To estimate f (vµ(y)) - f (v(y)) at points y of D, we recall that f = f1 + f2 with f 1 locally Lipschitz continuous and f2 nondecreasing. By the continuity of v, there is a number M = M(c, v, µ)

4.2 Four lemmas about reflection of v

119

such that Iv(y)I < M and Iv(yµ)I < M whenever y E D; hence there is a Lipschitz constant A = A(M) > 0 such that, in D,

f1(vµ) - fl(v) < Alvµ - vj = -Aw

[since w < 0].

Also,

fz(vµ) - fz(v) < 0,

because vµ < v and f2 is non-decreasing. By addition of these bounds and by (4.13),

O=f {-V .Vw+w[f(vµ)-f(v)]} - 0; in other words, w is a generalized subsolution relative to A - A and D. Now suppose that w(c) = 0 = SUPD w at some c E Y. Then the strong maximum principle, Theorem 2.13, states that wID = 0, and this shows that the zero set of w, say X :_ { y e Y w(y) = 0 }, is open in Y. Since w is continuous, X is also closed in Y. Then, since Y is connected, I

either X = Y or X = 0; our hypothesis that w(c) = 0 at some c E Y implies that X = Y, which contradicts hypothesis (b) of the lemma. Consequently, X must be empty, and this proves (III.a). (iii) Application of the boundary-point lemma for balls. Let p E Tµ be given. Define

B :_ -4 ((pl - p, p"), p)

for any p E (0,1].

We know from (ii) that w is a generalized subsolution relative to A - A and B, and that w < 0 in B. In addition, w(p) = 0 because pµ = p. Then the boundary-point lemma for balls, Lemma 2.12, states that (al w)(p) > 0. Now, (01 w) (p) = -2 (01 v) (p) because

81 w(x) = xj { v(2µ - x1, x") - v(xi, x") } , so that (01v) (p) < 0. This proves (III.b).

Lemma 4.8 The set I ,u > 0

1

(Ill.a) holds : y E Y(µ)

v(yµ) < v(y)} is open in R. (IV)

120


Proof Assume that the lemma is false. Then, for some value y = 2a > 0 at which (III.a) holds, there is a sequence (µn) in (a, 3a) such that (III.a) fails at each µn and µn 2a as n --+ oo. This failure of (III.a) must be for IyI < R(a), because (I) ensures that

it ? a, y E Y(µ) and IyI > R(a)

v(Yµ) < v(Y),

where R(µ) < R(a). Thus there are sequences (µn) and (yn) [here Yn is not a co-ordinate, but a point of RN] such that µn -* 2a, Yn E Y(µn), ynl < R(a)

and v (yn) >_ v(Yn)

By the relative compactness of (yn), and resort to a subsequence if need be, we may suppose that yn tends to some point q in . (0, R(oc)) as n - oo. Then q E Y(2oc) and v(g2a) > v(q). Since (III.a) holds at µ = 2a, we must have q E Tea. But (Ill.b) holds at µ = 2a [because (III.a) does]; hence (8iv)(q) < 0, and this contradicts v (yn^) >- v(yn) when n is sufficiently large.

4.3 Proof of Theorem 4.2 and a corollary

Proposition 4.9 The results (I) to (IV) imply that v is spherically symmetric

and that dv/dr < 0 for r > 0; that is, they imply Theorem 4.2.

Proof (i) Let (m, oo) be the largest open interval of µ in which (III.a) holds. This m exists, by (II) and (IV).

Assume that m > 0. We show that the hypotheses preceding (III.a) hold for y = m. First, yEY(m)=v(Y'n) m and by continuity: fix y E Y(m) and let µ ] m. Second, by (I),

y E Y(m) and IyI > R(m)

v(Ym) < v(Y)

Therefore (III.a) holds at u = m. But then (IV) states that (III.a) holds in (m - e, oo) for some e > 0, so that (m, oo) is not maximal for (III.a). This contradiction shows that m = 0. (ii) We now have

v(xl, x") < v(-xl, x") whenever xl >- 0,

by (Ill.a) for µ > 0 and by continuity: fix -x1 < 0 and let p 10. [For xl = 0 we have equality]

4.3 Proof of Theorem 4.2 and a corollary

121

Keeping the co-ordinates xt = zt = x k and x" = z", which are aligned with k, we repeat the argument for the unit vector -k; this yields

v(xt, x") >- v(-xi,x")

whenever xl >- 0.

Thus v" is an even function of zt for arbitrary unit vector k, hence v is spherically symmetric, by Lemma 1.8.

That dv/dr < 0 for r > 0 is shown by (Ill.b).

The method also yields symmetry relative to a single co-ordinate, in place of spherical symmetry, for a wider class of functions f. In the following corollary there is only one system of co-ordinates (no alignment with an arbitrary unit vector). We still write "

The remark about f, (., x") means that, for each compact set E C R, there is a constant A(E) independent of x" such that If 1(s, x") - f 1(t, x") I < A(E)Is - tI whenever s, t E E and x" E RN-t Corollary 4.10 Suppose that u : RN -> R satisfies condition (a) of Theorem 4.2 and that, in condition (b), f (u(x)) is replaced by f (u(x), x"), where f now has a decomposition f = ft + f2 such that f, (., x") : IR --> R is locally Lipschitz continuous uniformly over x" E RN-1, while f2(.,x") : R -> R is non-decreasing for each x" E RN-t. In case (C), these conditions need hold only on [0, oo). Again define q and v by (4.2). Then v(-xl, x") = v(xt, x") for all x E RN, and 7tv(x) < 0 whenever x1 > 0.

Proof The only changes are these. (i) The function v satisfies not (4.1) but, for all ip E C,(RN), { -Vip(x) Vv(x) + W(x) F (v(x), x") } dx = 0,

(4.14)

LN

where

F(t, x") := f (t, x" + q")

for all t E I[8,

or for all t E [0, oo) in case (C). This follows from step (iii) of Lemma 4.3. Then the decomposition f = f t + f2 implies a decomposition F = F1 + F2 with the same properties. (ii) The results (I) to (IV) are derived not for an arbitrary unit vector k, but only for k = (1,0,. .. , 0) [relative to the original co-ordinate system]. Proposition 4.9 is restricted similarly.


122

4.4 Application to some Newtonian potentials Notation As in (A.18) of Appendix A, the Newtonian kernel for RN is

-ZIxI

K(x) = K(x; N) :_

I

log

KN

Ixl

in

118,

in

1182 \ {0},

N > 3,

in RN \ {0},

IxIN_2

where KN =

Nx"2

1

(N - 2)1a-4N(0,1)1

(N

aMN(0,1)I

2),

=

(N12)!

The formula for the surface area of the unit sphere is a result of Exercise 1.14.

The characteristic function XA of any subset A of a universal set U is defined by 1 1 if x E A, XAx)

if x E U \ A.

0

We define the Heaviside function to be fH := X(oM, so that fH(t)

1

if t>0,

0

if t < 0.

(The value fH(O) = 0 is a little unorthodox.) Recall Theorem 1.1: if G is a bounded open subset of RN (N > 1), A is a positive constant, u(x) := A

f c

K(x - ) di;

for all x E R 1,

(4.15)

and

ul aG

= constant = /3,

(4.16)

then G is a ball. The truth of this will emerge from the more general Theorem 4.13, but it is worthwhile to anticipate here the reasons that (4.15) and (4.16) allow application of Theorem 4.2. First, results in Appendix A show that u E C'(RN) and that the asymptotic behaviour of u (as r -> co) is admissible. Second, we shall infer from the maximum

principle that u(x) > /3 if and only if x E G; therefore, in place of the classical result (Appendix A) that

Au =

-2 in G, 0

in RN

4.4 Application to some Newtonian potentials

123

we shall find that u is a generalized solution of the equation ,Lu + Afx(u - /3) = 0 in RN

(4.17)

In other words, u satisfies (4.1) with f = 2fH(.-f), and this f has a suitable decomposition: choose f l = 0. We consider, in the first instance, the Newtonian potential of a given density function F on a given set G; let

w(x) := fG K(x -) F(l;) d

for all x E RN, F E LP(G) with p > N.

where G is open and bounded in RN,

(4.18)

Lemma 4.11 The potential w in (4.18) enjoys the following properties. (i) w E C1(RN) (ii) Extend F to be zero outside G; then

for all cpEC`°(RN).

,RN

(4.19)

(iii) If fG F > 0, then the asymptotic behaviour of w is admissible in the sense of Definition 4.1; for N = 1 we have case (A) with m = 1, for N = 2 we have case (B), and for N 3 we have case (C) with m = N - 2.

Proof (i), (ii) Theorem A.11 establishes somewhat more than the C' property of w and also shows that w is a generalized solution of -Ow = F in RN; in other words, shows that (4.19) holds. (iii) Let R,v be so large that G c -4(0, -12R,,), and let r := IxI > R, For N = 1, we observe from (A.56), or else directly from the definition (4.18), that

w(x) = -Kr + cxr-1

(N = 1, r > Rw),

where K

1J 2

F( )

2

G

d

G

For N = 2, Theorem A.12 states that w(x) = K log

1

r

+ (c

x)r_2

+ h(x)

(N = 2, r it R,,),

where K

G

27r

d

(j = 1, 2),

124


Ih(x)I < const. r 2,

IVh(x)I < const. r-3.

For N >- 3, Theorem A.12 states that w(x) = Kr N+2 + (c - x)r-N + h(x)

(N >- 3, r >-

where K = KN

f F( )

/

cj = (N - 2)KN J

(1 = 1,..., N),

d

IVh(x)l < const. r N-i

(h(x)I < const. r-N

Lemma 4.12 Suppose that in (4.18) we have F(I;) > 0 almost everywhere in G, and that wIaG = constant=#. Then w(x) > /3 if and only if x E G; equivalently,

XG(x) = fH(w(x) - /3) for all x E RN.

Proof It is enough to prove that x E G

(4.20)

w(x) > /3 and that x E

RN \G=> w(x)< f3. (i) Consider points of G. By Lemma 4.11, w E C'(-G) and

- J Vq Vw = - 0.

(4.21b)

Thus w is a generalized supersolution relative to A and G. By the weak maximum principle, Theorem 2.11 with c = 0, we have w >- /3 in G. Now apply the strong maximum principle, Theorem 2.13, to any component [maximal connected subset] Go of G. If w(xo) = /3 at some point xo E Go, then w = /3 in Go, hence Vw = 0 in Go. This contradicts (4.21a) if we choose cp to be such that supp cp c Go, p >- 0 everywhere and T > 0 somewhere in Go.

(ii) Given any point y E RN \ G, we prove that w(y)

Let

0R(O,R)\G, with R so large that y E SZ and w < f3 on 09(0, R). This last is possible

because, if N = 1 or 2, then w(x) -+ -oo as r -> oo; if N > 3, then /3=wlaG>0andw(x)-->0asr--*oo. Now in (4.19)wehaveF=Oin 0; hence w E C 1(fi) and

-Jn

for all pEC10°(f)),


125

so that w is a generalized solution relative to A and S2. [In fact, subsolution would suffice.] By the weak maximum principle, Theorem 2.11 with c = 0, maxjj w = maxan w = P. 11

Accordingly, the density function of the potential u in (4.15) and (4.16)

is F = .1xG = *fH(u(.) - f) on IRN, and, as explained earlier, Theorem 1.1 follows from Theorem 4.2. However, we need not restrict attention to density functions that are constant in G. In the next theorem we consider certain density functions F = g o u that are positive and may vary with

the potential u itself; if u is constant on the boundary 8G, then G is a ball once more. Whether such functions u exist, and whether, if they do, they are of interest, will be considered briefly after the theorem has been proved.

Theorem 4.13 Let G be a bounded open subset of IRN (N >- 1), and suppose that a measurable function u : RN -p R satisfies

u(x) = fK(x - ) g(u()) dfor all x E RN,

(4.22)

uIaG = constant = f,

(4.23)

where g : R - R is as follows. (a) 0 < g(t) < M for all t E R and for some constant M. (b) g is continuous except at finitely many points. (c) The restriction of g to [f, oo) has a decomposition g 91 + 92 such that gl(f) = 0 and gl is locally Lipschitz continuous, while $2(f) > 0 and $2 is non-decreasing. Then G is a ball, there is a point q E RN such that the definition v(x)

u(x + q) makes v spherically symmetric, and dv/dr < 0 for r > 0. Proof We verify that u satisfies the hypotheses of Theorem 4.2. (i) u E C1(1RN). This will follow from Lemma 4.11 if g o u E LP(G) with

p> N. In fact, g o u E L,,(G) if g o u is measurable, because hypothesis (a) ensures that g o u is bounded. To prove that g o u is measurable, we let F denote the finite set of points at which g is discontinuous and let h gIR\F. If W is open in R, then g-1(W) = E U h-1(W) for some set E F, and h-1(W) is open in R because it can be written U(hm)-1(W), where each restriction h. of h is continuous. Thus g-1(W) is a Borel set

126


in R, which makes g Borel-measurable; by a standard result (Kingman & Taylor 1966, p.108; Rudin 1970, p.31), g o u is then measurable.

(ii) That u has admissible asymptotic behaviour also follows from Lemma 4.11. Case(C) of such behaviour occurs if and only if N >- 3; then the positivity of K and of g imply that u > 0 on RN. (iii) The generalized Poisson equation (4.1) is established as follows. By (4.19),

f

N

{

Vu + coF } =0 for all

E CC°(RN),

where

F(x) = g (u(x)) XG(x) for all x E ][8N [by Lemma 4.12] = g(u(x)) fH(u(x) - /3)

= f (u(x)) if we set f (t) := g(t) fH(t - /3) for all t E R. Also, let

fi(t) := ge(t) fH(t -13)

for j= 1,2 and all t E R;

then fl : R -+ R is locally Lipschitz continuous, f2 : R -> R is nondecreasing, and f = f, + f2. (iv) Accordingly, Theorem 4.2 applies to u; there is a point q E RN such that u(x + q) = v(x) =: V(r), and

say

(r := IxI),

V'(r) < 0 for r > 0.

(4.24)

Since V (O) = max u > /3, there is exactly one number b (for the function

V in question) such that V(b) = /3, and since x + q E G if and only if V (r) = u(x + q) > /3 (by Lemma 4.12), we have G = 3(q, b).

El

Apart from the case in which g is a constant, are there functions u that satisfy both (4.22) and (4.23)? The answer is that such functions exist in profusion; we shall see a few of them in Example 4.15 and in Exercises 4.25 and 4.26. However, equations (4.22) and (4.23) do not form the best starting point for an existence proof; it is more usual and profitable to proceed from (4.1), constructing solutions u in a Sobolev space by means of some variational principle or other. This allows functions g that are less restricted than those in Theorem 4.13. Further properties of u then follow from the theory of Sobolev spaces and from regularity theory for


127

solutions of elliptic equations. We do not follow this course here because some of these steps are outside the range of the present book. The proof of Theorem 4.13 shows that it is essentially a particular case of Theorem 4.2. Whether functions u as in Theorem 4.2 are of interest or of importance is, of course, in the eye of the beholder. For N = 2 and N = 5 there are applications to steady vortex flows of an ideal fluid and to the problem in magnetohydrostatics of the confinement of a plasma in equilibrium; some of these are indicated by Exercises 4.18 to 4.24. Example 4.14 In Theorem 4.13, let

g(t) = const. = A > 0

for all t E R.

(This is a return to Theorem 1.1 and Exercise 1.12.) If N = 1, (4.22) implies that /3 < 0; if N >- 3, that /j > 0. In these cases there is a unique radius b = b(A, /3) such that G = R(q, b) for some q E RN. In fact, (A.23) shows that 11/2 (N# 2). b = { N(N - 2)/3/A. If N = 2, then (A.23) shows that the radius b of the ball G is determined by b2 log b2

=

4fl

(N = 2)

(4.25)

Consider the graph of b2log(1/b2) for b > 0. If /3 > 0, then (4.25) has no solution when 4/3/A > 1/e, one solution when 4/3/A = 1/e, and two solutions when 4/3/A < 1/e. If /3 < 0, then (4.25) has exactly one positive solution.

Note that, for a given solution u in Theorem 4.13, there is always a unique radius b such that G = JJ(q, b), as was mentioned in the proof. However, for given data g and l3, there may well be more than one solution V as in (4.24), and hence more than one radius b; this is the case in Example 4.14 if N = 2 and 0 < 4/3/A < 1/e. Example 4.15 In Theorem 4.13, let

N = 2, where

g(t) = A + (µ - A)fx(t - a) for all t E R, 0 < A < µ,

/3 < 0,

/3 0 and we call V1 a one-step solution of the present problem. It is a particular one of the solutions in Example 4.14 (with N = 2 and /3 < 0), and, by (A.23), 2

=

Vl (b)

V1(0) =

-2 log b = Q, 1

(4.27)

?b2 0

on (1, r12),

'('1) < 0 and "(j) > 0

on (1,'12]. (4.36)

In view of (4.33a), the next step is to define

h('1) :_ (r1)

+ q(log 11 - 1),

log (r1) + log k + k

1 < q < 112,

1

ll

(4.37)

and to contemplate all solutions of h(r1) = A. Here h('1) = h('1;k,B) and we have k > 0, B > 0 and A < B; the parameter A may have either sign or be zero.

Remark 1 For each k > 0 and B > 0, h'(r1i) < 0

and

limn1

2

h'(q) = co.

(4.38)

Proof The definition (4.37) implies that

l h'(q) = C'('1) { log 01) + log k + k + 1 } + log r1.

(4.39)


131

11

Fig. 4.6.

For h = ril, we have (ql) = kris and [by (4.34) and (4.35)]

k+1 -1, -1=(log ril)2ril log rii B

whence h(hl)=-(k+1)- (TO

As nT rig,

+11

+1 '(n2) _

(1k

112 )2n2

oo because

-1

- 1 < 0. 109 q2

Remark 2 There is exactly one point no E (ri1,1J2) such that h'(i o) = 0. At this stationary point rfo, the restriction of h to N1,112] has a strict local and global minimum.

Proof Remark 1 shows that hl and rig are not stationary points of h, and that there is at least one stationary point, say 1o, in 01, 0. Thus


132

B

h(7)) .17 112

B - a(B)

Fig. 4.7.

h'(r1o) = 0; since '(r1o) < 0 by (4.36), and log no > 0, we infer from (4.39)

that log o(no) + log

1

1

+ + 1 > 0.

Also, "(rio) > 0 by (4.36), so that (

2

log x('10)+log+k+1}+

>0. +X10

Thus at any stationary point in ('j1,112) the function h has a strict local minimum. Hence there can be at most one such stationary point, and we have seen that there is at least one. Remark 3 The definitions of (r1), 11,112 and h(r1) imply that

h(nl) = (k+1)r1ilogi1 = B, h(112)

= 112(1og 112 - 1) = B - a(B),

where a is the function defined after (4.28). (iv) We can now collect ingredients for a list of solutions.

(a) Remarks 1 to 3 show that h is essentially as in Figure 4.7. The figure is misleading in that B - a(B) may have either sign or be zero (in fact, it is negative, zero or positive according as B < e, B = e or B > e),

4.5 Exercises

133

and in that negative values h"(q) occur only for large values of k, but the figure is sufficient for counting solutions of the equation h(r1) = A.

(b) A two-step solution V is determined completely by the solution n E (ni, n2) of h(r1) = A, because is then given by (4.34), the radii a and b are given by (4.32), and V is given by (4.30).

(c) The necessary and sufficient condition for existence of a one-step solution Vl is [by (4.29) and (4.32)] A 5 B - v(B). (v) Conclusions

The results in (iv) imply the following list of all

possible solutions. Recall that B > 0 and A < B by hypothesis, and that h(q) = h(r1;k,B).

(a) For B - Q(B) < A < B, there is one solution V; it is a two-step solution. (b) For A = B - a (B), there are two solutions: one has 7 E 01, u2) and

is a two-step solution, the other has 7 ='12, Vi (0) = a and is a one-step solution.

(c) For min, h(r1) < A < B - Q(B), there are two distinct two-step solutions and a one-step solution with Vl (0) < a.

(d) For A = min, h(r1), there is a two-step solution and a one-step solution with V1(0) < a. (e) For A < min, h(j), there is a one-step solution with V1(0) < a.

4.5 Exercises Exercise 4.16 Prove Lemma 4.5 for case (A) of Lemma 4.3 with 0 < in < 2.

[You may find it helpful to separate estimates of g for 4µh > s2,

m-1-6 >0from those for4ph>s2,m-1-8 0. Because of the identity

(V V)v=(Vxv)xv+V(zwI2)I

4 Symmetry for the Non-linear Poisson Equation in JRN

134

the equations of motion, under an extraneous force field -V(F, are

divv := V v = 0,/ curly := V x\v (4.40)

xv=-V(P+(F+IIV12 P

//

one calls ( the vorticity and p/p + (F + !Iv I2 a Bernoulli function (see Batchelor 1967, pp.74 and 160; Lamb 1932, pp. 4-6). (i) Flow independent of x3, with stream function W(xl, X2). Show that,

for any given functions W E C1(R) and H E C'(R), the velocity field and Bernoulli function v = (-821p, 01 W, W (W))

and

P + (F +

1

2

Iv12

= H(1V )

satisfy equations (4.40) if lp E C2(R2) and OW(x) = H' (W(x)) - W (W (x)) W' (W(x))

(4.41)

at all points x = (xl, x2) E JR2.

(ii) Cylindrically symmetric flow, with Stokes stream function 'P(xl, s). Write x = (x1, s cos 9, s sin 9), s > 0, for points of 1R3, and let

el := (1, 0, 0), e' := (0, cos 9, sin 9), es := (0, - sin 9, COO).

Show that, for any given functions F E C1(R) and H E C'(R), the velocity field and Bernoulli function

v = s (_OTael +

e' + r('t`)es)

(s > 0)

and

PP +(D+2wj2=H('I') satisfy equations (4.40) for s > 0 if T E C2 (R x (0, oo)) and z

(axi +

2

as2

- sa

'F = szH'('t') - T('`)T'('`)

(4.42)

at all points (x1, s) E R x (0, co).

Exercise 4.19 Transformation of the Hicks equation (4.42). Show that under the transformation 'P(xl, s) = s2X(xl, s), in which X is Greek capital chi, (4.42) becomes 2

z 5X

2

= (8xi + aS2 + s as X = H'(S2X) -

s2 r(s2X)r1(S2X ),

(4.43)

4.5 Exercises

135

where s > 0 and A5X = { (8/402+ +(a/8y5)2 }X if we write xl = yi + y5)'/2 (cf. Exercises 1.18 and A.25). In the context of generalized solutions, H and F need not be in C'(R). Let and s = (y22 +

H(t) = -2tfH(t) + const.

and

IF(t) = ytfH(t)

(4.44)

for all t E IR; here A,,y are constants (A > 0, y > 0) and fH is the Heaviside function (Chapter 0, (v)). Also, write

U = const. > 0.

X(xi, s) = x(xl, s) - U, 2

This corresponds to `P = W - 2 Us2, where V will be required to `vanish at infinity' while -2 Us2 is the Stokes stream function of velocity (U, 0, 0). Show that (4.43) becomes

o5x=-{.1+Y2(x-IU) }.fH(x-2U) at all points (x 1, s) such that s > 0 and x(xl, s)

(4.45)

2 U.

Exercise 4.20 Let W be a C2-solution of (4.41) on 1 co, either Vip(x) = -(wxl, wx2) + (V2, -Vi) + O(r-t),

where w = const. > 0,

Vl = const.,

V2 = const.,

(this corresponds, for the fluid, to 'solid-body' rotation about the origin plus a uniform velocity plus something smaller), or K

X2

Vip(x) _ --- (X1, X2) - (ci,-C2)l r4 where

K = const. > 0,

x22

-

cl = const.,

2x

Y4 2 +O(r 3),

c2 = const.,

(this could be due to negative vorticity in a bounded set).

(i) Show that the level sets of w are the point q where W has its maximum, and circles with centre q. Find q in terms of the data for each of the two cases.

(ii) The streamlines in R3 are the solution curves of dx/dt = v(x), where now x e ][R3 and v is as in Exercise 4.18, (i). Show that a streamline is either the line A := { (q 1, q2, x3) -oo < x3 < oc }, or a circle in I

a plane of constant x3 and with centre q, or a helix contained in a cylindrical surface with axis A.

136


Exercise 4.21 Consider the elliptic disk D := { (x, y) E R2 axe + fy2 < 1 }, where a and /3 are positive constants, and consider the function I

1 -axe _ fl y2 2(a + '

Wo(x, Y)

(x, Y) E D.

Q)

Note that /Wo = -1 in D, WO = 0 on 8D. (Thus WO is the stream function of flow in D with vorticity -1 and with velocity tangential to 8D.) We seek an extension W of Wo as follows: W E C1(R2) fl C2(R2 \ 8D),

W = Wo on D, /W = 0 in R2 \ D, W < 0 in R2 \ D and the asymptotic behaviour of W, as I(x,y)I -+ oo, is admissible (Definition 4.1). Prove that such a function W exists if and only if a = /3. If D is replaced by an arbitrary bounded open set S c R2, with smooth boundary 852, is there an analogous result?

Exercise 4.22 Let E be the Hilbert space formed by completion of the set in the norm I II defined by hull :=

(u,u),

where (u,v)

:=f

Vu- Vv. 5

The space E is embedded in Llo/3(R5). Referring to Exercise 4.19, we discard the cylindrical symmetry there; we call w :R5 -+ R a finite-energy solution of (4.45) if w E E \ {0} and

l

{A+y2(w-ZU)}fH(w-ZU)] =0

for all p E C'(R5). From these two properties of w it follows (by methods

beyond us here) that the equivalence class w E E has a representative w E C1(R5); that w has admissible asymptotic behaviour (Definition

4.1) of type (C), with m = 3 and 6 = 1; that w > 0 on R5 ; and that w E C2(R5 \ S), where S { y E R5 w(y) = 2 U I. Given these facts, deduce that every finite-energy solution w of (4.45) is as follows, for some point q E R5.

(i) If 2 > 0 and y = 0, define a := (15U/22)1/2; then

1U(5-3a2) if ra,

where r := IyI (cf. Example 4.14 with N = 5 and /1 = 2 U).

4.5 Exercises

137

(ii) If A > 0 and y > 0, a unique radius a is determined by

0 < ya < /Jl = 4.49...

and

yaJ512(ya)

J3/2(ya)

-

3 Uy2 2 2

where the Jv are Bessel functions (Exercises 1.19, 1.21) and IJ1 is the smallest positive zero of J3/2i then

3U AJ3/2(32)-B+3 2

w(Y + q) = xl(r)

(yr) /

if ra,

2 U a3

where r :=1Y1,

A :=

(ya) 1/2

J5/2(ya)

,

B :_

J312(ya)

yaJ512(ya)

Exercise 4.23 The uniqueness of Hill's vortex and of Hicks's vortex. Let F be the set of functions p : 1[8 x [0, oo) - II8 defined by OYn,s) = s2u(Yl,s) = s2u(Y),

s = (Y2 + ... + Ys )1/2,

where u is a cylindrically symmetric function in the space E (Exercise 4.22). Show that, if (p, u and gyp, v are such pairs of functions, then aW

ay)

aW

ff 1s2{ ayl ayl + 8s 8s

OW

} s dy1 ds = 1 J Du(Y) Vv(y) dY

Rx(o,oo)

2n2

R5

It follows that F can be made a Hilbert space; also, that a flow in R3 with Stokes stream function W E F (and with no velocity in the direction es) has finite kinetic energy. [Take (p = W.] Now consider all generalized solutions `P = W - 1 Use, with W E F, of (4.42) with H and IF as in (4.44). Prove that, for some constant c E JR and with r := (x1 + s2)1/2, ip(x1 + C, S) _

I s2xo(r) s2X1(r)

if 2 > 0 and y = 0, if 2 > 0 and > 0,

where Xo is as in (4.46) and xl as in (4.47). These functions were discovered, without reference to 1R5, by M.J.M. Hill (1894, for y = 0) and W.M. Hicks (1899, for y > 0). The streamlines of Hill's vortex are shown in Figure 4.8; for those of Hicks's vortex, see Moffatt (1969).

138

4 Symmetry for the Non-linear Poisson Equation in RN S

U Ì' = const. < 0

D s xl

Fig. 4.8.

Exercise 4.24 The equations of magnetohydrostatics relate the magnetic induction B : R3 1R3 to the current density j : 1183 -> R3 and pressure p 1[83 --+ 118 of an electrically conducting fluid in equilibrium. Under an idealization that is frequently made (Ferraro & Plumpton 1966, p.35; Thompson 1962, pp.47 and 53), these equations are :

curl B=µj, divB = 0, j x B = V(p + p4)), where the constants µ > 0 and p > 0 are respectively the magnetic permeability and density of the fluid, and -VI is an extraneous force field.

Translate (as far as it interests you to do so) the results of Exercises 4.18 to 4.23 into the language of magnetohydrostatics. Exercise 4.25 In Theorem 4.13, let

N = 1, g(t) = A + (µ - WO - a) for all t e I[8, where

0 a, the definition of a ensures that Y (a) c S2. Therefore X E S2, as desired.

The significance of the next little theorem is that the inclusion Y (p, k) c S2 extends from u > a(k) to y = a(k).

Theorem 5.3 If 92 is admissible for a given direction k, then for that direction the reflection in TT(k) of the alpha cap is in 0. In other words,

Y(a(k),k) c n. Proof Omitting the label k again, we wish to show that, if z E Z(a), then z" E S2. This follows from Lemma 5.2 with it = a. The remainder of this §5.2 leads to the result that a bounded region f2 with On of class C1 is admissible for every direction k. At the same time we establish properties of the outward unit normal n (to the boundary of such a set) that become useful in the context of Theorem 5.12, which concerns equation (5.2). This material may be of interest in its own right, but it is not needed for Theorems 5.10 and 5.11, which concern equation (5.1).

Definition 5.4 Let 8S2 be of class Cc for some I E No and let p e 852. As in Chapter 0, (viii), the set f2 is defined globally in terms of co-ordinates xj, while yj are `local' co-ordinates such that 8S2 has a representation YN = h(y'),

where y' := (Yl, ... , YN-1),

(5.3)

near the boundary point p. (a) The co-ordinate transformation will now be written y = Yp(x) := A(p)(x - p),

(5.4)

where A(p) is an orthogonal N x N matrix depending on p.

(b) Let Q'(0, p) :_ { y' E

RN-1 I

-p < yi < p, j = 1, ... , N - 1 }

denote a cube about the origin in RN-1, with edges of length 2p > 0. We define (Figure 5.2)

'rp,a(p):_{yERN Y EQ'(0,p), I

-a h(y').

(5.5b)

(The word rhomboid is used faute de mieux and because *-PAP) is approximately rhomboid when N = 2, e >- 1 and p is sufficiently small.) Exercise 5.5 Let 8S2 be of class Cc for some f >_ 1, let p E 892 and let the outward unit normal v(y') = A(p)n(x),

where y = A(p)(x - p),

146

5 Monotonicity of Positive Solutions in a Bounded Set S2

be as in equation (0.2). Abbreviating A(p) to A and writing Vx (a/ax,, ... , a/aXN), define new local co-ordinates z1, ... , zN by z = By = BA(x - p), where B is an orthogonal N x N matrix such that N

n(p) ' VXZN = V(0) ' V yZN = E vj(0)BNj * 0.

(5.6)

i=1

Prove the following.

(a) There is a set W, open in RN and containing p, such that aft fl W has a representation ZN = g(z'), z' E H, g E C'(H),

(5.7)

where H is open in JN-1 and convex. (b) There exist new rhomboid neighbourhoods, say Va,b(p) and its ximage Ua,b(p), characterized by z' E Q'(0, a) and by IZN - g(z')I < b, such that a statement analogous to (5.5a, b) holds when n(p) ' VXZN < 0(c) If the matrix B satisfies BN1 = -vj(0) for j = 1, ... , N, then ZN = -n(p) ' (x - p) for all x E ]RN,

(a1g)(0) = 0 for j = 1, ... , N - 1, (5.8)

so that the hyperplane { Z ZN = 0 } is tangent to aft at x = p (at z = 0), as is shown in Figure 5.3. I

[For a form of the implicit-function theorem that yields the Cc property of g in (5.7), and not merely C' smoothness, see Dieudonne 1969, p.272, or Fleming 1965, p.117]

Notation Until the contrary is stated, the unit vector k will be fixed; we adopt variables aligned with k (Definition 4.4). That is, we rotate the co-ordinate axes and use

xl =x'k, x" =(x2,...,xN)I Q:={5 EGRN

I xE1}, Z(µ):={zEsz z,>µ M:=sup{., I zE1 }

and so on. Again we omit the tilde of these aligned variables as long as there is no danger of confusion.

5.2 On the geometry of caps and reflected caps

147

x1

Fig. 5.3.

Lemma 5.6 Let n be a bounded region with M of class C'. Define

F:= { x E 3c

I

x1 =M1.

Let p E F and let l;j be the local co-ordinates called zj in Exercise 5.5, (c), so that the cN-axis is along the inward normal at p, and afl is described

near p by N = g('). Then

n(p) = el := (1,0,...,0) and

0

(5.9)

for all ' in the domain of g. Proof Observe first that M is the supremum of the continuous function x --> x1 not merely over fl, but also over the compact set S2 (Exercise 1.4); this supremum is attained in S2, but not in the open set 92. Thus F is not empty and is closed in RN. It is immediate that n(p) = e1 implies 0: if n(p) = e1, then

xl - M = -i;N = -g(l;') for x E ail and x near p; if g(l;') < 0 for some l;', then the definition of M is contradicted. That n(p) = el also seems self-evident: if n(p) * el (Figure 5.4), then there are points x in S2 near p for which x1 > M, so that the definition of

148

5 Monotonicity of Positive Solutions in a Bounded Set C1

TM

Fig. 5.4.

M is contradicted again. However, an analytical form of this argument is worthwhile.

= C(x - p), so that C is an

Write the co-ordinate transformation as orthogonal N x N matrix. Then

-(CNI, CN2, ... , CNN)

n(p) =

(5.10)

and

N-1

xl - M =

Cil j + CNIg( ')

and x near p. (5.11)

for x E 3f

i=1

Here g(l;') = o(Il;'I) as

- 0, because

g() - g(0) _ .f r

(t

l

dt = Jo

r

(Vg)(tr) dt

with g(0) = 0, (Vg)(0) = 0 by (5.8) and with Vg continuous by (5.7).

Assume that C1 0 for some m c {1,. .. , N - 11. Set l;' = (.... )in(5.11),with ,>0ifCl 0 such that, if µ E (M - 6, M), then (a) Y (p) =!Q, which implies that n is admissible for the arbitrary direction k,

(b) nl(x) > 0 for all x E I'µ.

Proof (i) Let F be as in Lemma 5.6. We shall use Exercise 5.5 again. The rhomboid neighbourhoods Ua,b(P) of all points p e F form an open cover of the compact set F. [That is, each set Ua,b(P) is open and F is a subset of the union of all of them.] We extract a finite subcover, say { Ua,b(pr)

I

r = 1,2,...,s },

where a = a(pr) and b = b(pr). [When F is a finite set, this extraction is unnecessary.] Figure 5.5 shows one set of the finite subcover; the local co-ordinates l;j are as in Lemma 5.6. Now define G

U Ua,b(Pr), r=1

M-s:=sup{x1 I xE J\G}=max{x, I xES2\G,}, 6 :=min{ s,2b(pl),...,Zb(ps) In describing M - e, we have used Exercise 1.4 once more. Note that e > 0, otherwise the definition of F would be contradicted, since G covers F.

(ii) Let p E (M - 6, M). We wish to show that Y (p) c S2; equivalently, that

z E 0 and z, > µ

z" E .Q.

150

5 Monotonicity of Positive Solutions in a Bounded Set 92

aUa b(Pr)

Fig. 5.5.

Now the conditions zl > It, u > M - 6 and 6 < e imply that zi > M - s, so that z ¢ 0 \ G. Therefore Z E S n G, whence z E S2 n Ua,b(pr) for some

r E {1,...,s}. Returning to the co-ordinate transformation = C(p)(x-p) used in Lemma 5.6, we define

( := C(pr)(z - pr),

j := C(pr)(z' - pr).

(5.12)

We shall show that g(0 < ?IN < b, where now g = g(. , pr) and b = b(pr) for the particular r introduced before (5.12); then zµ E f2. By Lemma 5.6, N = M - x1 when p E F; therefore reflection in Tµ is reflection in { I N = M - u }. Accordingly,

qN = 2(M - µ) - N,

(5.13)

in which 2(M - p) < 26 g(Z'') >- 0, so that 1N < b. Next, by (5.13) and repeated use of the inequality N = M-z1 < M-p, '1N > M - µ > SN > gW) = g(11'), since n' by (5.12). The results '1N g('1') now show (as we noted above) that zµ E S2, and hence that Y (y) c S2.

(iii) It remains to consider x E Fµ for P E (M - 6, M) and to show that n1(x) > 0. [Recall that F. := 3Z(µ) \ Tµ and that the identity Z(p) = S2 n { x x1 > p } implies that 8Z(µ) (-- Of) U Tµ; therefore f1, c 8S2 and here x E Of).] Since 6 p. > M - s; then x 0 0 \ G, by the definition of M - s. Consequently x E 3) n G, I

5.2 On the geometry of caps and reflected caps

151

so that x E OS2 n Ua,b(p') for some r E {1,...,s}. Equivalently, under the transformation = C(p') (x - pr) we have E Va,b(p'), N = g = g(., p'), and

nl(x)=-VN()=

1

{ (alg)(')2 + ... + (aN-ig)(')2 +

>0.

111/2

The condition established in Theorem 5.7 that nI(x) > 0 for all x E I'µ is a useful property of the right-hand boundary of a cap. It prompts the following definition, which is phrased in terms of our original variables rather than in terms of those aligned with k. Definition 5.8 Let 92 be a bounded region with On of class C1. For every unit vector k E RN we define

1(k) := inf{ v I µ E (o, M(k))

n(x) k > 0 for all x E T'N,(k)

/3(k) := max{ a(k), 1(k) },

and call Z (/3(k), k) the beta cap for the direction k.

Figure 5.6 illustrates this definition for k = e1. The set in part (a) of the figure is 521 :=

2

x E 1182

I

(XI - cxi) + x2 < 1, xl < 1/2c }

,

c = const. E (0, 4) , (5.14)

and we define \/ l

4c

(5.15)

.

The reader should verify that, for the set 521 and the direction e1,

the critical values of p are a (e') = y > 0 and 1 (e') = 0, so that

Q(el)=a(e') The set in part (b) of Figure 5.6 is 922

x E 1[82

I

(xl + cx21)2 + x2 < 1, x1 > -1/2c

},

c = const. E (0, 4) ,

(5.16)

and now a (el) = 0 and A (e') = 0. In part (c) of Figure 5.6 a small protuberance has been added to 922 near the point (M,0), where M := sup{ xl X E 922 }. This can be done I

152

5 Monotonicity of Positive Solutions in a Bounded Set Q

Fig. 5.6.

in such a way that a (el) is unchanged, 2 (e') is greatly increased, 00 remains of class Coo and 00 is unchanged for x1
A. Hence x E I'µ for some y > 2 [for example, it = 1(x, + A)] ; the definition of 2 now ensures that n, (x) > 0.

(ii) Assume for contradiction that p E F,, and n,(p) < 0. By this last, z := p+Eel E 0 and y := p-eel V SZ for sufficiently small s > 0. Choose ,u = p1; then It > a, z E Z(µ) and zµ = y V S). This contradicts the definition of a.

5.3 Monotonicity in f2

153

Remark 5.9 implies that, when A(k) > a(k), we have n(x) k > 0 for all x E I'fiki(k) and n(x) k >- 0 for all x E I'a(k)(k). Although this distinction may seem slight, it makes certain proofs much easier for beta caps than for alpha caps.

5.3 Monotonicity in S2

Our first monotonicity theorem, for sets f2 that may lack symmetry, is an

extension of Theorem 3.6 that takes longer to state than to prove. The reader who finds the proof somewhat glib is invited (in Exercise 5.25) to spell out the corresponding extension of Theorem 3.3. This smaller task may illuminate the present proof. Theorem 5.10 Let 0 be a bounded region that is admissible for the direction

el (Definition 5.1); we abbreviate T. (e'), Z (p,el), a (et),... to Tµ, Z(p), a,.... Assume that u : S2 -+ l1 has the following properties. (a) u E C(92) n C1(f2), u > 0 in f2, u = 0 on I'a. (b) For all (p E C,"°(S2)),

{-V

Vu+cpf(u)} =0,

(5.17)

where f has a decomposition f = f 1 + f2 such that f 1

: [0, oo) -* R is locally Lipschitz continuous, while f2 : [0, oo) -+ d8 is non-decreasing and is identically 0 on [0, ic] for some 1c > 0.

Then the conclusions of Theorem 3.6 hold in the alpha cap:

u(z) a and x E f2.

(5.19)

Proof Scrutiny of the proof of Theorem 3.6 shows that there we used only conditions that have counterparts here: rather than the full Steiner symmetry of SZ relative to To, we used only inclusions implied here by Lemma 5.2; rather than the full boundary condition u = 0 on 8f2, we used only that u = 0 on F0. (Of course, the full conditions were used for Corollary 3.9.) Therefore the proof of Theorem 3.6 extends to the present situation, provided that the earlier interval (0, M) is replaced by the present (a, M). Theorem 5.11 Let f2 and u be as in Theorem 5.10. Suppose that, in addition,

(01u)(xo) = 0 at some point xo E T,, ma.

(5.20)

154

5 Monotonicity of Positive Solutions in a Bounded Set C1

Let Zo(a) be the component of Z(a) that contains xo+Eel for sufficiently small e > 0, and let Yo(a) be the reflection of Zo(a) in T. Then u(xa) S2

= u(x) =

for all x E Zo(a),

Yo(a) U Zo(a).

(5.21) (5.22)

Proof (i) Once again let w(., u) := u-uµ on Z(µ), this time for µ E [a, M). Fix Z E Z(a) and let µ 1 a. Since u(z) < u(zµ) for µ E (a, z1), by (5.18), and since u(zµ) varies continuously for fixed z and varying p, we have w(z, a) < 0. Lemma 3.8 now states that Aw - Aw >- 0

in Z(a) in the generalized sense,

(5.23)

where w = w(z, a) and the constant A is described in Lemma 3.8.

(ii) Suppose for contradiction that w(zo, a) 0 at some point zo E Zo(a). Then w(., a) < 0 everywhere in Zo(a) [because, if the supremum 0 of w(., a) were attained in Zo(a), then w(., a) would be the zero function

in Zo(a), by (5.23) and the strong maximum principle, whereas our hypothesis is that w(zo, a) * 0]. Now consider the point xo in (5.20), observing that w(xo, a) = 0 because

xo E T. We apply the boundary-point lemma for balls (Lemma 2.12) to the function w(., a) and the ball B := -4 (xo + pet, p), with p so small that B c Zo(a). It follows that (01 w) (xo, a) < 0, which contradicts (5.20) because (81w)(xo,a) = 2(81u)(xo). Accordingly, w(z,a) = 0 first for all z E Zo(oc), and then, by continuity, for all z e Zo(a); thus we have proved (5.21). (iii) It remains to prove (5.22); let q E 8Yo(a) \ T. Then q" E 8Zo(a) \ Ta c F., so that u (q") = 0 by hypothesis (a) of Theorem 5.10. It now follows from (5.21), with x = q", that u(q) = u (q") = 0. But then q E Ofl [otherwise we would have u(q) > 0]; in other words,

8Yo(a) \ T,,

852.

(5.24)

This implies, since 0 is connected, that S2 = Yo(a) U Zo(a). Some readers may regard this implication of (5.24) as self-evident; others may consider our claim to be too intuitive or too condensed. A detailed proof is offered in §5.4.

We turn now to an analogue of Theorem 5.10 for the equation

Au+b181u+f(u)=0 in 0,

(5.25)

5.3 Monotonicity in 52

155

in which b1 = const. >- 0. For equation (5.2) with b * 0, we have chosen

variables aligned with b/jbj; in any case, the unit vector k = el. For the moment we suppose that f is locally Lipschitz continuous. The new feature is this: upon setting w(., p) := u - uµ on Z(p) for p E [a, M), we obtain the equation ,Lw - b181 w + y(z, p)w = -2b1(31u) (z)

for z E Z(p),

(5.26)

in place of (3.6); again w = w(z, p) and y is as in (3.5). When b1 > 0, we can infer that w(. , p) is a subsolution, relative to 0 - b181 + y and Z (P), only if we know that 81 u < 0 in Z (p). It is sufficient in the first instance to know that 81u < 0 in some neighbourhood of F,, but to prove even this is not easy if Fis not smooth, or if IF,, is smooth but contains points x at which n1(x) = 0, or if f (0) < 0. Therefore we restrict attention to boundaries slightly smoother than those of class C1, to beta caps and their right-hand boundaries Tp, and to functions f with f (0) >- 0. Theorem 5.12 Let SZ be a bounded region with 852 of class C' and with the interior-ball property (Definition 2.14) at every point of rp. Here I'p is the right-hand boundary of the beta cap Z(/3) for the direction e1 (Definitions 5.1 and 5.8) and T,, (e'), Z (µ,e1), /3 (e1) ,... are abbreviated to Tµ, Zµ, .... Assume that u : S2 --> R has the following properties.

(a)uEC1(4), u>0inn, u=0onF#. (b) For all cp E C'(52),

-VT - Vu+cpb181u+cpf(u)} =0, where b, = const.

(5.27)

0, f (0) >- 0 and f is otherwise as in Theorems 3.6 and

5.10.

Then the usual monotonicity conditions hold in the beta cap: u(z) /3

and

x E 52.

(5.28) (5.29)

Setting w(. , p) := u - uµ on Z (p) for y E [/3, M), we wish to prove that

z E Z(p)

(01u) (z) < 0 and w(z,,u) := u(z) - u (zµ) < 0

(**)

whenever p E (/3, M). The proof of Theorem 3.6 involved a boundary strip Zh(p), a subset of Z(p) having width h independent of p, with the property that u(x) < iC for x E Zh(p). That h was independent of p was a luxury rather than an essential ingredient of the proof. This time, we shall use a boundary strip WI(p), still within Z(p), with the property that

156

5 Monotonicity of Positive Solutions in a Bounded Set 0

both a1u(x) < 0 and u(x) < lc whenever x E W,(/.t); as far as we know a priori, it might be that the width q(p) 10 asp . f3. This prospect may be uncomfortable, but we shall find that it is not catastrophic. In fact, Lemmas 5.13 and 5.14 will almost reduce the proof of Theorem 5.12 to that of Theorem 3.6. Notation Once again y(., p)

f1(u) - f 1(uµ)

at points z where u(z)

0

at points z where u(z) = uµ(z).

u - uµ

uµ(z),

(5.30)

Since u vanishes on I'p and is uniformly continuous on n,

3 h > 0 such that

dist (x,rp) < h and x E 12

u(x) < K. (5.31)

We write

w, (/,,) := Z(u) n { x I dist(x,r) < q(p) }

(5.32)

for the boundary strip, to be constructed in Lemma 5.13, with the additional property that a1u < 0 in W,(p). Lemma 5.13 For each p E (f, M) there is a number ri(p) > 0 such that

(atu) (z) < 0 and u(z) < K.

z E WW(p)

(5.33)

It follows that ,Lw - b181w + y(z, p)w >- 0

in W,, (,u) in the generalized sense,

(5.34)

where w = w(z, p) and y is defined by (5.30).

Proof (i) Let p E (fl, M) and p E I'µ. Then p E fp; since /1 >- A, Remark

5.9 shows that nt(p) > 0. We shall use the boundary-point lemma to prove that (0 1u) (p) < 0. By hypothesis there is a ball B c Q such that pEaB;then, forallcpEC,11°(B)with 'p 0,

-

f{-Vco-Vu+coblalul

= - f cp{[ft(u)-ft(0)]+.ft(0)+f2(u)} < fco{Au_fi(o)-f2(u)},

where A is our usual Lipschitz constant for f t on the interval [0, supra u], so that A >- 0. Since f 1(0) >- 0 and f 2(u) >- 0 by hypothesis,

u(p) = 0 for all x E B, and el is an outward unit vector at p because ni(p) > 0. Thus Lemma 2.12 implies that (alu)(p) < 0. (ii) Let µ E [a, M). Calculating as in the proof of Lemma 3.7, step (i), and abbreviating Z(p) to Z, one finds that, for all (p E C,(Z),

J

{ -VVw-Wb1aiw+w[fi(u)-.fi(ua)] } dz =

+ [f2(u) - f2(uµ)] } dz, z where w = w(z, µ), p = lp(z), ai = alazi and so on. -Jip{2b,

(5.35)

(iii) Let µ E (/3,M). We have shown in (i) that aiu < 0 on the compact set t ; by hypothesis, aiu is (uniformly) continuous on S2. Hence a, u< -c(µ) on ,,, for some c(µ) > 0, and then there is a number

p(µ) > 0 such that (aiu)(z) < 0 if z E Z(µ) and dist(z,f) < p(µ). Choose ii(y) to be the smaller of p(µ) and the h in (5.31), observing that dist(z,I'f) < dist(z,r,) because f,, c I'p. With this choice of q(µ), condition (5.33) holds.

Now restrict (5.35) to test functions cp E C°°(W,(µ)) with 9 > 0. On

the right-hand side, (aiu)(z) < 0 and f2(u(z)) = 0 for z E WI(µ), by (5.33), while f2(ul,(z)) >- 0 because f2 >- 0 on [0,00). Thus the right-hand member of (5.35) is non-negative, which implies (5.34).

Lemma 5.14 If µ e [/3,M) and if, for all z E Z(µ), both (01u)(z) < 0 and w(z, µ) < 0, then

Lw - b1a1w - Aw >- 0

in Z(µ) in the generalized sense;

(5.36)

here w = w(z, µ) and A is as in Lemma 3.8.

Proof If cp > 0 in (5.35), then -cobiaiu >- 0 in Z(µ) by hypothesis. The rest follows from the proof of Lemma 3.8.

Proof of Theorem 5.12 (i) In order to prove that (**) holds for all sufficiently small, positive M - µ, we modify the proof of Theorem 3.6, step (i), as follows. First,

M-I

2M I< < M

dist(z,I'/)
_

M

an d

2

I

Z( p)I

m. Second, we obtain w(z,m) < 0 for z E Z(m) by fixing z E Z(m) and letting y 1 m. Then Lemma 5.14 and the strong maximum principle imply that w(z, m) < 0

for z E Z (m).

(5.37)

The condition alu < 0 now extends to T,,, fl 52 by application of the boundary-point lemma to w(.,m) and to balls in Z(m) with boundary meeting Tm; therefore,

a1u < 0

on Z(m) \ M.

(5.38)

(b) Writing wE := w(., m - e) for 0 < E < Eo, we shall obtain a contradiction by showing that a1u < 0 and wE < 0 in Z(m - E) if Eo is sufficiently small. Provided that m - eo > fl, we already have that a1 u < 0 in W,1(m - E), by Lemma 5.13. The only changes from the proof of Theorem 3.6 are that Ow-b1a1w replaces Lw in the two lemmas and that now we must so choose E, and

F that

GE :=Z(m-s)\(EEUF) e W,,(m-s).

5.4 A little topology

159

For this last, the main step is a suitable choice of the cross-section S of the cylindrical set E. To make this choice, we refer to (3.29), replace the h there by rl (2 (/3 + m)), and demand that the conditions on so include

m-Eo>- 2(Q+m). Then rl( z) > rl (z (/3 + m)) for m -to < p < M [by the second observation in (i)].

We already have alu < 0 on F, by (5.38), and the earlier construction gives a1 u < 0 on EE ; that w, < 0 on EE U F is argued as before. Theorem 5.15 Let 91 and u be as in Theorem 5.12. Suppose that, in addition,

(a1u)(xo) = 0

at some point

xo E Tp n Q.

(5.39)

Let Zo(/3) be the component of Z(/3) that contains xo + sel for sufficiently small s > 0, and let Yo(/3) be the reflection of Zo(f3) in Tp. Then

u(x'3) = u(x) for all

x E Zo(/3),

K2 = Yo (fl) U Zo(/3),

(5.40) (5.41)

and the coefficient b1 = 0.

Proof The proof of Theorem 5.11 remains valid here if we replace the interval (a, M) by (fl, M), Theorem 5.10 by Theorem 5.12, and Lemma 3.8 by Lemma 5.14.

It remains to prove that b1 = 0. Setting µ = /3 in (5.35), we observe from (5.40) and (5.41) that u = uil on Z(/3) and hence w(.,/3) = 0 on Z(/3); therefore (5.35) with µ = /3 reduces to ipb,Oju = 0

for all

cp c=

z (a)

Since 31u < 0 in Z(/3), by (5.29), it follows from a result of Exercise 1.16 that b1 = 0.

5.4 A little topology Remark 5.16 This section is concerned only with a full proof of (5.22) in Theorem 5.11. In addition to the basic fact that S2 is open, bounded and connected, we have the following data. (A) Zo := Zo(a) is a component of the alpha cap Z := Z(a).

160

5 Monotonicity of Positive Solutions in a Bounded Set E2

(B) Yo := Yo(a) is the reflection in Ta of Zo (so that Yo c S) by Theorem 5.3).

(C)8Yo\Tj c8S2. The question is whether these conditions imply that 0 = Yo U Zo. We recall that, for sets A and B in a metric space,

AUB=AUB, AnBcAnB, 8(AUB) c 8AU3B, 8(AnB) c 8AU0B.

Lemma 5.17 Let C be a connected, non-empty set in a metric space (M, d).

If A c C and C n 8A is empty, then either A = 0 or A = C.

Proof The pair (C,dI c.c) is also a metric space, and C = into A U 8cA U extc A,

where intc, 8c and extc denote respectively the interior, boundary and exterior relative to C. Given the hint in Exercise 5.26, one checks without difficulty that 8cA c C n 8A (where 8A is the boundary of A relative to M), so that 8cA is empty. Since C is connected, it cannot be the union of two disjoint, non-empty sets that are open relative to C; one of intc A and extc A must be empty. If intc A is empty, then

AcC=extcA=intc(C\A) cC\A

A=0.

If extc A is empty, then

AcC=intcAcA

A=C.

Lemma 5.18 Let D and E be sets in a metric space, with D connected, non-empty and open. If D intersects E but not 8E, then D c E. Proof Let A := D n E. Then A c D and D n 8A is empty because

DnOA cDn (8DU8E) =DnBE =0. Accordingly, we may apply Lemma 5.17 with C = D : either D n E = 0 or D n E = D. The former is contrary to hypothesis; the latter implies

that D c E.

5.4 A little topology

161

Lemma 5.19 If { Gt t E T } is a family of disjoint open sets in a metric space and G := UtET Gt, then each aGt c aG. I

Proof Let x E 3Gs for some s E T. Then X E G, x

GS and, since

every open set containing x intersects Gs, we have x 0 G, if t * s. Thus x E G \ G. But G \ G = aG because G is open. O Proposition 5.20 With 0, Zo and Yo as in Remark 5.16, define

A := Yo uZou (S2naZo) u (S2n3Yo).

(5.42)

Then A = 0, which implies that S2 = A = Yo U Zo.

Proof (i) That A = Yo U Zo is immediate from the definition (5.42). We

shall prove that A = Q by means of Lemma 5.17 (with C = 0 and M = RN). First, A c 0 because Yo c n by (B) of Remark 5.16 and Zo Q by (A). Second, we shall prove in (ii) that A is open in RN ; therefore,

aA=A\A = Y0UZ0\{Y0UZ0U(nnaZ0)U(12naYo)} = a(YouZo) \ { (12nazo) u (On OYO) }

c

(aYO u aZo) \ { (s2 n azo) u (s2 n aYo) }

,

which shows that S2 n OA is empty. Thus Lemma 5.17 applies; since A is not empty, we have A = 0.

(ii) To prove that A is open in RN, we consider separately three types of point in A. (a) Let X E Yo U Zo. Then X E int A because Yo U Zo is open. (b) Let p E 0 n aZ0. Now aZo c OZ by Lemma 5.19, and OZ c OOUTa because Z = 0 n { x I xl > a 1; since 0 and Of) are disjoint, p E s2 n Ta.

Hence there exists a radius p = p(p) > 0 such that 9(p, p) c 0. To show that M(p, p) c A, we consider B+ S

B_

.(p,p)n{x xl>a}, M(P, p) n T«,

P(p,p)n{x x, 0 and x E 92, and that u (xl, x2) = u (xl, -x2) for all x E SZ. Establish the signs of 81 u and 02u throughout S2, except for the sign of alu in the open subset { x E 92 I -y < xl < 0 }, where y is as in (5.15). Infer that the maximum of u can occur only in the line segment (-y, 0) x {0}. Exercise 5.24 Let 0 and .1 = 2 (el) be as in Definition 5.8. Prove that there exists a point xo E TT n 00 such that nI(xo) < 0; moreover, such that nl(xo) = 0 or nl(xo) = -1. Exercise 5.25 State a result that extends Theorem 3.3 in the way that Theorem 5.10 extends Theorem 3.6. Prove your result by adjusting the notation in the proof of Theorem 3.3.

Exercise 5.26 Let A c B c M, where (M, d) is a metric space and B is not empty. Write 8A and BBA, respectively, for the boundary of A in M

and the boundary of A relative to B. Prove that 8BA c B n 8A, with equality if B is open in M. Give an example of inequality. [A set D c B is open relative to B if and only if D = B n G for some G open in M. For a set S in a metric space (X, d), a point p belongs to 7S if and only if every open set containing p intersects both S and X \ S.]

Exercise 5.27 Define an annulus or spherical shell by A

x E RN

a< IxJ 2, 0 < a < b. Suppose that u E C(A) n C2(A),

u > 0 in A and u = 0 on 8.(0, b), Du + f (u) = 0

where f

:

in

A,

[0, oo) -> l[8 is locally Lipschitz continuous.

(5.43a) (5.43b)

Let 8/8r

(x/IxJ) . V.

Prove that, if x, y c A with xJ E ((a + b)/2, b), y/lyl = x/IxI and JyI+IxI =a+b, then au

(x) < 0 and u(x) < u(y);

(5.44a)

also that Or ( x

)- 0).

Show that, for this case, (5.44a, b) are satisfied amply when alb is small, but are satisfied only by a hair's breadth when alb is very close to 1. [A small part of Exercise 1.21 is relevant.] Exercise 5.29 This exercise prepares for the description, in Exercise 5.30, of a solution of (5.43a, b, c) that is not spherically symmetric. It is to be hoped that the essence of the argument emerges, even though parts of it are beyond the range of this book. With A as in Exercise 5.27, let H be the real Hilbert space formed by completion of the set C,(A) in the norm II . 11 defined by IlvII :=

(v,w) :=

where

(v,v),

fVv

Vw.

(i) Derive, or improve on, the following inequalities. (a) If V E H and is spherically symmetric, say r := lxl and v(r) := v(x), then IIvIIZ

D(r)2

q(r),

where q(r)

r-a

if a < r < z(a+b),

b - r if 2(a+b) 0,

1

1

Ix + cl

Ix - cl

(A.9)

'

c = (y, 0, 0) with y > 0,

and by then showing that this cp is the velocity potential of flow past a certain body E called a Rankine solid (Figure A.3). Note that the three terms in (A.9) are the potentials, respectively, of a uniform stream with velocity (U, 0, 0), of a point source at -c of strength q and of a point source at c of strength -q. Thus cp is defined on R3 \ {-c, c}, indeed, cp E C110(R3 \ {-c,c}), but we are interested mainly in the velocity -VT on and outside E. It is far from obvious that the formula (A.9) implies

the picture in Figure A.3; before taking up this question in (vi), we consider a limiting case. Set q = k/2y, and let y --> 0 with k fixed and positive, and with x * 0. Then q

1

1

47r

Ix + cl

Ix - cl

_

l

k ((al 1)

4n 2y 47

xl /

xj

2y + 0(72) }

47r IX13

J

(A.10)

172

Appendix A. On the Newtonian Potential s = (x2 + x3 )1'2

W- x1

tit --Ot-

a

Fig. A.3.

This is the potential of a dipole (or doublet) at the origin, of strength k and type (-1,0,0), or of strength -k and type (1,0,0); the type states the direction from negative source to positive source before the limit is taken. This function satisfies the Laplace equation in R3 \ {0}, because A and 01 commute when acting on C3 functions. Now define spherical co-ordinates r, 0, 2 by x = (xl, x2i x3) = (r cos 0, r sin 0 cos A, r sin 0 sin A),

0, 050Sn, -it 0; (A.17) dr = l a_4N(0, r) I

-

andK(x;N) is to vanishforr=Oif N= 1,forr=1 if N=2, and as r --> co if N > 3. Accordingly,

-ZIxI

K(x;N) .--

in

118,

in R2 \ {0},

2 log I

(A.18a)

1

where NJG "2

1

KN

(N - 2) Ia.N(O,1)I

(N

(N/2)!. (A.18b)

2),

The formula for the surface area of the unit sphere is a result of Exercise 1.14. Note that the formula stated for N >- 3, in (A.18a), is also valid for N = 1. The basic properties (A.2) and (A.3) of (D are unchanged, and are proved exactly as before.

AK(x - c; N) = 0

for x E RN \ {c}.

(A.19)

If 52 is a bounded open set in 1[8N that contains the point c, and a52 is of class C1, then

8K (x-c;N) = 1

(A.20)

here n denotes the outward unit normal on 852.

The Newtonian potential Let G be a bounded open set in RN and let

Appendix A. On the Newtonian Potential

176

f:G

R be a given function. Provided that the integral exists for all

x E RN, we call u(x)

:=.f K(x-l;;N) c

d

(x E RN, d = dg1... di;N) (A.21)

the Newtonian potential of the density function f . In classical physics the following hypothesis is made. The rule that

11:K(x-i;j;3)Qj, K

xE][83

J=1

is the electric potential at x of charges Q j at the points J, extends to the rule that (1/1c)u(x) is the potential, at all x E R3, of charge density f. Here f is measured in units of charge per volume. Essentially the same

hypothesis is made for the gravitational potential of a density f (now measured in units of mass per volume). One often calls cj or a source point, and x afield point. The new feature, that the potential is defined at points where there is charge or mass, prompts the question: is there a simple relationship between u and f at points of G? A short answer is that the Poisson equation -L u = f holds in G, provided that f is sufficiently smooth; this will be proved in §A.5. In this section we shall give an answer that is less direct but is both more general and easier to prove: if f has a certain integrability property, then a weak form of the Poisson equation holds in G.

Notation Henceforth we omit the label N when its absence causes no confusion. For some purposes the notation in u(xo) = jK(xo - x) f (x) dx,

xo E RN,

(A.22a)

is more convenient than that in (A.21); accompanying symbols are r := IxI,

ro := Ixol,

R := Ixo - xI,

(A.22b)

so that, for example, K(x - xo) = KNR-N+2 if N * 2.

There are a few cases for which the integral defining the Newtonian potential can be evaluated explicitly (for which the potential can be found ìn closed form'). Of these, the case of constant density in a ball is the simplest and perhaps the most valuable; we state the result and then explain its derivation in a sequence of remarks.

A.2 The Newtonian potential: first steps

177

The potential of unit density in a ball Let B :_ 9(0, a); then

ux

JB

K x-

cN(a), r< a,

2N

d

(A.23a)

r >: a,

IBIK(x),

where IBI = ir N12aN/(N/2)! denotes the volume of B, and

- i1a 2 cN(a) =

a2

if N = 1, 1

if N = 2,

(log a + 2 I

(A.23b)

Remarks 1. For N = 1 the result (A.23) follows from any easy exercise

in integration. The same is true for N = 2 if we use the notation in (A.22) and the series log

1

1

Iz

Re log = log z - zo - zol =

1

ro

°°

1

+En

r

cos n(B - 80), (-)o

r < ro, ) (A.24)

where z = xl + ix2 = re'0. [If r > ro, we interchange z and zo.] For N = 3 we choose, for source points x, spherical co-ordinates r, 0, A such

that the axis {0 = 0} passes through the field point xo (Figure A.5). The Fubini theorem and a theorem about co-ordinate transformations (Apostol 1974, p.421; Weir 1973, p.158) allow us to write " 2 / n (r2 sin g de uB (xo) = 47r , r dr J 2 + ro - 2rro cos 0) 1/2 0 0

-

f

" rz

d. ,

and this is an elementary integral. These integrations yield (A.23) for N = 1, 2,3 (the properly sceptical reader is invited to verify this). The function UB for N = 3 is shown in Figure A.6. One easily checks that, for all N < 3, both UB and duB / dr are continuous at r = a; hence UB E C'(]RN) for N< 3.

2. Observe that for r >- a the potential UB is merely that of a point source at the origin of strength IBI, and that - ILuB =

(02+...+02N)x1+ 0

at least for N < 3.

+xN 2N

-1 ifra,

(A.25)


178

Fig. A.S.

a2/2

a213

a

Fig. A.6.

We now proceed backwards. The result (A.25) can be regarded as an ordinary differential equation in terms of r, and one can use this equation to construct, with relatively little labour, a tentative formula for

all values of N. However, we shall exclude the case N = 1, because a third condition àt infinity' would be required in what follows. Using a formula in Exercise 1.18 for A acting on functions of r alone, we seek V(r) such that

LV=

d

rN 1 dr

(rN_1 dr

V=

-1, 00 as r-3oo if N=2, V(r)-30 as r ->oo if N>-3,


179

and such that V E C1 [0, oo) n C2([0, oo) \ {a}). This last condition fixes cer-

tain constants that are unknown initially and rules out singular behaviour at the origin. An elementary calculation now yields z

... AT-7 .f'_

Vlvlk, I -

l

//

J -4 +2a2( Aloga+2 I, r a;

r2

for N>-3:

2N+2a N-2' r 4, where uB(x)

denotes the integral in (A.23a). This follows from Theorem A.2 below, which gives a second characterization of the Newtonian potential when f and aG are exceptionally smooth. One easily verifies that, with G = B and v(x) = V(r), the hypotheses of Theorem A.2 are satisfied. The theorem has other applications, because one may well ask: when is a given function v the Newtonian potential of -AvIG? 4. We note the following in connection with condition (c) of Theorem A.2. If, as r := xI -> oo, IVw(x)I = 0(r-1-m) and w(x) -> 0, where the constant m > 0, then w(x) = 0(r-m) as r -+ oo. The reason is this: there are positive constants C1 and C2 such that, for r > C1, 00 dw(tx)

1w(X)1 = 1

dt

dt

r <J

00

co

x (Vw)(tx) dt C2r-m

IxIC2t1-mjxj-1-m

1

dt =

m

In Theorem A.2 the condition Av E Lp(G), p > N/2, is needed because condition (a) does not restrict sufficiently the behaviour of 5.

(Lv)(x) as x approaches

G.

Theorem A.2 Let G be a bounded open subset of RN with N > 2; suppose

that either 8G is of class C1 or G is listed in Remark D.4. Consider a function v : RN -+ R with the following properties.


180

(a) v E C1(RN) n C2(1RN \ aG). (b) Lv E LP(G) for some p > N/2, and L v = 0 in RN \ G. (c) As r := lxi --* oo,

Vv(x) = O(r) and v(x) -> 0 if N

3,

V{ v(x) - cK(x) } = O(r2) and v(x) - cK(x) ---> 0 if N = 2, for some constant c. T h e n v is the Newtonian potential o f -Ov I G

Proof (i) In this first step, v need not be the function specified above. Let 0 be a bounded open set with 3f of class C1, or else a set listed in Remark D.4, and let v and w be in C2(52). Then the divergence theorem for the vector field vVw states that

j{Vv.Vw+vL.w}=fv. Interchanging v and w, then subtracting the result, we obtain the Green identity

fn

{vtw-wpv}=fn {van-wan}. a

(A.26)

JJ

(ii) Let xo be a given field point in RN \ aG, arbitrary apart from x0

aG, but fixed henceforth. If xo E G, we define (Figure A.7) GE := G \ °(xo, s),

HA := -4(0, A) \ G,

where E is so small that °R(xo, E) c G and A is so large that G c -4(0, A).

If xo E R' \ G, we define HA as before, but with A now so large that G U {xo} c -4(0, A), and use the sets G and HA,c := HA \ Mxo, e),

where a is so small that -V(xo, e) c HA. From now until step (v), we suppose that xo E G; for the other case, the argument is changed only slightly.

(iii) Let GE,,,, and HA,,,, be mth approximations to G. and HA, respectively, in the sense of Theorem D.9. We apply the Green identity (A.26) to GE,,,, and HA,,,,, choosing v to be as in the statement of the present theorem, and choosing w(x) = K(x - xo) =: Ko(x), say. [We cannot apply the Green identity immediately to GE and HA, because v need not


181

a-4 (0, A)

Fig. A.7.

be in C2(Ge) and in C2(HA).] Since AK° = 0 in RN \ {x°} and Av = 0 in RN \ G, there results

-

/

K°

v

f I

Gem

0 =

OK°

8v

8n

- K 0 an

8K° 8n

- K°

v

V

HA,

(A . 27a)

'

8v 1

8n j

.

(A . 27b)

In order to apply Theorem D.9, we must verify that (a)

vajK° - K°a v E C(GE) n C(HA) for each j E { 1, ... , N},

(J3) K°w E Ll(GE). Now, K° is infinitely smooth outside -4(xo, e), so that the hypothesis v E C1(RN) implies (a), and hypothesis (b) implies more than (fl) [namely,

that K°Lv is in Lp(GE) with p > N/2, which is better than membership of L1(Ge) because GE is bounded; see Chapter 0, (xiv)].

Letting m -+ oo, using Theorem D.9, and adding the two identities which result from (A.27a,b), we find that the two contributions of 8G cancel each other, because v and K° are in C1(RN \ {x°}) and because the outward unit normals n to aGE and aHA are in opposite directions at each point of 8G where they exist. Accordingly,

avf, J KoAv=J +JA v 8n -K08n GE

aBE

s

where BE := 1(xo, 8), SA := a9(0, A) and n points towards x° on BBe.

(iv) Now let 6 --+ 0. Define Xe to be the characteristic function of GE


182

[equal to 1 on GE, and to 0 on TRN \ GE] ; then

Jf KoAv

= J XEKoAv G

GE

JG

KoAv

by the Lebesgue dominated convergence theorem, since, with S dist(xo, aG), we have Ko integrable and Ov bounded on Bb, while Ko is bounded and Lv is integrable on G \ Bb. Next, c

I

aKo n, an

- v(xo) _

{v(x) - v(xo)} LBE

Ko(x) an

dS(x) --0,

because aKo/an = 1/I3BEl on aBE by the definitive property (A.17) of the Newtonian kernel and because v(x) -+ v(xo) when x E OBE. Finally, Koav -+ 0 fB, an because Koav/an = o(e N+1) uniformly on aBE, while IaBEI = Thus (A.28) yields

-

/ / (aKo av 1 JGKow = v(xo) + JsA tv an - Ko an J

const.cN-1

(A.29)

(v) We now show that the last integral in (A.29) tends to zero as A -* oo, for each fixed x0 E RN \ 0G. First, Ko(x) is close to K(x) when r is sufficiently large; in fact, for r >_ 2ro and for each j we have (ajK)(x - xo) - (a1K)(x)I - 3, Ko(x) is O(r-N+2) by Remark 4. We proceed to apply hypothesis (c), Remark 4 and the observation just made about K0.

For N >- 3 and r = A -> oc, both terms vOKo/ar and Koav/ar are 0 (A-err+s), while the surface area ISA I is O (AN-1) ; then the integral over

SA is O (A-N+2) and tends to zero. For N = 2 and r = A

aK v

oo

- Ko ar

= {cK(x) + O (r-1) } {

oo,

d(x) + 0 (r-2) }

{dK(x) -{K(x)+O(r-1)} c+O(r 2) d

}

= 0 (A-2log A) , because the dominant terms cancel. Since ISAI = 21rA, the integral over SA is O (A-' log A) and tends to zero. Therefore (A.29) reduces to

v(xo) = -

JG

Koty

(xo E RN \ OG)

.

(A.30)


183

aN

Fig. A.8.

(vi) Finally, we use continuity to extend the result (A.30) to points xo E 8G. The function v is more than continuous on RN, by hypothesis.

Theorem A.6 will show (independently of the present theorem) that the condition AV E LP(G), p > N/2, makes the integral in (A.30) a continuous function of xo on RN. Therefore the function v + fG KoLv, being zero outside aG and continuous on 1RN, must vanish also on OG.

To make further progress, we need (a) a notation that enables us to write partial derivatives of arbitrary order without undue labour, (b) bounds for all derivatives of the Newtonian kernel K. Definition A.3 We write as

:= a1l a2z ... aN,

xa := xal xaz 2 1

xaN . N

the N-tuple a :_ (a1, a2, ... , aN) of integers aj >_ 0 is called a multi-index of length N; its order is lal := a1 + 062 + + N. (Figure A.8 shows a set of multi-indices of fixed order, and also illustrates the partial ordering

a < fJ introduced in Exercise A.23.) It is to be understood that x° = 1

even when xi = 0. The label Jul = m, when appended to E, means summation over all multi-indices of order m, the length N being implied by the context. For example, if N = 2, Caxa = C(3,0)xl + C(2 1)xlx2 + C(1 2)xlx2 + C(0,3)x2. Ia1=3


184

Lemma A.4 For all multi-indices /1 if N * 2, or for I#I > 1 if N = 2,

X# 0,

Cplxl-N+2-IRI,

(A.31)

where the constant C1q depends only on f (the length of which species N).

Proof We shall prove by induction that (for

as above and r := IxI)

aflK(x; N) = r N+2-21ftIPp(x),

x

0,

(A.32a)

where P p is a homogenous polynomial in x1, ... , xN of degree 1$1 (possibly the zero polynomial): cp,a x".

Pp(x) =

(A.32b)

Iai=1R1

[Many of the coefficients cp, a are zero; for example,

P(10) _ -2x1 = P(l,o)(x) = -xl/21C, 1

-12x,

P(o,1,o)(x) = -x2/47t,

P(2)(x) = 0, P(1,1)(x) = xlx2/1t,

P(o,0,2)(x) = (-xi - x2 + 2x2)/41.]

Assume that (A.32) is true for IP1 = m (it is true for 1P1 = 0 if N or for 1P1 = 1 if N = 2). Then

2,

a;a#K(x; N)

_ (-N + 2 -

2m)r-N+1-2m(xj

/r)P#(x) +

= r-N-2m{ (-N + 2 - 2m)xjPp(x) + (xi +

r-N+2-2majPp(x)

+ xN)aiPp(x) },

where ajPp is a homogenous polynomial in x1, ... , xN of degree m - 1. r-N-2m is a homogeneous polynomial in x1, ... , xN Hence the factor of of degree m + 1. This proves (A.32) for I Q 1 = m + 1 and hence for all f (except $ = 0 if N = 2). Finally, (A.32) implies (A.31) because IPp(x)l
- 2. If f E Lp(G) for some p > N/2, then the Newtonian potential u of f is uniformly continuous on IIBN.


188

DR (x0,21hI

Fig. A.9.

Proof (i) Let q be the Holder conjugate of p [that is, 1/p + 1/q = 1]; since p > N/2, we have -Nq + 2q + N > 0. The integral in the definition K (xo - x) f (x) dx,

14(X0)

X o E 1[8N

G

exists because the integrand is measurable and the integral can be bounded by means of the Holder inequality; the calculations in step (ii) imply that K(xo-.) E Lq(G). (ii) To prove the continuity of u, we introduce the difference kernel

a(xo, x, h) := K(xo + h - x) - K(xo - x), so that Iu(xo + h) - u(xo)I

=
N, then

for all xo E RN.

(BK) (xo - x) f (x) dx

(A.52)

Proof (i) We proceed very much as in the proof of Theorem A.6. Since now p > N, the Holder conjugate q of p [such that 1/p + 1/q = 1] now satisfies -Nq + q + N > 0. The integral in (A.52) exists because the integrand is measurable and

because the Holder inequality provides a bound; this bound can be inferred from the calculations in step (ii).

(ii) To justify the formula (A.52), we may set j = 1 without any real loss of generality, as the proof of Theorem A.5 shows. As in (A.37), let h := (h1,0,...,0), h1 * 0, and define k by k(xo, x, h) :=

1

hl

{ K(xo + h - x) - K(xo - x) - h1(81K)(xo - x) }.

Then I u(xo + hh) - u(xo)

-

/'

(01K)(xo - x) .f (x) dx

fk(xo,x,h)f(x) dx

IIf LP(G)II, and the lemma will follow if this norm of k tends to zero as h1 -* 0. In Iik(xo,.,h)

I

Lq(G)II

I

A.3 Continuity of the force field Vu

195

fact, we shall prove more: that the norm of k(xo,.,h) in Lq(RN) tends to zero, uniformly over xO E RN. We use the small-ball technique. (a) R < 2IhI, N * 2. Since K(xo + h - x) = +FRh N+2 (minus only for N = 1), the definition of k and the triangle inequality for the norm of Lq yield 1/q

fR Ixo - xI - IhI ? 1R,

so that, by (A.31), Ig"(t)I = Ihi (02K)(xo + th - x)I < hi { maxlfl1=2 CR } (IR) N , and

Ik(xo,x,h)I

=

hi{g(1)-g(0)-g'(0)} 1

hi

1

j(1_t)g"(t)dt

1, the next integral will converge at infinity and yield a simple estimate. In fact, 1/q

1/q

f

J R>2Ihj

R is Holder continuous at the point c if and only if there are constants A E (0, 1], A and 6 > 0 such that Iv(x) - v(c)l < AIx - cI2

whenever x E B(c, 6).

The number A is a Holder exponent and A is a Holder constant. When A = 1, the name Lipschitz often replaces the name Holder. Here is an example. If, for some a E R,

1+ar 1 + I log rI'

W(X)

to,

r := Ixl > 0, r = 0,

then w E C(RN). But w E Cb(RN) only if a = 0, and w is not Holder continuous at the origin, whatever a may be. In the following definition, we demand uniform Holder continuity on Q. [We take suprema over S2, rather than S2, because this is sufficient in the definition and is traditional] Definition A.10 (a) The normed linear space Cb (i) consists of the norm defined by IIv

I CbMII := sup,En Iv(x)I

and of the set

{v:S2-*JRI vEC( ),

IIvICb())II R is [w]A :=sup

1 1w(X) _

x,y E S2, 0 < Ix - yl < 1

I

,

in which the final constant 1 may be replaced by any positive constant characteristic of 92 or of the problem in hand. (c) The normed linear space Cb'2 (S2) consists of the norm defined by v

Cb'1(n)

:= supxEsi Iv(x)I + [vIA

and of the set { v E Cb (S2)

I

[V]A 2 because its analogue for one dimension is immediate: if N = 1 and (a, b) G, then the definition of u implies that

-Ii x f f() -lx u(x) = l x 2

f

G

f f

2

2

x> b,

G

(A.56) a. x - 2, f E Li(G) and IxI > 2a, then

u(x) =

6"(-1)I"18"K(x) + Ek(x), 0 2a,

ICI < a,

G

0< t< 1. Then

and set g(t) := K(x (m) %

(mt

= %() -i i

()t

m

K(x-t),

j=1

where x and are fixed, and a j means, as always, differentiation with respect to the jth argument of the operand. By the multinomial theorem (Exercise A.22), m

N

E j aj E

=

j=1

M

"a"'

IaI=m

whence g(m)(t)

= (-1)m E a 11 (811K)(x

-

Ial=m

Accordingly [Taylor's theorem or integration by parts], k-1

K(x -

g(1) _

M)

g(mi0)

+

m=0

J

1 (1

t1 )i1 g(k)(t) dt

(-1)1«1 a a"K (x) + Ak (x, ), 0 2, the condition f E C(G) fails to imply that u E C2(G); this is shown by Exercise A.29. The theme of this section is that Holder

continuity of f on G ensures not merely that u E C2(G), but that second derivatives of u are Holder continuous in G with the same Holder exponent as f. In symbols, f E C6'µ(G)

a,aju E C6'µ()

whenever p E (0, 1) and f2 c G. }

The reason for the condition p < 1 will be mentioned after the condition

has been used. [Of course, if f E C6'1(G), then f E C6'µ (G) for each p E (0, 1).] Except when G is a ball, we shall not pursue the more difficult

question of regularity up to the boundary: if f E C6'µ(G), does 8'8juIG have an extension to G that is in C (G) or perhaps in C6'µ (G) ? Exercise A.30 shows that the answer can be No when f is constant on G and 8G has a corner; a positive answer requires a condition on the smoothness of 8G and more analysis than is appropriate here.


204

For N = 1, the analogue of Theorems A.15 and A.16 is again very simple; the definition of u implies that

- u"(x) = f (x) whenever x E G and f E C (G).

(A.59)

Notation Following convention, we often omit the subscript b from Cb (S2)

and Cb'2 (S2) when fl is bounded, because then C (52) contains only bounded functions. The ambiguity, that C (S2) may or may not be not be normed, causes little difficulty. Lemma A.13 Assume that N >- 2. Let B := 9(0, a), let f E Co'u (B) for some p E (0, 1), and let

v(xo) :_ f K(xo - x) f (x) dx,

xo E IR' .

B

Extend f to be zero outside B. Then [for all i and j in { 1, ... , N}] (a,ajv)(xo)

= fs

(a,ajK)(xo - x) { f (x) - f (xo) } dx -S,JfN)

(A.60)

whenever x0 E RN \ aB.

Proof If xo E RN \ B, then f (xo) = 0 and (A.60) is a particular case of (A.33). Therefore we suppose in the remainder of this proof that xo E B. The existence of the integral in (A.60) will be implied once again by the calculations that follow. It is enough to prove the result for i = 1; let h = (h1, 0,..., 0),

0 < IhI < 1,

IhI < dist(xo, aB).

Note first that, by Lemma A.9 and (A.23a),

J

{ (ajK) (xo + h - x) - (ajK)(xo - x) } dx

_ (ajuB)(xo + h) - (ajuB)(xo) _ -N { (xo + h)j - xoj } h1

=-81;N. Hence, by a second application of Lemma A.9,

(ajv)(xo + h) - (ajv)(xo) =

J

{ (a;K)(xo + h - x) - (a;K)(xo - x) } x {.f (x) - f(xo)} dx - 6I j N f(xo)

205

A.5 Second derivatives of u at points in G

We are now on a familiar path. Fix j and define

hl { (a,K) (xo + h - x) - (8jK) (xo - x)

k(xo, x, h)

-hi (01ajK)(xo-x)}; then (aiv)(xo + h) - (aiv)(xo) h1

f (ai7iK)(xo-x){f(x)-f(xo)} dx+81jfN

f

B

_

)

k(xo, x, h) { f (x) - f (xo) } dx.

B

If this last integral tends to zero as h1 --+ 0, then the lemma will be IIf I COP(R)II and R = Ix - xol; the following coarse

proved. Let Ilf II

bound will be useful.

if R < 1,

[f]µRµ

If (W )-f(xo)I 2IIf

I

if R

c(B)IIR'`

21fI R.

(A.61)

1

In the inevitable application of the small-ball technique, constants F now depend on N and µ. We recall the notation Rh = Ix - xo - hl. (a) R < 2IhI. By the definition of k, by (A.31) and by (A.61), k{ f (x) - f (xo) } dx JR- 2, and let f E C°'µ (G) for some p e (0, 1). Then each second derivative of the Newtonian

potential u off has the property whenever S2 c G,

8i8ju E C°'µ (52)

(A.74)

and u is a C2-solution of -Au = f in G.

Proof In this proof (. ),j := a,a j(.) for functions (but not for the Kronecker delta), and If 11 = IIf C°'µ (G) (I. (i) Let an (open) set fl, such that 12 = G, be given. Then dist(12, aG) > 0 because S2 and aG are compact and disjoint; let p := 1 dist(S2,UG). For each c E S2 and all xo E RN, define B := R(c, p) and I

v(xo)

:=

f K(xo - x) f (x) dx, a

K(xo - x) f (x) dx.

w(xo)

(A.75a) (A.75b)

G\B

[The symbols v(xo, c) and w(xo, c) would be more accurate but more clumsy.] Then B c G and u = v + w. Application of (A.60) to v and of


212

(A.33) to w shows that, whenever c E S2 and xo E B, v1i( xo)

=

JB w (xo) = f

K ,j ( xo

- x ){ f (x ) - f (xo ) } dx - S,j fN

K1(xo - x) f (x) dx .

,

(A . 76 a) (A.76b)

\B

In particular, when xo E S2 is given, we may choose c = x0, so that the field point is at the centre of B. [This is legitimate because (A.76) holds for every c E S2 and every xo E -4(c, p), but it must be remembered that the differentiation with respect to xoi and xoi was for fixed c and before the choice c = xo was made.] (ii) To bound uij(xo) independently of xo E S2, we choose c = x0, adapt (A.70) to the present situation, obtaining Ivij(xo)I < rllf ll (Pµ + 1),

and infer from (A.76b) that, with D := diam G, Iwij(xo)I < IF

D

R-N

Ilf II

RN-1

dR = F IIf II log P

(iii) To prove uniform Holder continuity of uij on S2, we consider field points x0 and x0 + h in S2, with 0 < h i < p14 [recall Remark 4] ; again we choose c = xo. The identity (A.71) and estimates like those leading from (A.71) to (A.73), again with H := -V(xo,21hl) but now with B = -4(x0,p)1 with 0 < Ihl < p/4 and with (A.64) replacing Lemma A.14, yield Ivij(xo + h) - vij(xo)I < rllf II Ihlµ whenever xo, xo + h E S2 and 0 < IIt I < p14.

Next, our standard method of estimating for R > 21hl gives I wig (xo + h) - wil(xo) I =

J \B{Kid(xo+h-x) -Kij(xo-x)}f(x) dx

< r f R-N-1 IhI IIfll

RN-1 dR

P

= IIIf1IIhi/P whenever xo, xo + h E S2 and 0 < IhI < p/4. Thus uij E C°°µ(5l).

(iv) The result (A.74) implies that u E CZ(G) [because for each point xo E G there is an open set 0 = O(xo) such that xo E S and 52 c G]. To prove that -(L u) (xo) = f(xo) at a given point xo E G, choose any ball

A.6 Exercises

213

B such that xo E B and B c G; in (A.75) and (A.76), let B denote this new ball. It follows that u = v + w, that -(Av) (xo) = f(xo) and that (Aw) (xo) = 0. Thus u is a C2-solution of the Poisson equation in G. A.6 Exercises

Exercise A.17 Considering the Newtonian potential in one dimension (N = 1), (i) derive the formula (A.23) for the potential of unit density in (-a, a); (ii) prove that, if f E C (G), then -u" = f in G.

Exercise A.18 Show that in terms of spherical co-ordinates r, 0,A the definition (A.14) of the Stokes stream function becomes 1

CWr,

r Po) = r sin 0 C- r

(PrJ

Derive (modulo an additive constant) the Stokes stream function WO in (A.16) from the potential To in (A.12).

Exercise A.19 Let z = x + iy = re'B denote an arbitrary point of the complex plane C. [The point (x, y) E 1[82 and the point x + iy E C can be considered one and the same for many purposes, such as the definition of balls, of open sets and of continuity] If x is holomorphic (or analytic) in an open set n c C, if x = (p + iW, and if (p is regarded as a potential function, then x is called a complex potential, W a stream function, and - dx(z)/ dz a complex velocity (or complex electric field or ... ). (i) How is dx(z)/ dz related to V(p(x, y) and to VW(x, y)? Show that the vector field VT is tangential to the level sets of W. [Recall that the zero vector is both normal and parallel to every vector] (ii) The complex potential of a unit source at the origin is -(log z)/2n, after some suitable restriction of 0. Use this to find the analogues in the plane (JR2 or C) of the (po and WO in §A.1 for `flow past a closed ball'.

Exercise A.20 In the analogue for JR2 of the problem (A.8), one allows any bad behaviour of cp that is not passed on to V(p. (i) Referring to Exercise A.19, (ii), exhibit infinitely many solutions for flow past a closed ball in R2 under the condition V P(x, y) + (U, 0) _ 0 (r-1) as r -* oo. [Contemplate the complex potential i log z, z * 0.] (ii) State and prove an analogue, for flow in 1R2, of the uniqueness result in Theorem A.I.


214

Exercise A.21 (i) Show that in the plane a multipole at the origin, of unit strength and of type a with lal >- 1, has complex potential iaz

Pa(z) =

(_)1

iaz

1

d

dzlog

27L

1)!

z

z-I«I

(z

0).

(ii) For a = (1,0), find equations describing the field lines; sketch these level sets of W(1,O).

(iii) Explain, and demonstrate by an example, how your picture in (ii)

can be combined with a picture of the conformal map ( = z", where n E { 2,3,4.... } and 0< arg z < 2ir/n, to yield a sketch of any multipole in the plane with dal > 2. [To draw a picture of the map C = z", write z = re'B and draw corresponding level sets of r and of 0 in the z-plane and in the C-plane.]

Exercise A.22 Derive the multinomial theorem: for any positive integer m,

(XI + ... + XN)m

m!

=

xa,

where a! := al ! ... aN

!

dal=m

[One method is induction on N; a second is a combinatorial argument; a third proceeds from 00

(1-x1-...-xN)E(x1+...+xN)m=1

(Ix1I+...+.IxNI - a,

n=0

where (if 0 < r < a, z = rei0 and Re denotes the real part) si n 2/i z 2 log a + cos 2# Re z2 4 r2 , g(r, 0) := si 2# Re z 2 [log a + i n , fl < 0 < 2 ir

-

n

g(r, 0)

=

s in 2/3 41T

2

a

r log r cos 20 +

r2 it

40

E do cos nO

for all 0 E Il8,

n=0

and where /f

co = 2 log

1

a

/3

+4,

1

1

c2 = 4 sin 2/3 - 4 /f cos 2/3,

sin n/i

cn = - n2(n - 2)

if n = 1 or n >- 3, do = - 9 , 4

d2 = -

3

sin 2/3 +

1

/3 cos 2/3,

6 if n=1 or4 n>3.

do =

n

2 (n2n

4)

-

A.6 Exercises

219

n/2. (ii) Given that in Theorem A.2 the hypotheses v E C2 (RN \ G) and Av = 0 in RN \ G are implied by the conditions v E C1(RN) and (i) Show that laiu(x)l --> oo as x -> 0 in 1182\aSS, provided that

f

for all (pEC,°(RN\G), N

verify that the foregoing formulae for u do indeed represent the Newtonian potential of unit density in Sp.

Exercise A.31 In order to discuss the Newtonian potential of a density function f on an unbounded open set G c RN, we demand that

where W(x) :=

W f E L1(G),

l+lxl

if N=1,

log(e + 21x1)

if N = 2, if N>_3.

1

(A.78)

As preparation for extending certain theorems, prove that, if (A.78) holds, if a is a multi-index and if 6 E (0, 1], then

f

(aK)(xo-x)f(x)dx\f(xo,d)

if N =1, loci = 0,

(1 + Ixol) Il w.f ll

{log1+62Ixol+11

IIWfll

1

Cab-N+2-iai

11111

J

if N=2, Ial=0, if N or N

(A.79)

3, loci = 0 1,

loci

1,

L,(G)11, the constant Ca is as in (A.31), and G may be . 11 = II . unbounded. Prove also that, if f E Lp(G) for some p > 1, then (A.78) ensures that f c- LS(G) for l < s < p. [Observe that, with the notation (A.22b), we have log(1/R) < log(116) where 11

I

if 6 < R < 1; also, log R < log 2ro if R > 1 and r < ro ;

finally,

log R < log 2r if R >_ 1 and ro < r.]

Exercise A.32 Use the results of Exercise A.31, and inspection of proofs for a bounded set G, to establish the following results for an unbounded open set G c RN. (i) Theorem A.5 remains valid if the hypothesis f E L1(G) is replaced by (A.78).


220

(ii) Theorems A.6 and A.8, Lemma A.9 and Theorem A.11 remain valid if the hypothesis (A.78) is added. (iii) Theorem A.16 remains valid if the symbol CO,,"(.) is replaced by Cb'µ(. ), and the hypothesis (A.78) is added.

Exercise A.33 This exercise concerns an extension of Theorem A.16 to very smooth density functions f. (i) Let G be a bounded open subset of RN, N >- 2; assume that either aG is of class C1 or G is listed in Remark D.4. Let u be the Newtonian potential of a density function f E C°° (G) and let x° E G. Prove that

(aiu)(x°) _

JO ni(x)K(x-x°) f(x) dS(x)+J K(x-x°) (aif)(x) dx, c

c

(A.80)

and that, for each multi-index a, in terms of integers i = i(a, j) and multi-indices p = p(a, j) and a = ar(a, j) that need not be specified precisely, I"I

(a"u)(x°) =

L(-1)1«l+.i-1

f n(x) (aPK)(x-x°) (f)(x) dS(x)

+ J K(x - x°) (a"f) (x) dx,

(A.81)

G

where IPI + Ioi = lal -1 and the sum over j is zero when Joel = 0. (If a" is written as ai,, a;Za;,, where m = Joel, then a precise but labyrinthine form of (A.81) can be found). (ii) Now let f E Ck (G) and let Of E C°'` (G) if Jf I = k; here k E No, µ E (0,1) and y is the same for all Of of order k. Prove that a7u E Co," (S2) if IyI = k + 2 and SZ c G, and that u E Ck+2 (G).

Appendix B. Rudimentary Facts about Harmonic Functions and the Poisson Equation

B.1 Real-analytic functions

The formula W(t) :_

expl\ 11

t2

-1 0 such that Rk(x) - 0 for all x E (-p, p) as k -> oo. If there were such a number p E (0, 1), say, then we would have 0"

Zp)

n=0

j

(n)(1)(-2p)n n!

which is a contradiction because W(1 - Zp) > 0, while W(n)(1) = 0 for each n.

Real-analytic functions are real-valued functions that lack this flaw; 221

Appendix B. Rudimentary Facts

222

for a real-analytic function f and for every point p in its domain, the Taylor series of f about p has a positive radius of convergence. Here we make this statement precise for functions defined on open subsets of RN; then we establish a test for real-analyticity that is often more convenient than the definition. Notation As always, S is an open, non-empty subset of RN; unless the

contrary is stated, N is any positive integer. We use multi-indices a (Definition A.3) with a! := a1! a2! ... aN!. Theorem B.1 (Taylor's formula). If f E Ck(fl) and if the line segment { (1 - t)p + tx 10 < t < 1 } is a subset of 52, then

f(x) _

(aaf)(P) (x

a!p)a+E(0-f)((1-0)p+Ox) (x

lal 0 such that Rk(h) -+ 0 as k - oo whenever IhI < p. First, let p Zdp, where dp := dist(p,Of)) if fl * j N, or dp := 2


224

if 0 = RN. Then E := M(p, Zdp) is suitable for application of (B.6); if A and B are the constants corresponding to this set, we have

0 when Q # RN [because E is compact, On is closed and the two are disjoint], choose a bounded open set G such that E c G and G c 0, and define A := supYEG u(Y)I = maxyec1 u(Y)

By Theorem B.9, applied to the set G rather than to 52, I7u(x) 0 when 52 * RN. Therefore (B.19) holds if we choose B := N/dist(E, 7G). Theorem B.11 (a Liouville theorem). If u is smoothly harmonic in RN and u(x) = o(r) as r := IxI -> oo, then u is a constant.

Proof We shall prove that (7iu) (xo) = 0 for each xo E RN and each

j E {1,...,N} by applying Theorem B.9 with lal = 1 and with S = _V(O,Rm). Here (Rm) is an increasing sequence such that Rl > 21xol for given x0 E RN and such that Rm oo as m -> oo. Then (7Ju)(xo)j I u(ym) I, then lym+1I > lyml; if Iu(ym+1) I = Iu(ym) I, then we either have, or may take, ym+1 = ym. Thus the sequence (I ym I) is non-decreasing. The theorem now follows from (B.20) because I u(ym) I/ Rm , 0 as m --> oo. [If I ym I -* oo, then u (ym) = o (I.ym I) and I ym l < Rm ; if I ym I is bounded,

then so is u(ym).]

B.3 The Kelvin transformation

This section concerns a conformal map of RN \ {0}, and a related transformation of functions, under which smoothly harmonic functions remain smoothly harmonic. Definition B.12 Given a sphere 8-4(0, a) in RN, we call 2

Sx := 72x a

(x E RN \ {0}, r := lxi)

(B.21)

the reflection or inverse point of x relative to 0-4(0,a). The Kelvin transform T f relative to 8-4(0, a), of a function f : S2 -> R, is then defined by

a N-2

(Tf)(x) := (p)

f (Sx)

(Sx E S2),

(B.22)

provided that 92 c RN \ {0}.

To calculate with S and T, we use points l; E RN with p = Sx, then pr = a2 and

r2 (a2 I Y2 x S = p2 = a2 a2

If

x,

so that S is its own inverse operator: S-1 = S. If

= Sx and qp = T f,

then (ar

Ox) =

N-2 f(Sx)

N-2

f( )_ \p)

so that T is also its own inverse operator: T-1 = T. As regards the geometry of the transformation = Sx, it is obvious that 0-4(0,a), the sphere of inversion, is mapped onto itself, and that the sets -4(0, a) \ {0} and RN \ (0, a) are mapped onto each other. The Frechet or total or linear derivative of the transformation is 2

S'(x) _

(ax

(x)) =

r2

M(x),

2 where Mij(x) := Sid - r2 j.

(13.23)

B.3 The Kelvin transformation

233

This shows that the matrix M(x) is orthogonal as well as symmetric, because one checks easily that M(x)M(x) = I, the identity matrix. The orthogonality of M(x) implies that, for all h and k in RN \ {0}, the angle between S'(x)h and S'(x)k equals that between h and k. Consequently, the map S is conformal in the sense that, if two smooth arcs intersect

at a point, then the angle between them at that point is conserved by the mapping. [For, let x = f (t) and x = g(t) be descriptions of two smooth arcs that intersect at xo = f (to) = g(to). The orthogonality of M(xo) ensures that the angle between S'(xo)f'(to) and S'(xo)g'(to) equals that between f(to) and g'(to); the latter angle is that between tangents at xo to the original arcs x = f (t) and x = g(t), while the former angle is that between tangents at Sx0 to the transformed arcs = S(f(t)) and

= S require a small definition.

Definition B.13 By a half-space in RN we mean a set { x E RN where k E RN \ {0} and p E R.

I

>,u

Exercise B.14 Let S be the reflection operator in (B.21), and let G be a ball or a half-space in RN. (i) Prove that S(G) is a ball if 0 G, that S(G) is a half-space if 0 E 8G, and that S(G \ {0}) is the complement of a closed ball if 0 E G.

(ii) Prove that, for N > 2, S(G) = G if and only if 8G intersects 0-4(0, a) orthogonally.

[Orthogonality means here that normals nG to 7G and nB to 8,4(0, a) satisfy nG(y) nB(y) = 0 at a point y of intersection.]

The next theorem shows that, if u is smoothly harmonic in 1, then its Kelvin transform Tu is smoothly harmonic in S(fl); Definition 2.20 and Exercise B.37 deal with an extension to merely subharmonic functions. It is this property of the Kelvin transform that makes it useful. Indeed, almost everything that can be done with holomorphic (or complex analytic) functions in the complex plane C by means of a Mobius (or `bilinear') transformation, can be done with smoothly harmonic functions in R' by a composition of translation (u H u(. +c), where c E RN), dilation (u'--* u(2. ), where A E R \ {0}) and the Kelvin transformation. The proof of Theorem B.15 is not the shortest route to the formula (B.24), but includes details that are useful in applications of the transformation.


234

Fig. B.3.

Theorem B.15 Let f E C2(ST ), where fl c RN \ {0}, and let T f be its Kelvin transform, as in (B.22). Then, for x E S(c2),

A(Tf)(x) =

(ar)N+2

(L.f)(Sx).

(B.24)

Proof (i) Let = Sx, x * 0, so that x = 0. Consider two copies of RN \ {01: the first has Cartesian co-ordinates c,. .. , N ; the second (Figure B.3) is the image of the first by S and has Cartesian co-ordinates x1, ... , XN, while 1, ... , N are curvilinear co-ordinates there.

Let ax/ak := (akS)(). In the second copy of RN \ {0}, the vector ax/ask at any given point is tangential to a curve along which k increases while the other j are fixed. The vectors ax/al,...,ax/aN at the given point are mutually orthogonal because they are images, under the conformal map S, of vectors along the co-ordinate lines in the first copy of RN \ {0}. To verify this orthogonality of ax/a 1, ... , ax/a N, we infer from (B.23) that ax

ax -

aj ak

a4 N

a4

= P4 E Mpj( )Mpk( ) = P4 Sjk.

(B.25)

Thus 1,...1;N are orthogonal curvilinear co-ordinates in the second copy of RN \ {0}.

(ii) We use arc-length functions h j defined by az

axJ

2'

B.4 On the Dirichlet and Neumann problems

235

and basis vectors bj := for j = 1,...,N. The rules (Kellogg 1929, pp. 181 and 183; Spiegel 1959, pp. 148 and 151) a

...

axl a2

ax1

+ ... +

a

_b' a

axN

h1

a2

1

axN

h1... hN a

bN a

(B.26a)

hN

(h2... hN a

a h,

... hN_1

+ aCN

hN

+...

T

h1

)

a

(B.26b)

NN-

simplify here because all the h j are equal.

(iii) Let g(x) := f(Sx) and recall that Or-N+2 = 0 in RN \ {0} for all N E N; then, by the definition (B.22) of T f and the Leibniz rule for repeated differentiation of products,

A(T f)(x) = 2

v

(ar

(ar)N-2

N-2

Og(x),

Vg(x) +

where air = p/a and g(x) = f (Sx) = f

Application of (B.26a, b) now

gives

O(T f)(x) N

= 2(N - 2)a-aN-2pN Y j

af( )

j=1=1

+a-N-2

P

3N-2

1` j(-2N + 4)p-2N+2 j ON) j=1

p N+2

=

+ p_2N+4

J(Z >

j

(a

and this is the desired formula (B.24).

B.4 On the Dirichlet and Neumann problems In Appendix A we encountered the Poisson equation, -Au = f in G, as an equation satisfied by the Newtonian potential u of a smooth density

function f on G. Here we regard the Poisson equation as one to be solved when a boundary condition for u is specified.

Let f : S2 -> R and g : au -> R be given functions. The Dirichlet problem for -A in S2 is to find u E C(S2) n C2(SZ) such that

- Au = f in u,

ul an = g.

(B.27)

236


The Neumann problem for -,L in 0 is to find u e CI(S2) n C2(0) such that

-Du=f in

au 12,

an an

= g,

(B.28)

where n denotes the unit normal outward from S2 and a/an := n V. These statements omit a great deal. Only pointwise solutions (C2solutions) have been mentioned. We have not said what smoothness

the data f, g and aS must have in order that the solution u have the continuity properties that we have demanded. When 0 is bounded, we can hope to solve the Neumann problem only if the compatibility condition

-J nf =Jang

(B.29)

holds, because both sides of (B.29) equal fan au/an for a smooth solution of (B.28) with smooth data. In fact, if 52 is not connected, (B.29) must hold for each component (each maximal connected subset) of fl. When S2 is unbounded, growth conditions (or decay conditions), specifying how large Iu(x)I and perhaps IVu(x)I are allowed to become as IxI -+ co, must

be added to the statements of the Dirichlet and Neumann problems. Some of these gaps will be filled as we proceed.

Reduction to the Laplace equation or to zero boundary data (i) Often one can remove the term f from the Poisson equation in (B.27) and (B.28) by the substitution u = Uf+v, where Uf is the Newtonian potential off. For example, if 0 is bounded and f E C°'µ(S2) for some p E (0, 1) [Definition A.10, with the suffix b omitted from Cy°" because S2 is bounded], then

Theorems A.11 and A.16 ensure that Uf E CI(S2) n C2(f) and that -,L Uf = f pointwise in S2. The Dirichlet problem is then to find v E C(S2) n C2(S2) such that

Ov = 0 in n,

vI

an = g - U f Ion;

(B.30)

the Neumann problem is to find v E CI(S2) n C2(S2) such that

Ov = 0 in

S2,

av

=g-

an an

a Uf an

.

(B.31)

an

In both cases, v (if it exists) will be not merely in C2 (K2) but in C°°(S2) (Theorem B.6), indeed, it will be real-analytic (Theorem B.10). Note that, if the compatibility condition (B.29) holds for f and g in the Neumann

case, then it holds also for 0 and g - 0Uf/an.


237

(ii) Alternatively, if one can find a sufficiently smooth function h : S2

R such that h = g on OS2 in the Dirichlet case, or Oh/an = g on OS2 in the Neumann case, then the substitution u = h + w yields a problem for w with boundary condition w = 0 on OS2 or aw/an = 0 on OS2. However, for a Poisson equation with forcing function in C°4'(S2), it is necessary that Ah E C°°µ(S2); to find such a function h may not be easy if g and On are less than beautiful. We begin work on the Dirichlet and Neumann problems by proving uniqueness of solutions in some cases. Then any method of solution becomes acceptable in those cases, however contrived or squalid the method may seem, because what it produces is the solution. Theorem B.16 The Dirichlet problem for -L in n has at most one solution if either (a) S2 is bounded, or (b) S2 is a half-space (Definition B.13) and we add the growth condition:

u(x)=o(r)for xES2and r:=xj-+oc. Proof Let v := u1 - u2 be the difference of two solutions; then v E C(S2) n C2(S2), we have Av = 0 in n, and v = 0 on On. The additional smoothness implied by Theorems B.6 and B.10 is not needed here. (a) If 92 is bounded, then Theorem 2.5 (our first and simplest maximum

principle) states that v < 0 on fl, because v is a C2-subsolution relative to A and n, and that v >- 0 on n, because -v is also a C2-subsolution. Thus v = 0 on S2, as desired. (b) If 92 is a half-space, we so choose co-ordinates that n = { x E ][8N

XN > 0 }, and note that v(x) = o(r) as r -+ oc. For N = 1, the equation Lv = 0 implies that v(x) = co + clx, where co and cl are constants; then

co = 0 because v(0) = 0, and cl = 0 because v(x) = o(x) as x

oo.

For N >- 2, we apply Theorem 2.30 (a relatively advanced maximum principle). The hypotheses of that theorem are satisfied by both v and -v (with room to spare in both smoothness and growth condition); hence supra v = 0 and info v = 0, so that v = 0 on S once again. For the Neumann problem stated in (B.28), uniqueness in the strictest sense is impossible: if we add to a solution u any function k that

is constant on each component [on each maximal connected subset] of S2, then u + k is again a solution. However, in many applications only uniqueness of Vu is required, and we have already encountered an example of such uniqueness in Theorem A.1 and the remark following it.


238

Theorem B.17 The Neumann problem for -A in S2 admits at most one function Vu for solutions u if (a) S2 is bounded and as2 is of class C', or (b) 92 is bounded and is listed in Remark D.4, or (c) 0 is a half-space and we add the growth condition: u(x) = o(r) for

xES2and r:=lxl-+oo. Proof The difference v := u1-u2 of two solutions is now in C'(S2)r1C2(S2)

and satisfies Av = 0 in 0, av/an = 0 on M. (a),(b) If 5 is bounded, and either as2 is of class C' or S2 is listed in Remark D.4, then we have both a divergence theorem and approximations S2to u as in Theorem D.9. Applying the divergence theorem to the vector field vVv and one of these sets [which is legitimate because V E C2(1 m)], and observing that V (vVv) = IVv12 because Av = 0 in S2 [a fortiori in 52,,,], we obtain IvvI2 = fa.

van anm an

Let m -> oo; by Theorem D.9 and because v E C'(S2), av/an = 0 on aft, IVv12 = 0,

and this implies that IDvI = 0 on S2, because v E C'(S2).

(c) If 0 is a half-space, we choose co-ordinates again that make 0 = { x E RN

XN > 0 }, and notice again that v(x) = o(r) as r -+ oo. Let nn := { x E RN XN > 1 /m } for m E N. Integration by parts along co-ordinate lines, very much as in Remark 7 of §A.2, shows that

f(Lp)v = -.f m

a.m

{ (N(p)v

NV} + f Cty if (p E C(RN); m

the last integral vanishes because Av = 0 in 0. Let m -+ oo and recall that v E C'(S2,) and aNV = 0 on 00; then

(A(P)v = -

f

(aN(p)v

if (p E CC°(RN).

(B.32)

asp

Now extend v to RN as an even function of XN by setting v(x', -XN) :_ v(x), where x' := (x1,...,XN_1) and XN > 0. Let G := {x E R"' XN < 0 }; repeating forfG G the steps that led to (B.32), we obtain (

= Jac

if

E C(RN).

(B.33)


239

Also, V E C(RN). (In fact, v E C'(RN), but we do not need this.) Add (B.32) and (B.33) for the same function q ; the boundary terms cancel and the result shows, in view of Definition B.5, that v is harmonic in RN. Since also v is continuous, it is smoothly harmonic in RN, by the remark following Definition B.7, and the extended function v is still o(r) as r co. Therefore Theorem B.11 states that v is a constant. Regarding existence of solutions, we shall consider only some aspects of the method of Green functions, which is perhaps the most classical and constructive of the various approaches that are now available. Our limited treatment will yield results only for balls and half-spaces, with

a slender hint of how the method proceeds for other sets. The main ingredients of the method are the notion of a fundamental solution, and the representation formula (B.35) below. Definition B.18 Let K be the Newtonian kernel introduced as K(. ; N) in (A.18). A function F defined by F(xo, x) := K(xo - x) + q(xo, x)

for xo E S2, x E f2

and xo * x

is a fundamental solution of -A in 0 if, for each fixed x0 E 12,

q(xo,.) E C'(n) n C2(f), Aq(xo, x) = 0

for all x E S2,

(B.34)

where A is with respect to x. The function q will be called the non-singular

part of F. Theorems B.6 and B.10 imply once again that q(xo,.) is real-analytic in n; we have written C2 (fl) as a demand for a pointwise solution, not as a result. Obviously q is not determined uniquely by (B.34); presently we shall add one of several boundary conditions. Exercise B.19 In both the following situations F is a fundamental solution of -A in S2, and u E C1(S2) n C2(S) with Du E L1(S2). (i) Assume that 12 is bounded and that either 812 is of class C' or 92 is listed in Remark D.4. Prove that u(xo)

= - in F(xo, x) (Lu)(x) dx + I {F(xo, x) au(x)

7F(xn, x)

n

where xo E S2 and a/an is with respect to x.

u(x)} dS(x),

(B.35)


240

(ii) Assume that fl is a half-space and that (in addition to the foregoing hypotheses about F and u) one or other of the following growth conditions holds. For x E Sl and r := lxl -+ oo, for multi-indices a (Definition A.3) of order dal = 0 or 1, for each fixed xo E 0 and for some constant 6 > 0, either a

a (_)F(xox) = 0 (rN+1)

aU(x) = 0 (rl)

,

,

(B.36a,b)

or

a Ox

s

F(xo, x) =

0 (r

N+2-lal)

if N 2, if N = 2,

l 0 (r lal log r)

aaU(x) = O (r-b-l al

.

(B.37a,b)

Prove that (B.35) still holds. [For both (i) and (ii), adapt the proof of Theorem A.2, which contains all essential steps.]

Now we specify boundary conditions for q(xo,.) and hence for F(xo,. ).

In order to have a fundamental solution that is useful for the Dirichlet problem, we demand that

q(xo, x) = -K(xo - x)

if xo E Sl

and X E Of).

(B.38)

In other words, F(xo,.) is to vanish on afl; then on the right-hand side of (B.35) only the terms involving L u and ul an remain, and these functions are prescribed in the Dirichlet problem. By Theorem B.16, there is at most one solution of (B.34) and (B.38) if 1 is a bounded set or a half-space, provided that for a half-space we add the growth condition: q(xo, x) = o(r) as r := lxi --> oc with x0 fixed in Q. For the Neumann problem, if fl is bounded and either at) is of class C1 or f is listed in Remark D.4, a suitable boundary condition is aq(xo, x) _ aK(x - xo) 1 if xo E S and x E Of); (B.39) an

an

laslI

here a/an is with respect to x and Ia

l

is the surface area of M. We

must not demand that aF(xo, x)/an = 0, because (A.20) and (B.34) imply that

P aK(x - xo) Jan

an

- -1

and

f

n

aq(xo, x) = 0

an

for fixed x0 E Q. For the Neumann problem with Sl a half-space, we do demand that aq(xo, x) an

_ aK(x - xo) an

if x o E Sl

and x E asl;

(B.40)


241

in this case, whatever emerges from the point source at x0 can escape to infinity.

Exercise B.20 Let 0 be bounded and either have 8Q of class C1 or be listed in Remark D.4. (a) Assume that the solution q(xo,.) of (B.34) and (B.38) exists, and

denote the corresponding fundamental solution by G. Show that, if the solution u of the Dirichlet problem (B.27) exists and belongs to C1(52) n C2(52) with Au E L1(52), then it is given by

u(xo) = f G(xo, x) f (x) dx -

Jon

7G(xo, x) g(x) dS(x),

(B.41)

On

where x0 E SZ and 8/8n is with respect to x.

(b) Assume that a solution q(xo,.) of (B.34) and (B.39) exists, and denote the corresponding fundamental solution by H. Show that, if a solution u of the Neumann problem (B.28) exists and belongs to C1(S2) n C2(52) with Au E L1O), then it is given by u(xo) =

Jn

H(xo, x) f (x) dx +

Jan

H(xo, x) g(x) dS(x) + k,

(B.42)

where xo E 0 and the constant k is the mean value of u over 852.

Remarks 1. A fundamental solution that satisfies a useful boundary condition, and leads to a representation formula like (B.41) or (B.42), is called a Green function. This phrase can be lengthened; for example, the function G in Exercise B.20, (a), is (if it exists) the Green function of the Dirichlet problem for -A in 52. Green functions exist for differential operators other than A and for boundary conditions other than those of the Dirichlet and Neumann problems. 2. At this stage of the theory, the formula (B.41) rests on two assumptions :

that the non-singular part q of the Green function G, and the

solution u of the Dirichlet problem, both exist. This state of affairs can be improved.

(a) For a few simple sets 52, such as a ball in R1, there is an explicit formula for the Green function G. (In the case of the half-space, this formula is an obvious one, but at the moment we are considering bounded sets 52.) For these particular sets 52, the first assumption is not needed, and we can dispose of the second by a change of direction. Instead

of proceeding from the Dirichlet problem (B.27) to the representation formula (B.41) by means of assumptions about the solution u, we can


242

show (when G is known explicitly) that the function u defined by (B.41) satisfies equations (B.27) for suitable data f and g. This will be done for a ball in §B.5. (b) For an arbitrary set 0, one cannot expect an explicit formula for the Green function (however smooth the boundary 8S2 may be). Rather, a more abstract argument is needed to prove existence of solutions of the Dirichlet problem. This material is outside the range of this book, but we remark that one can cast the Dirichlet problem into a form to which the Fredholm alternatives apply. This means, in effect, that uniqueness of solutions implies their existence, so that Theorem B.16 is again a corner stone. For a pleasant boundary On, the existence theory for the Dirichlet problem implies existence of q(xo,.) and hence of the Green function; then the formula (B.41) has the advantage of representing the solution u for all admissible data f and g. Similar remarks apply to the Neumann problem. 3. For the Neumann problem, we observed before Theorem B.17 that a constant can always be added, on each component of S2, to a solution u. For the function q(xo,.) this arbitrary constant becomes an arbitrary function of x0 that can be added to q(xo,.) and hence to

F(xo,. ). To remedy this partly, we demand that, for a Green function F of the Neumann problem for -,L in a bounded set S2 (with On of class C1 or with f listed in Remark D.4),

I F(xo, x) dS(x) = c

Ion

for all xo E 0,

(B.43)

where c is independent of x0 but is otherwise arbitrary.

Given a fundamental solution F* suitable for the present Neumann problem in that its non-singular part satisfies (B.39), and such that fan F* (xo,.) = y* (xo) for all xo E 0, we must now define c - Y* (xo)

F(xo x) = F (xo x) .4 and here only c is arbitrary.

4. In our discussion up to now of fundamental solutions F, the field point xo has been fixed, although an arbitrary point of 0. However, when we come to use the representation formulae (B.41) and (B.42), the dependence on xo of the Green functions there (that is, the behaviour of G(xo, x) and H(xo, x) as xo varies) will be of importance. This dependence on the first variable is given to us cheaply by the next theorem, which


243

ensures that results postulated or established for F(p,. ), with p fixed in S2, apply also to F(. , p). Theorem B.21 Let 52 be bounded and either have as2 of class C1 or be listed

in Remark D.4. Assume the existence of Green functions of the Dirichlet and Neumann problems for -,L in 52. In the Dirichlet case, the non-singular part is to satisfy (B.34) and (B.38); in the Neumann case, conditions (B.34), (B.39) and (B.43) are to hold. Then, after extension by continuity, both Green functions have the symmetry property

F(y,z)=F(z,y)

if (y,z)E(52xS2)U(52x52) and y*z.

Proof (i) We need prove only that F(y, z) = F(z, y) for (y, z) E 52 x 5 and y * z. For, suppose that this has been shown and that y E 852, z E 0. There is a sequence (y") in 52 such that, for each n, y" * z and such that y" -+ y as n -> oo. Then F(y", z) = F(z, y") for each n, and F(z, y") -* F(z, y) by the continuity of F(z,.) on 52 \ {z}. Extending F(. , z) by continuity, we have F(y, z) := limy ,y F(y", z) = F(z, y). (ii) To prove symmetry for (y, z) E 52 x n and y z, it will be sufficient to consider the Neumann case; the proof for the Dirichlet case is similar but a little easier. We apply the Green identity (A.26),

fA

OW {vLw-w/ v}= fOA{va

n

Ov

-w8n

to v = F(y,. ), w = F(z,.) and A = 52,"

4(y, s) U (z, s) }, where is the mth approximation to n in Theorem D.9; once again this

approximation is required in order that v E C2(A) and W E C2(A). Given distinct points y and z in 52, we can choose m E N so large, and e > 0 so small, that 4-(Y18) and ,,t7(z, e) are within and are disjoint. The integral over A vanishes because Lv = L w = 0 in A. Dealing with the integrals over a9(y, s) and 02(z, s) as we dealt with the integral over a. (xo, e) in the proof of Theorem A.2, we find that

0 = F'(y, z) - F(z, y) + fan.

aF(yan, x)

aF(8zn, x)

F(y, x)

- F(z' x)

dS(x),

where a/an is with respect to x. Suppose that our choice of m was m >- M. Since F(y,.) and F(z,.) are in C 1(S2 \ 52,M ), it follows from Theorem D.9 that the integral over 3 m tends to the corresponding integral over a52


244

as m -> oo. By (B.39),

F(y, z) - F(z, y) = lal

J

n{ fly, x) - F(z, x) } dS(x),

(B.44)

and this integral over Of) vanishes for all y and z in SZ if and only if (B.43) holds. Thus F(y, z) = F(z, y).

We turn to particular cases: the Green functions of the Dirichlet and Neumann problems for half-space and ball. The ordering of this material is: notation, formulae for the half-space (presented first because of their simplicity), formulae for the ball, explanations and comments. Notation Let D := { x E RN XN > 0 } and B := M(0, a) in RN, for all N E N in both cases. The reflection in 8D of any point x E RN will be written x* := (x', -xN), where x' := (X1, ... , XN_1). The reflection in 8B, or inverse point, of any point x E 1RN\101 will be written Sx := (a2/r2)x, where r := IxI, as in Definition B.12. The Newtonian kernel, introduced as K(. ; N) in (A.18), continues to be denoted by K. Field points will often be denoted by y instead of xo. I

Statement B.22 The following formulae define Green functions on S2 x S2 \ { (y, x)

I

x = y }, where either ) = D or 12 = B. The non-singular

parts are defined on

S2x52\{(y,x) I this domain is bigger than that demanded in Definition B.18, and bigger than that established by extension in Theorem B.21. The reader should

verify that, if x E D, y E D and x = y*, then x = y E OD; also that, if

xEB,yEBandx=Sy,then x=yEBB. (i) The Green function of the Dirichlet problem for -,L in D is

GD(y, X) = K(y - x) - K(y* - x),

(B.45)

where

IY* - XI = { Iv' - x'IZ + (YN + xN)2

}112

= IY - x* I.

(B.46)

(ii) A Green function of the Neumann problem for -A in D is HD(y, x) = K(y - x) + K(y* - x).

(B.47)

(iii) The Green function of the Dirichlet problem for -A in B is

GB(Y,x)=K(y-x)-K (J-t(sy - x) I,

(B.48)


245

where IYI

a ISy - xI =

a

2 2 f a4 - 2a yz x +IYI IxI

}

1/2

IY - SxI. = IxI a

(B.49)

It is to be understood that limiting values are taken for y = 0 or x = 0, both here and, later, in formulae for derivatives of GB.

(iv) A Green function of the Neumann problem for -A in B is

HB(Y,x)=

K(y - x)

if N = 1,

K(y - x) + K (1X-t(sy - x) I

if N = 2,

K(y-x)+K (!i(Sy_x)) + E(y, 2 if N> 3, (B.50a,b,c)

where UN := Ia.N(0,1)I and E(y,x) is defined as follows. Let r := IxI, s := IyI and rs cos l :=x-y; then

fly, x)

f {PN_3(P2_2Prcos2+r2)_( N-2)

-1

dp,

(B.51a)

P

a2/s

00

J

tN-3 (2

-

2t rs cos A a2

+ r 2s2 1 -z(N-2) a4 J

-_

1 t

dt, (B.51 b)

00

f {PN_3(p2_2PscOsl+s2)_N_2)_ 1 } dp.

J

(B.51c)

P

a2/r

Again it is to be understood that limiting values are taken for y = 0 or

x=0. Remarks 5. Construction of these Green functions. (a) Definition B.18

obliges us always to begin with the potential K(y-.) [equally, K(. -y)] of a unit source at y. The artifice of making GD(Y, x) an odd function of xN (hence vanishing for xN = 0) by adding a negative unit source at the reflected point y*, and of making HD(y,x) an even function of xN (hence satisfying 8HD(y,x)/8xN = 0 for xN = 0) by adding a positive unit source at y*, is an application of a standard device in mathematics and physics. A shadow of this method can be found in virtually all formulae for Green functions.

(b) In the case of GB, there is an obvious resemblance of Sy to y*, but the factor IyI/a, accompanying Sy - x in the formula (B.48), requires

246


Sy

aB

Fig. B.4.

explanation. The situation is this: with y a fixed parameter and y E B, we seek the function -q(y,.) that is smoothly harmonic in B and equals K(y-.) on B. These conditions suggest the Kelvin transform, relative to 7B, ofK(y-.)IRN\B. [Here it is the dot that is restricted to RN\B, and with respect to which the transform is taken] By Definition B.12 and Theorem B.15, this Kelvin transform is defined on B \ {0}, is smoothly harmonic in B \ {0} and equals K(y-.) on B. If N # 2, we extend it to R by continuity to obtain -q(y,.). If N = 2, this Kelvin transform tends to minus infinity at the origin and is not symmetrical in . and y, but addition of the function x H (1 /27r) log(a/r) corrects both faults. (c) For the Green function HB of the Neumann problem, addition of the point singularity K ((IyI/a)(Sy-. )) is completely successful only for N = 2. When N = 1, this term is not needed for the boundary condition 8HB/8n = -1/IaBI; when N >- 3, it is not enough. Equation (B.51a) shows that the field -VE(y,.) of the extra term (for N >- 3 and y 0) is that of sources distributed on the radial line outward from Sy (Figure B.4); their strength (charge per length or mass per length) is proportional to pN-3 at the point distant p from the origin. The term -1/p in (B.51a) is required for convergence of the integral defining the potential E(y,. ), but makes no contribution when we differentiate with respect to x. 6.

Verification of basic properties. To check that the Green functions

in Statement B.22 have non-singular parts q(y,.) that are smoothly harmonic in S2, and satisfy appropriate auxiliary conditions, is a matter of inspection and direct calculation that is left mainly to the reader. Here are a few details that may be helpful.


247

(a) That the non-singular part of GB(y,.) is smoothly harmonic in B, for fixed parameter y E B (hence for fixed Sy B), is shown by the formulae yl

C

N+2

a

if N

K(x - Sy)

KNa N+2

K

(ii(SY_x)) _

if

K(x-Sy)+ I

I

2, y

0,

N*2, y=0,

logy if N=2, y*0, IYI

if N=2, y = 0,

og 1

a

271

(B.52)

where the values for y = 0 are limits of those for y # 0 [because JSyj = a2/lyl when y

0].

(b) That the function E(y,.), occurring in HB(y,.) for N > 3,

is

smoothly harmonic in B, for fixed parameter y E B, is established as follows. For y * 0, we use (B.51a); differentiation with respect to x under the integral sign is legitimate and is justified very much as in the proof of Theorem A.S. For y = 0, we observe from (B.51b) that E(0,x) = 0 for

allxEB. (c) To verify that HB, for N > 3, satisfies the boundary condition (B.39), we use the notation introduced before (B.51), write R

r2

- 2rs cos A + s2 }

1/2,

r2s2 l 1/2 A

{ a2 - 2rs cos A +

}

a2

JJJ

and use (B.51c) for the function E. Then (for r # 0) BHB( y, x)

N J R-N(r - s cos A) + A-N I/ a2

2

- s cos 2.

- a-N+2) } . - I (A-N+2 l setting r = a and observing a pleasing cancellation, we obtain 8HB(y, x) Or

1

r=a

aN aN-I '

(d) The function HB satisfies the normalization condition (B.43) because it has the symmetry property HB(y,z) = HB(z, y); the equivalence of these two conditions was established by (B.44).


248

(e) For the half-space D, the Green function GD satisfies the growth condition (B.36a), while HD satisfies (B.37a). This shows that the hypotheses leading to the representation formula (B.35) for the half-space were not artificial.

7. On uniqueness of the Green functions. (a) As was remarked after (B.38), Theorem B.16 implies uniqueness of GD and GB, provided that, in the case of D, the non-singular part gD(xo, x) = o(r) as r := Ixl -+ oo. This

growth condition is not satisfied for N = 1. However, if the condition is changed to GD(xo, x) = o(r) as r - oo, then this is satisfied even for N = 1, and the proof of Theorem B.16 still applies to the difference of two Green functions, because this difference has no singularity at

x=xoE0. (b) The function HD is unique apart from an additive constant, provided that for N = 1 we demand symmetry: HD(x,y) = HD(y,x), and provided that for N >- 2 we impose the growth condition: HD(xo, x) = o(r)

or gD(xo, x) = o(r) as r -* oo. For N = 1, this uniqueness follows from elementary consideration of what may be added to the function HD in (B.47); for N > 2, it follows from Theorem B.17. (c) One can concoct a partial uniqueness result for the function HB, but this is hardly worthwhile, because there are alternatives to the conditions (B.39) and (B.43), and because it is the uniqueness statement in Theorem B.17 that is the important one.

B.5 The solution of the Dirichlet problem for a ball

According to the representation formula (B.41), the solution of the Dirichlet problem for the ball B := -4(0, a) is u(xo) =

,fB

0G(xo, x)

G(xo, x) f (x) dx JOB OB

On

g(x) dS(x),

x0 E SZ,

(B.53)

where G = GB is now given explicitly by (B.48). In this section we discard

the strong assumptions on u under which (B.41) was derived. Rather, we impose various conditions on the data f and g, and show that (B.53) yields various kinds of solution of the Dirichlet problem.

Notation Throughout this §B.5 we write r := Ixl, ro := 1X01, UN I09N(0,1)1 and [as in §B.3] Sx := (a2/r2)x, Sx0 := (a2/ro)xo.

It is helpful to decompose the right-hand member of (B.53) into three


249

parts by the definitions

uf(xo) :_ f G(xo, x) f (x) dx = v(xo) - w(xo),

(B.54)

where

v(xo) := w(xo) :=

JaB

f

JB

fK(xo - x) f (x) dx,

K (r (xo - Sx)) f (x) dx,

(B.55)

xo E

(B.56)

if xo E B,

(B.57a)

if xo E 3B,

(B.57b)

a

P(xo, x) g(x) dS(x)

g(xo)

-aGxn, x)

P(xo x)

xo E RN,

.-a

Ixo z-z r°xN UN a

(xo E B, x E aB). (B.58)

Here of may be called the Green volume potential of the ball B and the density function f ; the function ug is the Poisson integral for B of the boundary-value function g; and P is the Poisson kernel of B. The two parts of the volume potential are the Newtonian potential

v of f and a function -w that has two useful representations. The first form of w results from changing the variable of integration to := Sx = (a2/r2)x; the Jacobian determinant of the transformation is det(ax;/al;j) = by (B.23). Then, with the notation a2

a

N+2

f*O :=f (2,) I -

p

(B.59)

and apart from an added constant when N = 2, w(xo) is the Newtonian potential at xo E B of the density function f*, which has support outside B. That is,

forN#2, w(xo) =

Jp>a

K(xo - )

d

(xo E B),

(B.60a)


250

for N = 2, w(xo) =

f

C=

where

a

K(xo - )f) d +C (xo E W),, f

1J 27r

p>a

log P

do

a

(B.60b)

.

The second form of w results from the identity (B.49), now written as (r/a)jxo - Sxj = (ro/a)ISxo - xl. This implies that, apart from an added (a/ro)N-2 times the Newtonian harmonic function when N = 2, w(xo) is potential, at the field point o := Sxo outside B, of the original density function f. That is, w(xo) _ B

K ( PO a PO

a )

x)) f (x) dx

(Po := I'o j ? a)

N-2

vo

v(o) + I log ao f f (x) dx

if N = 2.

(B.61b)

B

The case N = 1 of these formulae is discussed in Remark 1.5 and Exercise B.32; it is to be understood in the remainder of this §B.5 that the dimension N >- 2. Theorem B.23 Let u denote the Green volume potential called u f in (B.54).

(i) If f E Lp(B) for some p > N/2, then u E C(W), ulaB = 0 and u is a distributional solution (Definition A.7) of -Au = f in B; it is the only function with these three properties.

(ii) If f E LP(B) for some p > N, then u E C' (K), ul aB = 0 and u is a generalized solution (Definition A.7) of -Au = f in B; again it is the only function with these three properties. Proof We use the decomposition u = v - w described above, and results in Appendix A. (i) (a) First, v c C(W) when f e Lp(B) with p > N/2 because v E C(RN) by Theorem A.6. The same is true for w, by the extension of Theorem A.6 in Exercise A.32, if the function f * introduced in (B.59) has two

properties: f* E Lp(RN \ B) with p > N/2, and Wf* E L1(RN \ W), where W is the weight function in (A.78). For the first of these we use


251

the same p as in f E Lp(B); then P

a

J

>a

P(n't2)

d

P

()2N

P

Q= lB

P

If (x)Ip dx,

because p(N + 2) > z N2 + N > 2N. For the L1 property of W f*, first let N = 2. Then f * (l;) = f (x)(r/a)4 and d = (a/r)4 dx, so that

log (e+ p>a

If(x)I dx

Ixl

B

1 implies that q < oo, hence that e E Lq(B). For N > 3, the L1 property of W f* = f* is established again by means of the Holder inequality. Thus w E C(B). (b) That UI aB = 0 follows from (B.61), which shows that wI aB =

vI

aB

(c) Theorem A.8 establishes v as a distributional solution of -Av = f in B, while Aw = 0 in B pointwise by (B.60), which shows B to be outside the support of f*, and by the extension of Theorem A.5 in Exercise A.32.

(d) Let uo be the difference of two functions u having the three properties stated in (i) of the theorem. Then uo E C(B), uoI aB = 0 and both uo and -uo are distributional subsolutions (Definition 2.10) relative to A and B. It follows from Theorem 2.11 that uo = 0 on B.

(ii) The proof for f E Lp(B) with p > N is entirely analogous to the foregoing proof for p > N/2. In place of Theorems A.6 and A.8 we now use Theorem A. 11.

It is to be expected that, if f E C°'µ(B) for some p E (0, 1), then of is the

pointwise solution of the Dirichlet problem for -A in B with boundary condition u = 0 on OB. To prove this, we need a long (but straightforward) calculation like that in Lemma A.14. Reverting to the symbol q for the non-singular part of a fundamental solution, we now write q(xo, x)

q `j(xo,x)

where again

_

a2q(xo, x)

:= Sx and p = I

-K (a (xo - Sx)) , (a)_N+2

-P I.

(aiaj K)

(B.62a)

(xo - ),

(B.62b)


252

Lemma B.24 Let qij be as in (B.62) and let H :_ -4(xo,21h1) with xo E B and 0 < Ihl < !a. Then 2

E IB\H

x) dx

< const.,

where the constant depends only on N.

Proof To a large extent the proof follows that of Lemma A. 14. Although

gij(xo, x) has no singularity when xo E B and x E B, it is unbounded because Ixo - I can be arbitrarily small when xo and x are near 8B and near each other.

(i) In contrast to the function Kij(xo-.) in the earlier proof, gij(xo,.) is integrable on B and on HnB, so that the integrals over these two sets can be calculated separately. Moreover, gij(xo,.) E C00(B) and is smoothly harmonic in B [because q(xo,.) has these properties by (B.49) and (B.52), and because the operators 82/7xoi8xoj and A commute]. Therefore we can use the mean-value property of harmonic functions, Theorem B.8, to integrate gij(xo,.) over any ball contained in B. In particular, for all i, j E { 1,...,N},

fqiJ(xox) dx = IBlgij(xo,0) = 0,

(B.63)

because (B.62b) shows that qi.j (xo, x) = O(p2) as x -+ 0 and p - oo. (ii) Under a rotation of the co-ordinate frame, again written x = T(xo)x and xo = T (xo)zo, where T(xo) is an orthogonal N x N matrix depending on xo, we also have = T(xo)Z, where Z = Sz = (Sx), and qij transforms exactly as Kid did in the proof of Lemma A.14. Again we so choose T(xo) that zo = (Ixol, 0, ... , 0), and use these special co-ordinates zj in the rest of this proof, but omit the tilde for ease of writing. For the same reasons as before,

L\H

f

\H

if i * j,

gij(xo, x) dx = 0

gJJ(xo, x) dx,

ql l (xo, x) dx = j=2

B\H

so that we need bound only the integrals of qjj for j = 2,..., N, and these integrals are all equal.

(iii) For H c B, we evaluate the integral of qjj(xo,. ), j# 1, over H as


253

0

Fig. B.5.

follows. Set Ixol =: ta, 0 < t < 1, and apply the mean-value property of harmonic functions once more; there results

= IHI Igjj(xo,xo)I

.f gjj(xo,x) dx H

=

t2)-N

6N(21hl)N a-Nt2(1 -

N

(J

6N

1)

Since H c B, we have 21hl < a - Ixol = all - t) and 0< t < 1, so that gjj(xo,x) dx
0.

(ii) The inequality lu(x)I 0 for x0 E B and X E aB, and the result (B.76); accordingly,

fs P(xo,x)g(x) dS(x) < Ilgll JOB

,

f

P(xo,x) dS(x) = Ilgll

aB

(iii) It remains to prove that u E C(B \ A). Since u E C0°(B) and


259

u E C (OB \ A), we need prove only this: for given p E OB \ A ands > 0, there is a number S = 6(p, e) > 0 such that Iu(xo) - u(p) I < e

whenever x0 E B and Ixo - pI < 6.

In the following argument, (B.76) and the positivity of P are used repeatedly; we also use an abbreviated (but self-explanatory) notation for subsets of aB. Observe that, for every p > 0,

- u(P)I

Iu(xo)

= -
- 3 is valid for all N E N. [For N = 1, this function differs by a constant from that in (B.50a).]

Exercise B.40 Let B :=.N(0, a), let r := jxj and write 0/8r := (x/r) 0 if x * 0. Consider for N > 2 the Neumann problem of finding u such that u E C(B) n C2(B),

Au = 0 in B,

8u/ar E C(B \ {0}), 8u Or as

= g,

(B.89)

where g E C(8B) and fB g = 0 (the compatibility condition). The solution of Exercise 2.42 shows (even though less smoothness is demanded here) that any two solutions of (B.89) differ only by a constant.


266

Let HB be the Green function in (B.50), and define

g() dS()

u(xo) :=

for xo E

(B.90)

LB

Given the result that this function u e C(aB), prove that it is a solution of (B.89).

[One can verify the boundary condition, and prove that au/ar E C(B \ {0}), by means of the identity ro

allB

(xo,

) = a PB (x0, ) -

for x0 E B \ {0},

6N aN_2

COB,

Or oo

(B.91)

and by use of Theorem B.29; here PB denotes the Poisson kernel called P in (B.58). The proof that u e C(aB) is not difficult but not short.]

Exercise B.41 Show by an example that a function u satisfying (B.89) need not belong to C1(B). [One possibility is to choose N = 2, to write (x, y) for points of R2 and z = x + iy for points of C, and to contemplate the real part of z+a

w(z) := 3ial; log(logwhere

3a

and a convenient branch of arg(log ) is chosen.]

Exercise B.42 (i) Show that the function u in (B.90) may be written

u=v+wif,forN-2 andxoEB, K (xo - )

v(xo) := 2

(B.92)

LB

w(xo)

__

a

aN

rN-3

-

f>a Ixo - xIN-2 where

1

r

}

g()

rN-1

dx

:= ax. r

r := IxI,

}

(B.93)

Note that w = 0 (the zero function) if N = 2. (ii) By a lateral derivative of u at x we mean any directional derivative

m(x) Vu(x)

with x E B \ {0},

m(x)

0,

x m(x) = 0.

Show that every lateral derivative of u at x is a linear combination of N - 1 lateral derivatives of form u;;(x) := x j (aju) (x) - xi (a ju) (x) if xi

0, i is fixed and j E { 1, 2, ... , N} \ {i}.

B.6 Exercises

267

Caution This (. )ij must not be confused with that in (A.65) and (B.62b).

(iii) Prove that lateral derivatives of the v in (B.92) and w in (B.93) are given by (B.94)

g(oo) }

vii(xo) = 2 fs Kii (xo, ) { a

wij(xo) = a(N - 2)

/'

Jr>a

Kij (xo, x)

gO -g(oo) r2

dx,

(B.95)

where xo E B \ {0}, o := (a/ro)xo, i:# j and, for x E RN \ {xo}, Kid (xo, x) := xoi (aiK) (xo - x) - xoi (ajK) (xo - x).

(B.96)

Exercise B.43 This exercise concerns the following result. If in the Neumann problem (B.89) we have g E C°°µ(aB) for some u E (0,1), and LB g = 0, then not only does the solution u in (B.90) belong to C' (B) [cf. Exercise B.41], but also Vu E C°^µ(B,RN).

Give the main parts of the proof by showing that the functions x H rau(x)/ar and vii, all defined to be zero at the origin, belong to C°'µ(B). Here vij(xo) is as in (B.94) for X0 E B \ {0}.

[The uniform Holder continuity of rau/ar follows from (B.91) and For vii, the proof of Theorem B.30 is relevant, but perhaps the key step is a decomposition of vi j (xo + h) - vii (xo) that

Theorem B.30.

resembles (A.71), despite the different meaning here of (.)ij. The proof for wig, which is not demanded here, is similar to that for vii but easier.]

Exercise B.44 Consider for N > 2 the Green volume potential U f (xo) := in GD (xo, x) f (x) dx,

xo E D,

(B.97)

D

of the half-space D := { x c RN xN > 0 1; here, GD is the Green function in (B.45). Assume that f is measurable and that, for some I

6 E (0,1), IfI

I

I := ess supXED(l + r)'+a I f (x)I < oo

(r := Ix1).

(B.98)

Prove that, for all xo E D, I (a"u1) (xo) I -< 1111f I I (1 + ro) I

if (a1 = 0 or

1,

(B.99)

where the constants f depend only on N, 6 and a. Note that (B.98) fails to imply that f E L1(D) [since N > 2], hence


268

fails to imply the condition (A.78) assumed for f in Exercise A.31, but that (B.98) is enough to imply the growth conditions assumed for u in Theorem B.16 and in (B.36b) of Exercise B.19.

[It may be helpful to separate the cases ro < 1 and rp >- 1; and, for ro > 1, to integrate separately over the subsets of D in which (a) R := Ix - x0 l < 12 ro, (b) R > 12 ro and r < 2ro, (c) r > 2ro.]

Exercise B.45 For the half-space D and for N > 2, the Poisson kernel -0GD/7n is 2XON

-2(aNK)

PD (xp,

with xo E D,

YY

6N Ix0 - b IN

E OD,

and we shall also need the Marcel Riesz kernels

M'M1(xp,

-2 ajK)(xo

)

-

2(xo.i- 0) ) = 6N

Ixo - SIN

with j E {1,...,N - 1}, xoED \

E 10D.

The corresponding Poisson and Riesz integrals are, respectively, g

f

)

PD (xo, )

if xo E 8D,

and

Mj (xo,

if xoED,

aJD Mi(xo,

ID

if xo E 3D,

where m is a 'mollifier': m E C°° [0, oo), m(t) = 1 for 0 < t < 1, m(t) = 0 for t > 2, and m is non-increasing.

Still for N > 2, prove (or merely accept, in order to do Exercises B.46 and B.47) that the map g H ug is a bounded linear operator from C °'µ(OD) into C°'µ (D), and that g --> vg,j is a bounded linear operator from C°'µ(7D) n Lp(0D) into C°'µ(D) ; here µ E (0,1), p E [l, oo) and 7D may be identified with R' [For ug, the proof resembles that of Theorem B.30. Let o := (x', 0) ; 1

then vg,.i

(xo)

ID

for all xo E D.

B.6 Exercises

269

The uniform Holder continuity of vg,j follows once again from a decomposition like (A.71).]

Exercise B.46 Use the results of Exercise B.45 to prove an analogue, for the Neumann problem in the half-space D with N > 3, of the theorem in Exercise B.43.

u(xo) :=

More precisely, defining

JaD

for xo E D, N> 3,

HD(xo, )

where HD is the Green function in (B.47) and g c Cb'µ(aD) n Lp(aD) for some p E (0, 1) and p c- [1, N - 1), prove that u E C' (D) n C00(D), that Du E Ce''` (D, RN) and that

Du = 0 in

D,

-aNUI aD = g'

Exercise B.47 Here we are concerned with the smoothness on D of the Poisson integral u,y, defined in Exercise B.45, when ip E Cc°(aD). The compact support of ip could be the result of applying a partition of unity to a more general function in C0°(3D). Derive the (formulae (aN uw) (xo)

(aN+tuw)(xo)

= JfD PD (xo =

)

2JD(0'K)(xo-(0

+ aN_, and O' :_ (al, ... , ON-I)For xo E D, write any derivative (a"uy,) (xo) as an integral with kernel PD or kernel 2V'K. Use the results of Exercise B.45 to show that u. E C`° (D).

in which k E No, xo E D, A' := a, +

Appendix C. Construction of the Primary Function of Siegel Type

In this appendix we use the Poisson integral for a ball, which is the subject of Theorems B.29 and B.30, to prove Theorem 2.27, which describes the

primary function of Siegel type. The dimension N > 2. There is a clash of notation between §B.5 and §2.5; the primary function of Siegel type, denoted by g in Theorem 2.27, will be called v in this appendix. We recall from (B.57) and (B.58) that the Poisson integral for the ball B :_ R(0, a) may be written

fP(x, )

if x E B,

(Cl.a)

if x E aB,

(Cl.b)

B

g(x)

the Poisson kernel P being defined by P(x,

a

alx 2

)

r2IN

(r := lxi, x E B,

E aB).

(C.2)

QN

The symbol , used consistently in Appendix B for the inverse point (a2/r2)x of x, is now merely a variable of integration. Theorem B.29 shows that, if g E C(aB), then u E C(B)nC°°(B) and the formula (C.1) describes the unique pointwise solution of the particular

Dirichlet problem Au = 0 in B, u = g on

B.

The theorem also

establishes properties of the function u in (C.1) when g is discontinuous but bounded. In the present application, however, we shall have

g(x) = xN on aB \ E,

where E := aB n { x E I[8N I xN = 0 } , (C.3)

so that g is neither continuous nor bounded on aB; something beyond the results of Theorem B.29 will be needed. Our first task is to examine 270

Appendic C. The Primary Function of Siegel Type

271

the particular form of Poisson's integral for B when g is an odd function of its Nth argument.

Notation For upper and lower hemispheres we write

5+{xE18N Ixl=a, xN>0}, Ixl=a, xN- 2.

If g is an odd function of its Nth argument, so that g(x*) = -g(x), then the substitution

= rl* yields

.f P(x, )

fs+ P(x, i*) g(n) ds(n);

s

) s+

u(x) :=

Q(x,

)

g(x)

if x E B,

(C.4a)

if x E aB,

(C.4b)

where

Q(x,) := P(x,) - P(x, * ) a 2-r2

1

1

UN a

Ix - SIN

Ix - *IN

(C.5) '

Lemma C.1 The kernel Q(., ) is an odd function of its Nth argument. If XN >- 0 (and x e B, D E S+), then

0- 0; then 0 < R < R* and R* - R2 = 4xNSN

R* - R = R* +R

2XNSN

R*+R -

R

Accordingly, for m E N,

R* - Rm

0 < R-m - R-m _ *

(R *

- R)

2

(R* R)-

R Rm + R2 Rm-1

+ ... +

1

N

(C . 8)

Rm+2

The definition (C.5) of Q, and the inequality (C.8) with m = N, imply (C.6).

(iii) To derive (C.7), we begin with an application of the Leibniz rule (Exercise A.23) to obtain

l

/

z

_ UNa

/

Y

(Y-N)!N!

{()Y

a )' (R-N-RN Y

(ax)

(a2

- r2)

2, but a harmless one. (iv) For derivatives of R-N - R N, we use a slight variant of (A.32) : aflr-N = r-N-21RIpp(x)

(x * 0),

Appendic C. The Primary Function of Siegel Type

273

where pp is a homogeneous polynomial in x1, ... , xN of degree I# I ; say

pp(x) = E

C#'-X11.

1"1=1f1

It follows that le

a

ax C-/

(R_N

- R-N)

*J

=

(R_N_21P1

-R -N-21flI ) pp(x -

+R*N 211{pp(x-

pp(x- *)}.(C.11)

In this proof, const. will denote positive numbers independent of x, and a. By (C.8), and because Ipp(x - )I < const.RlB', (R_N_2Ifh

-

R*N-2IQ1)

lpfl(x - )I < const.

Next, with the notation a' RN-21,1I pp(x

aN_1),

- ) - pp(x - * )I

= RN-21#1

Cp a(x - 0", 1 (XN - N)"N - (XN + N)"N I I"I=1p1

< const. a1B1-1 NR-N-21BI ; RI" I because we have ignored the favourable inequality I (x' -')" there are terms with Cp," 0, Ia'I = 0 and aN = IQI Insertion of the last two estimates into (C.11) yields

C aX

I

(R-N - R*N) < const.

RI#I-2 + aI#I-2) R-N-21 1

(C.12)

where RIPI-2 + a1#1-2 < const. a' 2 if Ia I > 2, because R < 2a; the term R1#1-2 will be retained only for I#I < 1. '

(v) Finally, we return to (C.9) and use (C.10), (C.12) and the remark following (C.12); there results

(a)Y

Q(x,

) ICI

const. N

a2-IYIR-N-2

+

21QI

a IPI_

C2

IYIR-N

B-Y

(-)

Since 1 < 2a/R < oo, the terms with I#I = 1 dominate the first sum, and

274

Appendix C. The Primary Function of Siegel Type

that with $ = y dominates the second; therefore Q(x,

)

< const. N { a3-IvI R-N-3 + a1Y1 R-N-21Y1 } .

(C.13)

For 1vI = 1, the first term on the right of (C.13) dominates [because 1 < 2a/R < oo] ; for ly I >- 2, the second term does.

The primary function of Siegel type is now defined by

J

v(x) = v(x; a) :=

Q(x,

dS(g)

)

if x E B,

(C.14a)

if x E aB \ E,

(C.14b)

N

a/xN

where E denotes the equator of B, as in (C.3). Recall that the signum

function sgn is defined by sgn t = -1, 0,1 for t < 0, t = 0, t > 0 respectively.

Theorem C.2 The function v = v(. ;a) defined by (C.14) belongs to C(B \ E) n C00(B). Also,

Lv=0 for x E

E,

in B;

v(x', -xN) = -v(x', XN)

and sgn v(x) = sgn XN;

v(x; a) depends only on x/a; and

Iv(x)I 3 pN 1,

S2 := aB \ S1,

A:=aBn{x I IXNI=3pN}, J a/xN on Si, 0

on S2,

g2(x)

0

on Si U E,

a/xN

on S2 \ E,

and, for j = 1, 2, ve(x) :=

Then v=v1+v2 on

Q(x,) g1O dSO if x E B, s gj(x) if x E aB. E.

Since g1 E L,,,(aB)nC(aB\A), Theorem B.29 ensures that v1 E C(B\A); certainly v1 satisfies (C.15). Now consider v2 on B \ S2, observing that

S2 is closed. Adapting remarks in (i), we see that (a/ax)YQ(x, )$2()

276

Appendix C. The Primary Function of Siegel Type

is bounded in terms of dist(x, S2) for each y, and that v2 E C°°(B \ S2).

(Lemma C.1 extends to points x E OB, provided that Ix - I > 0 .) In particular, (C.7a) shows that for points x and x + h in B, such that I xN I > 3 PN and I xN + hNI > 3 PN, we have

Iv2(x + h) - V2(x)I < const. aN+2 (3 pN)-N-3 IhI,

with a constant that depends only on N. Thus v2 satisfies (C.15) with room to spare. (iv) To prove that v(x; a) depends only on x/a, we write Q(x, ; a) for the function defined by (C.5) and set x =: az, a in the definition (C.14a); there results

v(x; a) = fQ(;1) (1/iN)

if x E B,

I = 1, N > 0 }. This proves the claim for x E B; the result is immediate from (C.14b) if the integral being over the unit hemisphere {

I

I

xEBB\E. (v) It remains to bound Iv(x)I for xl < a/2. Now, Ix - I >- a/2 when IxI < a/2 and c E S+; then the definition of v and the bound (C.6) for Q imply that 1V(x)1
h(y').

(D3.a) (D3.b)

The functions Wm in the following lemma are an example of a partition of unity on S2. They are a key to our proof of the divergence theorem because we use them to reduce the proof to one for functions of small support in convenient neighbourhoods. [The support of a function

D. 1 A first divergence theorem

281

is defined in Chapter 0, (iv); essentially, it is the smallest closed set outside which the function equals zero.] The hatched set in Figure D.1 represents the support of a function W. when p = pm in the figure.

Lemma D.2 Let n be bounded and 3) of class C. Then for some integer M there exist functions P1,1P2,. .. ,'PM as follows. (a) im 1 ipm(x) = 1 whenever x E 52. (b) W. E C°°(S2) and 0 < lPm(x) < 1 for each m E { 1, ... , M} and all

xE52. (c) The functions 1m are of two kinds: there is an integer K < M such that, if m E {1,...,K - 11, then Pm E Cc(S2), while, if m e {K, ... , M}, then supp'WVm intersects 852. In the second case supp Wm c S2nW(pm), where

W(pm) is a quasi-cubical neighbourhood of a point pm E 852.

Proof (i) Assign to every point c E S2 a ball .1(c, p) as follows. If c E 52, choose p = p(c) so small that 4(c, p) c 52; if c E 852, choose p = p(c) so small that _4(c, p) c W(c), where W(c) is a quasi-cubical neighbourhood of c. The family {M(c, p) c E S2} is then an open cover of S2 [in other words, S2 is a subset of the union of all these open balls]. Since S2 is compact, there is a finite subfamily, say {M(cm, pm) I m = 1, ... , M}, that covers S2, and we so label this subfamily that cm E 52 if m < K, while cm EOn if m K. I

-

(ii) Define for each m E {1, ... , M} (see Figure D.2) exp -

0-(x)

1

if s := ifs>-1,

1 - S2

0

iVmx)

Pk(x)

Pm(X)

Ix - cmI

< 1,

Pm

if x E S2 ,

(D.4)

(D.5)

k=1

and let pm := cm when cm E On (when m > K). We observe that

>M 1 (pk(x) > 0 when x E S2, because then x E *c", Pk) for at least one k, and that (pm E Cc (RN) for each m, by Exercise 1.15. The properties (a) to (c) of the functions W, now follow directly from the definitions (D.4) and (D.5).

Notation We shall use the abbreviations

j g :=

Jn

g(x) dx

g := j g(x) dS(x),

and sZ

sp

282 Appendix D. On the Divergence Theorem and Related Matters

an Fig. D.2.

where dx := dxl dx2... dxN and dS(x) is as in Chapter 0, (viii), unless a transformation of co-ordinates or detailed evaluation of an integral should demand a fuller display. Theorem D.3 (a first divergence theorem). Let 91 be bounded, with On of class C1, and let n = (n1i...,nN) denote the outward unit normal on On. Then

f alf = fn n.if

(D.6)

whenever f E C 1(Sl) and j E{ I,-, N}.

Proof (i) Let p',. .. , W m be as in Lemma D.2, and define fn := f Wm for each m E { 1, ... , M}. By (a) of Lemma D.2, M

M

f .(x) = f (x) E Wm(x) = f (x) m=1

for x E S2,

(D.7)

m=1

and each fm E C1(S2) by (b) of the lemma. (If f should happen to be smoother than C', then so is fm.) Since

f ajfm = m=1

I fm) = f 8jf and inn al (\m=1 ,

m=1

.f nlf, Ja4 n.ifm = 84

it is sufficient to prove (D.6) for each fm; the result for f then follows by summation over m. Let Bm := e(cm, pm), where cm and pm are as in (D.4). There are two cases: if cm E S (if m < K), then supp fm c Bm c f ; if cm E 00 (if m >_ K), then pm := cm and supp fm C S2 n Bm, where Bm c WW(pm).

D.1 A first divergence theorem

283

(ii) Let cm E 52 (let m < K). We shall prove /that

in

ajfm = 0 = f

ajfm

(cm E n).

(D.8)

The right-hand equality is immediate because fm = 0 on 852. To prove the left-hand equality, we define fm to be zero not merely on S2 \ Bm but on RN \ Bm; then in ajfm = j ajfm m

for any cube Qm (in RN) that contains Bm. Suppose that j = 1 and that Q. =: (a,b) x Q", where Q" is a cube in RN-1, and let x" := (x2,...,XN). By Fubini's theorem,

JQ

m

alfm = f

l

Qf(f)(")

l dx1 } dx" JJJ

f {fm(b,x")-fm(a,x")} dx" = 0 because fm = 0 on BQm. The argument is similar for j E 12,..., NJ: the Fubini theorem allows us to replace the volume integral by a repeated integral, and we integrate 8 j f m first with respect to x j.

(iii) Let pm = cm E 852 (let m Z K). With m fixed, we abbreviate pm to p, adopt the notation in Definition D.1, and let fm(y) := fm(x) under the transformation y = Yp(x). Define

V+ := { y E K(p) I YN > h(y') 1,

F := { y E 'K(p) I YN = h(y') } ,

and recall that fm vanishes in S2 \ B. Then the desired result,

fJfm =

whenever j E {1,...,N},

(D.9)

is equivalent to the statement

f+1m)dy = f vk(y') fm(y) d(y') whenever k E {1,...,N}; r

(D.10)

here v and da denote [as in Chapter 0, (viii)] the outward unit normal and element of surface area, respectively, in terms of the local co-ordinates. [To pass from (D.9) to (D.10), multiply (D.9) by Akj(p) and sum over j; to pass from (D.10) to (D.9), multiply (D.10) by Akj(p) and sum over k.] Upon use of the formulae (0.2) and (0.3) for v and da, equation (D.10)

284 Appendix D. On the Divergence Theorem and Related Matters becomes

J(akh)(Y')Jm(Y'h(Y')) dy' if kE{1,...,N-1},

J v+

Q'(o,P)

(akjm)(Y) dy =

if k = N.

ffm(Y'h(Y')) dy'

(D.11)

(iv) In order to prove (D.11), we make the further co-ordinate transformation z

'=Y

zN = YN - h(Y)

,

(Y E "i(p)),

which maps Y" (p) onto the cube Q'(0, /3) x (-/3, /3) shown in Figure D.3. Then

if k E {1,...,N- 1},

aYk = aak - (kh)(Y )aN

aN aY

= a

,

N

and the Jacobian determinant is 1

0

0

1

0

-(aih)(Y')

-(a2h)(Y')

1

z;= detGyi) I

...

0

=1. 1

Let W+ := Q'(0, /3) x (0, /3) and let fm(z) := Jm(y); by the rule for subjecting integrals to co-ordinate transformations (Apostol 1974, p.421; Weir 1973, p.158), the desired result (D{.11) becomes

JW' { (akim)z) - (akh)(z) (aNfm)(z) } dz

J

(akh)(z') fm(z',0) dz'

if k E {1,...,N-1},

(D.12a)

,(o,a)

fm(z', 0) dz'

if k = N.

(D.12b)

Now in (D.12a) we have (ak.fm)(z) dz = 0

if k E {1,...,N - 1},

(D.13)

fw+

because, very much as in step (ii), Fubini's theorem allows us to replace the integral over W+ by a repeated integral and to integrate akfm first

D.2 Extension to some sets with edges and vertices

285

ZN

aW+

ZI

Fig. D.3.

with respect to xk; moreover, fm(z) = 0 on 8W+ \ { z (D.13) established, (D.12) is seen to be true because 0J'

I

ZN = 0 }. With

(aNfm) (Z', ZN) dZN = -fmW, 0),

and because we may use repeated integrals again, integrating aNfm first with respect to ZN.

D.2 Extension to some sets with edges and vertices

In many applications one needs a divergence theorem for sets f) with boundaries M that are not of class C' because they have edges and vertices. A general treatment of such boundaries would lead us into deep water, but results for some particular cases are useful and not difficult.

Remark D.4 We shall derive a divergence theorem for the following bounded open sets in RN (N (i) An open interval, say I

2). As elsewhere, x" := (x2, ... , xN). x (aN, bN). (al, b1) x (a2, b2) x

(ii) A circular cylinder, say C := { x E RN I a < x1 < b, Ix"I b cos a },

where a E (0, 7C).

(iv) A finite sector in R2, or circular cone in RN with N >_ 3, possibly

truncated near its vertex: in terms of r := IxI and 01 such that x1 =

286 Appendix D. On the Divergence Theorem and Related Matters r cos 01 and Ix"I = r sin 01, say

K :={xERN I a1 an

andw>0in5:. Proof We shall use the local representations YN = h(y', p) of 00 in Definition D.1; the transformations (D.2) to local co-ordinates will now be written y = Y(x,p),

where Y = (Y', YN)

and Y' :_ (Y1,..., YN_1),

and the yN-axis points into 0. We shall also use the partition functions , WM of the second kind in Lemma D.2. Since the functions WK, , WK-1 of the first kind all vanish on On, those of the second kind W1, form a partition of unity on an: M

EWk(x)=1

if xE00;

k=K

also, Wk E C%2), suppWk S2 n °u(pk) and 0 < Wk (X) < 1 for each k E {K_., M} and all x E S2. We define M

xE

w(x) E Wk(x)11k(x),

(D.26a)

k=K

where

kx

( YN(x, pk) - h(Y'(x, pk), pk) 1I

0

if x E S2 n &(pk), elsewhere in

S2.

D.26b (

)

[The definition of nk(x) for x 0 suppWk is unimportant, because Wk(x)Y1k(x) = 0 outside suppWk under any definition of Ilk Note that, if x E S2 n oh(pk) and y = Y(x, pk), then t1k(x) = YN - h(y, pk )

R.]

D.3 Interior approximations to the boundary 00

295

therefore, at points x E as2 n 1(pk) and with the abbreviation h(y) h(y', pk ),

=

N

- an (x)

aye

>vj(Y'){h(Y')-YN}

N-1

ah = vJ(Y')a

=

.(Y)-vN(Y')

y'

I=1

+... + (ON-1h)(Y )2 + 1 }1/2

{ (01h) (Y')2

(D.27)

upon application of the formula (0.2) for v(y'). The function w E C I (S2) because each 1pk E C'(0), each h(. , pk) E C1(Qk), where Qk := Q1(0,/3(pk)), and each Y(.,pk) is an affine function (that is, a linear function plus a constant). Moreover, wlan = 0 because nk an = 0 for all k ; and the definition (D.26) also shows that w > 0 in 0. Finally, for each k and all x E an n °h(pk) we have -a,)k(x)/an >_ 1 by (D.27); since also 11k I an = 0, this yields I

Ow an an

M Wk

k=K

M

_

E Wk

On

k=K

00

an

1'

as desired.

To construct nn, we shall choose a suitably small number al > 0, set a,n := al/m for m E N, and define to be the part of the level set {x E 12 w(x) = a,n} that lies close to 00. It will save trouble later to I

record here some implications of Lemma D.7.

Remark D.8 (i) Since Vw is continuous on the compact set S2, it

is

uniformly continuous: for every a > 0 there is a number SE > 0 such that x,

E SZ

and Ix - I < 8E

I(Vw)(x) -

E.

(D.28)

(ii) On as2 the vector field Vw is normal to as2 because w I an = 0, and it

points inwards to n because -Ow/On > 0. Therefore IVwl = -Ow/On > 1 on an, whence x E f2, p E as2

and Ix - pI < 8E

I(Vw)(x)l > 1 - E.

(D.29)

(iii) Define

d(x) := dist(x, an), if x E SZ and d(x) < 61/2 e(x) := Vw(x)/IVw(x)I

(D.30)

296 Appendix D. On the Divergence Theorem and Related Matters

Fig. D.7.

[This last ensures, in view of (D.29), that IVw(x)I > 2.] The direction field (or unit-vector field) e is a continuous extension of the inward unit

normal field -n on 852; that is, le(x)I = 1 and eI au = -n. Condition (D.28) now implies the existence of a number 6* > 0 as follows: if x E 52,

p E % and Ix - pI < 6*, then

e(p) e(x) >

2 = cos 6

and

e(p) Ow(x) > 5.

(D.31)

(A calculation, outlined in Exercise D.14, shows that 51 4 can serve as 6*

(iv) We use (D.31) to construct a useful cylinder D(p) about any boundary point p E 852. Take local co-ordinates zj such that z := Zv(x) := C(p)(x - p), (CNI(h))N

where

OxZN =

J=1 = e(p) = -n(p),

(D.32)

and where C(p) is an orthogonal N x N matrix depending on p. Define (Figure D.7) 6*

fl:_

,

E:_{zERN I IZ'Io

or d(x)>/3},

equivalently,

S2\52,,,_{xES2 I w(x) 0. Prove that aB is of class C1. [The boundary aB is the union of 2N - 1 pieces as follows. Call

a boundary piece of type k if, for some subset J := { jl, j2, ... , jk } of {1,2,...,N}, the piece is

{xERN I

=p2, x1 > -0 if jEJ, xm 1.]

Exercise D.15 Let G(S) := { x E 1 dist(x, 8s2) > S } for a given (open) set 52 and given number S > 0. Prove that I

(a)G(S)={xEl[R' 14(xS)cS2}; (b) if 52 is convex [Apostol 1974, p.66; Rudin 1976, p.31], then G(S) is also convex.

Exercise D.16 Let x = (x, y) and

n) denote points of R2. Given

g E C 1(R), define

52:={xER2 I xER, y>g(x)}, (-91(t),1)

(t, S) := (t, g(t)) + S

g(t + 1

t E R,

S = const. > 0,

and let P(S) t E ][R }. Note that P(S) is the set of points g(t, 6) called q(.,S) in (D.25); also that P(S), as the range of the continuous function g(., S) : R R2, may be called a path (or curve or arc). (i) Show that P(S) is parallel to 852 in the sense that I

dy

dx

PS)

_

11r(t, S)

-S(t)

at any point t such that g"(t) happens to exist and jt, S) (ii) Prove that, if G(S) is as in Exercise D.15, then 8G(6)

0.

P(S).

Exercise D.17 Consider the particular case of Exercise D.16 in which g(x) = 3 IxI3/2 for all x E JR. Show that (a) each point (0, y) E 52 is the point of intersection of three straight lines normal to 852; (b) the path P(S) is qualitatively as in Figure D.9, the numbers to = to(o) and

tl = t1(S) being uniquely determined by (to,S) = 0, to > 0 and by l;Jtl, S) = 0, t1 > 0; (c) neither P(S) nor 8G(S) is a boundary of class C1.

D.4 Exercises

303

Y

Fig. D.9.

Exercise D.18 Spherical co-ordinates r,01,.--,ON-1 for RN, N >_ 2, are defined as follows. If N = 2, x1 = r cos 01, x2 = r sin 01i

if N = 3, x1 = r cos 01, x2 = r sin 01. cos 02,

x3 = r sin 01. sin 02i

if N = 4, x, = rcos01, x2 = r sin 01. cos 02, x3 = r sin 01. sin 02. cos 03,

x4 = rsin01.sin02.sin03i if N >- 5, x1 = r cos 01, x2 = r sin 01. cos 02,

XN_1 = r sin 01 ... sin 0N_2 cos ON-1,

XN = r sin 01 ... sin 0N_2 sin ON-1

Write this transformation as x = f (v), v E E, where v := (r, 01,..., ON-1) and E := (0, oo) x (0, 7C)N-2 x (-n, it), with the understanding that E (0, oo) x (-7G, 7C) if N = 2.

304 Appendix D. On the Divergence Theorem and Related Matters Prove the following.

(i) xi + + xN = r2 for all v E E. [Begin with xN + 4_i.] (ii) xN + xN_1 >0 for N > 3 if and only if r > 0 and 0 < Oj < n for

j=1,...,N-2. (iii) If f is restricted to

(0, 00) x (0, n)N-2 x (-n, n], then its range

is

f (E) = { x E RN I x,

0 },

and f It is injective (one-to-one). (iv) Spherical co-ordinates are orthogonal co-ordinates: for v E Ox

Ox

avi

avj

j,

i

fax/avjl are given by

and the arc-length functions hj hi(v) = 1,

if

=0

h2(V)=r, h3(v) = r sin O1, ... , hN(v) = r sin 91 ... sin ON-2.

[This can be proved by induction, since y E RN+1 can be represented as

y = (xi, .... XN-i, XN COS ON, XN sin ON),

where x E RN and xj = f j(r, 01, ... , ON-1) for j = 1, ... , N.]

(v) The Jacobian determinant of the transformation is J(v)

:= det (avxij) =

h1h2...hN

rN-1 (sin 01)N-2 (sin 02)

N-3

... (sin ON-2).

(vi) The Laplace operator becomes, for v E E, hih3 ... hN a _ 1 a N-i a 1 a r

rN-i ar

a +aeN i

ar

(

+ J(v)

a01

hih2 ... hN_i

a

hN

aeN i

h2

aai

+

[See, for example, Kellogg 1929, p.183; or Spiegel 1959, p.151]

Appendix E. The Edge-Point Lemma

E.1 Preliminaries In this appendix we shall try to maintain a certain similarity between the boundary-point lemma (Lemma 2.12 and Theorem 2.15) and the edgepoint lemma (Theorems E.8 and E.9 below), but a greater complexity of the present situation cannot be avoided. Example E.5 will illustrate this, after we have established an appropriate terminology. The edge-point lemma will be described loosely after Example E.5. Definition E.1 By a neighbourhood of a point p E RN we mean a set that contains p and is open in RN.

A point p E 811 will be called an edge point of S2 if, for some neighbourhood U of p, (a) 52 n U = (521 n 5211) n U, where 521 and 011 are open subsets of RN having boundaries of class C2 and outward unit normals n' and n11, respectively; (b)

p E 8521 n 85211

and

-1 < n1(p) n11(p) < 1,

(E.1)

as is shown in Figure E.1.

A connected set of edge points of 52 will be called an edge of Q.

If, for example, 52 := R3 \ [0, oo) x [0, oo) x R (the set obtained by removing two octants from R3), then the origin is not an edge point, even though there is a corner there. This set 52 is a union of sets with smooth boundaries, rather than an intersection. Also, this set .0 has the interior-ball property at the origin (Definition 2.14), whereas there is never an interior ball at an edge point. 305


306

as21

Fig. E.1.

Definitions E.1 to E.3 would still have meaning if 8f21 and 89211 were

only of class C' and if cpi and cpj, were only in C'(U). However, C2 smoothness of all these objects will be required in Theorems E.8 and E.9, and it seems simpler to adopt this smoothness ab initio.

Definition E.2 We shall say that cpi is an admissible function describing 921 near p if, for some neighbourhood U of p, (a) (pi E C2(U); (b) IVgpi(x)I > 0 for all x E U; (c) for x E U, the value (pi(x) is negative, zero or positive according

asxE01,xE8921 or Admissible functions describing 9211 near p are defined similarly.

Definition E.3 Let p be an edge point of 92 (with 92 = 921 n 9211 near p); let 9, and cpii be admissible functions describing 921 and 9211, respectively,

in a neighbourhood U of p; and let a = (a, ), now defined on S2, be the matrix of leading coefficients of an elliptic operator L of order two. Then the function B defined by B(x)

:= V pi(x) . a(x) . V pj1(x) N

a1W1(x) a11(x) 8icpii(x),

x E s2 n U,

(E.2)

i,j=1

will be called a bluntness function for the edge 8921 n 89211 relative to the operator L.

We use the word bluntness because at an edge point B(x) is a weighted form of ni(x) n1I(x), and this latter varies from -1 at a perfectly sharp

E.1 Preliminaries

307

edge to 1 at a perfectly blunt edge. Of course, both extremes are excluded by (E.1).

In §E.3 we shall encounter the following alternative bluntness conditions at an edge point p E 8521 n On,,. B(p) > 0.

(E.3)

B(p) = 0 and (aB/OT)(p) = 0 for every differential operator O/aT tangential to On, n as111 at p.

(E.4)

The coefficients aid are to be sufficiently smooth for continuity of B at p in the case of (E.3), and for continuity of VB in ?in U in the case of (E.4). The derivatives (aB/OT)(p) may be evaluated either as (a) lim,- (T VB)(x), where x E n n U and T is a non-zero vector satisfying T n'(p) = 0 and T n11(p) = 0 [equivalently, (T V p1)(p) = 0 and (T - V pji)(p) = 0] ; or as

0 < t < 1 } is an arc in '(t)I > 0 on [0,1] and (to) = p for

(b) dB (fi(t)) / dt at t = to, where { (t) as21 n aQii for which some to E [0, 1].

E C' [0, 1],

1

Remark E.4 Each of the bluntness conditions (E.3) and (E.4) has a meaning independent of the choice of cp1 and T jj, provided that these are admissible in the sense of Definition E.2. To demonstrate this, we define

/3(x) := n1(x) a(x)

n11(x)

and

g(x) := Iow1(x)I

for x E (as21 n af)11) n U, and adopt the evaluation (b) of (aB/aT)(p). Then B(x) = f3(x)g(x),

with /3 independent of the choice of cp1 and p11, while g(x) > 0. Thus (E.3)

is equivalent to /3(p) > 0, and the pair of conditions (E.4) is equivalent to the pair /3(p) = 0 and (8/i/3T)(p) = 0. Example E.5 Consider the sector

S2 :_ {(rcosO,rsin0)ER2

I 0 0 in H,

(III')

where L denotes whichever of LO, L and Ll is considered in hypothesis (b).

Consider the values of u+bV on OH. On aH\F we have u< M, V = 0 and hence u + b V < M, with equality at p. On F, which is a subset of 0, we have u < M by hypothesis (c); since F is compact and u continuous, u < M - a on F for some constant a > 0. Choose b = Za/ maxr V ; then

u+6V<Mon F, so that maxaH (u + 6 V) = M.


314

Condition (III) now follows from (III') and one of our three versions of the weak maximum principle, in precisely the way that was spelled out in the proof of Lemma 2.12. (iii) In order to construct the set H and the function V, we simplify the geometry by means of co-ordinate transformations. First, let Xj be local co-ordinates of the usual kind (resulting from a translation and rotation of axes as in Exercise 5.5) such that (a) X = 0 at x = p; n'(p) + n"(p) (b) oxXN = InI(p) + nII(p)I

(c) the (two-dimensional) plane spanned by VxX1 and VxXN is that spanned by n'(p) and n"(p). Conditions (E.1) and (b) imply that n'(p) oxXN > 0 for J = I, II; then, by Exercise 5.5, 0921 and 75211 have local representations XN = gi(X')

(for J = I, 11) such that XN < gj(X') when x is near p and x E f1j. Moreover, conditions (a) to (c) and the C2 nature of the boundaries ffij (Definition E.1) allow us to write gi(X') = yXi + r1(X'), rI E CZ(S), gII(X') _ -yX1 +rII(X'), rII E C2(S),

(E.12)

where the constant y > 0; the set S is a convex neighbourhood of 0 in RN-1; and, as X' -+ 0, rj(X') = O(IX'12) and Vrj(X') = O(IX'I) for

J=I,II. The next co-ordinate transformation is YI

= XI + I {rI(X') - rII(X')},

yA

= XA

Y

for each A E {2,._, N -

11,

(E.13)

YN = XN - 2 { r1(X') + rII(X') }. If we write this as y = f(X), then the Freshet or total or linear derivative

f'(0) = I, the identity operator RN

RN.

Thus f'(0) is certainly

invertible (is certainly a linear homeomorphism) and the inverse-function theorem ensures that f is a C2 diffeomorphism on some neighbourhood of 0 in RN. [For a form of the inverse-function theorem that yields the C2 property of f -1, see Dieudonne 1969, p.273.] Define open half-spaces 61

{yERN I YN 0 (which will be a curvature), let GI G11

y E RN I YN < yYi -

K2IY'I2

y E RN I yN < -yY1 -

K2IY1I2

set G := GI n G11 (Figure E.3) and define

H :=GnR(0,p)

(E.15)

for some small p > 0 that remains to be chosen. The inverse image of k under the C2 diffeomorphism x -+ y is to be the set H specified in step (i).

(v) To construct the comparison function V, we first define (p(Y)

YN - yYl +

w(Y) := YN + yY1 +

K2IY,I2,

P(y)

8 KW(Y) - 1,

(E.16)

K2IY I

,

Q(v) := e-KW(Y) -

1,

for all y E RN and for some large K > 0 that also remains to be chosen. Observe that cp and -P are admissible functions describing GI (Definition E.2) and that ip and -Q are admissible functions describing GII; in particular, P(y) > 0 and Q(y) > 0 if y E G, and P, Q vanish on 8G1i aG11 respectively.


316

Now define V E C00(RN) by V := PQ, and let y = Y(x) denote the C2 diffeomorphism that we have constructed on a neighbourhood of p. Denoting this neighbourhood by #(p), we set V(x) := V (Y (x)),

x E ./#(P).

(E.17)

Then V E C2(H) if H c .gy(p), and we proceed to show that V satisfies (I'), (Ila,b) and (III') if p and K are chosen suitably. This will complete the proof. (vi) Condition (I') holds if we define IF to be the inverse image of G n 89(0, p) under the map Y [because PQ vanishes on 0G]. For (Ila,b), let a unit vector m, outward from H at p, be given; define, for r E {1,..., N}, N

mr(Y) i=1

a

a Y,

axi (x) mi,

am

N

= E mr(Y) r=1

a ay'.

Then (aV/am)(p) = (aV/ain)(0) and m(0) is outward from if at 0, by Remark E.7, (i) and (ii). Now VV = (VP)Q+P(VQ) and P(0) = Q(0) = 0, so that certainly (aV/am)(p) = (aV/am)(0) = 0. Next, using once more the condition T(O) W(0) = P(0) = Q(0) = 0, we obtain am

(p) =

a z01V (0)

8P

ainn

am

=2

(0)

8Q amp aye (0) = 2K2 (0) (0) > 0, am am am

since m is outward from GI and from GII at 0.

(vii) It remains to prove that V satisfies (III') if p and K are chosen suitably; equivalently, that LV > 0 in fl, where L is the operator to which L is transformed (§E.2) by the map y = Y(x). Let air(y), ba(y) and c(y) denote the coefficients of L. Then

LV = (LP)Q + 2VP a VQ + P(LQ) - cPQ.

(E.18)

Since P > 0 and Q > 0 in ft, the essentially new term relative to Lemma 2.12 is

2VP where

BG, say, is a bluntness function for the edge 0GInaGII

relative to L. By hypothesis (d) and the two kinds of invariance of the condition B(p) > 0 [Remark E.4 and (E.10)], we have BG(0) > 0, and BG is continuous at 0 by hypothesis (e) and the definitions of (p and W. Therefore, if p is chosen sufficiently small in the definition (E.15) of ft, then

in ft.

E.3 Two stages of the edge-point lemma

317

Consider the remaining terms in (E.18). We have -cPQ >- 0 in H, and

make LP > 0 and LQ > 0 in fl by choosing K sufficiently large, very much as in the proof of Lemma 2.12. In fact, LP

=

e-Kw

K2 E aid (a,cp)(ai(,)

-K

aii ajaiq ii

i,i

-K

- c,

bi 8icp +

(E.19)

where -c >- 0 and

=, = (-Y + KYI, KY2, ..., KYN-1, 1),

(aj(p(Y))N

a,aj(o(Y)

K

if i=jE{1,...,N-1},

0

otherwise.

(E.20a)

(E.20b)

Since uniform ellipticity, say with modulus L > 0 [Remark E.7, (v)], implies that a,j (0,(p)(ai(P) >- oIV(P12 > 1o

in fl,

(E.21)

we can certainly choose K so large that LP > 0 in ft, and similarly for Q. Theorem E.9 Suppose that hypotheses (a) to (c) of Theorem E.8 stand,

while (d) and (e) are replaced by (d') a bluntness function B, for the edge 8521 n a011 relative to the op-

erator considered in (b), satisfies B(p) = 0 and (8B/ar)(p) = 0 for every differential operator 0/0,r tangential to 8521 n 001, at p; (e') all coefficients aij E C2 (S2 n 1(p, a)) for some a > 0. Then the conclusion of Theorem E.8 remains true : we have either (a) or (fl) there.

Proof Steps (i) to (iii) in the proof of Theorem E.8 are not changed. However, in order to have LV >- 0 in H, we introduce here: (a) a further co-ordinate transformation, which will simplify the matrix a(y) of leading coefficients; (b) a further parameter, in the definitions of H and V, such that LV >- 0 in H when the new parameter is sufficiently large. (iv') Proceeding from the co-ordinates y defined by (E. 13) and the set C2 defined by (E. 14) and S2 := C21nC211, we transform linearly to co-ordinates


318

z, obtaining a set A and coefficients ai j(z), b j(z), c(z) such that 15 = C2 and

aAN(0) = O for A E {2,...,N -1}.

Notation In this proof, Greek subscripts take only the values 2,...,N-1; `for each 2' will mean for each A E {2,...,N - 1}; and E. := EN 2

Define, for constants CA that we are about to determine, Z1 = yt,

ZA = YA + CA YN for each A,

then aZA aYyi

= h + CA 6M,

a

ZN = YN;

(E.22)

aZN

= SNj,

Yj

so that 8ZA

aAN(Z) _ i,j

ayi

aij(Y)

-?N

ayj

= aAN(Y) + CA aNN(Y)

Here aNN(O) >- .10 > 0 by the uniform ellipticity of L with modulus .1o and

with test vector (0,...,0,1); therefore we choose CA = -aAN(0)/aNN(0). The transformation (E.22) is invertible [since yA = zA - CAZN for each A]; its only effect is that aAN(0) = 0

for each 2;

(E.23)

by (E.14) and (E.22), z E S2 := C2 if and only if y E C2. Now define a new set W := WI n W11 (Figure E.4) by WI

:=

{zEI''

ZN < YZl - 2 Zl - 2 E ZA j A

W11

:=

{ZERN

ZN < -yZ1 - 2Z1 - 2

} ZA

, >

1

where K > 0 and A > K/y. Any K > 0 will serve, as before, but the main choice of the large parameter A has yet to be made. Next, let

k := W n 9(0, p)

(E.24)

for some small p > 0 that also remains to be chosen. The inverse image of H under the C2 diffeomorphism x --* z is to be the set H specified in step (i).

E.3 Two stages of the edge-point lemma ZN

319

ZN

edge of S2

SID

edge F of W

Fig. E.4.

(v') To construct the comparison function V, we first define [cf. (E.16)]

f(z) := ZN - Yz1 + 2zI + 2

z22,

g(z) = ZN + YZ1 + 2 Zi + 2 E zA,

(F(z) := e-/{f(z) - 1

T(z)

e-Kg(z)

- 1, (E.25)

for all z E RN and for some large K > 0 that will be chosen after A has been fixed. We note that f and -4) are admissible functions describing WI, while g and -'F are admissible functions describing W11. Now define V E C°°(RN) by V := (F'F, and let z = Z(x) denote the C2 diffeomorphism that we have constructed on a neighbourhood, say ,*'(p), of p. We set V(x) := V (Z(x)),

X E .N(p);

(E.26)

then V E C2(H) if H c .K(p).

(vi') The argument in step (vi) of the earlier proof shows that the function V in (E.26) satisfies conditions (I') and (Ita,b) [stated in steps (ii) and (i), respectively]. Condition (III'), or equivalently that LV > 0 in H, requires further labour. Here L denotes whichever of Lo, L and L1 is considered in hypothesis (b); L is the operator to which L is transformed by the map z = Z(x); and the coefficients of L are aij(z), bi(z) and c(z). Accordingly,

LP = (L(D)'' + 2v(D . a . V'Y + 4)(L') - i^(F'P,

(E.27)

and the troublesome term is 2e -Kf-Kg K

where

Bw := V f a Vg on H.

1

(E.28)


320

The bluntness hypothesis (d') concerns not the edge of W and the function Bw, but [in view of Remark E.4 and the transformation law (E.10)] the edge of 6 and the function

(-y,0,...,0,1)'a(z)'(Y,O,...,0,1)

BA(z)

_

-Y2a11(Z) + aNN(Z)

(z E S2,

IzI

small),

in which the alN-terms have cancelled each other. Thus hypothesis (d') implies that and

-Y2all(0) + aNN(O) = 0, -y2 (&Aal1) (0) + (BAaNN) (0) = 0

for each A.

(vii') We begin with the restriction of Bw to the edge F := a Wl n a Wit of W ; more precisely, to F n.4(0, p* ), the radius p* begin so small that (E.30)

A0, p*) - Z (gy(p) n °M(p, a)),

where X(p) is the domain of Z and a is as in hypothesis (e'). The edge F of W has a representation 2A IC 12),

Z = ZF(O := (0,C,-

RN-2

C :_ (b2,...,CN-1) E

Let aij(C) := aiJ (ZF(c)) and let

BF(C) := Bw (zF(O) for 1C1 (A2.o - const.A - const.) 112

ICI- 0 when zo E FnR(0,1/A), by (E.32), there is a constant C2 > 0 such that

Bw(z) > Bw(z) - Bw(zo) > -C2IZ - Zol = -C2dF(z) for all z E ff. By the inequality (E.33),

Bw(z) >- -C1C2h(z) = 1CiC2{f(z)+g(z)}

for all z E ft,

(E.37)

provided that p< 1/2A. (x') It remains to choose K, and to make a final choice of p, in order that LP > 0 in f7. First we observe [cf. (E.19) to (E.21)] that

L) > e-Kf

ail alai f - K

K 2 .o I Vf 12 - K i,i

>-

ZeKf K2 ,o

bi ai f + c

i

in fi

if we make a first, sufficiently large choice of K and if p < 1/2A. The function LT is treated similarly. The formula (E.27) for LV now yields

LP > Ze

Kg-1}+2e Kf-KgK2Bw

+1e-KQK2 o { eKf - 1 }

= ZeKf-KgK2 o l 2 - eKf -eKg + (4/),o)Bw 1 > ZeKf-KgK2 o { 2 -eKf -eKg + C3f + C3g } , (E.38)

E.3 Two stages of the edge-point lemma

323

where C3 := (2/.10) Ci C2 and we have used (E.37). The result (E.38) holds in ft if p < 1/2A. Now define

q(t):=1-eKt-C3t

for

0- C3e; then q(t) > 0 for 0 < t < 1/K. If necessary, decrease p below

1/2A so that -f(z) < 1/K and -g(z) < 1/K when z E ff. Then (E.38) implies that, in fl,

LV > 2i e Kr-Kg K2 o {q(-f) + q(-g)} > 0, as desired.

Notes on Sources

I apologize for the inadequacy of these notes. They describe material that I used, rather than all the material that I should have used. The notes also point to further theorems that I happen to have encountered; they do not amount to a sketch of the present state of the subject. Chapter 1 It is worth mention that Aharonov, Schiffer & Zalcman (1981) obtained the conclusion of Theorem 1.1 from quite different hypotheses.

Their first result concerns a compact set P in RN which, inter alia, is connected and such that ][8N \ P is connected. They proved that, if in a (non-empty) open set outside P the Newtonian potential of constant density in P is equal to the potential of a point mass, then P is a ball. In using a flower to illustrate a symmetry group, I have followed Weyl (1952, p.66), who used vinca herbacea for the cyclic group C5 and used a geranium for the dihedral group D5. Weyl's book is full of charming examples; every page reflects the erudition and humanity of a master from an age less barbarous than ours. For problems that allow symmetry of solutions and that can be formulated as exercises in the calculus of variations, there is an approach to proofs of symmetry that is older than, and quite different from, the methods set out in this book. This approach is based on various geometrical operations (devised by J. Steiner and H.A. Schwarz in the nineteenth century) called symmetrization with respect to a plane, a line or a point. Typically, such an operation assigns to a given set A c R3 (which may be the set below the graph of a function of two variables) a set A* that has a specified symmetry and has the same cross-sectional area as A for each of a family of cross-sections. In particular, symmetrization with respect to a hyperplane, also called Steiner symmetrization, yields sets of the form called Steiner symmetric in Definition 3.2. The books of Polya 324

Notes on Sources

325

and Szego (1951), of Bandle (1980) and of Kawohl (1985) explain these procedures fully.

Chapter 2 In §§2.2 and 2.3 the weak maximum principle, the boundarypoint lemma and the strong maximum principle for C2-subsolutions are taken from Gilbarg & Trudinger's book (1983, Chapter 3); of course, with expanded proofs and some differences of detail. The boundary-point lemma in this form is due to E. Hopf (1952b) and to Oleinik (1952); the others are somewhat older. Theorem 2.11 (the weak maximum principle

for distributional subsolutions relative to the operator L1, which has constant coefficients) is a very particular case of maximum principles due to Littman (1959, 1963). The elementary proof of Theorem 2.11 seems (to me) to be new, but may well occur somewhere in the literature of the 1950s or 1960s; my colleague Buffoni suggested the use of (2.16) after I had used (2.17).

The maximum principle for thin sets (Theorem 2.19) is a very restricted version, although not quite a particular case, of theorems due to Berestycki & Nirenberg (1991, Proposition 1.1), to Dancer (1992, Lemma 1) and to Healey, Kielhofer & Stuart (1994, Theorem 3.2). It seems that, for theorems of this kind, the basic observation appeared first in work of Bakelman preparatory to publication of his book (1994) and in unpublished remarks of S.R.S. Varadhan. Theorem 2.19 is more naive than its counterparts elsewhere because the constant coefficients of the operator L10 make it easy to avoid derivatives of the subsolution (and thus to avoid Sobolev spaces). The Phragmen-Lindelof theory in §§2.4 and 2.6 is a mixture based on my reading of Protter & Weinberger (1984, §2.9) and of Stein & Weiss (1971, §11.5.2), but with different comparison functions. As is mentioned in §2.6, the primary comparison function g, described by Theorem 2.27, is an extension to all dimensions N >_ 2 of a function introduced by D. Siegel (1988) for N = 2. Recently, Siegel has abandoned this function in favour of a more elaborate one, defined on polytopes for all N >_ 2 (Siegel & Talvila 1996, Theorem 4.1), that allows larger rates of growth of a subharmonic function when lxi --* oo and xN/Ixi -> 0 in the halfspace { x E RN I XN > 0 }. I have not seen previously the comparison functions ge and $2, constructed from g in §2.6 by means of the Kelvin

transformation; the trio g, ge and $2 is intended to be a compromise between the simplest comparison functions and difficult ones that yield results close to best possible. Chapter 2 provides only an introduction to the subject of maximum

326

Notes on Sources

principles for elliptic equations. For example, information about positive solutions of Lu + f (u) = 0 in a bounded set S2 can be gleaned by means of P-functions; these are of form IVuI2g(u)+h(u) and, for suitable choices of g and h, attain their supremum either on On or at a stationary point of u. The construction and use of such functions are the main topics of the book by Sperb (1981). In another direction, Bakelman (1994) considers fully non-linear equations, such as those of Monge-Ampere type, and derives for such equations maximum principles (Chapter 8 of his book) that make those in our Chapter 2 look like toys.

Chapter 3 For boundaries 00 of class C2, monotonicity and symmetry results like Theorem 3.3 and its corollaries, but for more general elliptic equations, were proved by Gidas, Ni & Nirenberg (1979), building on work of Alexandrov (see H. Hopf 1956, p.147) and of Serrin (1971). A maximum principle for narrow sets ), and use of the compact set called F in the proof of Theorem 3.3, were introduced into problems of this kind by Berestycki & Nirenberg (1991); thereby the need for smoothness of On was removed and proofs were shortened. Parallel steps were taken by Dancer (1992), who made more use of Sobolev spaces and proved monotonicity results like those in Theorem 3.3 under hypotheses on u that are weaker than those of Berestycki & Nirenberg (and considerably weaker than those in Theorem 3.3). Theorem 3.6 and Corollary 3.9 have their roots in the paper of Gidas, Ni & Nirenberg (1979), in the work of Berestycki & Nirenberg (1991) and in an extension of two results of the 1979 paper to the case of discontinuous f (Amick & Fraenkel 1986). The combination of discontinuous f and a set ) that is merely Steiner symmetric seems to be new. Exercises 3.18 and 3.19 (which show that one cannot relax significantly

the conditions on f used in most of the book) extend a construction of Gidas, Ni & Nirenberg (1979, p.220) in a way demonstrated for ordinary differential equations by work of C.A. Stuart (1976) and by unpublished examples of J.F. Toland. Gidas, Ni & Nirenberg (1979, p.220) raised the question of whether results like those of Theorem 3.3 remain true if the hypothesis u > 0 in

n is changed to u >- 0 in n with u > 0 somewhere in n. They observed that for N = 1 (for Q c ll8) the answer is No because u(x) = 1 + cos xx

u" + m2(u - 1) = 0 in (-3,3),

u(±3) = 0,

and u' has the wrong sign in (1,2); they proved that the answer is Yes for all N E N if f(0) >_ 0. Castro & Shivaji (1989) have shown that for

Notes on Sources

327

f (0) < 0 the wider hypothesis is allowable in Theorem 3.3 if c2 is a ball and N > 2. Note that extensions of the functions u in Exercises 3.10 and 3.11 are consistent with all this. Allegretto & Siegel (1995) have considered the problem Au + A p(x) g(u) = 0

in K2,

u=0,

where f2 := (-1, 1) x n' is a cylinder, S2' being open and bounded in RN-1, and p(-x1, x") = p(x1, x"). Using Sobolev-space methods, a family of non-negative eigenfunctions defined on caps of fl, and corresponding hypotheses on p and g, they avoid pointwise estimates near 092 and prove symmetry under hypotheses on u slightly weaker than those of Berestycki & Nirenberg (1991).

Chapter 4 Regarding the behaviour of u(x) as r := IxI --f oo: by now there are many different sets of conditions that lead to symmetry for solutions u of suitable equations in RN. Gidas, Ni & Nirenberg (1979, Theorem 4) considered positive solutions of rather general equations and introduced hypothesis (C) of Definition 4.1, but with three explicit terms before a remainder. Caffarelli & Friedman (1980, Theorem 7.1) observed for the equation Au + f (u) = 0 in R2 that positivity of u could be abandoned, that logarithmic growth at infinity could be tolerated and that the asymptotic approximation could be shortened. Definition 4.1, and the sufficiency of its conditions for Theorem 4.2, evolved from these

two sources in the course of my lectures at Bath, given in the years 1989-1992.

The approach of Gidas, Ni & Nirenberg in their 1981 paper is quite different: for positive C2-solutions of Au + f (u) = 0 in IRN, with N z 2 in some cases and N >- 3 in others, f is specified rather precisely but very little is demanded of u as r -+ oo; for example, merely that u(x) = O(r-') with m > 0. More detailed asymptotic behaviour is then deduced. Li & Ni (1993) have taken this further, even for fully non-linear equations, by demanding only that u(x) -> 0 as r -> oo; no rate of decay is prescribed.

Of course, the function f or its fully non-linear counterpart must be specified carefully. The reader who is bewildered but attracted by remarks in the text about

Sobolev spaces and variational principles (for example, after Theorem 4.13 and in Exercises 4.22, 5.29 and 5.30) will find lucid and satisfying explanations in the books by Berger (1977), by Brezis (1983) and by Friedman (1982).

Notes on Sources

328

Regarding steady vortex motions. Turkington (1983), considering various steady vortex flows in two dimensions, used reflection in hyperplanes

(in this case reflection in lines) to prove that intense vortices of small diameter are approximately circular and, suitably scaled, become exactly circular in a certain limit. Concerning Hill's vortex and Hicks's vortex: the solutions of Exercises 4.22 and 4.23 can be found, to a large extent, in papers by Amick & Fraenkel (1986) and by Fraenkel (1992). Uniqueness of the vortex pair in R2 that corresponds to Hill's vortex has been proved by Burton (1996).

Two conspicuous omissions from this book are (a) overdetermined problems, by which we mean here that two boundary conditions ul an

= constant

and

8u

an

= constant an

accompany an elliptic equation of order two; (b) problems on an exterior domain S2, by which we mean a connected set in RN the complement

of which is compact. The first of these omissions is unfortunate in that overdetermined problems in a bounded set 0 were the subject of Serrin's early and fruitful use of reflection in hyperplanes (Serrin 1971); the conclusion was that 92 must be a ball and the solution u spherically symmetric and monotonic. Reichel (1996, 1997) has considered overdetermined problems on ex-

terior domains for the equation Au + f (u, IVul) = 0 and for certain fully non-linear equations, recovering Serrin's conclusions under suitable

hypotheses on f (u, JVuj) or on its fully non-linear counterpart. These hypotheses, however, exclude the case f (u, JVul) = fo(u) with fo positive and increasing on (0, oo). This defect has been made good by Aftalion & Busca (1998), who combined skilful use of the Kelvin transformation with a maximum principle for thin sets.

Chapter 5 Most authors seem to regard results like Theorem 5.7 (two pleasant properties of boundaries of class C1) and Proposition 5.20 (resolution of a topological question) as obvious; however, colleagues attending my lectures at Bath insisted on proof of such matters. Theorem 5.7 is taken from Amick & Fraenkel (1986, Lemma A.1) and from Keady

& Kloeden (1987, Lemma 2.1); Proposition 5.20 appears to be new. Rhomboid neighbourhoods (Definition 5.4) have a long history in the theory of Sobolev spaces, although they are seldom given a name; their treatment in Chapter 5 follows that of Fraenkel (1979). References to their earlier use are given in that paper.

Notes on Sources

329

The monotonicity results in §5.3 are due to Gidas, Ni & Nirenberg (1979). In our version, somewhat more attention is given to the components of a cap, and less smoothness is required of 852, f and u. On the other hand, our equations (5.1) and (5.2) are far less general than those considered by Gidas, Ni & Nirenberg. The result in Exercise 5.30, that symmetry results for a ball do not extend to an annulus or spherical shell A, is not new; examples of positive solutions in A that lack spherical symmetry have been given by Brezis & Nirenberg (1983, p.453) and by Coffman (1984). Possibly the example in Exercise 5.30 is simpler than previous ones; it is a modified form of an example suggested to me by G. Keady. The annulus or spherical shell A has also been used by Dancer (1997)

to prove that positive solutions u E C2(S2), of Au + f(u) = 0 in 52 with u = 0 on 852, can have stationary points arbitrarily close to 052. Dancer used 52 = A and a family of functions f = f (., e) to construct spherically symmetric, positive solutions u(. , e) having a stationary point

that approaches the inner part of aA as a --* 0. Thereby he answered a question of Gidas, Ni & Nirenberg (1979, p.223), who had gathered evidence for the opposite conclusion.

Keady & Kloeden (1987) applied monotonicity results like those in Theorem 5.10 to discuss solutions of

Au + 2fH(u - k) = 0 in a bounded set 52 c > k }, which they called the core, is connected. In { x E 52 particular, there is a branch of solutions along which diam C -> 0 and I

oo. (When Q is a ball, all solutions are known explicitly; see Exercises 3.15 and 3.16, in which the same equation occurs and 52 is a ball in I[8 or 2

in 1185.)

Two important techniques, beyond those in Chapters 3 to 5, were introduced by Berestycki & Nirenberg (1988, 1990) and used to prove monotonicity, symmetry or anti-symmetry in a variety of problems with boundary conditions more general than ul an = 0. These techniques were also simplified and extended in their 1991 paper. First, equations more general than Du + f (u) = 0 may lead to Lw + b(x, u) Vw + c(x, µ)w - 2 /3(x, µ) aW ->O for x E Z (µ), µ

(j')

Notes on Sources

330

in place of our (3.6); here l3(x,µ) > 0 and the identity 8x1

u(2µ - xI, x") = 1 2

a

aµ

{ u(x) - u(2µ - xi, x")} =

1 8w 2 aµ

(x µ)

has been used. Now, if µ is regarded as a time-like variable, then (t) is an inequality for a parabolic partial differential operator (although a degenerate one wherever /3(x,µ) = 0); somewhat surprisingly, boundary conditions that suit such an inequality also arise in the (basically elliptic) problems considered by Berestycki & Nirenberg. Thus maximum principles for parabolic operators play an important part. Second, the method of sliding domains was introduced and used to good effect. In this method, one compares the graph of a function not with a reflection of that graph, but with a translation of it, defining, for example, 92T:={xE11RN

w(x, T) := u(x) - u(x - Tel)

I x-Te'ES2}, for T > 0 and x E 12 n f'T.

When the boundary conditions suggest it, monotonicity in the whole of S2 is often established by an application of maximum principles that is not unlike that used for reflection and symmetry. In the same direction and exhibiting further developments, the work of Craig & Sternberg (1988, 1992) established the symmetry of certain

gravity waves in hydrodynamics; in particular, the symmetry of the classical solitary wave. This is of interest because, when surface tension is included, there is no shortage of asymmetric solitary waves. Appendix A Little need be said about this material except to acknowledge that I first learned the main results, some thirty years ago, from the book of Gunter (1967). However, the resemblance of Appendix A to Ginter's

book now seems rather faint; for example, in the context of Theorems A.15 and A.16, Gunter (1967, p.308) was willing to accept hIµlog(1/1hl) in place of the jhV L for which we work so hard. Exercise A.25 is a fragment of generalized axially symmetric potential theory (Weinstein 1953).

Appendix B In the proof of Theorem B.6, the choice of test function is that in Sobolev's monograph (1963, p.92). Theorem B.9 is taken from Gilbarg & Trudinger (1983, p.23). Theorems B.29 and B.30, on the Poisson integral for a ball, should really be accompanied by a description

Notes on Sources

331

of how boundary values are attained when the boundary-value function g is merely in Lp(8B); Folland (1976, p.124) demonstrates this beautifully. Of course, my hope is that Appendices A and B may prepare beginners

for the more concise and extensive material in, say, Dautray & Lions (1990), Folland (1976) and Gilbarg & Trudinger (1983).

Appendix E As was remarked in the Preface, the edge-point lemma (or

boundary-point lemma at a corner) is due to Serrin (1971) and was extended by Gidas, Ni & Nirenberg (1979, pp.214 and 237-243), who called it Lemma S. The version in Appendix E is essentially Lemma S, with slight changes intended to increase the resemblance to our boundarypoint lemma, and with a vastly expanded proof.

References

Aftalion, A. & Busca, J. (1998). Radial symmetry of overdetermined boundary value problems in exterior domains. Archive for Rational Mechanics and Analysis, 143, 195-206. Aharonov, D., Schiffer, M.M. & Zalcman, L. (1981). Potato Kugel. Israel Journal of Mathematics, 40, 331-339. Allegretto, W. & Siegel, D. (1995). Picone's identity and the moving plane procedure. Electronic Journal of Differential Equations, 1995, 1-13. Amick, C.J. & Fraenkel, L.E. (1986). The uniqueness of Hill's spherical vortex. Archive for Rational Mechanics and Analysis, 92, 91-119. Apostol, T.M. (1974). Mathematical analysis. 2nd ed. Reading, Massachusetts: Addison-Wesley.

Arnold, V.I. (1990). Huygens and Barrow, Newton and Hooke. Basel: Birkhauser Verlag.

Bakelman, I.J. (1994). Convex analysis and nonlinear geometric elliptic equations. Berlin: Springer. Bandle, C. (1980). Isoperimetric inequalities and applications. London: Pitman. Batchelor, G.K. (1967). An introduction to fluid dynamics. Cambridge University Press.

Berestycki, H. & Nirenberg, L. (1988). Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations. Journal of Geometry and Physics, 5, 237-275. Berestycki, H. & Nirenberg, L. (1990). Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains. In Analysis, et cetera,

edited by P.H. Rabinowitz & E. Zehnder. New York: Academic Press, pp. 115-164.

Berestycki, H. & Nirenberg, L. (1991). On the method of moving planes and the sliding method. Boletim Sociedade Brasileira de Matematica, 22, 1-37. Berger, M.S. (1977). Nonlinearity and functional analysis. New York: Academic Press.

Brezis, H. (1983). Analyse fonctionnelle, theorie et applications. Paris: Masson. Brezis, H. & Nirenberg, L. (1983). Positive solutions of nonlinear elliptic equations

involving critical Sobolev exponents. Communications on Pure and Applied Mathematics, 36, 437-477. 332

References

333

Burkill, J.C. (1975). The theory of ordinary differential equations. 3rd ed. London: Longman.

Burkill, J.C. & Burkill, H. (1970). A second course in mathematical analysis. Cambridge University Press. Burton, G.R. (1996). Uniqueness for the circular vortex-pair in a uniform flow. Proceedings of the Royal Society of London, A 452, 2343-2350. Caffarelli, L.A. & Friedman, A. (1980). Asymptotic estimates for the plasma problem. Duke Mathematical Journal, 47, 705-742. Cartan, H. (1971). Differential calculus. Paris: Hermann; also Boston: Houghton Mifflin.

Castro, A. & Shivaji, R. (1989). Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric. Communications in Partial Differential Equations, 14, 1091-1100. Coffman, C.V. ( 1984). A non-linear boundary value problem with many positive solutions. Journal of Differential Equations, 54, 429-437. Coxeter, H.S.M. (1963). Regular polytopes. 2nd ed. New York: Macmillan. Coxeter, H.S.M. (1981). Introduction to geometry. 2nd ed. New York: Wiley. Craig, W. & Sternberg, P. (1988). Symmetry of solitary waves. Communications in Partial Differential Equations, 13, 603-633. Craig, W. & Sternberg, P. (1992). Symmetry of free-surface flows. Archive for Rational Mechanics and Analysis, 118, 1-36. Dancer, E.N. (1992). Some notes on the method of moving planes. Bulletin of the Australian Mathematical Society, 46, 425-434. Dancer, E.N. (1997). Some singularly perturbed problems on annuli and a counterexample to a problem of Gidas, Ni and Nirenberg. Bulletin of the London Mathematical Society, 29, 322-326. Dautray, R. & Lions, J.-L. (1990). Mathematical analysis and numerical methods for science and technology. Vol. 1, Physical origins and classical methods. Berlin: Springer. Dieudonne, J. (1969). Foundations of modern analysis. New York: Academic Press.

Evans, L.C. & Gariepy, R.F. (1992). Measure theory and fine properties of functions. Boca Raton, Florida: CRC Press. Fauvel, J., Flood, R., Shorland, M. & Wilson, R. (1988). Let Newton be! Oxford University Press. Ferraro, V.C.A. & Plumpton, C. (1966). An introduction to magneto fluid mechanics. 2nd ed. Oxford University Press. Fleming, W.H. (1965). Functions of several variables. Reading, Massachusetts: Addison-Wesley. Folland, G.B. (1976). Introduction to partial differential equations. Princeton University Press.

Fraenkel, L.E. (1979). On regularity of the boundary in the theory of Sobolev spaces. Proceedings of the London Mathematical Society (3), 39, 385-427. Fraenkel, L.E. (1992). On steady vortex rings with swirl and a Sobolev inequality. In Progress in partial differential equations: calculus of variations, applications, edited by C. Bandle et al. Harlow, Essex: Longman; also New York: Wiley, pp. 13-26.

334

References

Friedman, A. (1982). Variational principles and free-boundary problems. New York: Wiley. Gidas, B., Ni, W.-M. & Nirenberg, L. (1979). Symmetry and related properties via the maximum principle. Communications in Mathematical Physics, 68, 209-243. Gidas, B., Ni, W: M. & Nirenberg, L. (1981). Symmetry of positive solutions of nonlinear elliptic equations in R". In Mathematical analysis and applications,

part A, Advances in mathematical supplementary studies, 7A, edited by L. Nachbin. New York: Academic Press, pp. 369-402. Gilbarg, D. (1952). The Phragmen-Lindelof theorem for elliptic partial differential equations. Journal of Rational Mechanics and Analysis, 1, 411-417.

Gilbarg, D. & Trudinger, N.S. (1983). Elliptic partial differential equations of second order, 2nd ed. Berlin: Springer. Gunter, N.M. (1967). Potential theory. New York: Ungar. Healey, T.J., Kielhofer, H. & Stuart, C.A. (1994). Global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains. Proceedings of the Royal Society of Edinburgh, 124A, 371-388. Hicks, W.M. (1899). Researches in vortex motion. Part III. On spiral or gyrostatic vortex aggregates. Philosophical Transactions of the Royal Society of London, A 192, 33-99. Hill, M.J.M. (1894). On a spherical vortex. Philosophical Transactions of the Royal Society of London, A 185, 213-245.

Hille, E. (1973). Analytic function theory, vols. I and II. 2nd ed. New York: Chelsea.

Hopf, E. (1952a). Remarks on the preceding paper by D. Gilbarg. Journal of Rational Mechanics and Analysis, 1, 419-424.

Hopf, E. (1952b). A remark on linear elliptic differential equations of second order. Proceedings of the American Mathematical Society, 3, 791-793. Hopf, H. (1946, 1956). Differential geometry in the large. Seminar lectures New York University 1946 and Stanford University 1956. Re-issued as Lecture Notes in Mathematics, 1000, by Springer, Berlin, 1983. Kawohl, B. (1985). Rearrangements and convexity of level sets in PDE. Lecture Notes in Mathematics, 1150. Berlin: Springer. Keady, G. & Kloeden, P.E. (1987). An elliptic boundary-value problem with a discontinuous nonlinearity, II. Proceedings of the Royal Society of Edinburgh, 105A, 23-36. Kellogg, O.D. (1929). Foundations of potential theory. Berlin: Springer. Re-issued by Dover, New York, 1954. Kingman, J.F.C. & Taylor, S.J. (1966). Introduction to measure and probability. Cambridge University Press. Lamb, H. (1932). Hydrodynamics. 6th ed. Cambridge University Press. Li, Y. & Ni, W.-M. (1993). Radial symmetry of positive solutions of nonlinear elliptic equations in W. Communications in Partial Differential Equations, 18, 1043-1054.

Littman, W. (1959). A strong maximum principle for weakly L-subharmonic functions. Journal of Mathematics and Mechanics, 8, 761-770.

Littman, W. (1963). Generalized subharmonic functions: monotonic approximations and an improved maximum principle. Annali della Scuola Normale

References

335

Superiore di Pisa (3), 17, 207-222. Maz'ja, V.G. (1985). Sobolev spaces. Berlin: Springer.

Moffat, H.K. (1969). The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics, 35, 117-129.

Oleinik, O.A. (1952). On properties of solutions of certain boundary problems for equations of elliptic type. Matematichesky Sbornik (N.S.), 30, 695-702. Polya, G. & Szego, G. (1951). Isoperimetric inequalities in mathematical physics. Princeton University Press.

Protter, M.H. & Weinberger, H.F. (1984). Maximum principles in differential equations. 2nd ed. Berlin: Springer. Rees, E.G. (1983). Notes on geometry. Berlin: Springer. Reichel, W. (1996). Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains. Zeitschrift fur Analysis and ihre Anwendungen, 16, 619-635.

Reichel, W. (1997). Radial symmetry for elliptic boundary value problems on exterior domains. Archive for Rational Mechanics and Analysis, 137, 381-394. Rudin, W. (1970). Real and complex analysis. New York: McGraw-Hill.

Rudin, W. (1976). Principles of mathematical analysis. 3rd ed. New York: McGraw-Hill. Serrin, J. (1971). A symmetry problem in potential theory. Archive for Rational Mechanics and Analysis, 43, 304-318.

Siegel, D. (1988). The Dirichlet problem in a half-space and a new PhragmenLindelof principle. In Maximum principles and eigenvalue problems in partial differential equations, edited by P.W. Schaefer. Harlow, Essex: Longman; also New York: Wiley, pp. 208-217. Siegel, D. & Talvila, E.O. (1996). Uniqueness for the n-dimensional half space Dirichlet problem. Pacific Journal of Mathematics, 175, 571-587. Simmons, G.F. (1972). Differential equations with applications and historical notes.

New York: McGraw-Hill. Sobolev, S.L. (1963). Applications of functional analysis in mathematical physics. Providence: American Mathematical Society. Sobolev, S.L. (1964). Partial differential equations of mathematical physics. Oxford: Pergamon Press. Re-issued by Dover, New York, 1989. Sperb, R.P. (1981). Maximum principles and their applications. New York: Academic Press.

Spiegel, M.R. (1959). Vector analysis. Schaum's outline series. New York: McGraw-Hill. Stein, E.M. & Weiss, G. (1971). Introduction to Fourier analysis on Euclidean spaces. Princeton University Press.

Stuart, C.A. (1976). Differential equations with discontinuous non-linearities. Archive for Rational Mechanics and Analysis, 63, 59-75. Thompson, W.B. (1962). An introduction to plasma physics. 2nd ed. Oxford: Pergamon Press. Titchmarsh, E.C. (1932). The theory of functions. Oxford University Press.

Turkington, B. (1983). On steady vortex flow in two dimensions, I and II. Communications in Partial Differential Equations, 8, 999-1030 and 1031-1071.

336

References

Weinstein, A. (1953). Generalized axially symmetric potential theory. Bulletin of the American Mathematical Society, 59, 20-38. Weir, A.J. (1973). Lebesgue integration and measure. Cambridge University Press. Weyl, H. (1952). Symmetry. Princeton University Press. Whittaker, E.T. & Watson, G.N. (1927). A course of modern analysis. 4th ed. Cambridge University Press. Ziemer, W.P. (1989). Weakly differentiable functions. Berlin: Springer.

Index

admissible asymptotic behaviour, 107, 122 admissible for direction k, 142 admissible function describing Q1, 306 affine, 28

affinely independent, 28 aligned with k, 111 almost everywhere, 11, 225 alpha cap, 142, 153 annulus, 163, 329 averaging functions in L,(IRN), 38 averaging kernel, 37 ball, 3 Bernoulli function, 134 Bessel function, 35, 36, 101, 137 beta cap, 151 bluntness conditions, 307 bluntness function, 306 boundary, 4 boundary of class Ck, 7 162 boundary of class

boundary-point lemma for balls, 50 boundary-point theorem for 52, 54 bounded continuous functions, 196 Buniakowsky's inequality, 13 C2-solution, 44, 190 C2-subsolution, 41 C2-supersolution, 43 cap, 88, 142 carissa grandii lora, 28 Cartesian product, 2 centre of charge, 202 centre of mass, 202 characteristic function, 4, 122 closure, 4 co-domain, 4 co-ordinate transformation, 309 compact, 7 compact support, 7

comparison function of first kind, 64 of second kind, 68 of Siegel type, 72-77 compatibility condition, 236 complex potential, 213 complex velocity, 213 component, 4 component of alpha cap, 159 conformal map, 79, 81, 233 connected, 4 conservation under co-ordinate transformation of bluntness, 311 of uniform ellipticity, 311 constants 1', 187 continuity of force field, 194 continuity of translation, 38 continuously differentiable functions, 6 contravariant, 310 convex, 8 Coulomb, 168 covariant, 310 cyclic group, 30 cylindrical co-ordinates, 173 cylindrical symmetry, 199 cylindrically symmetric, 173

decreasing, 2 diameter, 4 diffeomorphism, 309 dihedral group, 30 dipole, 172, 199, 202 Dirichlet problem for L, 44

for -0, 235-248

for -0 in a ball, 248-262 for -A in a half-space, 263, 267, 269 distance function, 3 distributional sense, 57 337

338

distributional solution, 46, 191 distributional subsolution, 46 distributional supersolution, 46 divergence of a vector field, 10 divergence theorem, 9, 279-292 domain, 4 downward jumps, 94 dyadic cube, 91, 99, 102 dyadic square, 99 edge, 286 edge point, 55, 287, 305 edge-point lemma, 305-323 eigenfunctions, 19, 45 electric field, 169 electric potential, 176 electrostatics, 168 elliptic disk, 33, 136, 162 elliptic operators, 39 LO, Land L1, 41 equator, 64, 72 essential supremum, 11 Euclidean space, 3 extended Minkowski inequality, 14 extended real-number system, 2 extension, 5 exterior, 3 exterior domain, 328 exterior function of Siegel type, 73 exterior-ball property, 54

factorial function, 33, 223 field lines, 199, 214 field point, 176 first divergence theorem, 282 function, 4 fundamental solution, 239 gamma function, 33 generalized sense, 94 generalized solution, 191 generalized subsolution, 45 generalized supersolution, 46 geometry of caps, 142 geometry of reflected caps, 142 gradient operator, 6 gravitation, 169 gravitational potential, 176 Green function, 241-248 for ball, 245 for half-space, 244-245 of Dirichlet problem, 241-244 of Neumann problem, 241-244, 265 symmetry property, 243 Green identity, 180 Green volume potential, 249, 255, 267 group isomorphism, 30

Index

growth condition, 65, 236-238

Holder conjugate, 12 Holder constant, 196 Holder continuity, 196, 203, 265 Holder continuous, 94 Holder exponent, 196 Holder's inequality, 13 half-space, 61 half-space in RN, 233 half-strip, 80 harmonic, 62, 224 harmonic polynomials, 62 heat equation, 40 Heaviside function, 6, 122 Hicks equation, 134 Hicks's vortex, 137 Hill's vortex, 137 Hill's vortex in a ball, 103 Hooke, 169 hydrodynamics, 133, 170, 216 hyperplane, 24 image, 4 implicit-function theorem, 146 increasing, 2 inequality for parabolic operator, 330 integrable, 8 integrals, 8 interior, 3 interior approximations to boundary 852, 293

interior-ball property, 54, 84 interpolation between LP spaces, 15 interval in R2, 300 invariance, 310 invariance group, 31 invariance transformation, 30 inverse image, 4 inverse-square law, 169 isometry, 28 Kelvin transform, 232 Kelvin transformation, 72, 232, 264 Kronecker delta, 5

Landau symbols O(.) and o(.), 5 Laplace operator, 6 lateral derivative, 266 Lebesgue spaces LP(S2), 10 Lebesgue spine, 85 Legendre function, 36 Leibniz rule, 214 level set, 4, 173 limit point, 4 Liouville theorem, 231 Lipschitz constant, 87

Index

Lipschitz continuous, 87, 196 locally, 87 uniformly, 87 local co-ordinates, 8 local maximum, 20 locally integrable, 224 logarithmic growth at infinity, 327

339

of density function with unbounded support, 219 of density r' in spherical shell, 217 of unit density in a ball, 177 of unit density in sector, 218 second characterization, 179 solves Poisson equation, 193, 197, 210, 211

magnetohydrostatics, 138 map, 4 mapping, 4 matrix multiplication, 5 maximum, 20 maximum principles, 19-24, 39-84 boundary-point for balls, 50 boundary-point for Q, 54 edge point, 311, 317 for thin sets n, 60 Phragmen-Lindelof for bad boundary point, 69, 81, 82, 86 Phragmen-Lindelof for unbounded S2,

67,77-80,86 strong, 53 weak for Lo, 41 weak for L, 43 weak for Lt, 46 mean function, 37 mean-value property of harmonic functions, 228 measurable, 9 measure zero, 10 metric space, 3 Minkowski's inequality, 13 modified Newtonian kernel, 56 modified Newtonian potential, 57 moduli of ellipticity, 39 monotonicity of positive solutions, 141 multi-index, 183 multinomial theorem, 214 multipole, 199, 214

neighbourhood, 305 Neumann problem for L, 85

for -A, 235-248, 236 for -A in a ball, 262, 265, 267 for -A in a half-space, 269 Newton, 169 Newtonian kernel, 122, 174 Newtonian potential, 174-220 behaviour outside support of density function, 185 continuity of first derivatives, 197 continuity of potential, 188 continuity of second derivatives, 210 far field, 200

symmetry of potential, 17, 122-133, 137-139 non-increasing, 2 non-negative part, 5 non-positive part, 5 non-singular part of F, 239 normed linear spaces, 10 one-step solution, 128 open cover, 149 open interval in RN, 10 orthogonal curvilinear co-ordinates, 234 overdetermined problems, 328

partial derivatives, 6 partition of unity, 280 pathwise connected, 53 Phragmen-Lindelof theory, 61 for bad boundary point, 69, 81, 82, 86 for unbounded 4, 67, 77-80, 86 point charge, 168 point mass, 169 point source, 167 pointwise modulus of ellipticity, 40 Poisson equation, 176 Poisson integral, 249, 257, 270 Poisson kernel, 63, 249, 268 positive, 2 potential of unit density in a ball, 177 potential of unit source, 167 primary function of Siegel type, 72, 270 proper subset, 2 protuberance, 151

quadrupole, 199 quasi-cubical neighbourhood, 280 range, 4 Rankine solid, 171, 216 real-analytic, 62, 223 real-analytic functions, 221 reflected cap, 88, 142 reflection in hyperplane of a function, 25 of a point, 25 reflection in hyperplanes (method), 24-28 reflection operator, 32 reflection relative to 8&(0, a), 232 region, 3

340

regularization, 37 repeated integrals, 9 representation formula, 239, 241 restriction, 5 rhomboid neighbourhood, 145, 146 Riesz kernel, 268 right-hand boundary of cap, 142 Schwarz inequality, 13 sector, 79 semi-norm, 10 separated, 4 sets with edges and vertices, 285 sign of potential functions, 169 signum function, 72 sliding domain, 330 small-ball technique, 187 smoothing kernel, 37 smoothing operation, 37 smoothing radius, 37 smoothly harmonic, 228 solution, 21 source of unit strength, 167 source point, 176 space Cb(S2), 196 space Cb°' (52), 197 spherical co-ordinates, 172, 303 spherical shell, 163, 329 spherically symmetric, 19, 25 Steiner symmetric, 88, 101 Stokes stream function, 134, 173, 213, 214 stream function, 134, 213 streamlines, 135 strict meaning, 2 subharmonic, 62, 225 subset, 2 subsolution, 21 superharmonic, 62, 225 supersolution, 21 support, 4, 185

Index

supremum, 20 surface area, 8 surface area of sphere, 33 symmetrization with respect to a line, 324 with respect to a plane, 324 with respect to a point, 324 symmetry group, 29 symmetry of potential, 18 symmetry transformation, 28 Taylor's theorem, 222 test functions, 190 test particle, 168, 169 thin set, 56 Tricomi operator, 40 truncated sector, 300

two-ball function of Siegel type, 76 two-step solution, 128, 138, 139 uniform Holder continuity, 196 uniformly elliptic, 39, 41 uniqueness of solutions Dirichlet problem, 44, 237, 250 idealized flow past body, 170, 213 Neumann problem, 85, 238 unit normal, 8 upward jumps, 94 vanishes, 4 variables aligned with k, 111 velocity potential, 170 vinca herbacea, 324 volume of ball, 33 vorticity, 134

wave equation, 40 Weyl, H., 324 Young's inequality, 12

An Introduction to Maximum Principles and Symmetry in Elliptic Problems (Cambridge Tracts in Mathematics)

An introduction to maximum principles and symmetry in elliptic problems

Divisors (Cambridge Tracts in Mathematics)

Convexity (Cambridge Tracts in Mathematics)

Divisors (Cambridge Tracts in Mathematics)

The Lebesgue Integral (Cambridge Tracts in Mathematics)

The Lévy Laplacian (Cambridge Tracts in Mathematics)

Hopf Algebras (Cambridge Tracts in Mathematics)

Simple Noetherian Rings (Cambridge Tracts in Mathematics)

Quasi-Frobenius Rings (Cambridge Tracts in Mathematics)

The Lebesgue Integral (Cambridge Tracts in Mathematics)

An Introduction to Diophantine Approximation (Cambridge tracts in mathematics and mathematical physics)

Simple Noetherian Rings (Cambridge Tracts in Mathematics)

Proximity Spaces (Cambridge Tracts in Mathematics 59)

Forcing Idealized (Cambridge Tracts in Mathematics)

Totally Positive Matrices (Cambridge Tracts in Mathematics)

Multiple Forcing (Cambridge Tracts in Mathematics 88)

Totally Positive Matrices (Cambridge Tracts in Mathematics)

Sets of Multiples (Cambridge Tracts in Mathematics)

Forcing Idealized (Cambridge Tracts in Mathematics)

Asymptotic Expansions (Cambridge Tracts in Mathematics)

Fixed Point Theorems (Cambridge Tracts in Mathematics)

An introduction to nonlinear functional analysis and elliptic problems

An Introduction to Nonlinear Functional Analysis and Elliptic Problems

Jordan Structures in Geometry and Analysis (Cambridge Tracts in Mathematics)

Convexity: An Analytic Viewpoint (Cambridge Tracts in Mathematics)

An Introduction to Magnetohydrodynamics (Cambridge Texts in Applied Mathematics)

An Introduction to Contact Topology (Cambridge Studies in Advanced Mathematics)

An Introduction to Random Matrices (Cambridge Studies in Advanced Mathematics)

Maximum Principles in Differential Equations

Maximum Principles in Differential Equations

An Introduction to Maximum Principles and Symmetry in Elliptic Problems (Cambridge Tracts in Mathematics)