Degree Theory in Analysis and Applications (Oxford Lecture Series in Mathematics and Its Applications, No 2)

OXFORD LECTURE SERIES IN MATHEMATICS A\1) ITS AI'I'IJCA'I'IONS 2

Degree Theory in Analysis and Applications IRENE FONSECA

and WILFRID GANGBO

OXFORD SCIENCE PUBLICATIONS

0

Oxford Lecture Series in Mathematics and its Applications 2 Series editors John Ball Dominic Welsh

OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS 1. John C. Baez (ed): Knots and quantum gravity 2. Irene Fonseca and Wilfrid Gangbo: Degree theory in analysis and applications

Degree Theory in Analysis and Applications Irene Fonseca and

Wilfrid Gangbo Department of Mathematics Carnegie Mellon University

CLARENDON PRESS 1995

OXFORD

Oxford University Press, Walton Street. Oxford OX2 6DP Oxford New York

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0 L M. Q. C. do Fonseca and W. Gangho. 1995 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press. Within the UK, exceptions are allowed in respect of any fair dealing for the purpose of research or private study, or criticism or review, as permitted tinder the Copyright. Designs and Patents Act, 1988, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms and in other countries should he sent to the Rights Department. Oxford University Press, at the address above. This book i: sold subject to the condition that it shall not.

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PREFACE In these lecture notes we study degree theory and some of its applications in analysis. We will focus on the recent developments of this theory for Sobolev functions which cannot be found in the existing literature dealing with the notion of topological degree for continuous functions (see e.g. Deimling 1985, Gold'sthein and Reshetnyak 1990, Lloyd 1978, and Schwartz 1969). In recent years, the need

to extend the notion of degree to nonsmooth functions was motivated in part by applications to nonlinear analysis and, specifically, to nonlinear elasticity (see Ball 1978, Fonseca and Gangbo 1995, Muller et at. 1994, Sverak 1988, and Tang

1988). As an example, in Chapter 6 we illustrate how a change of variables formula for Sobolev functions yields a local invertibility theorem for Sobolev functions which, in turn, allows us to prove a lower semicontinuity theorem for energy functionals involving variation of the domain. Our primary goal was to assemble the literature that is nowadays scattered on papers, manuscripts, and sometimes private communications, hopefully rendering these results readily accessible to analysts with a reasonable background in infinitesimal real analysis. In order to keep these notes as self contained as possible, in Chapter 1 and following the work of Schwartz (1969) we define the degree for differentiable functions and then extend the definition to continuous functions. In Chapter 2 we give some properties of the degree for continuous functions and we study the continuity of d(4), D, p) with respect to 0, D, p. We show that the degree d(4), D, p) gives a topological characterization of ¢IaD. Indeed, let BN = {x E RN : lxi < 1} and let SN-1 be the boundary of BN. If W: SN-1 _ SN-1 is a continuous mapping, then the following assertions are equivalent (see Corollary 2.13): p is not homotopic to a constant. RN admits a zero. Every continuous extension ¢ : BN Every continuous extension 0 : BN -, RN verifies d(4), BN, 0) 36 0. In Chapter 3 we present some classical results as applications of degree theory in topology, namely the Brouwer Fixed Point Theorem, the Borsuh-Ulam Theorem, the Jordan Separation Theorem, the Invariance of Domain Theorem (Open Mapping Theorem), the Perron-Frobenuis Theorem. In Chapter 4 we briefly review the theory of Sobolev spaces, introducing the necessary background for Chapter 5, where we give some properties of degree for Sobolev functions 0 E W1,P. Following Marcus and Mizel (1973), Gold'sthein and Reshetnyak (1990), and Sverak (1988), we state and prove change of variables formulae using the notion of degree. The case p > N is due to Marcus and Mizel (1973), the case p = N was treated by Gol'dshtein and Reshetnyak (1990), and

the case N - 1 < p < N was first studied by Sverak (1988). These change of

vi

PREFACE

variables formulae rely on the fact that d(4, B(x, r), ¢(x)) = sgn(det OO(x)) for r small enough and whenever det V0(x) 0. In Chapter 6 we give some applications of degree theory in Sobolev spaces

and we prove a local inverse function theorem for u E W"'(D)N under the condition det Vu(x) > 0 -or almost every x E D C RN (see Fonseca and Gangbo 1995).

Finally, in Chapter 7 we extend the notion of degree to infinite-dimensional spaces. We prove that most of the properties of degree discussed in Chapter 3 still hold in this context and we present some applications of degree theory to ordinary and partial differential equations. This work was supported by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis at Carnegie Mellon University. Also, the work of the first author, Irene Fonseca, was partially supported by the National Science Foundation under Grant No. DMS-9201215. The authors would like to thank J. Ball and J. Manfredi for their very hepful suggestions. Our thanks also go to Mrs E. Gangbo who skilfully typed the manuscript.

Pittsburgh March 1995

I.F., W.G.

CONTENTS

Introduction 1

Degree theory for continuous functions

1.3

Topological degree for Cl functions Topological degree for continuous functions Generalization of the degree

1.4

Exercises

1.1 1.2

2

1

Degree theory in finite-dimensional spaces

5 5 16

20 25 30 30

2.3 2.4 2.5

Dependence of the degree on ¢ and p Dependence of the degree on the domain D The multiplication theorem An application of Hopf's theorem Degree and winding number

2.6

Exercises

46

2.1 2.2

3 3.1

3.2 3.3 3.4

4 4.1

4.2 4.3 4.4

5 5.1

Some applications of the degree theory to topology The Brouwer Fixed Point Theorem Odd mappings The Jordan Separation Theorem Exercises

Measure theory and Sobolev spaces Rpyipw of measure theory Raucdorff mpasnrpc Overview of Sobolev spaces p-capacity

Properties of the degree for Sobolev functions Results of weakly differential mappings

5.2 Weakly monotone functions 5.3 5.4 5.5

6 6.1

6.2

Change of variables via the multiplicity function Change of variables via the degree Change of variables for Sobolev functions

Local invertibility of Sobolev functions and applications Local invertihility in W1,N Energy functionals involving variation of the domain

32 35

39 41

48 48 54 64 71

74

I4

is 87 92 106 107 11

131

135

140

149 144

160

CONTENTS

viii

7

Degree in infinite-dimensional spaces

172

172

7.2 7.3 7.4 7.5 7.6

Introduction to the Leray-Schauder degree Properties of the Leray-Schauder degree Fixed point theorems An application of the degree theory to ODEs First application of the degree theory to PDEs Second application of the degree theory to PDEs

7.7

Exercises

203

7.1

177 185

190 192 199

References

205

Index

209

INTRODUCTION The degree d(o, D, p) is a tool that describes the number of solutions for the

equation 0(x) = p in a given open set D C X, where 0 : D C X -+ X is a continuous function, p ¢ 0(8D), and X is a topological space, often a metric space. As it turns out, the degree is a generalization of the winding number of C when we identify the complex plane C to R2 (see Chapter 2). 0:DCC Assuming that A C {(0, D, p) : 0 : D

X is continuous, p V ¢(8D) },

we search for a function d : A -' Z satisfying the following properties: (Dl) If X = RN, D is open, bounded and if p E D, then

d(IID,D,P) = 1,

where I denotes the identity mapping of X. (D2) If d(0, D, p) 96 0, then there exists x E D such that 0(x) = p. (D3) If Dl f1 D2 = 0 and if p it 0(8DI U 8D2), then d(01 D,, D1, p) + d(0I D2, D2, P) = d(b, Dl U D2, P)

(D4) If h :

[0, 1]

C°(D)N is a Co homotopy such that p it h(t)((9D) for all

t E [0,1], then d(h(t), D, p) = d(h(0), D, p).

(D5) If p j9 0(8D), then d(0, D, p) = d(0 - p, D, 0). There is only one function d : A -+ Z verifying (D1)-(D5) (see Lloyd 1978 and Deimling 1985) and this is called the topological degree. In finite dimensions it is also designated by the Brouwer degree since the idea was first developed by Brouwer in 1912. There are many approaches to the introduction of the notion of degree and the background involved in each one of them may differ considerably. As an example, Alexandroff and Hopf (1935) and Dold (1972) made considerable use of concepts from algebraic topology and group theory, while the approach that we adopt here is more recent and uses only some basic analytical tools, such as the Implicit Function Theorem and Sard's Lemma. The analytic point of view was first given by Nagumo (1951). There is a very strong connection between the notions of topological degree and null Lagrangians (see Ball 1978). This was remarked upon by Tartar in 1974 1

INTRODUCTION

2

(see Tartar 1988). Null Lagrangians, such as det V4 and lower order minors of V0, depend only on the trace values of 0 on OD and are thus weakly continuous since the oscillations that may take place inside D are not felt by the functional

'0 - ID M (V (x)) dx, where M(VO) denotes the list of all minors of V0. As mentioned earlier, the degree d(4, D, p) is sought to count the number

of solutions for the equation O(x) = p, p ¢ q5(8D). Let us assume that D is an open, bounded, smooth subdomain of RN. In the scalar case, we consider D = (0, L), ' : (0, L) - R, p ¢ {0(0), 0(L)}. Note that information on the boundary values of 0 may help, in same cases, to decide whether or not y(x) = p admits a solution. Indeed, if

(0(0) - p) . (0(L) - p) < 0,

then there is at least one zero of 0(x) - p in D. We may count the number of such solutions as if 0(0) < p < m(L)

11 d(0, D, p) =

-1 if 0(0) > p > 0(L) otherwise.

0

In turn, this formula reduces to

d(m,D,p) = E Sgn(m'(x)),

(0.1)

xE4- ' (P)

whenever 0 E C' and ¢-1({p}) C {x E D : O'(x) 96 0}. This is in agreement with Definition 1.2. In order to extend this notion to higher dimensions, let 0 E C1(D)N,v E C(RN) and define A,(0)

fD(0(x))4(x)dx1

where J4(x) := det 00(x). We show that, for t' E C1(D)N, Iv(*)

if 0 = t!, on 8D.

(0.2)

Heuristically, this can easily be seen in the case where 0 and t/, are invertible, as, then, using the change of variables formula, we would have Iv(0) =

f

v(y)dy

4(D)

f

v(y) dy (D)

(0.3)

INTRODUCTION

3

= I,, (tb)

In general, we assume first that v E C1(RN), 0, 4 E C2(D)N and we remark that

dtI°(0+t(tI'-0)) = f v(O+t(O-0))CHo(z,t) vdHN-1(x),

(0.4)

8D

where HN-1 stands for the N - 1-dimensional Hausdorff measure, Ho(x, t) := ¢(x) + t(,y(x) - ¢(x)) and the kth component of CH(y,t), for a given function H(x, t), is defined by

det

(.'H

(x, t), ...

,

8'H

(x,

1

t), at (x, t), axH 1 (x, t), ... , a N (x, t)J

Thus, if ip = 0 on OD we obtain (0.2) for smooth v, 0, io and the result now follows for v E C(RN), 0, ip E C1(D)N via a density argument. Using (0.4) it can also be seen that if 0, t/' are homotopic (see Definition 1.11), precisely if there exists a smooth function H : D x [0, 11 -- RN such that

H(x,0) = O(x), H(x, 1) = P(x)

(0.5)

and v(H(x, t)) = 0 whenever (x, t) E OD x [0, 11,

then 0.6)

It suffices to perform the differentiation

d Wt-, t)) =

J8D'

vdHN-1(x) = 0.

Assume that 0 E C(D)N and that O(OD) n sptv = 0. It is possible to find On E C1(D)N such that On

0 uniformly on D

and

On, Om are homotopic in the sense of (0.5)

for n, m large enough. Hence by (0.6) we have

I (¢n) constant for n large and we set

INTRODUCTION

4

I"(O) := n Jim Iv(0n).

Based on this property, if 0 E C(D)' and if p

(0.7)

m(OD), then for n large enough,

q5(0D) (1 spt Vn = 0,

where Vn(y)

nNp(n(y - p)), p E C,,(RN), J p(y) dy = 1, spt p = 2(0, 1), N

and so, due to (0.7), I,,,, (0) is well defined and we set d(4', D, p) :=

lim n-.}oo I,,,, (O).

(0.8)

We remark that this is in agreement with Proposition 1.7. From (0.2), (0.7), and (0.8), it follows that

d(¢, D, p) = d(ti, D, p) if 0 = 0 on OD (see also Theorem 2.4). Using Sard's Lemma 1.4, it is also possible to show that if -0 E C1(D)N and if p if 0(,9D), .0-1({p}) C {x E D : J,0 (x) 54 0}, then (0.1) becomes d(O, D,p) = E sgn(J4(x)).

(0.9)

=Em- I (p)

Properties (D1)-(D5) follow from the definition of topological degree (see also Theorems 2.1, 2.3, Proposition 2.5, and Theorem 2.7). In addition, degree is a constant integer on every connected component of D \ O(OD) (see Theorem 2.3), and the change of variables formula (0.3) can now be written as (see Theorem 5.31) V(,O(x))det VO(x)dx

J

_

N

v(y)d(0, D, y) dy.

Often, to determine the degree d(¢, D, p) we construct a homotopy between 0 and a simple C' function to which formula (0.9) applies and we use the invariance property of the degree under homotopies. Such is the case of Lemma 5.9, where

to show that a continuous function, differentiable at xo, with J4(xo) 96 0, has degree equal to sgn(J4(xo)) near xo, we construct a homotopy between 0 and the linear function '+,(x) = O(xo) +VO(xo)(x - xo). This result is used later in Chapter 6, Lemma 6.5.

1

DEGREE THEORY FOR CONTINUOUS FUNCTIONS In the first three chapters, unless the contrary is explicitly stated, D is a bounded, open subset of RN. Also, in the sequel we will use the following notation:

x = (XI, ... , XN) E RN, the infinity norm of x is IxI := max{Ix;I : i = 1, ... , N}, and the euclidean norm of x is Ix12 := (x?, ... , 4)'I . We set p(x, y) := Ix - yI and dist(x, y) := Ix - y12- If x = (xl, ... , XN) E RN and y = (y1, , tN) E RN, we denote by x - y the inner product EN, xiyi. We identify the set RMxN of all M x N matrices with RMN and if A is an M x N matrix we denote by IAI (resp. IAI2) the infinity norm (resp. the euclidean norm) of A. If S C RN, the distance of x to S is given by inf { p(x, y) : y E S} and is denoted by p(x, S). Q(x, r) :_ {y E RN : p(x, y) < r}. B(x,r) := {y E RN : dist(x, y) < r}. If S C RN, g stands for the closure of S, int S its interior, 8S its boundary, and Sc its complement. C(D)N {f : D -- RN : f is continuous} and Iif II sup( If(x)I : x E D}. (f E C(D)N: spt (f) CC D}, where we write A CC B if A is CC(D)N a compact subset of B. 0). Co(RN)N := {f E C(RN)N I f (x)I = If f E C1(D)N, V f (x) and Jj(x) := det V f (x).

Cl(D)N denotes the set of functions f E C(D)N such that f admits an extension f to an open set D(f) containing D, and VI is continuous on D(f). If f E C1(D)N, then Ilf1I, := IIf1I + IJVf1I. If p E RN, x E 1) is said to be a p-point of 0 if O(x) = p. GN stands for the Lebesgue measure in RN (see Definition 4.1). n(A) denotes the number of distinct elements of A when A is a finite set, and n(A) if A is a set of finite cardinality O(A) +oo if A is a set of infinite cardinality.

Topological degree for C' functions Definition 1.1 Let 0 E C1(D)N,x E D. We say that x is a critical point of 0 1.1

if JJ(x) = 0. In this case O(x) is a critical value of 0. We define Zm := {x E 5

DEGREE THEORY FOR CONTINUOUS FUNCTIONS

6

Ch.

1

D : J,0 (x) = 0}, and q(Z,) is called the crease of 4. If pit 4(Z,,), then p is said to be a regular value of 0. Definition 1.2 Let -0 E C'(D)N and p ¢ (Q(Zo) U O(8D)). The degree of 0 at p with respect to D is defined by

d(O,D,p) := E sgn(Jo(x)),

(1.1)

zE0-' (p)

where sgn(t) = 1 fort > 0 and sgn(t) = -1 fort < 0. Notice that since p ¢ 4i(Z©), ¢-1(p) is finite and so (1.1) is well defined (see Exercise 1.2). The task ahead will be to relax the conditions p ¢ ¢(ZZ) and 0 E C1(D)N imposed in Definition 1.2.

Proposition 1.3 Let ¢ E C1(D)N and p i! 0(Zb) U y5((9D). Then there exists 6 > 0 such that if ry E C'(D)N and 1146 - 0II1 S 6, then p ¢ t/,(Z,,) u 0(8D) and d(0, D, p) = d(t), D, p).

Proof As 0-1(p) is either finite or empty (see Exercise 1.2), we consider these two cases separately. Firstly we assume that 0-'(p) = 0. Then 6 = i p(p, ¢(D)) > 0 and, given E (p) = 0. Thus p ¢ O(Z,) U ty(8D) C1(D)N such that I I - tLI I1 < 6, we have and

d(0,D,p) =d(10,D,p) =0. Next, suppose that 0-'(p) = {a1, ... , ak}. We show that there exist r > 0, 6 > 0, such that whenever tP E C' (D)N, Im - T/'

< 6, then tG admits exactly

one p-point in Q(a r), i = 1, ... , k. Indeed, fix ro such that 0 < ro < min I p(a 3 aj) : i 0 i, i, j = 1, ... , k ro

<minlp(a;,OD UZ4)

3

I,...,k}.

Set

Q(r) := Q(a1, r) U ... U Q(ak, r) and

c := min{ IJ#(ai)I : i =I,-, k}. We have c > 0 and since J0 is continuous on D, there exists 0 < r1 < ro such that IJ4(x)I > 2c for every x E Q(r1). Choosing 61 > 0 such that

sup{IJ#(x) - Jo(x)I : x E D} < 3c

TOPOLOGICAL DEGREE FOR C' FUNCTIONS

§1.1

whenever 11 0

7

- ipIIi 5 b1i we deduce that sup{IJ,0(x)I : x E Q(rl)} > 3c

if I I0 - 1GI I1 0({O-1(0)}),

for every n > 1.

(1.6)

TOPOLOGICAL DEGREE FOR C' FUNCTIONS

§1.1

15

(2) By Sard's Lemma the set of points for which (1.6) holds has measure zero.

Definition 1.11 Let 0,,p E Cl(D)N and H : D x [0,11 -+ RN. We say that H is a C' homotopy between 0 and ii if (i) Hi E C'(D)N for every t E 10, 11,

(ii) 1imt, IIHt - H.111 = 0 for every s E [0,11, (iii) Ho(x) = O(x), Hi(x) = ii(x) for every x E D, where we set Ht(x) _ H(x, t), x E D, t E [0, 1].

Theorem 1.12 Let 0 E C1(D)N. (i) d(O, D,.) is constant on each connected component of RN \ O(W). (ii) If p ¢ cb(8D), then there exists e > 0 such that I I ' - 0I l i e implies that p if tp(8D) and d(O, D, p) = d(t1', D, p). (iii) If H is a C' homotopy between 0 and ' and p ¢ Ht(8D) for every t E [0, 1), then d(0, D, p) = d(,4, D, p). (iv) If p ¢ . (8D), then d(4 + a, D, p + a) = d(q5, D, p) for every a E RN.

Proof (1) Let S1 be a connected component of RN \ 0(8D) and pi, p2 E 0. If pl ¢ O(Zo) we set q, = pl, and, if not, by Sard's Lemma we choose qi ¢ O(Z.0) such that Ipl - q, I < p(pl,¢(8D)). It is obvious that ql E S2 since q, E B(pl, p(pi, 0((9D))) C RN \ O(8D) and similarly, we select q2 E 0 such that q2 E B(p2, p(p2, 0, 8D)). By Proposition 1.8 d(o, D, ql) = d(O, D, q2) and the result now follows from Definition 1.9.

(ii) By Sard's Lemma there exists q E RN \ ¢(8D) such that q ¢ 0(ZZ) and Iq - p[ < z p(p, 4(8D)). Hence p and q belong to the same component of RN \ 4(8D) and, by Proposition 1.3, there exists 0 < co = Eo(q, ¢) < I p(p, 4'(8D)) such that

q ¢ O(Z,) UP(OD) and d(O, D, q) = d(¢, D, q) whenever 110 - 16111 < co. For every x E 8D we have

I VI(x) - P1 ? 10(x) - pI - I V,(x) - 4'(x)[ > Ip - qI,

therefore

Ip - qI < 2 p(p, 4'(8D)) < p(p, 0(,9D))

and so p, q belong to the same connected component of RN \ t/ (8D). By (i) we conclude that d(O, D, p) = d(0, D, q) = d(¢, D, q) = d(4', D, p), where we have used Proposition 1.3 and Definition 1.9.

DEGREE THEORY FOR CONTINUOUS FUNCTIONS

16

Ch. I

(iii) Define u : [0,1] - Z as u(t) = d(Ht, D, p). We show that u is continuous. Fix t E [0,1]. By (ii) there exists e > 0 such that IIHt - H,II1 < e implies d(Ht, D, p) = d(H,, D, q). Since limt--, I IHt - H. I I1 = 0 there exists b >

0 such that It - sI < b implies IIHt - H,II1 1

is a positive symmetric mollifier if the constant C is chosen so that fRN 8(x) dx = 1 (see Exercise 1.1).

Lemma 1.17 Let D C RN be an open, bounded set and let f : D -+ R be a continuous function. Then for every e > 0 there exists f E COO(RN) such that 11(x) - f(x)12 < E. for every x E D.

Proof Let g : RN

R be a continuous extension of f given by the Tietze

Extension Theorem. Define 0,.: RN R by 8r(x) convolution f,.:= 9,. * g.

-0(*) and introduce the

Clearly, fr E COO(RN) for every r > 0. Since g is uniformly continuous on the

compact set K = {x E RN : p(x, D) < 11, there exists 0 < r < 1 such that y,zEKand 1Y Z12:5 r imply 19(y) - 9(z)I < GN (K)'

For every xEDwe have If,-(x) - f(x)1 = JRN 0,-(x - y)(9(y) - 9(x))dyl < e

and it suffices to set i:= f,..

0

Definition 1.18 Let 0 E C(D)N and P E RN \O(8D). We define d(4, D, p), the degree of m at p with respect to D, to be d(,/i, D, p) for any 0 E C1(D)N such that 10(x) - O(x) I < p(p, 0(8D)) for every x e D.

DEGREE THEORY FOR CONTINUOUS FUNCTIONS

18

Ch. I

Justification. Applying Lemma 1.17 to each component 0;, we may find IJi E Cl (D)N such that I0(x) - O(x) I < p(p, O(8D)) for every x E D. Since p 95 .O(8D), 0 E C1(D)N, and I,0(x) -'(x)I < p(p,0(8D)) for every x E D, we have p ¢(8D). Next we claim that, if V11, 02 E C1(D)N are such that (>li1(x) - ¢(x) I, I02(x) O(x)I < p(p, 45(8D)) for every x E D, then d(ti1i D, p) = d(y2i D, p). Indeed, let

H(x, t) := t1i1(x) + (1 - t)>ii2(x), x E D, t E [1, 0].

H is a C' homotopy between fit and 02 and p ¢ H((9D, t) for every t E [0,1] because for every x E D we have

I H(x, t) - (x)I = It(+'1(x) - i(x)) + (1tl (+G1(x)

- -O(x))I + (1 - t)I(T'2(x) - 45(x))I

< tp(p,O(8D)) + (1 - t)p(p,q((9D)) = p(p, O(OD))

By Theorem 1.12 we conclude that d(01, D, p) = d(02, D, P)

Proposition 1.19 In Definition 1.18 the function 0 can be chosen such that P V O(Zo)-

Proof Take X E C1(D)N such that p(p, O(8D))

110 - XI I < 2

and by Sard's Lemma choose q E RN such that q f x(Zx) and IP - qI < 2P(P, c(8D)). Set 1G(x) := X(x) + P - q.

Clearly, 0 E C' (D) and I+'(x) - O(x)I 2P,O(8D)), yielding a contradiction.

We show that the degree does not change under composition with a diffeomorphism. We recall that f : RN -+ RN is a C' diffeomorphism if f-1 exists

and both f and f'1 are C'. Theorem 1.20 Let f : RN RN be a C' diffeomorphism and let E C RN be a bounded, open set such that f (E) = D. Let p ¢ O(8D), q = f'(p), and let 0 = f -1 o 0 o f, where -0 E C(D, RN). Then d(5, D, p) = d(tp, E, q).

Proof Since f is a diffeomorphism we have f (8E) = OD and so

q E t/,(8E) q f' 1(p) E f -1 o 0 o f (8E) a p E 0(8D), which implies that q it t)(8E). Step 1. Assume that 0 E C' (D)" and p V 0(ZZ). Clearly, q ¢ tp(Z ,) and

d(+', E, q) _ E sgn(J+,(y)) >G(v)=a

-

sgn [det (00(f (y)))

det Of (y)

(detVf((f- o0of)(y))l/

sgndet (V0(f (y))) (00f)(y)=p

E sgndet (V0(x)) O(z)=p

= d(0, D, p)

Step 2. If p E 0(Zm), by Sard's Lemma we approximate p by {p,,} C 0(Z,,)`; we C 0(Z,,)° and the result now follows from Step 1 obtain that and Theorem 1.12 (i). Step 8. If 0 E C(D)N, we approximate 0 by 0 E Cl(D)N converging uniformly to 0. Setting tai, := f'1 o0,a of, then iGn E C1(E)N, to converges uniformly to 0 and we conclude using Step 2 and Theorem 1.12 (ii).

Remark 1.21 The theory developed in Section 1.2 is still valid if we replace RN by an oriented space of finite dimension with a fixed basis. By Theorem 1.20 the definition of degree does not depend on the choice of fixed basis.

20

DEGREE THEORY FOR CONTINUOUS FUNCTIONS

Ch.

1

1.3 Generalization of the degree In the previous sections we studied the degree for continuous functions 0: D C

RN -. RN. Here we indicate briefly how to construct the degree in a more general framework and, in particular, the following lemma indicates a natural way to introduce this concept for continuous functions 0 : D C RN RM, where M < N are positive integers. In what follows, we identify RM with {x E RM C RN, where RN : zM+1 = ... = xN = 0} and we consider 0: D C RN

OM+1=..._ON =0. We define 0 E C(D)N by OW := x + ¢(x)

and we set X:= +GInru1N

Lemma 1.22 If 0 E C(D)N and if p E RM \ 0(8D), then d(+G, D, p) = d(X, D n RM, P)

Proof We start by showing that t/,-1(p) = X-1(p). The inclusion x-1(p) C 0-1(p) is obvious. If z E 0-1(p), then p = O(x) _ x + O(x), which implies that x = p - 1(x) E R. Therefore, x E D n RM and since p ¢ Vi((9D) we have x E D n RM and so p = x(z). Next, we prove that if D n RM = 0, then d(O, D, p) = d(x, DM, p). Indeed, D n RM = 0, implies that d(x, D n RM, p) = 0, and we have

0 = (DnRM) nX-1(p) = DnV,-1(p) = Du>G-1(p), since p ¢ t/,(8D). Therefore, p V 0(b) and so d({', D, p) = 0.

Step 1. If 0 E C1(D)M, D n RM 96 0, and if p ¢ x(ZX), then d(o, D, p) _ d(X, D n RM, P) Indeed, we have

Vx(x) V '(x) = (\OM,N-M B IN-M where

B:= O=M+1

Therefore,

GENERALIZATION OF THE DEGREE

§1.3

21

sgn(J, (x)) = sgn(Jx(x)) for every x E D n RM, which, together with the fact that p V x(ZX) and also 0-1(p) = x-1(p), yields p it t'(ZZ) and sgn(JX(x)) = d(x, D n RM, p)

sgn(J>,(x)) =

d(V,, D, p) xE,+-1(p)

xEX-1(P)

Step 2. If 0 E C(D)M, then d(ty, D, p) = d(x, D n RM, p). Choose E C1(D)M such that

*x) - 0x)I < Ip(p,V)(8D)),

(1.7)

for every x E D, set t%i(x) := x + t(x), x E D,

and let k be the restriction of r% to D n RM. By Sard's Lemma there exists a E RM such that

lal < 2p(p,t(i(8D)), P+a¢X(ZZ) By (1.7) we have I

(x) - O(x)I < 2P(p,O(OD))

for every x E D and so (see Definition 1.18) d(t , D, p) = d(o, D, p)

(1.8)

d(z, D n RM,p) = d(x, D n RM,p).

(1.9)

and, in the same way,

By Step 1, Theorem 1.12 (iv) and as p + a ¢ X(ZX), we deduce that

d( +a,D,p+a) = d(X+a,DnRM,p+a), which implies

d(i , D, p) = d(X, D n RM, p)

This, together with (1.8) and (1.9), concludes the proof.

0

In the remaining part of this section, we briefly introduce the notion of topological degree on manifolds and for more details we refer the reader to Nirenberg (1974) and Spivak (1979).

22

DEGREE THEORY FOR CONTINUOUS FUNCTIONS

Ch.

1

In what follows M is a C°°, paracompact, oriented manifold of dimension N; we denote by d the operator of exterior differentiation on M and by n the exterior product operator between two differential forms on M. We recall that if w is a smooth (N - 1)-form on M, then w can be written as N

w = E(-1)'-'gg(y)dyi A... Adye-1 A dyJ+1 A ... A dyN j=1

in local coordinates (Y1, ... , YN) and so N

dwayg.L(y)dylA...AdyN.

7=1j

Consider a CO° N-form on M,

µ=fdylA...AdyN where f : M - R is a C°° mapping. Let St c RN be an open, connected set and let 0: Il - M be a CO° mapping. The pull-back of the form p by 0 is given by 0* (µ) := f(O(x))(detV (x)) dx1 n ... A dXN.

In general, when 0: P - M is a C°D mapping and P is a smooth N-dimensional manifold, then the pull-back 0'(µ) is an N-form on P. If spt (p) C 0(1l) and if ¢ is an orientation-preserving diffeomorphism, then we define

JM µ

j 0 *6u) = J f (O(x)) det V (x) dx. A

T

This definition does not depend on the choice of 0 (see Spivak 1979, Chapter 8, Theorem 5). By Stoke's Theorem it follows that if w is a smooth (N - 1)-form with compact support on M, then

JM

dw=0.

Lemma 1.23 [Bard's Lemma for Manifolds] Let F : M -. M be a C1 mapping. Then the set of critical values of F has measure zero in M. The proof is similar to that of Sard's Lemma in RN (see Lemma 1.4), and it can be found in Nirenberg (1974) and Spivak (1979). Lemma 1.24 Assume that p is a COO N -form on M with fm it = 0 and spt p C fl, where fl is a connected subset of M. Then there exists an (N - 1)-form w on M such that spt w C Sl and p = dw.

§1.3

GENERALIZATION OF THE DEGREE

23

For the proof of this result we refer the reader to Schwartz (1969). Motivated by Remark 1.14, we introduce the notion of degree of m at p E M.

Definition 1.25 Let M be a CO°, paracompact, oriented manifold of dimension N, let D C M be an open set with compact closure, let m E C(D; M) f1 C' (D; M), and p E M \ m(8D). Let 11 C M be the connected component of M \ m(8D) containing p and let p = f dy1 A... AdyN be a C°° N-form with support contained in tl such that fM µ = 1. We say that u is admissible for m and p and we define the degree of m at p with respect to D by d(m,D,p) :=

JD

m*(A)

Justification. Let µ1,112 be two admissible N-forms and set u := it, -µ2. Then spt µ C S2, fm p = 0, and, by Lemma 1.24, there exists an (N - 1)-form w such that spt w C 11 and µ = dw. Hence, ID

0%ul)

m*

-L2)

m* (µ) (aID

= 1D m'(dw)

=I

D

=0 by Stoke's Theorem, and so d(m, µ, p) is well defined. It follows immediately from Definition 1.25 that if p, q E CI, then d(O, D, p) = d(O, D, q)

and also, by Exercise 1.6, d(m, D, p) is an integer number. To extend the notion of degree to continuous mappings, we use the following facts: 1. limq_p d(O, D, q) = d(O, D, p) 2. If p is a regular value then d(m, D, p) = K'160- - W sgn(J#(x)). 3. The degree is invariant under C-1 homotopy.

Definition 1.26 Let M be a COO, paracompact, oriented manifold of dimension

N, let D C M be an open set with compact closure, let m E C(D, M), and p E M \ m(8D). Then d(O, D, p) :=

n

li d(mn, D, p)

where {mn}nEN C C(D, M)f1C1(D; M) is any sequence converging tom uniformly

on D.

DEGREE THEORY FOR CONTINUOUS FUNCTIONS

24

Ch.

1

We recall that BN is the unit ball in RN, SN-1 is the unit sphere, and Ix12

stands for VX_1.

Proposition 1.27 Let 0 E C(BN)N be such that O(SN-1) C RN \ {0}. Let r(i(x)

ro-

z s for x E SN-1. Then d(ti,SN_1,SN-1)

d(O,BN,0) =d(1,,SN-1,p) =: for every p E

SN-1

Proof We notice first that d(o, BN, 0) is well defined since 0 ¢ q(BBN). Without loss of generality, assume that 0 E Cl(BN)N and that p is not a critical value (this can be done by virtue of Sard's Lemma in RN). Considering the homotopy Ot(x)`

iO( IZ, sEB, 0 0 small enough such that po := e2p is a regular value of ' for all 0 < e < eo. We have G(a) = po q a E {ex1,...,Exk};

hence, by Exercise 1.6, d(16,SN-1,P)

k

_ >Sgn(JV, (xk)) i=1

EXERCISES

§1.4

25

and

d(G, BN, e2P) = r sgn(JG(exi)).

i-1 Noticing that sgn(J,,(xi)) = sgn(JG(exi)), we conclude that SN-1, SN-1),

d(c5, BN, 0) = d(G, BN, 0) = d(lIi,

where we have used the fact that, for a small, d(G, BN, e2p) = d(G, BN, 0).

This property can easily be deduced from Proposition 1.8 within the framework 0 of continuous functions. This will be proved in detail in Theorem 2.3. 1.4

Exercises

Exercise 1.1 Let 9 : RN

R be defined by

9(x) =

exp

1

to

1x12 < 1

IX12?

Prove that 9 is integrable in RN and 0 E CCO(RN). Solution 1.1. Let p : R --+ R be defined by

µ(t)

OXP(-)

iti > 1.

It is obvious that µ is infinitely differentiable at every point t 0 -1, 1. Note that for every n E N there exists a polynomial P,a(t) such that, for every Iti < 1, Pn(t)

{(n) (t)_ (t2-1)2nexP(

1

t2-1

and so lim µW(t) _'limiJA (n)(t) = 0.

Since p(')(t) = 0 for every It1 > 1 we deduce that p( )(-1) and µ(n)(1) exist and so p E COO(R). Considering, in addition, the fact that the function x - Ix12 is infinitely differentiable, we deduce that 9 E COO(RN). We also have spt (9) C {x E RN : Ix12 < 1}

and so 9 is integrable on RN.

Exercise 1.2 Let D C R^' be an open, bounded set, 0 E C1(D,RN), and p it e(ZZ). Prove that 0-1(p) is finite.

DEGREE THEORY FOR CONTINUOUS FUNCTIONS

26

Ch. I

Solution 1.2. Assume that there exists a sequence {xk} C 0-1(p) such that xk # x1 for every k # 1. Since D is a compact set we can assume, without loss

of generality, that there exists 2 E RN such that xk - 2. We obtain 2 E D and by the continuity of

E ¢-1(p). Using the fact that m is continuously

differentiable we have 0 = O(xk) - 0(2) = VO(x)(xk - 2) + (xk - 2)E(xk - 2),

(1.10)

where e is a real function satisfying lim=o E(t) = 0. Since 2 E 46-1(p) and p¢O(Z.0),wehave 0 0 (simply write f = f + f - ). Note that ID f (O(x)) det (V4 (x)) dx =

J f (O(x)) det (V (x)) dx, U

where U is the open set where det (V¢(x)) 96 0 and decompose U into a union of disjoint open sets U, such that det (V(x)) has a constant sign on each U,, and a set of zero measure. Apply the Inverse Function Theorem in each U; and, using the definition of the degree for noncritical values, deduce the result (see also Theorem 5.30 for a more general proof). Exercise 1.6 Let M be COO paracompact, oriented manifolds of dimension N, let D C M be an open set with compact closure, let 0 E C(D; M) n C1(D; M), and p E M \ (O(8D) U O(Zm)). Prove that d(i, D, p) = E=Em-'(n)sgn(Jo(x)). Solution 1.6 (see also Spivak 1979, Chapter 8, Theorem 2). Assume first that

p V O(D), and let f? be the connected component of M \ 0(8D). Let W be an open set contained in a coordinate patch of p and let µ be a COD N-form with support contained in W such that fiat p

1.

Then p = f dyl n ... A dyN and spt f n 0(D) = 0; hence

0=

f 0' (µ) = d(0,D,p)

Next, we assume that p E 0(D). Since p

0(Zm), we know that -'(p) must be

finite,

0-1(p) _ {xl,...,xk} , and we choose disjoint open sets N, C D, i = I,-, k, such that x; E Ni and 401N, is a diffeomorphism. Set

V :=

n

0(N1).

Then V is an open neighbourhood of p and we choose a C°O N-form on M with support contained in V such that

EXERCISES

§1.4

29

JM µ=1. We have

d(O, D, P) = ID '(µ) k

k

_ E sgn(J,6(xc)) {=1

_

f

O(N:)

sgn(J0(x,)) i=1

µ

2

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES In this chapter we prove those properties of d(cb, D, p) that we consider to be most useful in applications. 2.1 Dependence of the degree on 0 and p The first theorem is the basis of many of the applications of degree theory in analysis.

Theorem 2.1 Let Q E C(D)N, p ¢ O(8D) such that d(O, D, p) j4 0. Then there exists x E D such that m(x) = p.

Proof Assume that p ¢ O(D). Since p ¢ ¢(8D) we have p ¢ O(D), and as O(D) is a compact set we have p(p, b(D)) > 0. By Proposition 1.19, choose t/i E C1(D)N such that 110 - Oil < p(p,4'W(D)) and p ¢ t4i(Zo) U i (8D). We have p V ;U(D) and then, using Definition 1.18,

0 = d(t/i, D, p) = d(¢, D, p),

0

which yields a contradiction.

Definition 2.2 H : D x 10,11 -+ RN is a Co homotopy between 0, 0 E C(D)N if H is continuous on D x (0, 1), H(x, 0) = 0(x), and H(x,1) = V;(x) for every x E D. Theorem 2.3 Let 0 E C(D)N and let p ¢ O(8D). Then (1) for every 0 E C(D)N such that IIV) - 011 < p(p,O(8D)), we have d(O, D, p) = d(tG, D, p);

(2) if H(x, t) =: ht (x) is a C° homotopy between h°, h1 and p ¢ ht (OD) for every t E 10, 1], then d(ht, D, p) does not depend on t E (0,1]; (3) if pi, p2 belong to the same connected component of RN \ O(8D), then d(O, D, pi) = d(¢, D, p2).

Proof (1) Let X E C1(D)N be such that

II0-XII p(7, 4(8D)) - p (4(8D), *(OD)) > p(7, f1l) - p(4(8D), 0(8D)) > 0,

we have that -y C RN \ 0(8D) and so pi,p2 are in the same connected component of RN \ O(8D). By Proposition 1.8 we deduce that d(t,1, D, pi) = d('J,D,p2)

0

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

32

Ch. 2

Theorem 2.4 Let 0, ti, E C(D)N be such that 018D = tGIaD Then d(O, D, p) _ d(t/i, D, p) for every p it O(OD) -

Proof As c(8D) = ry(8D), d(¢, D, p) and d(ry, D, p) exist for every p ¢ c(8D). Set

H(x, t) := tO(x) + (1 - t) i(x), x E D, t E [0,1]. H is a CO homotopy between t l + 1. We obtain

d(O, D, p) _ E sgn(J,,(x)) xE,4-' (P)

_j i=1

_

E

sgn(Jv,(x))

zEt.& '(P)nD.

d(V,, Di, p) i=1 +00

_

d(,), Di, p) i=1 +00

_ Ed(O,Di,p) i=1

(2) Let i' E C1(D)N be such that 110 - 011 < p(p,O(K U 8D)) and p it t/i(Z,;,). It is clear that p it O(K) and so

d(1G,D,p) =d(t1',D\K,p).

Note that 8(D \ K) C 8D U K. AS III - 10II (D\K) < p(p,O(8D U K)) < p(p, 0(8D)) we have Also, I10

d(O,D,p) =d(O,D,p) II(D\K) < III - 011D < p(p,O(8D U K)) and so d(O, D \ K, p) = d(t', D \ K, p) = d(+,, D, p) = d(O, D, p).

0 Now we introduce the notion of the index of a p-point xo E D.

Definition 2.8 Let i E C(D)N, p ¢ 4(8D), and let xo E D be an isolated p-point of 0 (i.e. O(xo) = p and there exists a neighbourhood V of xo such that

34

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

Ch. 2

V n .0-1(x0) = 0). Let U be the collection of all open neighbourhoods V of x0 such that V does not contain another p-point of ¢. We define the index of 0 with respect to (X0' p) by i(0,xo,P) := d(i5,V,P)

for any V E U. Justification (1) For V E U we obtain that p ¢ 0((9V) and so d(¢, V, p) is well defined. (2) Let V1, V2 E U. We have V = V1 U V2 E U and setting K = V1 n V2 we have that K C V is a compact set and p O(K). By the excision property of the degree (see Theorem 2.7) we obtain that

d(,, V, p) = d(t, V \ K, p) = d(¢, V n (Vc u V2), p) = d(d, V2, p).

Using a similar argument, we have d(o, VI U V2i p) = d(y, V1, p); thus

d(4,V1,P) = d(o,V2,P)

Theorem 2.9 Let 0 E C(D)N and let p ¢(8D). (1) If 0-1(p) is finite, then d(0, D, p) = E E0-'(P) i(0,x, p). (2) If 0 E C1(D)N, a E g-1(p), and if J0 (a) 3k 0, then a is an isolated point and i(Q,a,p) = (-1)", where v is the number of real negative eigenvalues of V0(a), counting algebraic multiplicity. Warning. In general we do not have I i(0, x, p)1 < 1. As an example, consider O(x,y):= (x2 - y2,2xy),

where (x, y) E D:= (-1,1) x (-1,1). We have d

(0) andilo,l 0J,(0))

D, (c)) = 2, 0-1 10l =

=2.

Proof of Theorem 2.9 (1) Since 0-1(p) is finite, then every a E m-1(p) is an isolated point and so i(0, a, p) is well defined. Assume that

0-1(p) _ {al,...,ak}. Let V1.... , VK be open sets in D, mutually disjoint, and such that a1 E V1,

..., ak E VK. By Definition 2.8 we obtain that i(0, a;, p) = d(0, Vi, p).

THE MULTIPLICATION THEOREM

§2.3

35

Therefore, using the excision property of the degree (see Theorem 2.7), we have k

i(4,x,P) _ Ei(4,ai,P) t=1

X66-'(p)

k

_

d(4, VV, p)

ic1 k

= d(4, iOi Vi, P)

=d(4,D\K,p) = d(4, D, p),

where K=D\Uk,V. (2) Let V C D be an open set such that a E V, 4(a) = p and 4(x) 96 p for every x E V, x 96 a. By Definition 2.8 we have

i(4,ai,P) = d(4,V,P) = sgn(J4(a)) Let Ai, ... , A be the eigenvalues of V4(a). Then

JO(a)=ai...\,,. where the complex eigenvalues occur in conjugate pairs a, a such that ar > 0, and so

sgn(JJ(a)) = (-1)'. 0

2.3 The multiplication theorem Given two mappings 4 E Cl(D)N and 0 E Ci(4(D))N we want to compute d(t, o 4, D, p) in terms of d(i/), Di, p) and d(4, D, Di), where Di (i E N) are the connected components of RN \ 4(8D). This is obtained in the multiplication theorem.

Theorem 2.10 [Multiplication Theorem) Let 4 E C(D)N, let M C RN be an open set containing 4(D), 0 = M \ 4(8D), and let iii E C(M)N. Let (Di)iEN be the connected components of A and p ¢ 0 o 4(8D) U vi(8M). Then we have (1) p it O(80,) for every i E N. (2) d(4 o 4, D, P) _ `,EN d(,P, A, p)d(O, D, Di)

Proof Note that 80, C OM U 4(OD) for every i E N. Indeed, if q E 80i for some i E N, then q E 8Ai C TA-i C M. Assume that q ¢ 8M. We obtain

36

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

Ch. 2

that q E M = u,ENL;, thus for a suitable j E N we have q E A. and i contradicting ii n 0, = 0. Hence ODi C OM u m(OD)

j, (2.1)

and d(W, Di, p) is well defined. Also, p ¢ >G o O (OD) and so d(t,b o 0, D, p) is well

defined. Since Di is a connected component of M \ O(OD), we deduce that 0, is a connected set included in RN \ 0(8D) and so Di is a subset of a connected component of RN \ 0(8D), Di. As d(q, D, q) is a constant for q E Di, we denote this constant by d(q5, D, ii). We claim that there is only a finite number of i such that d(i/i, Di, p) 76 0. Indeed, take f E Cl (M)N such that I if - 011 < p(p,1,(8M)) and p ¢ f (Zj). Since there are only finitely many x such that f (x) = p and since

the sets Ai are mutually disjoint, there are finitely many i so that f (x) = p admits a solution in Di, say A,,- , Ok. We have d(f, Di, p) = 0 for every i > k. As

IIf - 01 IA. k. The proof of d(t& o 46, D,p) _

d(t,b, Di,p)d(0, D, Di)

is divided into four cases. Case 1. We assume that 0 E Cl(M)N, 0 E Cl(D)N, and p ¢ ik o gs(ZOo0). Then p ¢ V,(Z,,) and y if 0(Z.) for every y E 0-'(p), hence

E E

d(tG o 0, D, p) _

sgn(J,,oo(x))

zED, Oo4(x)=p

sgn(Jo (O(x)))sgn(JJ(x))

xED, Oom(x)=p

_E

E

sgn(J+,(y)) sgn(JJ(x))

YEA,*h(v)=p zED,b(x)=Y

E sgn(Jo(y)) d(0, D, y) LEA, O(V)=p

_EE

sgn(J4 (y)) d(0, D, ,&i)

As

d(,P, Di,p)d(O, D, Di)

Case 2. We assume that tJi E C1(M)N, 0 E Cl(D)N, and P E tG o g1(ZOom).

§2.3

THE MULTIPLICATION THEOREM

37

By Sard's Lemma there exists q ¢ ip o q5(Z*.,6) such that

IP - qI < min{p(P, V) o 0(8D)), p(p, 0(8M))}.

By (2.1) 80i C 8M U ¢(8D) for every i E N; hence IP - qI < p(P, +/'(8&))

and so p and q belong to the same connected component of RN \ ?P(C)0i). This yields d(0, Ai, p) = d(O, Di, q) for every i E N and so, by Case 1,

d(O o0,D,p) =d(V,o0,D,q) _ d(,G, 0., q)d(O, D, Ai) _

d(+G,

Ai, P)d(4, D, Di)

,

Case 3. We assume that

E C'(M)N and 0 E C(D)N.

Set

L:='0-'(P)' e := 1p(L, O(8D))

(2.2)

and let ¢ E C'(D)N be such that, for every x E D, I3(x) - O(x)I < min{b, a},

(2.3)

Ia - bI < b = hi(a) - 0(b) I < p(P, 0 o O(8D)).

(2.4)

where b > 0 is such that

Suppose, in addition, that (D) C M. We have

d(r/io0,D,p)=d(r1 o¢,D,p).

Let 0 = M \ (8D) and let

(2.5)

E N} be the family of connected components

of A. Set

A = {Di : Ai n L # 0, d(0, D, 0:) # 0} and

A={Di : DinL00,

0}.

Since L is a compact set, both A and A have finite cardinality. Write

A = {Ol,...,Ak}, and A = {Al...... i}. We claim that k = 1. Indeed, fix i = 1, ... , k, and y E .; n L. By (2.2) and (2.3) we obtain that I0(x) - O(x) I < p(y,O(8D)) for every x E D and so

38

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

Ch. 2 (2.6)

D, y) = d(O, D, y) 0 0.

We want to prove that Q(y, e) C RN \ (8D). Assume, on the contrary, that yI < E. We obtain

there exists x E OD such that

O(x)I < 2e < p(y, O(OD)),

yI +

I0(x) - yI 1.

D, ) is a constant on 0j (,) we have

Using (2.6) and the fact that degree d(O, D, ,&i) =

(2.8)

D, Or(s)).

Note also that, by (2.2) and (2.3), e
k,

which, together with (2.7), yields

(2.11)

1 = k.

We show that d(1, 0;, p) = d(+), 0;, p) for every i E N. Using the excision property of degree (see Theorem 2.7), with K = (0 \A;) U (0; \ At), we have d(O, A1, p) = d(+G, Di n ij, p) = d(,P, Di, p).

(2.12)

By (2.2), (2.4), (2.5), (2.8), (2.10), (2.11), (2.12), and the decomposition formula

for 0 and , we deduce that d(+Go0,D,p) =d(tio¢,D,p)

_

i

d(0, Af,

D,,&))

§2.4

AN APPLICATION OF HOPF'S THEOREM

_

39

d(t,b, A,,p)d(-O, D, D{)

Case 4. We assume that t/i E C(M)' and 0 E C(D)N. Let >u E C1(M)N be such that for every y E M 1t (y) - VG(y)I < min{p(p, 0 o 4(8D)), p(p, 0(8M))}.

For every x E .b we have ItG o O(x) - 10 o O(x)I < min{p(p, iG o ct(8D)), p(p, V,(8M))}

and by Proposition 1.19 and the decomposition formula for t/ ', ., obtained in Case 3, we have d(' o 0, D, p) _

d(O, ti, p)d(c, D, Di ) 11

2.4

An application of Hopf's theorem

In this section we state the well-known Hopf's Theorem. The idea is the following:

we have already proved that the degree is invariant under a CO homotopy (see Theorem 2.3) and the question now is whether functions with the same degree must be homotopic. Hopf proved that this is true for balls (see Hopf 1926), i.e. if two continuous mappings defined on the unit ball B(0,1) = BN have the same degree at the origin, then they are Co homotopic.

Theorem 2.11 [First Version of Hopf's Theorem] Let N > 2 and let ) = BN be the unit ball of centre 0 in RN. Assume that Q, 0 E C(1l)N, 0 ¢,0(8f2) U t,(8S2), and d(O, fl, 0) = d(i,, ), 0). Then there exists a continuous function H : [0, 1] x n RN such that 0 it H(t, 8S1) for every t E 10, 1] and H(0, ) _ , H(1, ) = iG We refer the reader to Guillemin and Pollack (1974) for an analytic proof of this result. We state a second version of Hopf's Theorem which can be easily deduced from the first version.

Theorem 2.12 [Second Version of Hopf's Theorem] Let N > 2 and let SN-1 = 8BN be the unit sphere in RN. Assume that V1,,p2 E SN-1, SN-1) = d(W2,

C(SN-1,SN-1)

SN-1, SN-1). Then there exists a continuous and d(Pl, cp2. Cpl, function H : SN-1 x [0,1] -+ SN-1 such that SN-1 . SN-1 be a continuous mapping. Then the Corollary 2.13 Let 'P following assertions are equivalent: (i) p is not homotopic to a constant. :

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

40

(ii) Every continuous extension 0 : BN (iii) Every continuous extension 0 : BN

Ch. 2

RN admits a zero. RN verifies d(O, BN, 0) y6 0.

Proof Step 1. We start by showing that if c E SN-1, then d(c, SN-1, SN-1) = 0. In fact, by Proposition 1.27, d(c, SN-1, SN-1) = d(c, BN, 0)

and by Theorem 2.1 d(c, BN,0) = 0. Step 2. Let .F:= {4 E C(BN; RN) :

First, we prove that (i)

0I SN

=

}.

(ii). By Hopf's Theorem, we have

d(co,SN-1,S"'-1)

0.

For every 0 E F, by Proposition 1.27 we have

d(5,BN,0)

=d(co,SN-1,SN-1) : 0

and by Theorem 2.1 we deduce that there exists x E BN such that O(x) = 0. Next, we show that (ii) (iii). Assume that W admits a continuous extension 01

: B - RN such that d(¢l, BN, 0) = 0. Then by Proposition 1.27 and by

SN-1, Hopf's Theorem V is homotopic to a constant, i.e. given a constant c E there exists a continuous mapping H : SN-1 x (0,11-. SN-1 such that, for every

xE

SN-1,

H(x,0) = c, H(x, l) = cp(x). Define 0 : BN - SN-1 by

0(x) = H ( I X12' IXI2)

,

x E BN.

It is clear that 0 is continuous at every x E BN\{0}. Using the uniform continuity

of H and the fact that H(x, 0) = c for every x E BN, we deduce that ' is continuous at 0. Therefore, we find 0 E F such that the equation 0(x) = 0 has no solution in BN. Finally, we prove that (iii) As before, setting 0(x)

.

(i). Assume that cp is homotopic to a constant.

H ( I!_,

x12)

,

x E BN,

it is easy to verify that 0 is a continuous extension of 'p, homotopic to a constant. 0 By Theorem 2.3 and Step 1 we conclude that d(0, BN, 0) = 0.

DEGREE AND WINDING NUMBER

§2.5

41

2.5 Degree and winding number In this section, we establish the relation between the degree and the winding number. In order to achieve this goal, we recall some definitions and properties of holomorphic functions. Throughout this section C denotes the set of complex numbers (which we identify with R2) i2 = -1, D is an open, bounded, subset of C, and the derivative Q of a function Q : D C is given by O(z + h) - 4(z) z =- hEC,h-»0 lim h

If 45'(z) exists for all z E D, then 0 is said to be holomorphic in D, and we write 45 E H(D).

Definition 2.14 Assume that OD is a C' closed curve, let

: D -i C be a

holomorphic function and let p it m(8D). We define the winding number of 0 at p with respect to OD by w(0, D, p) :=

21a

z)

L T0)

p

dz.

Lemma 2.15 Let

:D C be a holomorphic function, Z :_ {z E D I O(z) _ 0} and let E be the set of limit points of Z. Then E is both an open and a closed subset of D.

Remark 2.16 (i) Assume that D is a connected set. We recall that y5(z) = p then has an infinite number of solutions if and only if 0 is a constant p.

(ii) It is easy to prove that F :_ {z E D : Qtnl(z) = 0 for all n E N} is both closed and open in D. Indeed,

F= n

nENU{0}

{zED:O(n)(z)=0}

is a closed subset of D. Moreover, if zo E F by the analyticity of 0 there exists R > 0 such that +00 O(n)

O(z) = 1

(zo) (z - zo)n

n=0

for every z E B(zo, R), and so 0(z) =

(z) = ¢(2)(z) _ ... = 0

for every z E B(zo, R). Hence,

B(zo, R) C F

and F is an open set.

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

42

Ch. 2

Lemma 2.17 [Cauchy's Lemma] Let D be a convex set, zo E D, and let 0 E H(D \ {zo}) n C(D). Then there is a holomorphic function F E H(D) such that F = .

For the proof we refer the reader to Rudin (1966).

Lemma 2.18 Assume that,0 E H(D) is not constant, zo E D, and wo = 0(zo). Then

(i) there exists m E N such that zo is a zero of q5 - wo of order m, i.e. O(z) wo = (z - zo)mg(z), when g is holomorphic and g(zo) # 0; (ii) there is an open neighbourhood V of zo and a function cP E H(V) such that O(z) = wo + 6P(z))m

and W is a bijection from V into B(0, r) for a suitable r > 0.

Proof We may assume without loss of generality that +oo

D = B(zo, R1) and O(z) _ E an(z - zo)n n=0

for every z E B(zo, R1).

(i) Let O(zo) = wo. Since m is not constant, there is n so that an 96 0. Let m E N be such that

at =...=am-1 =0, a.960. We obtain

+oo

O(z) = (z - zo)n' K am+n(z - zo)n n=0

and setting +00

9(x) E am+n(z - zo)n n=0

we have O(z) - wo = (z - zo)mg(z), g E H(B(zo, R1)) and g(zo) iA 0.

(ii) Let 0 < R2 < R1 be such that g(z) # 0

for every z E B(zo, R2). Then 9 : is holomorphic in B(zo, R2) and, by Cauchy's Lemma, there exists h E H(B(zo, R2)) such that

DEGREE AND WINDING NUMBER

§2.5

43

h =9 9

We have

(g.exp(-h))' = g' exp(-h)-h g.exp(-h) = g.exp(--h) (_(i

- h') )

=0

and since B(zo, R2) is a connected set, we deduce that there exists a constant c E C such that

g exp(-h) = c for every z E B(zo, R2). Set

W(z) :_ (z - zo)cmexp \m/

.

We obtain that cp E H(B(zo, R2)) and (w(z))m = (z - zo)'c . exp(h) = g(z)(z - zo)m

and so ('P(z))n`

O(z) = wo +

for every z E B(zo, R2). Hence cp(zo) = 0 and, in addition, W '(z) = cm exp

(m )

\1 + z

z'o h (z)/

thus 0 (zo) = c-exp (h(' O)) 54 0. By the Inverse Function Theorem we deduce that there is an open set V containing zo such that

0. 0

Lemma 2.19 Let 0 E H(D) be such that 0(x + iy) = 01(x, y) + i02 (X, y) for every x, y E R. Set

JO(x,y):=det(j Tl. Then J,(x, y) > 0 for every x, y E R.

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

44

Ch. 2

Proof Since (zo) _ li .

O(zo + h) - 4(zo)

taking h = t E R (resp. h = it, t E R), we obtain (zo) =

(zo) = VOL + i

(resp.

- i L) which yield the Cauchy-Riemann equations 1901

amz

ax = ay

and

801

aO2

ay = ay

We conclude that

J2>0.

J-0

Theorem 2.20 Let D C C be a connected, bounded, open set such that 8D is a C' closed curve. Let 4 : D -+ C be a holomorphic function and let p it O(aD). Then

d(c, D, p) = w(5, D, p) =

tai

f

D

O() z) p

dz.

Proof Suppose, first, that 0 is constant, O(z) = c for every z E D. For every p E C \ O(8D) we have p E C \ O(D) and so d(m, D, p) = 0. Also, w(q, D, p) _ °-: = 0 and so 2 i f8D a-p d(O, D, p) = w(O, D, p) -

Now let 0 be a nonconstant holomorphic function in D. If p then d(O, D, p) = 0 and z : n is a holomorphic function on D. Therefore,

feDO:zpdz=0and d(O, D, p) = w(O, D, p)

Assume that p E O(D). Since O(z) is nonconstant, 0-'{p} must be finite (see Remark 2.16), say

¢-'{p} = {z1,...,zk}. By Lemma 2.18, for each j = I,-, k, there are R, > 0,

3

: B(zJ, R,) -+ C

such that

cps E H(B(zj, Rj)), B(z., Rj) CC D, : B(zi, R,) -+ vj (B (z.,, R,)) is a bijection, $,(z) - p = (z - z))m gj(z), for every z E B(z,, Rj), cps

DECREE AND WINDING NUMBER

§2.5

45

(,p.1)m = (z - zj)m gj(z) for every z E B(z7, Rj),

for some gj E H(B(zj, Rj)). If R := z min{Rl,... , Rk}, then w (0, D , p ) =

1J

(z)

.

27rt

1

k

(Z )

O(z)

OD

-P

dz =

27ri J8B(z,R)

j_1

O(z) - P

d z.

We now show that

¢ (z) dz

1

27ri f 8B(z,,R) O(z) - P

d z = mj.

Indeed,

O(Z) = P+ (z - zi),gi(z) and so z,)":,-1[mjg1(z)

4 (z) = (z -

+ (z - zj)9g(z))

This implies that (z) dz = 8B(z;,R) O(z) - P

mi +

J

Z - z0

9j (z) dz = 27rimj;

fm.,,,R) 9j (z)

therefore, k

w(O,D,P)=Emj. j=1

mj. It remains to show that d(o,D,p) Let C be the connected component of C \ 0(8D) containing p. It is obvious that C is an open set, hence there is 60 > 0 such that p+6 E C for every 161 < 60 and we have d(f, D, p) = d(¢, D, p + 6) for every 161 < 6o. The equation

/(z) = p + 6, z E B(zj, R) is equivalent to

p + (V(z))m' = p + 6, z E B(zj, R), which, in turn, is equivalent to (w(z))m' = Iblesxte,

= e2w'B and 9 E (0, If. As (pj is injective on B(z,, R), the equation (wj(z))'", = I6le2xie has exactly m j distinct solutions in B(z,,R) for every where

161 < 61, for some 60 > 61 > 0. Let z,',..., z,'j be these solutions. We have

46

DEGREE THEORY IN FINITE-DIMENSIONAL SPACES

Ch. 2

0' (z") = mj(' ,(z,))m3-lpj(z") 0 0.

Using the definition of the degree for C' mappings by Lemma 2.19, we have k

mj

k

sgn(J,(z )) _ E mu

d(O, D, p + b) = E

j=1

j=1 l=1

and we conclude that d(0, D, p) = w(0, D, p) -

0 2.6 Exercises Exercise 2.1 Let D = (-1,1) x (-1,1) and let 0 E C(D)2 be defined by 0(x1, x2) _ (max{Ixi I, Ix2I}, 0), for (xl, x2) E D.

Find d(0, D, p) at p = (0, 0).

Solution 2.1. First we note that p = (0, 0) ¢ 0(8D) and so d(0, D, p) is well defined. Let IP E C1(D)2 be defined by ?(xl,x2) = (1,0), for(xl,x2) E D.

We have cI8D = 01 aD and p 0 t/'(D). By Theorem 2.1 we have d(tb, D, p) = 0 and by Theorem 2.4 we deduce that d(0, D, p) = 0.

Exercise 2.2 Let D = (-1,1) x (-1,1) and let 0 E C(D)N be defined by O(xl, x2) = (x2 -

X31

, x2), (xl, x2) E D.

Find d(0, D, p) at p = (0, 0).

Solution 2.2. We have 0(x1, x2) = (0,0) if and only if (x1, x2) = (0,0)Therefore, (0,0) ¢ 0(8D) and so d(0, D, p) is well defined. Also, J4(xl, x2) _

-3xi and so (0,0) E 0(Z0). Setting q = (-11,0), then q and (0,0) belong to the same connected component of R2 \ 0(8D). Moreover, 0(x1, x2) = q if and only if (X 1, x2) _ (2', 0). Since q 0 0(8D), it is easy to see that d(0, D, q) _ Ese'_'(o) sgn(J.O(x))

= -1; thus

d(0, D, p) = -1.

Exercise 2.3 Let a, b E R be such that a < b and set D = (a, b). Assume that 0 E C(D) and p ¢ {0(a), 0(b)}. Prove that d(0, D, p) E (-1,0, 11.

EXERCISES

§2.6

47

Solution 2.3. We first note that p ¢ 0(8D) and so d(0, D, p) is well defined. Set

0(b) ?P(x)

- a(a) (x - a) + 0(a).

If 0(b) = O(a), then 0 is a constant and d(i,1, D, p) = 0. Assume that 0(b) 96 0(a). We obtain Ob _Q ° P E [0( a ) , 0(b)] sgn d(O,D,P) = 0

p V 10(a), c(b)]

and Since 0I8D ='I8D, by Theorem 2.4 we deduce that d(d, D, p) = d(t, D, p) and so d(O,D,p) E {-1,0,1}.

Exercise 2.4 [Poincard-Bohl] Let D C RN be an open, bounded set, 0, ) E C(D)N and p [O(x),Vi(x)] := {aO(x) + (1 - a)10(x) : a E [0,1]} for every x E 8D. Prove that d(0, D, p) = d(ib, D, p). Solution 2.4. It is obvious that d(¢, D, p) and d(o, D, p) exist. Set

H(x, t) := tO(x) + (1 - t)0(x), x E D, t E [0,1].

H is a C homotopy between tai and 0 and p E H(8D, t) for every t E [0, 1]. Therefore, Theorem 2.3 implies that and so d(O, D, p) = d(+L, D, p).

t), D, p) does not depend on t E [0,1]

3

SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY 3.1 The Brouwer Fixed Point Theorem In this section we give two versions of the Brouwer Fixed Point Theorem.

Definition 3.1 A set E C RN is said to be a convex set if for every x, y E E and for every A E (0, 1), Ax + (1 - A)y E E.

Definition 3.2 Let E be a convex set such that 0 E int E. We define the gauge of E (or Minkowski function) , PE, by

xERN.

PE(x):=inf{t>0 : t EE

Remark 3.3 Since 0 E int E, there is n > 0 such that B(0, 77) C E. As i E E it follows that {t > 0 : E E} 0 0 and so pE(x) is well for every t > defined.

Lemma 3.4 Let E C RN be a convex set containing 0 in its interior. Let PE be the gauge of E. Then

(1) XEE

pE(x) 0 and for every x E RN; (3) there exists m > 0 such that pE(x) 5 mIx12 for all x E RN; (4) PE (X + y) 5 PE (x) + PE (Y) for every x, y E RN; (5) PE is continuous in RN; (6) if E is bounded, then pE(x) > 0 for every x 96 0 and os = E E.

Proof (1) Clearly, if x = i E E, then inf{t > 0 : e E E} 5 1, i.e. pE(x) < 1. (2) Given A > 0,

PE(AX) =inf{t>O : = inf r As

fx

EE}

EE, s > 0 }

:

JJJ

=Ainf{s>0 = APE(x) 48

:

x EE}

THE BROUWER FIXED POINT THEOREM

§3.1

49

(3) Since 0 E E, there exists 77 > 0 such that B(0, q) C E. For every t _> ""

we obtain that i E E and so PEW < ' (4) Let x, y E RN. There exist two sequences {an }, 10n) C (0, oo) such that

PE (X) = Jim an,

x -an EE

and

PE(Y) =n-.+oo lim Qn, LEE. Since E is a convex set, we deduce that x an an + On an thus Qn+

On q qy E E; an + Yn Fin

E E and so an + i3n > PE (X + y). Passing to the limit we have PE(x) + PE(Y) >- PE(X + Y)-

(5) Let x, h E RN. By (3) and (4) we have PE(x + h) - pE(x) 5 PE(h) < Alhl2

PE(x + h) < PE(x) + PE(h)

and

-PE(x + h) + PE(x) 5 \lhl2-

PE(x) 5 PE(x + h) + pE(-h) Therefore,

IpE(x + h) - pE(x)I 5 A h12

and so PE is continuous in RN.

(6) Assume that E is bounded and let R > 0 be such that E C B(0, R). If i E E, then 41

I

R > 0 if x 0 0.

Also, ° E E for every t > 0 implies that PE(0) = 0. Finally, considering a sequence {a,, } C (0,oo) such that pE(x) = lim an

and -& E E, taking the limit as n goes to infinity, we conclude that x PE(x)

E E.

13

50 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3

Proposition 3.5 Let E C RN be a compact, convex set such that 0 E int E. There exists a homeomorphism a : RN : RN such that

a(E) = B, where B is the unit ball of RN with respect to the norm I

Proof

- 12

R

a(x) :=

I x x#0 0

x=0,

vsyy y0 /3(y)

10

y=0.

O we have ao/3(y) = a(1E y) = PEW 2 p_ y = y and ao/3(0) = 0. Thus ao,Q(y) = y for every y E RN and, vsimilarly,y f3oa(x) = x for every x E RN. Therefore, a : RN RN is a bijection. From Lemma 3.4 (5) it follows that a is continuous at every x 96 0. Moreover, For every y

Ia(x) - a(0)I2 = IPIE12)xl2 = PE(x); therefore lim a(x) = 0 and a is continuous at 0. Hence, a is continuous on RN. X-0 Similarly, #: RN -+ RN is continuous.

Next, we prove that a(E) = B. By Lemma 3.4 (1), x E E implies that Ia(x)I2 = PE(x) < 1, therefore a(E) C B(0,1). Conversely, let y E B(0,1). Then y = a(/3(y)) and pE(/3(y)) = Iyi2 5 1 and by Lemma 3.4 (6) we have WOW) E E. Since E is a convex set containing 0, we obtain (1 - PE o ft)) 0 + PE 0)3(Y)

0(y) E E, PE 00(y)

i.e. /3(y) E E. Thus y E a(E) and so 6(0,1) C a(E).

Remark 3.6 Using the gauge function it can be shown that if Cl c RN is a smooth domain (e.g. strongly Lipschitz) and if CN(0) = CN (B(0,1)), then there exists a Lipschitz map v : fl --+ B such that (i) v(fl) = B; (ii) det Vv = 1 CN a.e. x E 11; (iii) v E C'(U;) for some finite partition {U,}i=1.....M of Cl into smooth domains (e.g. strongly Lipschitz).

For a proof of this result, we refer the reader to Fonseca and Parry (1987). The next two results are known as Brouwer's Fixed Point Theorem, the first one being the most commonly stated, while the latter has wider applications.

§3.1

THE BROUWER FIXED POINT THEOREM

51

Theorem 3.7 [First Version of the Brouwer Fixed Point Theorem] Let D C RN be an open, bounded set such that D is homeomorphic to the closed unit ball B. Let 0 E C(D)N be such that ¢(D) C D. Then 0 has a fixed point in D. Proof Let a : D - B be a homeomorphism. Setting 1G := a o o o a-1, we show that t/i : B - B has a fixed point. Clearly, 0 is continuous and either there exists

x E 8B such that O(x) = x, in which case' admits a fixed point, or O(x) 96 x for every x E 8B. Define

H(x, t) := x - tO(x) for x E B, t E [0,1]. Then 0 ¢ H(8B, t) for every t E [1, 0] and so, by Theorem 2.3 (2), d(H(., t), B, 0) does not depend on t. This yields

d(I -0, B, 0) = =

B, 0) 0), B, 0)

= d(I, B, 0) = 1.

By Theorem 2.7 the equation (I - t/)(x) = 0 admits a solution in B, i.e. lk has a fixed point x E k Setting y = a''(x), we have y E D and ¢(y) = Y.

0

Corollary 3.8 [Second Version of Brouwer Fixed Point Theorem) Let K C RN be a compact, convex set such that the interior of K is not empty. Let ¢ E C(K)N be such that O(K) C K. Then 0 admits a fixed point.

Proof Let xo E int K and define T : RN RN by T(x) = x - xo. Then T(K) is a compact, convex set such that 0 E int T(K). By Proposition 3.5 there exists a homeomorphism a : T(K) B and we define

tp:=To0oT-1. Then t[i : T(K) - T(K) is continuous and by Theorem 3.7 1b has a fixed point

y E T(K). Let x E K be such that y = T(x). It is clear that x is a fixed point of 0.

We give some applications of the Brouwer Fixed Point Theorem.

Proposition 3.9 Assume that N is odd, 0 E D, and 0 E C(D)N is such that 0 0 q5(8D). Then there exist y E 8D, \ 0 0, such that ¢(y) = ay. Proof If for every y E 8D, A E R, we have 0(y) 4 Ay, then set H(x, t) := tx + (1 - t)O(x), x E D, t E [0,1]

52 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3 and

K(x, t) := -tx + (1 - t)O(x), x E D, t E [0,1]. We have 0 ¢ H(OD, t) for every t E [0, 1] and 0 it K(8D, t) for every t E [0,1]. Therefore, by Theorem 2.3, d(H(0, t), D, 0) and t), D, 0) are independent of t and so, as H(., 0) = 0),

1 = d(I, D, 0) = d(-1, D, 0) _ (-1)N = -1, which yields a contradiction. We conclude that there exist y E 8D and A E R 0 such that 0(y) = Ay and, since 0 E O(8D), we must have A 96 0.

Theorem 3.10 [Perron-Frobenius] Let A = (ate) be an N x N matrix such that ai, > 0 for all i, j. Then there exist A > 0, x # 0, such that x; > 0 for every i and Ax = Ax.

0

Proof See Exercice 3.1.

Remark 3.11 In Proposition 3.9 the condition `N is odd' is essential. Indeed, let N = 2 and define 0 : B -+ R2 in polar co-ordinates by 4(r, 9) _ (r, 0 + r). The equation (cos(9 + 1), sin(B + 1)) = A(cos 0, sin 0)

has no solution.

The next application of Proposition 3.9 provides periodic solutions for Lipschitz flows. We will need some standard results on ordinary differential equations, which we will state without proving them.

Theorem 3.12 Let f : B(0, r) x R -+ RN be a locally Lipschitz function. Consider the initial value problem

fx= f(x,t) j x(to) = xo, when to E R, xo E B(0, r). Then there exists a function x : U . RN, (t; to, xo) r+ x(t, to, xo),

when U := ((t, to, xo) (to, xo) E R x B(0, r), t E 1(to, xo) ), such that (i) for each (to, xo) E R x B(0, r), x(t; to, xo) is continuously differentiable with respect to t, and it satisfies (3.12) in the open interval I(to, xo); (ii) if y is a solution of (3.12) defined in (a, p), then (a, p) C 1 (to, xo) and y(t) = x(t, to, xo) for every t E (a, p); (lli) if I(to, xo) = (r, a), then either a = r or lim Ix(t, to, xo) I = r; t-V (iv) for every (to,xo) E R x B(0,r), them exists a 6 > 0 such that Iso - toI + :

Iyo - xoI < 6 implies that 1(to, xo) C 1(so, yo).

§3.1

THE BROUWER FIXED POINT THEOREM

53

Definition 3.13 Assume that there exist T > 0, b : R - R, such that = f (0, t), 0(t + T) = \O(t),

for every t E R and for some constant A 0 0. Then 0 is called a Floquet solution

of (3.12). Moreover, if A = 1, 0 is said to be periodic of the first kind and if A 9k 1, 0, 0 is said to be periodic of the second kind.

The task ahead will be to prove that if f (x, ) is periodic for every x E RN and if Ixo12 is small enough, then there exists a Floquet solution for the initial value problem

{ ¢ = f(o,t) 0(0) =

xo.

Theorem 3.14 Let f : RN x R

RN be a locally Lipschitz function such that t) is positively homogeneous of degree one (i.e. f (ryx, t) = -yf(x, t) for every ry > 0 and for every (x, t) E RN x R). Assume that f (x, ) is periodic, of period T > 0, for every x E RN. Then there exists a 6 > 0 such that for every 0 < a < 6 there exists yQ E B(0, a) such that

f

1 x = f(x,t)

ll x(0) =

y,,,

admits a Floquet solution. Futhermore, if N is odd, then ya can be taken on 8B(0, a). Proof First we suppose that N is odd. As f is continuous and positively homogeneous of degree one, we must have f (O, t) = 0 for every t E R. Therefore, by Theorem 3.12 (ii), x(t) - 0 for all t E R is the unique solution of the equation

i = f (x, t) X(0) =

0.

Let us denote by x(t, 0, c) the solution of the equation f i =f(x,t)

l x(0) =

(E)

c.

By Theorem 3.12 (iv) there exists 6 > 0 such that Ic12 < 6 implies that the equation (E) admits a unique solution defined in R. Fix 0 < a < 6 and define F(c) := x(T, 0, c).

By Theorem 3.12 F is continuous and by the uniqueness of the solutions of (E),

F(c) = 0 if and only if c = 0. Therefore, 0 it F(8B(0, a)) and, by Proposition 3.9, there are C. E 8B(0, a), A > 0, such that T(co) = Ac.. For t E R set 0(t) := Ax(t, 0, cQ)

54 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3 and

+/i(t) := x(t +T,0,c0). It suffices to prove that 0(t) =1G(t) for every t E R. Indeed, fi(t)

0, c0 )

= Af(x(t,0,C0,t)

=f

o, C.), t)

= f(Ot),t) and 0(0) = Axx(O,0,c0) = .ca. Also,

*(t) = z(t + T, 0, c o = f(x(t + T, 0,c0),t)

= f (x(t + T, 0, c,), t + T) = f (,/,(t), t + T) = f (+G(t), t) and

0(0) = x(T, O, c0) = Tca, = Ac0.

By the uniqueness of the solution of (E), we conclude that 4(t) = V1 (t) for every

tER.

Finally, if N is even we define g : RN+1 _, RN+l by 9(x, xN+1, t) :_ (f (x, t), xN+1) Since N + 1 is odd, there exists (y, yN+1) E R1+1, I (y, YN+ 1) 12 = a, for which s 1N+1 x(O)

( XN+1(0))

= _ 9(x, xN+1, t) y = (VN+1)

admits a Floquet solution. We conclude that (E) admits a Floquet solution for some initial value y, (y12 < a.

3.2 Odd mappings The aim of this section is to show that the degree of a continuous, odd function RN is an odd number (see Theorem 3.23). As a consequence of this 0 : fl result, if 0: B(0,1) C RN -, RN is an odd function then the equation O(x) = 0 admits a solution in B(0,1). Indeed, since d(O, B(0,1), 0) is odd, we obtain d(m, B(0,1), 0) 34 0

and using Theorem 2.1 we conclude that the equation m(x) = 0 admits a solution.

ODD MAPPINGS

§3.2

55

Definition 3.15 The set D C RN is said to be symmetric if for every x E D we have -x E D. The mapping : D -' RM is said to be an odd mapping if O(x) = -0(-x) for every x E D. Before stating the main result of this section, the Odd Mapping Theorem (Theorem 3.23), following the scheme of Schwartz (1969) we first make some remarks on the Tietze Extension Theorem (see Theorem 1.15) and we prove some lemmas which will be used in the sequel.

Remark 3.16 We may deduce from Tietze's Extension Theorem (Theorem 1.15) that if K, L C RN are two compact sets such that K C L and if f : K RM is a continuous function, then there is a continuous extension g : L -+ RN

of f such that If IK = I9IL = sup{Ig;IL : i =1, ... , M}.

This is Exercise 3.2.

Proposition 3.17 Let M > N be two integers, let K C RN be a compact set, and let 0 E Cl(K)M. Then 4(K) has measure zero in RM.

Proof We identify K with K' C RM and

: K -, RM with

: K' - RM,

where

K':={(x,0,...,0)EKxRM-N} and +)(x,0,...,0)

O(x)-

Since J ,(y) = 0 for every y E K', by Sard's Lemma 1.4 we have G"'(O(K)) = GM(tP(K')) = 0.

Theorem 3.18 Let K C L C RN be two compact, nonempty sets, let M > N be two integer numbers, and let 0 E C(K)M be a function such that 0 is nowhere zero. Then m can be extended to a mapping X E C(L)M which is nowhere zero.

Proof Let c := min{I0(x)I : x E K} > 0 and choose 0 < e < z. Claim 1. We claim that there exists IP E C'(L)M such that t/, is nowhere zero and 10(x) - O(x)I < e for every x E K. Indeed, by Tietze's Extension Theorem (see Theorem 1.15), there exists 01 E C(L)M such that 011K = 0-

Let 02 E Cl(L)M be such that I02(x) - 01(x)I < 2 for every x E L. By Proposition 3.17,

56 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3

GM(02(L)) = 0

and so there exists p ¢ 02(L) such that IpI < 2. Let

02(x) - p

O(x)

Clearly, ip is nowhere zero, 0 E Cl(L)M, and, for every x E K, I+G(x) - 02(x)I + I02(x) - m(x)1 < 2 + 2 = e.

Claim 2. There exists 4D E C(L)M such that I'(x) - O(x)I < e for every x E K for every x E L. Indeed, letz

and I4D(x)I >

1

rl(t)

2t

t>

t
2, then

t(x) = t/,(x) and so I1(x)I = k&(x)I > Z. Therefore, I4'(x)I > z for every x E L. Also, for every x E K, we have

10(0

10(0 - I0(x) - +Gx)I > c - 2 =

c 2'

which implies that fi(x) _ O(x) and so I -OW - O(x)1
0. This implies that m : = m __

61

which yields a contradiction. Therefore, H(x, t) i4 0 for every x E 8D and for every t E [0, 11, and we conclude that ,

d(O, D, 0) = d(t,i, D, 0).

Due to the first step, in the sequel we assume, in addition, that 0 is an odd mapping. Second step. We prove that we can assume, without loss of generality, that 0(x)

x in a neighbourhood of 0. Let e > 0 be such that U := B(0, e) C D and set

R'' by

DI := D \ U. We define 01 : U U OD

x

1fi(x)

xEU x E 8D.

Then ¢1 is a continuous function,

8D1 =ODu8U =ODu8B(0,e) and 01 is an odd mapping. Let 42 := 01IaD, .

As ¢2 is nowhere zero, by Lemma 3.21 there exists a continuous extension of 02,

03 Dl - RN, such that 0 is odd and nowhere zero on D1 n RN-1. Define 03(x)

04(x) := Ix

x E D1 x E U.

We show that 04 is well defined. Indeed, if x E U n Dl = U n D, then x E 8U and 03(x) _ 42(x) = x. It is also easy to show that 04 is an odd, continuous mapping. For every x E 8D, we have x E 8D1 and 04(x) = 03(x) = 02(x) = 01(x) = 0(x)

Therefore, d(04, D, 0) = d(0, D, 0), 0 ¢ 04(OD) and m z 96 m

for every

x E OD.

Third step. We show that d(03i D1,0) is an even number. Let K := D1 n RN-1. Then 0 11 03(K) and it is easy to see that K is a compact set included in D1. By the excision property of the degree (see Theorem 2.7 (2)) we have d(03, Dl, 0) = d(03, D1 \ K, 0) = d(03, Dl , uDi, 0),

where Di := D1 n R+ and Di := D1 n R. By the decomposition property of the degree (see Theorem 2.7 (1)) we deduce that

62 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3 d(03, D1, 0) = d(03, Dl , 0) + d(03, Di , 0)

and, using the invariance of the degree under a C' change of variables (see Theorem 1.20), we have that

d(43, Di , 0) = d((-I) o

o (-I), -I (Di ), -1(0)) = d(03, Di , 0).

Therefore, d(, Dl, 0) = 2d(¢3i Di , 0). Fourth step. We conclude that d(04, D, 0) is an odd number. Since 018D = 04 18D,

04laintu = I, 04ID1 = 031D1, and D \ U = Dl, we have by the excision and decomposition properties of the degree (see Theorem 2.7), d(O, D, 0) = d(04i D, 0) = d(04, D \ 8B(0, e), 0) = d(q54, int U U (D \ U), 0) = d(04,int U,0) + d(¢4i D \ U,0)

= d(¢4, int U U (D \ U), 0) = 1 + d(04, D \ U, 0) = 1 + d(cb3, D1, 0)

= 1 + 2d(03, Di , 0).

Theorem 3.25 [The Borsuk Fixed Point Theorem] Let D C RN be an open, bounded, symmetric set containing 0 and let 0 : 8D - RM be a continuous function, M < N. Then there exists x E 8D such that O(x) = m(-x).

Proof We identify RM with {x = (xl, ... , xN) E RN : xN+1 = ... xN = 0}. Define >G(x) := m(x) - 0(-x) for x E 8D. We must show that there exists x E OD

such that O(x) = 0. Assume, on the contrary, that tb(x) 96 0 for every x E OD. Since 10 : OD RM is a continuous, odd, nowhere zero mapping and

OW # V,(-x) 10W I

10(-x)1

for every x E 8D,

we obtain by Theorem 3.23 that d(t,, D, 0) is an odd number. By Theorem 2.3 there exists co > 0 such that, for every 0 < e < co, d(t,b, D, pE) = d(tl,, D, 0),

where P, = (0,...,U, e).

Therefore, d(o, D, pE) 96 0 and, by Theorem 2.1, pE E O(D) C RM for every 0 < e < co, which contradicts p. ¢ RM.

ODD MAPPINGS

§3.2

63

The following corollary is also known as the Borsuk-Ulam Theorem.

Corollary 3.26 [Borsuk-Ulam Theorem) Let SN C RN+1 be the N-sphere and let ¢ : SN -. RN be a continuous mapping. Then there exists x E SN such that 0(-x) = O(x). Proof The result follows immediately from Theorem 3.25, where we set D:= B(0,1) C RN+1 0

Corollary 3.27 Let SN be covered by N closed subsets A,,-, AN. Then some At must contain a pair of antipodal points, i.e. there exists a set A. and there

existsxEA, such that -x E At. Proof For x E SN we define f,(x) := p(x, A1) and

f(x) :_ (fl(x)...., fN(x)), and we recall that p(x, y) = max{ Ixj - yj I : j = 1, ... , N}. Clearly, the function f : SN - RN is continuous and by the Borsuk-Ulam Theorem (Corollary 3.26) there exists l; E SN such that f (l;) = f (-l;). Since {As}i cover SN, there exists

some i E {1, ... , N} such that l; E At and so f,(1;) = 0 = f,(-l;). Since At is

0

closed, we conclude that l;, -£ E At.

Theorem 3.28 [The Ham-Sandwich Theorem] Let Xl,... , XN C RN be N bounded, measurable sets. Then there exists an (N-1)-hyperplane Y C RN that bisects each of the sets Xl,... , XN.

Proof Without loss of generality, we assume that GN(X,) > 0,

i =1,...,N.

(3.1)

Recall that an (N-1)-hyperplane llb,Q C RN has the form 11b.a

l = {(x1,...,XN) E RN : xlbl +... +ZNbN = a}

where b E RN, b 0, and a E R. We identify RN with {(yl, ... , yN+l) E RN+1 yN+1 = 0}. Let c = (0, ... , 0,1) E RN} 1 and define f : SN RN by

f :_(f1,...,fN), fi(x):_.CN({yEXi : (y-c)-x>0})We show that f, is continuous. Fix x ESN and let {x,.} C RN be a sequence such that x,. -s- x when n tends to infinity. By Lebesgue's Dominated Convergence Theorem and since X, is a bounded set, we have X(yEX, :

X{yEx.: (y-c)-x>O}

64 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3

almost everywhere and so fi(x,.) -- f,(x). By the Borsuk-Ulam Theorem there exists a E SN such that

f(a) = f(-a), i.e. for every i = 1, ... , N,

CN({yEXi :

(3.2)

We claim that there exists j E { 1, ... , N} such that a. 0 0. Assume, on the contrary, that a1 = ... = aN = 0. Since a ESN we have aN+1 = ±1 and (3.2) implies that

CN({yEX, :

yN+1-1>0})=,CN({yEX;

YN+1 - 1 < 0))) .

As Xi C {y E RN+I : YN+1 = 0}, we obtain CN(X1) = 0, contradicting (3.1). Now (3.2) is equivalent to

CN({(yl,...,yN)EX; :

a y = a1y1 +... + aNYN. Setting Y:= {(y1,...,YN) we conclude that Y bisects each of X,.... , XN.

3.3 The Jordan Separation Theorem In R2 the Jordan Separation Theorem asserts that if C is a Jordan curve, then R2 \ C is a union of two connected, disjoint, open sets D1 and D2 such that D1 is bounded and D2 is unbounded. In this section we present a generalization of this theorem in RN. We recall that K, L C RN are said to be homeomorphic if there exists a bijection h : K - L such that h and h-1 are both continuous.

Theorem 3.29 [Jordan Separation Theorem] Let K, L C RN be two compact sets such that K and L are homeomorphic. Then either K` and L` have the same finite number of connected components or both have countably infinitely many connected components.

Proof Let {Ai : i E I) be the family of connected components of K`. The sets Di are mutually disjoint and, since K is closed, each Ai is open. Therefore, there are at most countably infinitely many Di, since each of them contains a point with rational co-ordinates. We may write I C N and assume without loss

of generality that if i E I and i > 2, then i - 1 E I. As K is a bounded set, exactly one of the connected components is unbounded, say Oo. By the same argument, let {D,: r E R} be the set of connected components of L` such that

R C N, and R has the property that r E R and r > 2 implies that r- 1 E R.

§3.3

THE JORDAN SEPARATION THEOREM

65

Let Do be the unique, unbounded, connected component. Let h : K -+ L be a homeomorphism. By the Tietze Extension Theorem (Theorem 1.15) there exist RN, and a continuous extension of a continuous extension of h, 0 : RN h-1,10: RN RN. We show that, for every i, j > 1, J R1

bij = E d(4, Di, Dk)d(iP, D,, ,&j), k=1 III

bij = L d(O, Ak, Di)d(,G, Dj, Ok) k=1

and conclude the theorem by means of an elementary argument of linear algebra. Here 6,, denotes the Kronecker symbol, equal to zero if i 54 j and one if i 54 j. Fix j E N and let {Gjj:I E Al be the set of connected components of (O(80,))c,

where A C N and A has the property that l E A and l> 2 implies l- 1 E A. Since 8,&j C K and is bounded, then O(80j) is a compact set and so exactly one of the Gi is unbounded, say Go. We have

K` = UjEI,, L` = U.ERDr, (4(&))C = UIELG. Claim 1. For every r E N, there exists I E N such that Dr C Gl,.

Indeed, fixing r E N, we observe that 80j C K implies that O(80j) C O(K) = L and so Lc C ¢(80j)c. Therefore, Dr C L° C o(8e,j)c and, since Dr is connected, there exists I E N such that Dr C G, . The collection {Dr} may be relabelled {D1 k} in such a way that

Uo =Do UDo.1UDo,2U...CGo ,

U1 =Di.1UDi.2u...CGil. Claim 2. d(tli, G) j', p) = F,+- d(t&, Di.k, p) for p E Di and I > 1. First we prove that d(ip, G31, p) is well defined. It suffices to prove that 8G)I C

L. Let x E 8G{ and assume that x it L. We obtain that x E A. for a suitable r and so, since Dr C Gi(r), we conclude that x E Gl(r). Thus Gi f1G;(r) 0 0 and so I = 1(r). Therefore, we have x E 8Gi f1 GJ,, which yields a contradiction. Hence 8G! C L and

0(8Gi) C O(L) = K. Finally, P .E Di implies that p ¢ K, which, in turn, implies that p ¢ b,(8G'i) and so d(>1, G21, p) is well defined.

Since 8DI k = 8D,. for a suitable r and LC = urEKD,., we deduce that MY, A; C L. Hence d(O, D%k, p) is well defined.

66 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3

Consider the compact set M := Gi \Ui and fix x E M. Then either x E 8GI' or x E GJ, \U'; therefore x ¢ Dr for every r E N. Thus X E L and t/i(x) E t6(L) = K.

Moreover, if p E Di, then p ¢ t/i(M) and by the excision property and the decomposition property of the domain (see Theorem 2.7) we obtain that +oo

d(t,&,GI,p) =d(t,b,G)l \M,p) =d(i,b,Uf,p) = Ed(V,,V1.k,p)k=1 +oo

Claim 3. We claim that bij = 1: d(1G, Dk, 0,)d(cb, 03, Dk) for i, j > 1. k=1

Let us recall that d(tk, Dk, ii) = d(cb, Ak, p) for all p E Ai and these are well

defined since Ai C K° = (cb(L))`. Fixing p E 0, and using the multiplication theorem, we have +00

d(1,b,G{,p)d(tu,03,G1)

d(tGo0,0j,p) _ l=1

and the summation is finite. Since D; k C Gi , we deduce that d(o, A., Cl) _ d(q5, Aj, D1 k) for every k, which, together with claim 2, yields +oo

d(tfio0,Oj,p) _ Ed(tI,,Gi,p)d(0,03,Gi) t= t

+00 +00

_

d(O, Dj',k, p)d(,0, A), D11,01=1

k=1

Recall that d(t/', 03, D03k) = d(o, Oj, Go) = 0 because Go is unbounded and so +00 +00 d(t/i o O, Aj, p)

_ E E d(0, D1,k1 &(0, 03, Pf",k) 1=0 k=1 +oo

_ E d(1G, Dr, p)d(O, I j, Dr), r=1

since {Dr : r > 1} _ {Df.k : l > 0, k > 1}. The last equality can be written as +00

d(tb o 0, 03, Di) = E d(O, Dr, Di)d(0, A3, Dr)r=1

Using the fact that 803 c K and tG o O(x) = x for every x E K, we have d(tG o 0, Aj, Di) = d(I, Aj, A,) = bij;

THE JORDAN SEPARATION THEOREM

§3.3 thus

67

too

bij _

fsi

d(i, Dr, ti)d(ct, ij, Dr).

(3.3)

Using a similar argument we have that +oo

bi, = E d(?,b, D1, Or)d(t, Or, Dj).

r-i

(3.4)

Claim 4. {Dr : r > 1,r E R} and {0j

1, j E I} are in bijection. We divide the proof of claim 4 in three cases.

First case. R < +oo and I < +oo. Let A E RRx I be defined by ar, := d(q5, Ar, D,)

and let B E RIxR be defined by

br, := d(', Dr, 0,). We have ABERRxRand

(AB)ij _

aikbkj = k-1

d(O, Di, Dk)d(tb, Dk, Ai) = bij k=1

hence AB is the identity matrix of RRxR and so it has rank R. We observe that

R = rank (AB) < rank A < min{R, I} < I. Similarly, BA E RI x I is the identity matrix of R"',

I = rank (BA) < rank B < min{R,1} S R. and we conclude that R = I. Second case. R = +oo and I = +oo. In this case, it is obvious that {D, : r E R} and {Di : i E I} are in bijection. Third case. R = +oo and I < +oo. Let

X = R[x]

be the set of real polynomials with the basis 1, x, x2, x3, .... Set

Y := span {1,x,x2,...,x1} and define two linear applications f : Y - X and g : X -+ Y by

68 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3 00

f(x') := 1: ar,2r,j = 1,...,I r=1

and

9(x1) :_00>ar,xr, j = 1,...,00. r=1

Since are = 0, except for finitely many r E N, j E {1, ..., I} , f and g are well defined. By (3.3) and (3.4), we obtain that

fo9=Ix and

9of =Iy, where IX stands for the identity matrix of X and Iy stands for the identity matrix of X. Hence

+oo = rank (fog) < rank f < I, which yields a contradiction.

Fourth case. R < +oo and I = +oo. The proof of this case is identical to the proof of the third case. We give an application of the Jordan Separation Theorem.

Theorem 3.30 (Invariance of Domain) Let D C RN be an open set and let 0: D - RN be a continuous, injective function. Then 0(D) is an open set.

Proof We observe that for every p E D there exists r(p) > 0 such that B(p,r(p)) CC D. Thus 0(D) =pEo O(B(p, r(p)))

To show that 0(D) is an open set, it suffices to show that Q(B(p, r(p))) is an open set for every p E D. Fix p E D, and set B := B(p, r(p)). Since 8B is a compact

set and 0 is injective, we obtain that 0: 8B - ¢(8B) is a homeomorphism and, by the Jordan Separation Theorem, (O(8B))C has two connected components, 01 open and bounded and A2 open and unbounded. We claim that 0(B) = 01. Indeed, as B is a compact set and 0 is injective, we have that 101g : B -. 0(B) is a homeomorphism and so, by the Jordan Separation Theorem, we conclude that (0(B))` is an unbounded, connected set.

Considering, in addition, the fact that (0(B))° C (O(8B))c, we deduce that either (O(B))C C Al or (003))c C O2. Since 01 is bounded and (0(B))° is not, we must have (O(B))` C 02.

§3.3

THE JORDAN SEPARATION THEOREM

69

This implies that Al U ¢(aB) = A2 CO(B) U 0(8B) and so, as 46 is an injection, we obtain that

01 C O(B).

(3.5)

Similarly, O(B) is a connected, bounded set, O(B) C (O(8B))`, and we conclude

that ¢(B) C 01 or 4(B) C 02, which, together with (3.5), yields

O(B) = 01.

0

We conclude that 0(B) is an open set.

Corollary 3.31 Let M < N be two integer numbers. Then there is no injective mapping 0: RN _ RM. Proof Assume that there exists an injective mapping 0 : RN - RM and define

tji:RN -RNby

'+)(x) :_ (0(2), a)

X E RN ,

where a = (0,. .. , 0) E RN-M. Then 0 is injective and, by Theorem 3.30, O(RN) = RM x {a} is an open set of RN. This yields a contradiction.

O

Corollary 3.32 Let N < M be two integer numbers and let 0 : RN -, RM be an injective mapping. Then (O(RN))` is dense in RM.

The proof of this result uses Baire's Theorem. We recall its statement and refer the reader to Eisenberg (1974) for a proof.

Theorem 3.33 [Baire's Theorem] Let X be a complete, metric space and let {U;, i E N} be a collection of open, dense subsets of X. Then l$ENUi is dense in X.

Proof of Corollary 3.32. We first prove that (O(Ik ))` is dense in RM, where Ik := [-k, k].

Assume, on the contrary, that there exist a E RM and r > 0 such that B(a, r) C O(I'). Setting K := 0-1(B(a,r)), we observe that K is a compact set and so, since 0 is injective,

9:K-B(a,r) is a homeomorphism, where 9:= 01K-

Then

70 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3

g- 1:B(a,r)CRM_KCRN is a homeomorphism, contradicting the Borsuk-Ulam Theorem (see Corollary 3.26).

Next we show that (O(RN))` is dense in RM. IN, we obtain Since RN = UkENIk (O(RN))c -kEN (O(Ik ))c.

We observe that IN are compact sets, 0 is continuous, and so O(IN) are compact sets. Hence (0(IF))` are open sets, each (0(IN))` is dense in Of, and thus by Baire's Theorem nkeN(0(Ik ))` is dense in 0

P.

Remark 3.34 Peano's Theorem states that there exists a continuous surjection f : [0, 1] -+ [0, 11 x [0, 1]. We refer the reader to Eisenberg (1974, pp.367-370) or Kuratowsky (1966, pp. 150-151). Actually, it is possible to prove that, for every N, M > 1, there exists a continuous surjection 0 : RN - RM (see Exercise 3.3).

Theorem 3.35 Let D C RN be an open, bounded set and let ¢ E C(D)N be an injective mapping. Then for every p E 0(D) we have

d(0,D,p) = ±1. Proof By Theorem 3.30, 0(D) is an open set and O-' : 0(D) D is a homeomorphism. Let p E 0(D). Then there exists r > 0 such that B := B(p,r) CC 0(D) and we set

A:= D \ 0-' (8B). 0(D) and We want to apply the Multiplication Theorem to 0 o O-' : B : j E J} be the countable family of the connected 0-1 : B .- D. Let {Aj components of A. We have

pit 0o0-'(8B)=8B and also,

p g O(OD).

Therefore, by the Multiplication Theorem (Theorem 2.10), +00

1 =d(I,B,p)

=d(0o0-1,B,p)

=

E,d(0,Di,p)d(0-1, B,0:).

(3.6)

i=1

Also, by the Jordan Separation Theorem, 0-'(8B)° has two open connected components D1 and D2 such that Dl is bounded and D2 is unbounded.

EXERCISES

§3.4

71

We claim that there exists io E J such that D1 = 0,0. Indeed, 0 = D \ /-1(8B) C 4' 1(8B)` and, by an argument similar to that of the proof of Theorem 3.30, we have that

D1 = 0-'(B) C D \ 0-'(8B) = A= UjEjAj Therefore, D1 C 0;o for some it, E J and as

A,,cicr-1(8B)`cD1uD2, we have 0,o C D1 and we conclude that Aio = D1. Hence A j C D2 for every i 31 io and so

d(O-1, B, 0;) = d(O-1, B, D2) = 0 for every i 0 it,, which, together with (3.6), yields 1 = d(O, Ob,p)d(O-1, B, Ob)

and so d(4, Dio, p) = ±1.

(3.7)

Now it suffices to show that d(O, i{o, p) = d(O, D, p). Let

K:=D\A,o. Since d(O, A,o, p) # 0, by Theorem 2.1 we deduce that p E O(A,o) and, as 0is injective, we deduce that

pV Using the excision property of the degree (see Theorem 2.7), we have d(O, O 0, p) = d(O, D \ K, p) = d(O, D, p),

which, together with (3.7), yields d(o, D, p) = t1.

0 3.4

Exercises

Exercise 3.1 [Perron-Frobenius Theorem] Let A = (ai j) be an N x N matrix such that a;, > 0 for all i, j. Prove that there exist A > 0, z 96 0 such that x, > 0 for every i and Ax = Ax.

72 SOME APPLICATIONS OF THE DEGREE THEORY TO TOPOLOGY Ch. 3

Solution 3.1.Let D:_Ix ERN:xi>O,i=1,...,N, 1NIxi=1}.If Ax = 0 for some x E D, then it suffices to set A = 0. Assume that Ax i4 0 for every x E D. Then Nj(Ax)j > a for everyx E D and for some a > 0, and the function f : D R defined by f (x) :=

EN1(Ax)i

is continuous in D. It is clear that N

E fi(x) i=1 and A(x) > 0,

for every x E D. Hence,

f (D) C D. By the Brouwer Fixed Point Theorem (see Corollary 3.8), there exists xo E D

such that f(xo) = xo. Setting \:= EN 1(Axo)i we obtain that Axo = Axo.

A generalization of this theorem can be found in Varga (1962).

Exercise 3.2 Let K, L C RN be two compact sets such that K C L. Assume that f : K -+ RM is a continuous function. Prove that there exists a continuous function 9 : L - RM which coincides with f on K such that IfIK = 191L = suP{I9jIL : i = 1, ... , M}. Solution 3.2. Since K is a compact set and f : K - RM is continuous, we deduce that f is bounded. Let I fIIK = sup{I fi(x)I : x E K). We recall that IfIK = sup{Ifil : i = 1, ... , M}. By Tietze's Extension Theorem, for each i = 1,... , m we obtain the existence of a continuous function gi : RN -+ R such

that inf g, = inf fi. 9i I K = fi, sup 9i = sup fi, and xEL xEK xEL

xEK

We have 191M>_191K=IfIK.Conversely 191L=max{I9i3L: i=1,...,M}= I9io IL for some io E {1,. .. , M} and, as L is a compact set, we obtain that I9io IL = I9io (.t) I for some z E L. There exist a, b E K such that fio(a) = minXEKfio(x) :5 gio(x) !5 m

fio(x) = f.0(b)

and either I9b(2)I 0, there exists a continuous surjection fk : [2k - 2,2k - 1] - [-k, k] x [-k, k]. Define gk [2k - 1, 2k] - R2 by gk(2k - 1 + t) := (1 - t) fk(2k - 1) + t fk+1(2k), t E [0, 1].

We observe that gk is continuous and defining 0: R - R2 as

0(x) :=

f, (0)

x < 0

fk(x)

x E [2k - 2,2k - 11 x E [2k - 1,2k),

9k(x)

then ¢ : R - R2 is a continuous surjection. We easily deduce that, for every N, M > 1, there exists a continuous surjection 0 : RN -+ RM.

Exercise 3.4 Let f : SN

SN be a continuous function such that f (x) 34 -x for every x E SN. Show that, if N is even, then f has a fixed point on SN. Solution 3.4. By Tietze's Extension Theorem (Theorem 1.15) there exists a continuous function F : RN+1 - RN+1 such that

F(x) = f (x) 0 -x for every x ESN. By Proposition 3.9, and as N + 1 is odd and F(SN) C SN, we conclude that there exist A > 0 and xo E SN such that F(xo) = Axo

f (xo) = Axo.

Since IxoI = 1 = If (xo)I and f (xo) 36 -xo, we must have f(xo) = xo.

4

MEASURE THEORY AND SOBOLEV SPACES In this chapter we recall some properties of Sobolev functions and measure theory. In order to measure the sets on which a Sobolev function is continuous, we introduce the notion of Hausdorff measures and p-capacities. For a detailed description of these topics, we refer the reader to Evans and Gariepy (1992) and Ziemer (1989). The Hausdorff measures were first introduced by Caratheodory (1914). He developed only the Hausdorff linear measure in RN and indicated how the k-dimensional measure could be defined for k E N. Later, motivated by the study of the dimension of the Cantor ternary set, Hausdorff (1919) developed the theory of the k-dimensional measure.

4.1 Review of measure theory We recall some definitions and well-known results used in measure theory, such as the Riesz Representation Theorem, the Radon-Nikodym Differentiation Theorem and Vitali's Covering Theorem. For more details and for the proofs of the results presented in this section, we refer the reader to Evans and Gariepy (1992).

Definition 4.1 (i) We say that I C RN is a closed N-interval if there exist a, < b;, i, ... , N, such that I = [a1, bi) x ... x [aN, bN].

We set m(I) :_ (b1 - al) ... (bN - aN). (ii) We define the Lebesgue outer measure GN(E) of a set E C RN by 00

GN(E)

inf { E m(Ik) : E 1k=1

CkU

Ik, Ik closed N-interval }

.

JJJ

Definition 4.2 Let X C RN be a nonempty set, let P(X) be the collection of the subsets of X, and let µ : P(X) -+ [0, oo).

(i) u is said to be an outer measure if 00

µ(A,)

µ(o) = 0, p(,U A.) O for all r > 0

p(B(x,r)) = Ofor somer > 0

and

Dµv(x)

lim inf

v(B(x, r))

r-O+ p(B(x, r))

p(B(x, r)) > 0 for all r > 0

u(B(x,r)) = Ofor somer > 0.

+00

Definition 4.12 Let p and v be two Radon measures on RN and let x E RN. If 5m-v(x) = D,v(x) < +oo, then we say that v is differentiable with respect to p and we write Dµv(x) = Dµv(x) = Dµv(x).

_

D,v(x) is called the derivative of v with respect to p.

Definition 4.13 Let p be a measure on RN. A function f : RN [-oo + oo) is said to be p-measurable if f -1((-oo, a)) is p-measurable for every a E R. When p is the Lebesgue measure, a p-measurable function is said to be measurable.

Definition 4.14 Let p be a measure on RN, let 1 < p:5 +oo, and let f : RN [-oo + oo) be a p -measurable function. We say that f E LP(RN, p) if /RN if I9 du < +oo for p < +oo,

and if p = +oo there exists M E R such that

If(x)I <M for ua.e.xERN. The following is also known as the Radon-Nikodym Theorem.

Theorem 4.15 Let p and v be two Radon measures on RN. Then for p almost every x E RN, we have that DP(x) exists and Dµv(x) < oo. lrthennore, Dµv is p-measurable.

REVIEW OF MEASURE THEORY

§4.1

77

Definition 4.16 Let p and v be two measures on RN. (i) We say that v is absolutely continuous with respect to µ, and we write

if p(A) = 0 implies v(A) = 0 for every A C RN. (ii) We say that v and u are mutually singular, and we write

vlp, if there exists a Bonel set B C RN such that

v(B) = p(RN \ B).

Theorem 4.17 [Differentiation Theorem for Radon measures) Let p and N. Then v = va + v where v be two Radon measures on RN va « µ, v3 .1. {A. Moreover,

D,,v = D,,va, D,,v, = 0 p a.e. and

v(A) = j D,,vdp+v,(A) for every Bonel set A C RN.

Theorem 4.18 [Lebesgue-Besicovitch Differentiation Theorem] Let p be a Radon measure on RN, 1 < p < oo, and let f E L ;c(RN, µ). Then /1 l µ(B(x, iO e))

B(s,c)

If (y) - f (x)IP dµ(y) = 0

(4.1)

for p a.e. x E RN. Definition 4.19 A point x for which (4.1) holds is called a p-Lebesgue point of f with respect to p. We say that a 1-Lebesgue point off with respect to GN is a Lebesgue point of f.

Theorem 4.20 [R.iesz Representation Theorem] Let L : Cc(RN)M - R be a linear functional such that

sup{L(f) : f E CC(RN)M, IfI2 < 1,spt(f) C K} < 00 for each compact set K C RN. Then there exists a Radon measure p on RN and a p measurable function o : RN RM such that (i) Io(x)I2 = 1 for p a.e. x E RN, (ii) L(f) = f R. f o dp, for all f E CC(RN)M.

Ch. 4

MEASURE THEORY AND SOBOLEV SPACES

78

4.2 Hausdorff measures The Hausdorff measure H'(E) of a set E C R2 was introduced for the purpose of generalizing the notion of length in the case where E is a smooth curve. As a first approximation, we define 00

inf { diam(A;) : E C$EN A, } 1.-

.

JJ

It is easy to see that if N = 2 and if E:= {(t, sin 1) : t E (0,1] }, then A(E) < oo, while the length of E is infinite. This is due to the fact that in the definition of A(E) the sets A; are not forced to follow the geometry of E. In view of this, Hk(E) will be defined as a limit of outer measures H6 (E) which will follow the geometry of E.

Definition 4.21 Let E C RN, 0 < 6 < oo. We define

H

inf

0 a(s) (diarnCs)s 2

E c u C;diam C, < 6 iEN

JJJ

where a(s) := r +I and r(s) := f o'

e-=x'-1 dx (0

< s < oo) is the usual

Gamma function.

R.emark 4.22 If 61 < 62, then H62 (E) < H6, (E) and so lim6_o H6 (E) _ sup6>0 H68 (E) exists.

Definition 4.23 Let E C RN and 0 < s < oo. We define the s-dimensional Hausdorff measure on RN by

H'(E) = aim H6(E).

Theorem 4.24 For each 0 < s < oo, H' is a Borel measure. The proof of this result can be found in Evans and Gariepy (1992).

Theorem 4.25 (i) Ho is the counting measure. (ii) H1 = H6 =41 on R for every b > 0. (iii) HI (E) = 0 for all E C RN and for all s > N. (iv) Hs(AE) = A'H'(E) for all E C RN and for all A > 0. (v) HN(L(E)) = H'(E) for all E C RN and for all affine isometry L : RN R

.

HAUSDORFF MEASURES

§4.2

79

Proof (i) We observe that a(0) = 1. Therefore, H°({a}) = 1 for every a E RN and

H6({al,...,ak}) > k for every mutually distinct a1,. .. , ak E RN and for every positive b such that 0 < 6 < I min{Iai - a.I : i 0 j}. Hence,

H°({a1i...,ak)) = k, i.e. H° is the counting measure. (ii) Fix 6 > O and E C R. We have 00

£'(E) = inf

la, - biI : E C,EN [ai,bi]} =1 00

diam(C*) : E

= inf

CiEN

ci}

00

U Ci, diam Ci < 6 } l -1 diam (Ci) : E C iEN

< inf {

JJ

= Hb''(E) On the other hand, setting Ik := [6k, 6(k + 1)], k E Z, for every C C R we observe that diam (C n Ik) < 6 and too

E diam (C n Ik) < diem (C). k=-oo

Therefore, we have 00

G'(E) = inf

diam (Ci) : E CiEN `i=1

C,I

00

> inf

diem (Ci n Ik) If, i=1 k=-oo

EC iEN Ci I

> H6(E) and this concludes (ii).

(iii) Let Q = (0,1)N and let m > 1 be an integer. For k = (kl,... , kN) E K:=

{0,...,m-1)Nset Qk :=

Lk-m', klm 1J

x ... x [kN,kN+1]. ,n

MEASURE THEORY AND SOBOLEV SPACES

80

Ch. 4

We observe that Q

kEK

Qk and diem (Qk) _

rN-

and so la

H (Q) < >2 a(s) kE K

( \Z/

= a(s)mN-'

Letting m - +oo we deduce that

H'(Q) = 0, which yields

H'(RN) = 0. (iv) Using the property that diam (AC) = A diam (C),it is easy to verify that A'H'(C) = H'(AC) for every A > 0 and for every C C RN. (v) This follows from the fact that diam (L(C)) = diam (C) for every affine isometry L : RN - RN and for every C C RN. 0

Remark 4.26 It turns out that the equality HN(E) = GN(E)

holds for every E C RN. The proof is not trivial and it uses the isodiamettic inequality

G'(E):5 a(N)

(diam E )'

which is valid for every E C RN. We remark that E does not have to be contained in a ball of diameter diam E.

Lemma 4.27 Let E C RN, 0 < 6 < oo, and let 0 < s < oo be such that H6 (E) = 0. Then

H'(E) = 0. Proof Ifs = 0, H6 (E) = 0 implies that E = 0 and so H°(E) = 0. Now assume that s > 0 and fix e > 0. There exist sets {C,},EN such that E C U,ENC$, diam C; < 6 for every i E N and 00

diam C. 2

< J

HAUSDORFF MEASURES

§4.2

We observe that a(s) (d'

81

)' < e for every i = 1, ... , oo and so

_: l(e).

diam Ci < 2 Cads)/ Hence,

Hi'(,) (E) < e

and letting e go to zero we deduce that H'(E) = 0. Lemma 4.28 Let E C RN and let 0 < s < t < oo be two real numbers.

(i) If H'(E) < +oo, then Ht(E) = 0. (ii) If Ht(E) > 0, then H'(E) = +oo. Proof Assume that H'(E) < +oo and fix b > 0. There exist sets {C,}IEN such that E C UieNC1, diam C; < b and 00

/diam Ci 2

i=1 Since

H6(E)

00a(t)

(diam C1)t 00

a/ s(8) 2'-t 1


s while Ht(E) = +oo for every t < s. The dimension Hdim(E) is not necessarily an integer and may be any number in [0, NJ.

Proposition 4.31 Let f : RN - RM be a Lipschitz function, E C RN, and let 0:5 s < oo. Then

H'(f(E)) < K'H'(E), where K := sup { 4 _y

y, x, y E RN }

82

MEASURE THEORY AND SOBOLEV SPACES

Ch. 4

Proof Without loss of generality, assume that H'(E) < +oo and fix b, e > 0. There exist sets {C,};EN such that E C U;ENC;, diam C, < 6 for every i E N, and

00a(s) gal

(diem C;

\

2

J

k, diam B < 2 diam B B' and 9 is a countable family since any collection of disjoint open sets must be countable. Corollary 4.35 Let.F be a fine cover of A by closed balls, i.e. inf {diam B: B E .F, x E B} = 0 for every x E A, and suppose, further, that sup{diem B : B E F} < +oo. There exists a countable family Q of disjoint balls in.F such that for each finite subset {B1, ... , CF we have B') CBE9\{s U

Proof By Vitali's Covering Theorem there exists a countable family g of disjoint balls in F such that

A CBE B

CU

B.

Fixing B1, ... , B E .F, if x ¢ A\(.U B;), then, as B, are dosed and .F is a fine =1 cover of A, there exists B E F with x E B, B n Bi = 0 for i = 1, ... , n. From the claim in the proof of Vitali's Theorem, it follows that there exists B' E Q with

B'nB00;hence

BCB'.

Proof of Proposition 4.33. It is clear that to prove (4.2) it suffices to show that H'(At) = 0

Ch. 4

MEASURE THEORY AND SOBOLEV SPACES

84

for every t > 0, where

At:= xERN\E:limsup H'(B(zr r) n E) >t 1 r-o+

Fix e > 0. As H' LE is a Radon measure, we may find a compact set K C E such that H'(E\K) < E. Thus At C (RN\K) and, for fixed 6 > 0, we consider

.F:=

r) n E) {.(xr) : B(x, r) C (R N \K), O < r < 6, H'(B(z, a(s)r'

>tI

Without loss of generality, we may assume that F j4 0; otherwise At = 0 and so

H'(At) = 0. Using Vitali's Covering Theorem we write At

CiU

B,

where {B(xi,r,}°=1 C F is a disjoint family of closed balls. Then 00

Hio6(At) 5 Eo(s)(5r,)' i=1

00

J:H'(Bi n E)


tI

Since H' LE is a Radon measure, there exists an open set U containing Be such

that

HAUSDORFF MEASURES

§4.2

85

H'(U n E):5 Hs(Bt) + e and we define, for fixed 6 > 0,

.F:= 11 B(x, r) : B(x, r) C U, 0 < r < b,

H'(B(x, r) n E) > t a(s)rs

By Corollary 4.35, for each m E N there exists a countable disjoint family of balls {Bi } 1 in F such that m

Bt CsUI

B,

00

s=m+1 Bi ,

where Bi = B(x,, r,). Then m

00

H106(Bt) < Eo(s)ri + > a(s)(5ri)' i=m+1

i=1

t

M

00

tEH'(BinE)+ 5' i=1

H'(BinE) i=m+1

< H'(U n E) + L H' (i=m+1 B n E)

.

Letting m -+ +oo we deduce that H1'106 (Bt)
1. 0 Remark 4.36 Under the hypotheses of Proposition 4.33, it can be shown that lim sup

c-0

H"

e) n E) > or(s)es

1

- 2s

for H' a.e. x0 E E. It is worth noting that it is possible to have

86

Ch. 4

MEASURE THEORY AND SOBOLEV SPACES

H"(B(xo, e) fl E) 01

Then H8 (A,) = 0.

Proof Without loss of generality, we may assume that f E L' (RN), as it suffices

to prove the result for f 'k, O k E CO(RN),

0 < 1Ik < 1, V)k(x) = 1, if

x E B(0, k). By the Lebesgue-Besicovitch Differentiation Theorem (see Theorem 4.18),

lim

GN (B(x, r))

JB(x,r) I f (y) I dy =I f (x)

for £Na.e.xERNandso lim 1

r-0 r"

f

If(y)I dy = 0.

B(x,r)

Thus CN(A,) = 0. Fix e,6,a > 0, and as f E L1(RN) choose q > 0 such that JuIf(x)Idxe}. JJJ

Since A; C A, we have GN(A") = 0

and so we may find an open set U D A, such that GN(U) < ti. We set

OVERVIEW OF SOBOLEV SPACES

§4.3

87

F:= B(x,r): xEA;,0 1) }

.

1

Remark 4.54 (i) If K C RN is a compact set, then, using mollification of the characteristic function Xk, one has Capp(K) = inf 1JR N

I

Vflpdt : f E

XK1.

(ii) It is clear that if E C F C RN, then Capp(E) < Capp(F). The following Sobolev-Niremberg-Gagliardo generalization will be used to prove that Capp is an outer measure.

Lemma 4.55 Let 1 < p < N. Then there exists C = C(N, p) such that /

{LN

l dx

1

l 1/p

/PIf(x)I"

C(N,p) If N Ivf(x)'I dx

for every f E KP.

Proof We start by constructing a sequence Wn E CCO°(RN), 0 < Pn < 1, converges increasingly to 1 for a.e. x E RN, and ft) If(x)I'*

dx}

1-p/p'

If

IVVn(x)Ip(j)' dx} N

and p (y) = N. Thus, p/p' N

If(z )VWn(x)I'

dxnj

and, due to the integrability of I f Ip*, we conclude the result.

Theorem 4.56 Let 1:5 p < N. Then Capp is an outer measure on R''. Proof Let {Ek, k E N} be a sequence of subsets of R1' and

E:= U Ek. kEN

We want to show that

0

94

MEASURE THEORY AND SOBOLEV SPACES

Ch. 4

00

Capp(E) k=1

Capp(Ek)

and so, we may assume, without loss of generality, that Cap,(Ek) < +oo. kEN

Fix e > 0. For each k E N, we choose a function fk E KP such that Ek c int{fk > 1} and

JRN

IVfk(x)I2 dx < Cap,(Ek) +

2k

.

Let g := supkEN fk. We observe that E C int{g > 1} and, by Lemma 4.55,

RN

g(x)p dx = JRN su p fk (x) dx 00


0, N-1 < p < N and let E C B(0,1) C RN. Assume that for each r E I there exists a unique x,. E 8B(0, r) such that xr E E. Then Capp(E) > 0.

Proof Let f : RN -+ R be defined by f (x) := Ix12. We observe that f is a Lipschitz mapping such that If (X) - f (0I2 0.

0 Next we show how the notion of p-capacity plays an important role in the study of the continuity property for Sobolev functions.

Definition 4.60 Let f : RN -+ R be a measurable function. f is said to be p-quasicontinuous if for every e > 0 there exists an open set V C RN such that f IRN \y is continuous and Capp(V) < e.

Theorem 4.61 Suppose that f E W 1p (RN) and 1 < p < N. (i) Then exists a Bowl set E C RN such that Capp(E) = 0 and

r-o LN(B(x, r)) JB(:,r) f (y) dy = f*(x) exists for every x E RN \ E.

(ii) In addition, r

N(B(x,r))

for every x E RN \ E.

IB(z,r) 11(y) - f*(x)1p dy = 0

Ch. 4

MEASURE THEORY AND SOBOLEV SPACES

98

(iii) f' is p-quasicontinuous. The proof of this result uses the following lemma.

Lemma 4.62 Let l < p < N, f E Kp, let e > 0, and

E:=SxERN: 11

CN

((I

))

B(=.r)

f(y)dy>eforsome r>0}. JJJ

Then

Capp(E)
_

ji))

J(.) f(y)dy > E a.e. in B,')

and, setting

h:=

i = 1, ... , k, j E N},

we have h > 0. We claim that h E KP. Indeed, h E L1(RN) because, by (4.8), k

JRN iJsup[h(x)]'dx

o0

i=1 j=1 k

JRN [h(x))adx ao

CzE

IVf(x)ID dx

j=1 fB,(.

e a.e. in E and as E is an open set we deduce that

V(f)

Capp(E)!5J RN

5 p{

x)IP

(

dx

I

f

IVf(x)IPdx+

RN

f IOh(x)IPdx R

which, together with (4.9) and (4.12), yields CapP(E) N - p and so

Hdim(E) < N - p.

(4.11)

On the other hand, (4.11) does not necessarily imply that HN-P(E) < +oo, which, in turn, yields Capp(E) = 0. (4.12)

by Proposition 4.58. Therefore, (4.12) may be a stronger statement than (4.11).

(ii) Recently, Maly sharpened Theorem 4.61 by showing that the quasicontinuous representative f' of f E W1,N(el;RN) is approximately Holder continuous, except on a set of Hausdorff dimension zero, i.e. if S is the set of all points of S1 at which f' is approximately Holder continuous, then Hdim(St\S) = 0.

Actually, in view of (i), Maly's precise statement asserts that for every e > 0, p < N, there exists an open set G C RN such that Capp(G) < e and f In\a is locally Holder continuous. Here f' is said to be approximately Holder continuous at xo if for every oY E (0,11 there exists a set M such that the Lebesgue density of M at xo is 1, i.e. £ N(M n B(xo, f)) = 1 lim 0+0; CN(B(xo,f))

Ch. 4

MEASURE THEORY AND SOBOLEV SPACES

102 and

I f (y) - f (x) I < +00.

lim sup

I y - xo la

Y-xo,YEM

Proof of Theorem 4.61 Let f E WI,P(R°),1 < p < N. (i) We want to show that there exists E C RN, Capp(E) = 0 such that rl 0* GN (B(x, r)) J B(x.r)

f (=J) dfJ - f *(x)

for every x E RN \E. We define

A:=I xE RN:limsup 1 r-.e+ rn

JB(x,r)

IVf(y)Indy>0}.

By Proposition 4.37, HIV-P(A) = 0 and so, by Theorem 4.57 (v), Cap9(A) = 0. By Poincare's Inequality we have e

li oGN (

IB(x.r) I f (y) - GN(B(x r)) fB(.,r) f (z)dz

(x,r))

B

Nrf

< C lim

r-.o+ r

P */p

B(=,r)

I Vf(y) I9

dy 1} n'/P

1

C r-.O lim {l r

IP*

N-P BJ(x,r) I Vf (y) IP dy

I

= 0

(4.13)

for every x ¢ A. Due to the density of smooth functions in WI-P, for each i E N we choose fi E WI,P(RN) n COO(RN) such that

JN

I Vf (x) - Vfi(x) In dx < 2(P1

and we set

B; := {XERN.

1

CN(B(x,r))

8(x,r)

If(y)-fi(y)I dy>

for some r > 0 } .

By Lemma 4.62 and also because, due to Theorem 4.44, if f E WI,P(RN),

then 111 K, we have

P-CAPACITY

g4.4

Cap (Bi) < C

103

I V f(x) - Vfi(x) I P dx JRN

C

2(P+I)i'

Capp(Bi)S2

(4.14)

Now 1

CN(B(x, r))

J (y,r) f(y)dy-fi(x)I If (y) - GN(B(x, f (z) dz r')) JB(x,r)

< CN(B(x, r) JB(z,r)

dy

I f (z) - fi(z) I dz

1

+ CN(B(x,r) B(zr) 1

+,CN(B(x, r)) B(x,r)

I fi(z) - fi(x) I dz.

As fi is smooth, the last integral in the above inequality converges to zero as r 0+, and so, due to (4.13), we conclude that lim sup

Lz,r)f(y)dy-fi(x) I

C N(B(x,r'))

for every x ¢ (A U B,). Set

Ek:=Au (.k Bj I.

/

Then, by Theorem 4.56 and (4.14), we have +"0

Capp(Ek) < Capp(A) + ECapp(Bj) j=k +00

< CEOI j=k

and if x E RN\Ek, i, j > k, then, by (4.15),

2'

(4.15)

Ch. 4

MEASURE THEORY AND SOBOLEV SPACES

104

If.(x) - fi(x)I S lim sup r-.0+

+ lim sup

1

f (y) dy - f: (x)

1

f (y) dy - f, (x)

GN(B(x,r)) L(z,r)

GN(B(x,r)) L(x,r)

1 1 Zt+23

We conclude that f,

g in LO°(RN\Ek) and g is a continuous function.

Also, lim su r-O+

x - GN( B (x, r)) J (x,r) f (y) dy 9() 1

S 1g(X) - A(x)1 + limsup f:(x - £N(B(x, r))

JB(z,r) f (y) dy

S Ig(x) - f:(x)I + 2, , where we have used (4.15). Thus, 9(x) = rll.0+

CN(B(x, r))

J (x.r) f (y) dy

f' (x) for every x E RN\Ek. Setting E:= lkENEk, then, by Theorem 4.57,

Capp(E) < lim Capp(Ek) k+00

=0

and f' (x) exists for every x E RN\E. (ii) We claim that lim

1

r0+ GN(B(x,r)) 1B(x,r)

I f (y) - f' (x) I p dy = 0.

Indeed, for every x it E, by definition of f' we have

r11.0 J CN(B(x, r)) JB(x,r) I f (y)

- f (x) Ip dy f(z)dzlp.

< r* LN(B(x,r)) fB (x,r)

If (y)

1

- GN(B(x,r)) JB(x,r)

dy

P-CAPACITY

§4.4

1

1

+ CN(B(x,r)) lira

1

B(x,r)

CN(B(x,r)) fB(x,r)

r- O+ ,CN(B(x,r)) fB(x,r)

105

f(z)dz - f*(x)

f(z)dz - f'(x)

= 0.

(iii) It remains to prove that f' is p-quasicontinuous. Fix e > 0 and choose k large enough so that CapP(Ek) < 2. By Theorem 4.57 (i) there exists an open set U D Ek such that CapP(U) < e. Then

fi - f' in L°°(RN\U) and so f' I(RN\U) is continuous. O

There are other notions of capacity, such as Bessel capacity, linear capacity, etc., and the interested reader may find a detailed description in Hayman and Kennedy (1976), Havin and Maz'ya (1972), Stein (1970), and Ziemer (1989). With the help of these concepts one can prove the following.

Theorem 4.65 Let f E Ll(RN) be a function with compact support, spt f C

KCCR".Let

k Ix - yl0

when a E [0, 1), and let Kl C RN be a compact set such that w(x) = +oo on Kl. Then CapN_a_E(Kl) = 0 for all o > e > 0. The proof of this result may be found in Hayman and Kennedy (1976), where it is stated more generally for measures of bounded variation in place of £N.

5

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS In this chapter we present a generalization of Sard's Lemma for Sobolev functions and the following change of variables formulae. Let D C RN be open and

bounded and assume that either p > N or N - 1 < p< N and J4(x) > 0 CN a.e. x E D. Then =

ID

JRN

v(y)N(O,D,y)dy

for every v E LOO(RN) and for every 0 E WI,P(D)N, where the multiplicity function of ¢ at y E RN with respect to D is defined by

N(0,D,y):=0{xED: t5(x)=y}. A second change of variables formula is

ID

O(x)J(x) dx = j v(y)d(q, D, y) N

for every v E LOO(RN) and for every 0 E W l.P(D)N, where d(o, D, y) stands for the topological degree of ¢ with respect to D at the point y.

As an immediate consequence of the latter change of variables formula, if a sequence {0n} C WI,P(D)N converges to 0 uniformly, then the sequence {d(on, D, y)} C Z converges to d(o, D, y) and so, under some additional assumptions, UD D v o 0.(x)J#,(x) dx} converges to fD v o O(x)J,(x) dx. Given an open set D C RN, we will use the following notation: if 1 < p:5 +oo,

f E LP(D), then IIftIp(D) := fD IfIPdx, if p < 00 and

11f1100(D) := inf {M > 0 : I f (x)I < M CN a.e. x E D}

If 0 E LP(D)M, then N

I I0I IP(D) :=

EJ

and if -0 E WI-P(D)M, then 106

D

I0.IP

,

if p = oo.

§5.1

RESULTS OF WEAKLY DIFFERENTIAL MAPPINGS

107

N

I1011i,p(D) := 11011p (D) +

(I

is=1 8xj

I lp(D)p

5.1 Results of weakly differential mappings In this section we present some results relating the notion of topological degree to weak differentiability conditions. As usual, we denote by B the unit ball centred at 0 of RN with respect to the norm I.12

Lemma 5.1 Let F E C(B)N and let L be a linear mapping of RN into RN. Then

CN(F(P)) < CN(L(B)) + O(e),

where P:= {x ED : IF(x) - L(x)I < e}.

Proof Fix e > 0. If det (L) = 0, then L maps B into a hyperplane P of dimension N - 1. Let 1:= max{IL(x)I : x E B}.

Then F(P) lies inside a parallelepiped which has (N-1) sides of length less th 21 + e and the Nth side of length less than 2e. Hence,

CN(F(P)) < (21 + e)N-12e = CN (L(B)) + 0(e)

(5.1)

because CN (L(B)) = 0. If det (L) A 0, then L is a homeomorphism of RN onto RN and L-1(F(P)) C {x E RN : p(x,B) 5 e11L-1I1},

where we recall that p(x, B) = inf{Ix - ti norm of L-1 in C(8B). Thus, CN(L_1(F(P)))

:

t E B} and 11L-I II stands for the

0 and, given z E (0,1), we choose mo E N such

that Em < 6(1 - z),

for every m>mo. Claim 1. We claim that

RESULTS OF WEAKLY DIFFERENTIAL MAPPINGS

§5.1

117

IYi-Y2I2>6(1-z)hm for every yi E L(8B(xo, h,,,)) and for every y2 E L(B(xo, zhm)). Indeed,

yi = V0(xo)(zi - xo) + s(xo) for some z1 such that 1z1 - xoI = h,,, and

y2 = V (xo)(z2 - xo) + O(xo) for some z2 such/ that Iz2 - xoI < zh,,,. We have Iy1-Y2I2 = IVO(xo)(zi-z2)I > 6121 -z2I

6(Izi-xoI-Ixo-z2I) > 6(1-z)hm.

Let

H(x, t) :_ (1 - t)o(x) + tL(x), x E B(xo, hm), t E [0, 1]. Claim 2. H(8B(xo, hm), t) C RN \L(B(xo, zhm)) for every m > mo(z) and every t E [0, 1].

Assume, on the contrary, that for some t E [0, 11, z2 E B(xo, zhm) and for

some z1 E 8B(xo,hm), we have H(zi,t) = L(z2), i.e. (1 - t)4(zi) + tL(zi) _ L(z2). Then

(1 - t)('(zl) - L(zi)) = L(z2)

- L(zi)

and by Claim 1 we have b(1 - z)hm < IL(z2)

-

L(zl)I2 :5

I0(zl) - L(zl)I2

Using (5.24) and the fact that zi = xo + hma for some a E 8B(0,1), we deduce that 6(1 - z)hm < Em, which yields a contradiction. Hence,

H(8B(xo, hm), t) C RN \ L(B(xo, zhm))

for every t E 10,11 and every m > mo(z). Thus t), B(xo, hm), y) is well defined for every y E L(B(xo, zhm)). Using the fact that L is a bijection and H is a C homotopy between 0 and L, for every y E L(B(xo, zhm)) we have d(0, B(xo, hm), y) = d(L, B(xo, hm), y) = s9n JL(xo) # 0

and so we obtain (5.22). Furthermore, by Theorem 2.1 we obtain that y E O(B(xo, hm)) thus

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

118

Ch. 5

L(B(xo, zhm)) C qS(B(Xo, hm))

for every z E (0, 1) and every m > mo(z), which, together with (5.25), yields GN(cb(B(xo,hm))) m +oo GN(B(xo, hm))

CN(L(B(xo,zhm)))

liminf

= zNIJ#(xo)I

C'(B(xo, hm))

for every z E (0, 1). Letting z go to 1 we conclude (5.23).

Theorem 5.11 Let D C RN be an open, bounded set and let 0 E C(D)N be a mapping which has a weak differential almost everywhere and has the N-property.

Assume that Jj E Lf (D). Then for every measurable set E C D we have JE

I Jm(x)I dx = j N(4), E, y) dy. N

Proof Recall that, by Theorem 5.5, N(¢, E, ) is measurable. Also, by Lemma 5.8 it suffices to show that ( I Jo(x)I dx I J#(x)I

Since

,CN(4)(B(x,

hm))) < in N(4), B(x, hm), y) dy = a

we have lim M-+00 oo

h.))

G(B(x, hm)) , IJm()I

hm)),

WEAKLY MONOTONE FUNCTIONS

§5.2

119

By the Differentiation Theorem for Radon measures, we deduce that 4)(x) > I JJ(x) I

almost everywhere in D and so JRN N(O, E, y) dy = L 4'(x)dx ? L IJm(x)I dx.

5.2 Weakly monotone functions In this section we study regularity properties for functions 0 E

sat-

isfying J,(x) > 0 Vv a.e. x E D, where D C RN is an open set. Following Gold'sthein and Vodopyanov (1977), we show that if p = N, then 0 is continuous and monotonic (see Theorems 5.14 and 5.17 ). The definition of monotone functions was introduced by Lebesgue (1907) and, heuristically, a function is said to be monotone if it satisfies certain weak maximum and minimum principles. The following class of functions was introduced by Ball (1978) in his fundamental work on nonlinear elasticity and it will be used at length here, ;,g(D) := f4) E WI.P(D)N : adj(V4)) E

L9(D)N,N,

Jm > 0 a.e. X E D}

where adj(V4)) is the matrix of the cofactors of V4), p > 1, and q >

.

9verak

(1988) proved that if p > N - 1, then the mappings of ..4 (D) are continuous except perhaps on a set of p-capacity zero and this set is empty if p = N (see Theorem 5.17 and Remark 5.18). Here, to obtain $verak's (1988) results we follow Manfredi's (1994) approach.

Definition 5.12 Let D C RN be an open set, 1:5 p < +oo and let f E Wl (D). We say that f is weakly monotone if for every Cl cc D open, bounded, connected set and for every pair of constants m < M such that

(m - f)+ E we have that

W0,P(Q)N

and (f - M)+ E W01.P(1)N,

m0 t+:-ft - 1o t0

-t t Pp l and let 0 E WI.P(D)N. Assume that Jm > 041v a.e. x E D and adj(V) E LQ(D)NxN.

Then for each i = 1, ... , N the ith component 0, is weakly monotone. In particular, if o E W1,N(D)N and if J,0 > 0 GN a.e. x E D, then 0 is continuous.

Proof Fix i = 1, ... , N, let f? CC D be an open, bounded, connected set and let m < M be a pair of constants such that (m - 0i)+ E Wo,P(c2)N and (0i - M)+ E Wo,P(n)N

Set g

(m - 0i)+ and let {or: r E N} C C°°(D) be a sequence such that

or -.0 in

WI.P(1)N.

By Exercise 1.3 and Theorem 4.42, we have N

E

k

(5.26)

(adj(V0r))ik = 0

and by Theorem 4.42 we obtain Jm dx

Jflr1(.<m}

N V'

dx = Jn(#;

E a"' (adJ(VOr))ik dx

J r kv

N r r limo

dx

Jk=1

= r llm o

r

N

/nk=1 F9

(adj(omr))ik dx

= 0.

Since JJ(x) > 0 LN a.e. X E 91, we deduce that GN({Oi < m}) = 0 and so

4,(x) > m a.e. X E ft

WEAKLY MONOTONE FUNCTIONS

§5.2

121

Using a similar argument we obtain that 0,(x) < M a.e. x E Q, thus ¢; is weakly monotone.

Recall that if X C RN is a Coo paracompact manifold of dimension r E N, then X can be covered by finitely many manifolds X; C X such that for each i there exists a C°D diffeomorphism u, : (0, 1)r - X. We say that f E W1'P(X) if and only if f ou; E W1-P((0, 1)1). Recall also that, given any ball B(xo, r) C RN, 8B(xo, r) is a C°O paracompact manifold of dimension N - 1.

Theorem 5.15 Let N-1 < p:5 N and let f E W'.P(8B(xo,r)) be a continuous function on the sphere 8B(xo, r). Then, (diam(f (8B(xo, r))))P < C(N,

p)r°+1-N

J8B(xo.r))

IVf(x)Ii dH N-',

where C(N, p) is a constant depending only on N and p.

Proof Since p > diam(8B(xo, r)) = N -1, by the Sobolev Imbedding Theorem expressed on 8B(xo, r), we have (diam(f (B(xo, r))))P < C(r, p, N) J

B(zo,r)

I Vf (x)IdHN1.

Using a rescaling argument, we conclude that

C(r,p, N) =

rP+1-NC(N,p)

In the sequel, given f E WI-P(D) for some open set D C RN, we denote by f' the p-quasicontinuous representative of f of Theorem 4.61 and by LP(8B(xo, r)) the trace operator for every r such that T,. : W 1-P(B(xo, R)) B(xo,r) CC D (see Theorem 4.50).

Theorem 5.16 Let D C RN be an open, bounded set, 1 < p < +00, f E W'.P(D). Then (i) f* 18B(..,,-) E W' P(8B(xo, r)) for H' almost every r such that B(xo, r) cc D;

(ii) f'(x) = Tr(x) for HN-1 almost every x E 8B(xo,r), for every r such that B(xo,r) CC D;

(iii) if p > N - 1 and if B(xo, h) cc D, then f admits a representative f E W l.P(B(xo, h)) such that f $8B(=o,r) is continuous on 8B(xo, r) for H1 almost every r E (0, h).

Proof We assume, without loss of generality, that f' - f.

122

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

Ch. 5

(i) Let xo E D and let h > 0 be such that B(xo, h) CC D. Since f E W1'P(D) there exists a sequence { fn : n E N) C C' (D) such that

L Ifn -fIdx 0 be such that that B(xo, h) CC D. By assertions (i) and (ii) and

the Sobolev Imbedding Theorem on 88(xo, r), there exists J C (0, h) such that £1(J) = 0 and Tr(f) has a continuous representative gr on 8B(xo, r) for every r E (0, h) \ J. Define f by f (x)

{

g;(x) f *(x)

if lx - xol = r E (0, h) \ J if lx -xoI = r E J.

We observe that

f(x) = f'(x) for GN almost every x E B(xo, h) and f has the property required in assertion (iii).

0

Theorem 5.17 Let D C RN be an open set, N - 1 < p < N, and let f E W"""(D) be a weakly monotone function. Then f admits a representative (still denoted by f) such that f I ors : D \ S -. R is continuous for some set S C D such that Capp_.(S) = 0

for every 0 < a < p. If p = N, then S = 0 and f is continuous in D. In particular, if f E W1.N(D)N and if Jj > 0 GN a.e. in D, then f is continuous in D. The argument we use in the proof below was introduced by Gold'sthein and Reshetnyak (1990) in the case where p > N, and later extended by Manfredi (1994) to the case where p > N - 1. In fact, in the theorem above we can choose the set S such that its Hausdorff dimension is equal to N - p.

Proof Let xo E D, let 0 < h < 2 be such that B(xo, h) cc D, and assume, without loss of generality, that

f =f, where f is the representative of f given by Theorem 5.16 (iii). Let J C (0, h) be such that V (J) = 0, f I8B(=,,r) is continuous and f I8B(:o,r) E W 1"(8B(xo, r))

for every r E (0, h) \ J. Set

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

124

Ch. 5

mr(xo) := inf{f(x) : x c- 8B(xo,r)} and

Me(xo) := sup( f (x) : x E 9B(xo, r)},

for every r E (0, h). Claim 1. mr(xo) < f(x) < Me(xo) for GN almost every x E B(xo,r) and for

every rE (0,h)\J Indeed, for every r E (0, h) \ J

Tr(mr(xo) - f)+ = 0 and

Tr(f - Mr(xo))+ = 0 and since f is weakly monotone on D, we deduce that mr(xo) < f(x) 5 Mr (x0)

for GN almost every x E B(xo, r) and for every r E (0, h) \ J. Define eSSOSCB(xo r) = esS SUp{ f (x) : x E B(xo, r)} - ess inf { f (x) : x E B(xo, r)}

c(r, xo ) and

d(r)

h

Jr

tP-Ndt

JBzo,r)

IVf I°dHN-1. a

Let C(N,p) be the constant in Theorem 5.15. Claim 2. (c(r, xo))P log < C(N, p)d(r) for every r E (0, h). Indeed, by TheoremT 5.15 and the fact that f = f, (Me(xo) - mt(xo))P
0 be such that

rC(N, P) ( IL loges

I Df (y) I i dy

1B(zo.26) Iy -xI2

"


0 such that < f (x)

£''(B(x, r)) B(z,r)

f (y) dy
N . Also, by Theorem

4.57, Hdtm(S) < N - p implies that Capp_a(S) = 0 for all 0 < a < p and we recover Theorem 4.57.

The following result is due to Reshetnyak (1989).

Corollary 5.19 Let D C RN be an open set, let K be a compact set, and let V be an open, bounded set such that K C V CC D. Let 0 E Wia (D)N be such that J0 (x) > 0 VV a.e. X E D. Then there exists a constant 6 >0050 such that I0(xl) - O(x2)I2 < C(N)M*6(Ixl - x212) for every X1, X2 E K such that Ixl - X212 < 6, where

M:=

fV

I Vv(x)I N dx, 6(t) :_

(

\lo 2

l/*

lim 0(t) = 0

e_Q+

and C(N) is the constant of Theorem 5.15.

Proof Fix i E {1, ... , N}and set f := 0;. Let d:= dist(K, RN \ V) and

6:=max{t: 0 0. For t > 0 set

Et:={XED:u(x)OandEg={xED\(AUM) :u(x) N. This was extended later by

Marcus and Mizel (1973) to functions 0 E W-P(D)', p > N, and N:5 m, and it was shown that, for every v : R' - R measurable and for every measurable set E C D, JE

v o O(x)JN(4(x)) dCN(x) = f(E) v(y)N(O, E, y) d.Cn(y) m

whenever one of the two sides is meaningful. Here we use the notation (JN(.O(x))12

=

N P 1: J:(F'i(x))2

where F is the matrix of the N x N minors of V (x) and P :=

m

As it turns out, Theorem 5.23 will generalize this result for m = N, since, by Theorems 5.21 and 5.28, if 0 E Wl,P(D)N and p > N, then is weakly differentiable and has the N-property.

Proof of Theorem 5.23 Without loss of generality, we may assume that v(y) > 0

for almost every y E RN. We divide the proof into three cases. Case 1. Assume that v = XF for some measurable set F C RN. By Proposition 5.22, 0-'(F) is a measurable set and so

0-'(F)nE=0-1(F\A)nE is measurable. By Theorem 5.11,

f-I(F)nE

I J.(x) I dx =

f

f voq5(x)IJ,0(x)I dx = E

N(O, 0-'(F) n E, y) dy,

RN

f

RN

N(O,E,y)v(y)dy.

(5.41)

It follows that (5.40) holds if v is a linear combination of a finite number of characteristic functions of measurable, bounded sets E C D. Case 2. Assume that E C D. Let {va : n E N} be a nondecreasing, nonnegative sequence of simple, measurable functions and let C C RN be a set of measure zero such that lim vn(y) n+oo

= v(y)

134

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

Ch. 5

for every y E RN \ C. Since E C D and Jm E L' (D), by Theorem 5.11 we deduce that

N(4, E, y) < +oo

for almost every y E RNand

vnN(4, E, ) ""' vN(4, E, ) in L1(RN). By Proposition 5.22,

,CN(4-'(C) n (E \ A)) = 0,

where A:= {x E D : J#(x) = 01, and as limn+,,. v 04(x) = v o 4(x) for every x E E \ 4' 1(C), we obtain lim V. O 4(x)IJm(x)I = v o 4(x)I Jm(x)I

n +00

for almost every x E E\A. On the other hand, by Case 1 for all n E N,

E\A

vn o 4(x)I JJ(x) I dx = j N(4, E \ A, y)vn(y) dy N

and since {vn o 4(x)I JJ(x)I } and {N(4, E, y)vn(y)} are nonnegative and nondecreasing, by the Beppo-Levi Theorem (or the Lebesgue Monotone Convergence Theorem), we deduce that

E\A

v o 4(x)I JJ(x)I dx =

JRN

N(4, E \ A, y), dy,

which, together with the fact that JJ = 0 a.e. in A, and using Lemma 5.8, implies

Lvo,1dx = f\A

v

E

=

r

N (4, E\A, y) dy N

f N(4,E,y) dy. RN

ase 3. General case, E C D and V E LOO(RN). Case

There exists a sequence {Kk : k E N} of compact sets such that

Kk C Kk+1 C E and GN(E \ Kk) < k,

CHANGE OF VARIABLES VIA THE DEGREE

§5.4

135

for every k E N. Since

JRN N(, Kk, y) dy

IK,,

for every k E N, passing to the limit in k and using the Beppo-Levi Theorem, we deduce that

Lvo1JI

= f N(O, E, y) dy Ht

0

5.4 Change of variables via the degree Proposition 5.25 Let D C RN be an open, bounded set and let 0 E C(D)N be a mapping which has a weak differential almost everywhere and has the Nproperty. Assume that Jm E Lloc(D). Then, for every bounded, open set G CC D satisfying GN(8G) = 0, we have (i) E L1(RN); (ii) fG J.(x) dx = ,fRN d(-O, G, y) dy.

Proof Fix G CC D. Given v E L°°(RN), set

1: No (v, G, y) =

v(x)

if N(0, G, y) < +oo

xEm-' (y)nG

if N(O, G, y) _ +oo.

+00

Claim 1. No (v, D, y) = No (w, D, y) EN a.e. y E RN and for every w E L°°(D) such that v(x) = w(x) CA( a.e. x E RN. Indeed, let

A :_ {y E RN : N(0, G, y) = +oo}. By Theorem 5.11 we obtain GN(A) = 0.

Let B C D be such that

CN(B) = 0 and v(x) = w(x) for every x E D \ B. Since 0 has the N-property, GN(4(B)) = 0, and for every

yERN\(4'(B)UA)wehave

E xE0-'(y)nG

v(x) _

w(x), xE0-'(y)nG

thus No (v, G, y) = No (w, G, y)

for almost every y E RN. Claim 2. E LI(RN).

136

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

Ch. 5

Let {vn : n E N} be a nondecreasing sequence of simple, measurable functions which converges pointwise to v, and let

k

vn :=

[anXE.. i=1

As

k No(vn, G, y) _ E anN(O, En, y) i=1

by Theorem 5.5, Nm(vn, G, ) is measurable. Furthermore, as {N'j(vn, n E N} is a nondecreasing sequence which converges pointwise to No (v, G, ), we conclude

that N,(v,G, ) is measurable. This, together with the fact that

E

LI(RN) (see Theorem 5.11) and the fact that N4 (v, G,

N(O, G, -) sup 1V(x)1' xED

yields

No(v, G, ) E L1(RN),

proving Claim 2. Now, by Theorem 5.11, we obtain

jv(x)IJ(x)ldx =

JRN

No(v,G, y) dy

and so by the Beppo-Levi Theorem we deduce that

Jv(x)I.4(x)Idx =

jN

No (v, G, y) dy.

(5.42)

In the sequel, we set v(x) = sgn (J,(x)) and (5.42) reduces to

14 J,(x) dx = JgN N4,(v, G, y) dy. Claim 3. We claim that N4,(v, G, y) = d(O, G, y) for almost every y E RN. Indeed, let

El :_ {x E D :.0 does not have a weak differential at x},

E2 :_ {x E D \ E, : Js(x) = 0},

(5.43)

§5.4

CHANGE OF VARIABLES VIA THE DEGREE

137

and

C:= A u 4,(E1) u 4,(E2) U 4,(8G).

Since GN(E1) = GN(8G) = 0 and 0 has the N-property, we deduce that 'CN(.O(El)) = £N(4,(8G)) = 0.

Since G C D is a compact set, by Theorem 5.6 we obtain ,CN(O(E2))

=0

and so

,CN(C)=0.

Let y E RN \ C. If 4-1(y) n G = 0, then 4-1(y) n G = 0 and by the excision property of the degree (see Theorem 2.7), we have d(4,, G, y) = 0 = N,(v, G, y).

If 4-1(y) n G = {a1, ... , ak }, by Lemma 5.10, we deduce that there exists r > 0 such that B(al, r), ... , B(ak, r) CC G are mutually disjoint and d(4,, B(ai, r), y) = sgn(J.(ai)) (i = 1, ... , k).

(5.44)

By the decomposition and the excision property of the degree (see Theorem 2.7) we obtain k

d(4,,B(ai,r),y) = No(v,G,y)

d(4,,G,y) _ i=1 and so

d(4,, G, y) = No (v, G, y)

for almost every y E RN. This, together with Claim 2, yields E L1(RN)

and using (5.43) and (5.45) we conclude that

fJ(x)dx = JRN d(4,, G, y) dy Remark 5.26 Let D, 0, and G be as in Proposition 5.25.

(5.45)

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

138

Ch. 5

(i) (5.44) yields d(O,G,y) _

sgn(JJ(x)) =Em-'

for almost every y E RN and so Id(0, G, y) 15 N(O, G, y)

for almost every y E RN.

(ii) We observe that if we assume in the proof that CN(o(8G)) = 0, without assuming that CI(8G) = 0, the conclusion of Proposition 5.25 still holds.

Next, we state and prove a change of variables formula using the degree function.

Theorem 5.27 Let D C RN be an open set, let 0 E C(D)N be such that 0 has the N-property, m has a weak differential almost everywhere, and JO E L' (D). Assume that G C D is a bounded, open set and that v E LOO (RN ). Then (i) y v(y)d(O, G, y) is integrable, (ii) v o 0 Jm is locally integrable, (iii) fc v o c(x)J,(x) dx = f R, v(y)d(c5, G, y) dy.

Proof By Remark 5.26, Id(¢, G, y)I < N(¢, G, y) CN a.e. y E R" and, together with Theorem 5.23, we conclude (ii). By Proposition 5.25, (i) and (iii) hold in the case where v is a characteristic function. Let v E LOO(RN) and we may assume without loss of generality that v > 041v a.e. Let {vn : n E N} be a nondecreasing sequence of simple functions, bounded in LOO(RN), and converging pointwise to

v. Let A C RN be such that CN(A) = 0 and lira vn(y) = v(y) n+oo

for every y E RN \ A. Setting

B:={xED: Jo(x)=0}, we observe that

m-'(A) = (4-'(A) n B) U (0-'(A) \ B) and by Theorem 5.11 we obtain

CN(4-'(A) \ B) = 0. Clearly,

lm vn o 4(x)Jm(x) = v o ¢(x)J©(x) n-oo

CHANGE OF VARIABLES VIA THE DEGREE

§5.4

139

for every x E (c-'(A) n B) U (D \ 4-'(A)), i.e. {v o OJo} converges to v o OJo almost everywhere. Since {vn} is a sequence bounded in LOO(RN) and Jo E we have vn o O(x)JO -y v o O(x)Jo in L' (G)

(5.46)

and, by Proposition 5.25, v. d(O, G,

v d(O, G, ) in L' (RN).

(5.47)

Suppose, first, that v = Xv, where V C D is an open set such that LN(8V) = 0. Set

F:=Gn0-'(V)nG.

Then F is an open set, F CC D, and GN(5(aF)) N be two positive integers, p > N, let D C RN be an open bounded set, and let 0 E W1,n(D)M. Assume that 0 coincides with its continuous representative. Then, for every A C RN measurable set, we have A

HN(O(A))

N

C(N, p),C (A)" VA I V0(x)IDdCN(x)l P

where a = 1 - v and C(N, p) is a constant depending only on N and p.

Proof We may assume, without loss of generality, that A is an open set and let {Q3 : j E N} be a partition of A into half-open cubes. Using the Sobolev Imbedding Theorem for W1"p (see Theorem 4.45), we obtain for each of these cubes the estimate P

ma I0i(x) - iI

C'(N,P)r1-1 [IQ ,V 1(x)Ip dGN(x)J

,

where ; is the mean value of 0; over Q, r is the edge length of Q, and C'(N,p) is a constant depending only on N and p. Therefore, ¢(Qj) is contained in the mM) and having edges of length cube centred at I.1 = 2C'(N,p)r

Vmj(x)II dGN(x)1 LJ Q

I

J

and so O(Qj) is contained in the ball B. of diameter P 1, = 2IRC'(N,p)r1 p [I IV0 (x)l 'dGN(x)I

.

Q

Since I, tends to 0 when r. tends to zero, we obtain

f

00

HN(O(A)) 5 C(N,P),r, =1

If

IVO(x)I°dGN(x)I 3

J

§5.5

1=

CHANGE OF VARIABLES FOR SOBOLEV FUNCTIONS 1_.1 v

00

< C(N, p)

Lv

oo

[iJ

[r]N

j=1

j=1

141

C(N,p)LN(A)1

VA

v

IVO(x)IP dGN(x)

IDm(x)I°dGN(x)P

.

Remark 5.29 If 0 E W1,P(D)N, p > N, then by the Sobolev Imbedding Theorem (Theorem 4.45) and by Theorems 5.21 and 5.27 we conclude that ¢ is continuous, it is weakly differentiable almost everywhere in D, and it has the N-property. The following versions of Theorems 5.23 and 5.27 are due to Marcus and Mizel (1973).

Theorem 5.30 Let D C RN be an open, bounded set, let p > N, let 0 E W1,P(D)N be equal to its continuous representative, and let v E LOO(RN). For every measurable set E C D, we have

Lv o (x)I J4(x)I dx = f v(y)N(4, E, y) dy.

(5.48)

R

Proof By The Sobolev Imbedding Theorem 0 is continuous (0 E CO,o(D)N, where a = 1 - ). By Theorem 5.28, ¢ has the N-property and by Theorem 5.21, Q has a weak v differential almost everywhere. The result now follows from 0 Theorem 5.23.

As mentioned in Remark 5.24, Marcus and Mizel (1973) proved Theorem

5.30 for 0EWl.P(D)m,m>N. Theorem 5.31 Let D C RN be an open, bounded set, let p > N, and let yh E be equal to its continuous representative. If v E LOO(RN), then, for every open set G C D such that GN(8G) = 0, we have

fvo(x)J(x)dx = JRN v(y)d(e, G, y) dy P roof The result follows from Remark 5.29 and Theorem 5.27.

0

We will focus now on functions in W1,N. Due to Corollary 5.19, in what follows we identify 0 E W1.N(D)N, J,,(x) > 0 a.e. x E D, with its continuous representative.

Theorem 5.32 Let D C RN be an open, bounded set and let 0 E be a mapping such that J#(x) > 0 GN a.e. x E D. Then 0 has the N- and the N-1-properties.

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

142

Ch. 5

Proof (i) We start by showing that 0 has the N-property. Let a E D, R. > 0, be such that QR. = (al

- Ra, al + Ra) x ... x (aN - Ra, aN + Ra) CC D.

Using the argument of the proof of Theorem 5.16, we observe that for each

i E {1,...,N} there exists I; C (-R.,R.) such that L1(I;) = 0 and 0IPr(a) E W1,N(Pr'(a))N (1C(P, (a))N

for every r E (-Ra, Ra) \ Ii, where Ra, a,-1 + Ra) x {a, + r} PT(a) = (a1 - Ra, a1 + Ra) x ... x x (a:+l - Ra, ai+l + Ra,) x ... x (aN - Ra, aN + Ra ).

By Theorem 5.28 we obtain HN-1(((P,'.(a))) < +oo

for every r E (-Ra, Ra) \ I, and so, by Lemma 4.28, and Remark 4.26, we deduce that

GN(0(P'(a))) = 0

(5.49)

for every r E (- R., Ra) \ 1;. Let 9 : RN -. R be a symmetric mollifier (see Definition 1.16), such that 0 < 9(x) for every x E RN,spt(9) C B(0,1), and fore > 0 define

0,(x):= -N 9 ('),X E RN, e > 0, and q,, := 9. * 0. Then 0n E COO(D),

0n _.0 in W1'N(D)N

and, by Corollary 5.19,

4 -4 0 uniformly in QR.. By (5.49) there exists I C (0, Ra) such that L1(I) = 0 and

LN(0(0Qr(a))) = 0

(5.50)

for every r E (0, R.) \ I. Fix r E (0, R0) \ I. Claim 1. For every b E ,(Qr(a)) \ 4,(8Qr(a)) we have d(4,, Qr(a), b) Let b E RN \4(8Qr(a)) and 0 < e < dist(b, ¢(8Qr(a))). Since 4,n converges uniformly to 0 in QR., for n large enough, and by Proposition 1.7 we have d(O,, Qr(a), b) = JRN 0,(on(x) - b)Jm. (x) dx.

CHANGE OF VARIABLES FOR SOBOLEV FUNCTIONS

§5.5

143

Letting n -' +oo and by Theorem 2.3 we deduce that d(4,Qr(a),b) = JRN e(O(x) - b)Jo(x) dx,

and since J41(x) > 0,CN a.e. x E D and d(o, Q, (a), b) is an integer number, by (5.50) we deduce that d(O, Qr (a), b) > 1

for L' almost every b E RN. Claim 2. Xn converges to X GN almost everywhere, where Xn is the characteristic function of 4'n(Qr(a)) and X is the characteristic function of O(Qr(a)) Indeed, let b E RN \ O(8Qr(a)). If b ¢ ¢(Qr(a)) there exists n(x, e) E N such that I0n(x) - O(x)12 < dist(b, 4(Qr(a)) for all x E Q,-(a)

for every n > n(x, e). Therefore, b ¢ On(Qr(a)) for every n > n(x, e) and so

lim Xn(b) = X(b) n-.+oo If b E O(Qr(a))\4(8Qr(a)), by the result of Claim 1 we have d(o, Qr(a), b) >

1 and by Theorem 2.1, X(b) = I. Using the fact that d(¢, Qr(a), b) _ d(On, Qr(a), b) for n large enough, and again by Theorem 2.1, we deduce that Xn(b) = 1 and so lim Xn(b) = X(b) n+oo

Recalling that £N(i)(8Qr(a))) = 0 (see (5.50)), we obtain that Xn converges to X GN almost everywhere. Claim S. GN(O(Q,.(a))) < fQ,(a) J,6(x) dx for every r E (0, Ra).

If r E (0, Ra) \ I, using Fatou's Lemma, Theorem 5.31, and the result of Claim 2, we obtain GNWQr(a))) =

x(x) dx JRN

< lim inf in X. (x) dx n-.+00

< lim inf

N

J

n-+oo Q,.(a)

Jo (x) dx

Jj (x) dx.

(5.51)

Q,4 i)

The result for arbitrary r E (0, Ra) follows from (5.51) and the Lebesgue Monotone Convergence Theorem, where it suffices to consider an increasing

144

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

sequence rk E (0, R.) \ I such that rk Claim 4. 0 has the N-property.

Ch. 5

r.

Let A C D be such that LN(A) = 0. We may assume without loss of generality that A Cc- D. Fix e > 0. Since Jo E L1(D) there exists an open

set 0 cc D cointaining A such that

Jo

Jo(x) dx < e

and there exists a collection {QI: j E N} of half-open cubes such that {Qj: j E N} is a partition of 0 and

O=u Qj iEN (see Rudin 1966, Section 2.19). By (5.51) we have

GN(q (A)) < CN(o(C)) 00

j:CN(0(Qj)) i=1

f Jo(x) dx v

= f JJ(x) dx < e. It suffices to let a -* 0+.

(ii) We show now that q5 has the N'1-property. In fact, by part (i) and by Theorems 5.21 and 5.11, we have

J J0 (x) dx =

JN

N(O, E, y) dy

(5.52)

for every E C D. Thus, setting E = /-1(F) where GN(F) = 0, we obtain

J -'(F) J0 (x) dx = =

JRN

f

N(O, 0-1(F'), y) dy

N(O,

4-1(F), y) dy

=0 and as J4,(x) > 0 for CN a.e. X E D, we conclude that GN(¢-1(F)) = 0.

0

§5.5

CHANGE OF VARIABLES FOR SOBOLEV FUNCTIONS

145

Remark 5.33 It can be shown that if 0 E WI,N(D)N is continuous, open and discrete (i.e. 4' 1 {y} is finite, for all y E RN), then 0 satisfies the N-property. For details we refer the reader to Martio and Ziemer (1992).

The two following theorems were proved by Gold'sthein and Reshetnyak (1990), where, as usual, due to Corollary 5.19 we assume that 4, coincides with its continuous representative.

Theorem 5.34 Let D C RN be an open, bounded set and let 4, E WI,N(D)N be a mapping such that Jm > 0 GN a.e. x E D. If V E L°°(RN) then, for every measurable set E C D, we have JE

v ° 0(x)jJm(x)j dx = j v(y)N(4,, E, y) dy ^'

Proof By Theorem 5.21 0 has a weak differential almost everywhere and, by Theorem 5.32, d has the N-property. The result now follows from Theorem 5.23.

0 Theorem 5.35 Let D C RN be an open, bounded set and let 0 E W1,N(D)N be a mapping such that J# > 0 GN a.e. x E D. If v E L°°(RN), then, for every open set G C D such that GN(8G) = 0, we have JC

v o 4,(x)Jm(x) dx = j v(y)d(4,, G, y) dy.

(5.53)

N

Proof By Corollary 5.19 and Theorem 5.21, 0 is continuous and has a weak differential almost everywhere and, by Theorem 5.32, 0 has the N-property. It suffices to apply Theorem 5.27. Recently, $verak (1988) and Miiller et al. (1994), generalized Theorems 5.34 and 5.35 to a class of Sobolev mappings not necessarily continuous but under some additional regularity hypotheses on the boundary. We recall that the notion of Lipschitz boundary was introduced in Definition 4.40. Although the space Ap,q(D) has already been defined in Section 5.2, for convenience we include its description below.

Definition 5.36 Let D C RN be an open set with Lipschitz boundary, let p > N - 1, q > f-iN-1. Then Ap,a(D) := JOE Wl,P(D)N : adj(VO) E LQ(D)NXN} and

4,q (D) = {¢ E Ap,q(D) : 4(x) > 0 GN a.e. x E D}.

We recall that if p > N - 1 mappings in 4.,(D) are continuous outside sets of Hausdorff dimension less than or equal to N - p (see Remark 5.18). Also, if 0 E A,,a(D), then Jm E LI(D). In fact, from the identity

PROPERTIES OF THE DEGREE FOR SOBOLEV FUNCTIONS

146

Ch. 5

J,(x)IN = V (x) adj (O(x)) we obtain I J.(x)I N= I JJ(x)I Idetadj(0(x))I S I JJ(x)I Iadj(0(x))IN

and so

E L'(D).

I Jo(x)I S I adj (-O(x))I'°

We use A to denote the exterior product (or wedge product) between vectors and we recall that the mapping

N-1) E

(RN)N-1

,

.

t1 A... AtN-1

is multilinear, alternate and, if {el, ... , eN } is the canonical basis of RN, then e1A...Ae;-1Ae:+1A...AeN=(-1)N-le,,

i=1,...,N.

It can easily be verified that if F is a N x N matrix, then ladj F) = sup {I(Fe;) A ... A (F1;N-1)I : I& A ... A tN-1I N - 1, and yo E RN \'(8D). By Theorem 1.12 and (5.55), d(0, 8D, yo) will be well defined and (5.55) will hold for ¢ if we show that we may find a sequence Tn E C°O(D)N such that %P,,18D -, 0I8D uniformly and W,a - ¢ in W1,p(8D)N. To construct the approximating sequence {'P, : n E NJ, using a partition of unity and a change of coordinates we may assume that there exist a, fl > 0 and a Lipschitz function a such that

a(0) = 0, and

loll N(N - 1), then w o v(x) = x for every x E D, v o w(y) = y for every y E B(yo, r), v is a local homeomorphism, and v is an open mapping

on 0 \ L for some set L C 12 of measure zero. In particular, if N = 2, then N(N -1) = N = 2 and v is a local homeomorphism at xo.

Remark 6.4 (i) By Theorem 5.32, if v E W 1,N(f2)N with det Vv(x) > 0 for GN a.e. X E fl,

then v satisfies the N- and the N'-properties. Also, by Theorem 5.32, if v E W 1.p(f2)N, p> N, then v satisfies the N-property. (ii) A function v E W1.N(O)N is said to be a mapping of bounded distortion if lVv(x)Iz < K(det Vv(x)) for 0 a.e. x E fl and for some constant K. It is well known that every mapping of bounded distortion v E W1.N(f2)N is a homeomorphism locally at almost every point xo E f2 (see Reshetnyak 1989, Theorem 6.6, p. 187). Moreover, mappings of bounded distortion are open mappings or are constant in ft (see Reshetnyak 1989, Theorem 6.4, p. 184). More generally, defining the dilatation K of v as et

lvtz)l ec m N

det Ve() W

1

det VO(x) = 0,

K(x) :=

0

v is said to have finite dilatation if 1 < K(x) < +oo for GN a.e. X E fl. Gold'sthein and Vodopyanov (1977) proved that a mapping v of finite dilatation is continuous and its components vi are weakly monotone. Recently, Heinonen and Koskela (1993) and Manfredi and Villamor (in press) proved that if V E W 1.N (f2)N has finite dilatation, if v is quasi-light (i.e. v'1({y}) is compact for all y E RN), and if K E LN-1+, for some e > 0,

then v is open and discrete (i.e. v-'({y}) is finite for all yin RN). In particular, by Remark 5.33 v satisfies the N-property. (iii) Note that, even if v E C1(St)N is such that det Vv(x) > y > 0 for all x E QZ, we cannot expect global invertibility of v without any regularity assumptions on the trace of v. As an example, consider f2 :_ {(x, y) E R2 : 1 < x2 + y2 < 2},

v(x, y) :_ (x2 - y2, 2xy).

For every (x, y) E fl we have det Vv(x, y) = 4(x2 + y2) > 4, although v(x, y) = v(-x, -y) (see also Ball 1981). (iv) Under the assumptions of Theorem 6.1, we cannot expect v to be locally invertible everywhere. The following example is provided by Ball (1981): let N > 3 and consider the cylinder

fl ={xERN :0 0 a.e. x E il. Therefore, to

prove Theorem 6.1 one cannot expect to approximate the function v by a sequence of locally invertible, smooth functions v,,. (vi) Note that for every bounded, open set fl C RN , there exist a measurable set E C fl of nonzero measure and a homeomorphism v E W1,O°(ci)N such that det Vv(x) = 0 for every x E E. The following example is provided by Martio and Ziemer (1992, Remarks 3.7). Let E C [0,1] be a Cantor set of positive one-dimensional measure 0 < 1 - a < 1 and write

vl(x) =v1(x1,...,XN) = f i x--- (t) dt 0

where XEC is the characteristic function of the complement of E in [0,1]. Setting v(x) := (vl (x), x2, ... , xN), it follows that v E W1 -00 (ci) N, v is a homeomorphism of [0, 1]N onto [0, a] x [0,11N-1, and det Vv(x) = 0 a.e. X E

Ex

[0,1]N-1.

(vii) Due to the previous remark, the assumption det Vv(x) 4 0 a.e. x E S2 in Corollary 6.2 is essential.

Lemma 6.5 Let 5t C RN be an open set, let v E W1,N((j)N be such that det Vv(x) > 0 a.e. x e St. Then for every xo E SZ such that v is differentiable at xo (in the classical sense) and det Vv(xo) > 0 there is Ro =- Ro(xo) such that for every 0 < R < Ro the following holds:

N(v, B(xo, R), y) = 1 for almost every y E CR, d(v, B(xo, R), y) =1 for every y E CR, d(v, B, y) = 1 for every y E v(B) \ v(BB),

(6.4) (6.5) (6.6)

for every nonempty, open set B C v 1(CR) n B(xo, R) such that CN(8B) = 0, where CR is the connected component of RN \ v(8B(xo, R)) containing yo v(xo) and N(v, E, y) is the cardinality of the set {x E E : v(x) = y}.

LOCAL INVERTIBILITY IN W'.^'

§6.1

153

Proof By Theorem 5.21, v is differentiable at almost every point x E 11. Fix xo E SZ such that v is differentiable at x0 and detVv(xo) > 0. By Lemma 5.9 there is Ro > 0 such that B(xo, Ro) cc SZ and d(v, B(xo, R), yo) = 1 for every 0 < R < Ro and (6.5) follows from Theorem 2.3. Since det Vv(x) > 0 a.e. x E SZ, Theorems 5.23, 5.27, along with (6.5), yield (6.4). Finally, we prove (6.6).

As v satisfies the N-property (see Remark 6.4) LN(v(cB(xo, Ro))) = 0 and GN(v(OB)) = 0. Since B is a nonempty open set, by Theorem 5.32 we have that GN(v(B)) 96 0 and so GN(v(B) \v(OB)) > 0. Let y E v(B) \v(OB) and let C be the connected component of RN\v(OB) containing y. As GN(v(8B(xo, R0))) = 0 and since d(v, B, -) is a constant on C, we may assume without loss of generality that y ¢ v(8B(xo, Ro)). Let p. E C°D(RN) be such that

00 2 < P" (y), for all yEB(0,2) spilt p, C B(0, e), for all e > 0

JRNp,(y)dy=1 for all e>0.

(6.7)

Since y E v(B) there exists x E B such that y = v(x) and by Theorem 2.3 we have aim

J p, (v(z) - y)det Vv(z) dz = d(v, B, y)

(6.8)

B

and using the continuity of v at x, we deduce that for every e > 0 there exists 6 > 0 such that Iv(z) - y12 < 1 for every z E B(x, 6). Recalling that det Vv(z) > 0 a.e. Z E B(x, 6), by (6.7) and (6.8) we obtain d(v, B, y) > 0.

(6.9)

Finally as the degree d(v, , y) is a nondecreasing function of the set, using (6.5) and the fact that B C v'1(CR) n B(xo, R) we obtain d(v, B, y) < d(v, B(xo, R), y) = 1

(6.10)

which, together with (6.9) and the fact that the degree is an integer number, yields (6.6).

Lemma 6.6 Let SZ, v, Ro, xo, be as in Lemma 6.5, (6.4), (6.5). Let C& be the connected component of RN \ v(aB(xo, R0)) containing yo := v(xo). Then for every r > 0 such that B(yo, r) CC C&,

v(O) = B(yo,r), v(aO) C av(O) = aB(yo, r), where 0:= v-1(B(yo, r)) n B(xo, Ro) CC B(xo, Ro).

154

LOCAL INVERTIBILITY OF SOBOLEV FUNCTIONS

Ch. 6

Proof It is clear that v(O) C B(yo, r). Conversely, if y E B(yo,r), by (6.5) d(v, B(xo, Ro), y) = 1 and so by Theorem 2.1, there exists x E B(xo, Ro) such that y = v(x), implying y E v(O). Let x E 80 and let {an} C 0, {bn} C B(xo, Ro) \ 0 be such that

lim an = n-.+oo lim bn = x.

n-+oo

We have v(an) E v(O) and, as v(O) = v(v-I(B(yo, r))) = B(yo, r), we observe that v(bn) ¢ v(O) = B(yo, r). Using the continuity of v at x, we have

v(x) =n-+oo lim v(an) = n-+oo lim v(bn) 0

and this gives x E 8v(O).

Lemma 6.7 Let v E W1'N(S2)N, detVv(x) > 0 a.e. x E S1 and let xo E D be such that v(x) 0 v(xo) for every x E B(xo, Ro) \ {xo}. For every 0 < R < Ro there exists r > 0 such that v-' (B(yo, r)) n B(xo, R) CC B(xo, R).

Proof Define d(6) := sup{fix - x012: x E B(xo, R), Iv(x) - v(xo)12 < 6}. Since v(x) 9& v(xo) for every x E B(xo, R) \ (xo) and v is uniformly continuous on B(xo, R), we have lmd(b)=0.

Now take r > 0 such that d(r) < a . We have v-' (B(yo, r)) n B(xo, R) C B ( xo, 2) CC B(xo, R). 0

Proof of Theorem 6.1 Let fly be the set of points xo E f2 such that v is completely differentiable at xo and detVv(xo) > 0. By Theorem 5.21 we have GN(cl \ fI) = 0. In the sequel, we fix xo E ff, we set yo = v(xo), and we show that v is locally invertible at xo. By Lemmas 5.9 and 6.5 there exists Ro > 0 such that B(xo, Ro) CC f2, N(v, B(xo, Ro), y) = 1 a.e. y E C&,

(6.11)

where Cp is the connected component of RN \ v(8B(xo, Ro)) containing yo, with N(v, B(xo, Ro), yo) = 1. By Lemma 6.7 we deduce that there exists r > 0 such that v-1(B(yo, r)) n B(xo, Ro) CC B(xo, Ro) (6.12) and

LOCAL INVERTIBILITY IN W'-N

§6.1

B(yo,r) CC Cp.

155

(6.13)

Setting D := v-1(B(yo,r)) fl B(xo,Ro), by (6.12) and (6.13) we have D C v-1(C,,) fl B(xo, Ra) and, by Lemma 6.6, v(D) = B(yo, r), v(OD) C 09v(D) = OB(yo, r).

(6.14)

By the N-'-property of v (see Theorem 5.32) and (6.14), we have CN(8D) = 0 which, together with (6.6), yields

d(v, D, y) =1 for ally E v(D) \ v(8D).

(6.15)

Using the definition of D, the fact that D C B(xo, Ro), (6.11), and (6.13), we obtain N(v, D, y) = 1 a.e. y E v(D). (6.16) Let E := {y E v(D) = B(yo,r) : N(v, D, y) 54 1}, so that CN(E) = 0. We define the candidate for the local inverse function, w, by

w(y) := x if y E v(D) \ E, and v(x) = y, x E D, w(y) := x if y E E, v(x) = y,

(6.17)

(6.18)

x E D being chosen by the axiom of choice. Claiml. w E L°O(B(yo, r))N. We have w(y) E D C 11 for every y E v(D) and so w is uniformly bounded in v(D). To prove that w is Lebesgue measurable, we fix a E R and show that

the set A:= {y E v(D) : wi(y) > a} is measurable, i = 1, ... , N. Clearly, A = Al U A2, where

Al :_ {y E v(D) \ E : wi(y) > a}, A2 :_ {y E E : wi(y) > a}. Since CN(A2) = 0 we deduce that A2 is measurable. Using the fact that the restriction of v to v-1(v(D) \ E) is one-to-one, one can see that

Al = {v(x) : x E v-r'(v(D) \ N), xi > a} _ (v(D) \ E) fl I U V ({x E B(xo, Ro) : a + n 5 xi < a + n + 11) J . Using the fact that for every n E N, {x E B(xo, Ro) : a + n < xi < a+n+1} is a compact set, v is a continuous function, and v(D) \ E is measurable we conclude that Al is measurable. Claim 2.

V 0 w(y) = y for every y E v(D) = B(yo, r),

(6.19)

156

LOCAL INVERTIBILITY OF SOBOLEV FUNCTIONS

W o v(x) = x for every x E D\ v-1(E).

Ch. 6 (6.20)

These follow immediately from (6.17) and (6.18). We notice that, due to Theorem 5.32, GN(v-1(N)) = 0.

Claim 3. f o w is measurable for every f : D -+ R measurable. Since every Lebesgue measurable set is a union of a Borel measurable set and a set of measure zero, to show that f o w is measurable, by Claim 1 it suffices to show that w-1(R) is measurable for every R C D such that C' (R) = 0. Indeed, by (6.19), w-1(R) C v(R),

and by the N-property of v we obtain that LN(w-1(R)) = 0. Thus w-1(R) is measurable. Let g : v(D) = B(yo, r)

R be defined by g(y) __ Iadjov(w(y))I2

detVv(w(y))

Claim 4. g E L'(v(D)). By Claim 3 g is measurable and by Theorem 5.23, Claim 2, and (6.16) we obtain

J N(v, D, y) Ig(y)I dy = N

f

D) lg(y)I dy

= L Ig o v(x)Idet vv(x) dx = fIadiVv(x)I2dx. Therefore, g E L1(v(D)). Claim 5. w E W1,1(v(D))`v and Vw(y) _ (a vv w y )T To prove this claim, we fix 0 E Cp (v(D)) and set K := spt0. We show that

(adJVv(w(Y)))'

f (D)wa(y)°j(y)dy=-f

a(y)dy,

(D)

detVv(w(y))

where Aa denotes the component of the jth row and the ath column of a matrix

A. Set 6 := dist(K,Ov(D)) > 0. Using the uniform continuity of v on D C B(xo, Ro) we choose e > 0 such that

Iv(x)-v(x')I2 N - 1, if v is continuous, adjVv E L7T(S2), and detVv(x) > 0 a.e. in S2, then there is local invertibility a.e. in 0, i.e. for a.e. xo E S2 there exists r > 0 such that VI B(xo,r) is almost everywhere injective with the inverse w E BV,c(vIv(B(xo,,)) )N and there

exists a set E C v(B(xo, r)) such that

E is an open set of v(B(xo, r)), GN(v(B(xo, r) \ E)) = 0, in E

W1.1(E)N,

v o w(y) = y a.e. y E v(B(xo, r)), w o v(x) = x a.e. X E B(xo, r).

By Theorem 5.21 v is weakly differentiable a.e. in S2 and by Lemma 5.10

d(v, B(zo, r), v(xo)) = I

for some r > 0. Let Co be the connected component of RN \ v(aB(xo, r)) which contains v(xo). Then d(v, B(xo, r), y) = 1 (6.24)

for every y e Co and so, if we choose 0 < r' < r such that B(xo, r') C B(xo, r) n v- I (CO),

then, by (6.24) and since det Vv > 0 a.e., repeating the argument used in (6.7)(6.10) we have d(v, B(xo, r'), y) < 1

for every y E RN \ v(8B(zo, r')). It suffices now to use Tang's results (1988, (1.3)-(1.5), (2.26), and Theorem 3.7 (i)). Note, however, that in Tang (1988) it

is assumed that adjVv E Lr, r > I and if N - 1 < q < N, then >p as was no te d by As it t urns ou t , T ang' s resu lts (1988 ) still h old for r Muller et al. (1994, Theorem 5.3).

Proof of Corollary 6.2. (a) We have V E W1'N(f 1)N,

det Vv(x) > 0 a.e. x E SZ1

and

v E W"N(02)N, det Vv(x) < 0 a.e. X E 522. It suffices to apply Theorem 6.1 to v in S21 and to Rov in 522i where Ro is a constant rotation with det Ro = -1.

LOCAL INVERTIBILITY IN W1,N

§6.1

159

(b) Now we assume that v E W1'Q(1l)N, q > N, det Vv(x) # 0 a.e. x E S1 and for almost every xo E Sl, v is locally almost injective in a neighbourhood of xo, in the sense that there is an open set D =_ D(xo) CC S2 and there is a function w : v(D) -i D such that

wov(x)=xa.e.xED. By a corollary of Vitali's Covering Theorem (see Corollary 4.35) there is a countable family of nonempty, open, mutually disjoint balls {B i : i E N) and there is a sequence of functions wi : v(Bi) St such that Bi C S and ,CN(S1\

iEN Bi) = 0,

w o v(x) = x a.e. x c Bi.

(6.25)

The task ahead will be to partition B, into three subsets B; , B?, and Ni such that B,, Bi are two open sets and Ni is a set of measure zero, det Vv(x) > 0 a.e. x E Bi ,

detVv(x) < 0 a.e. X E B. Using the fact that v E W1,9(Bi)N, q > N, by (6.25) and Theorem 5.21 we deduce that there is a set of measure zero A, C B, such that v is differentiable at every x E Bi \ A,, w o v(x) = x for every x E Bi \ Ai, 0 for every x E Bi \ Ai.

(6.26)

det Vv(x)

Let {Ci } be the countable collection of the (open) connected components of RN \ v(8Bi). By Theorem 5.32 we have ,CN(v-1(v(8B, U A,))) = 0.

We claim the following.

Claim 1. d(v, Bi, v(x)) = sgndetVv(x) for every x E Bi \ v-1(v(8B, U A,)). Fix x E Bi \ v-1(v(8Bi U A,)). As v is differentiable at x and det Vv(x) 54 0, by Lemma 5.9 there exists ro > 0 such that for every 0 < r < ro we have d(v, B(x, r), v(x)) = sgn det Vv(x).

On the other hand, setting K := Bi \ B(x, ro), K is a compact set included in Bi and by (6.26) v(x) ¢ v(K) because v(x) ¢ v(A,). By the excision property of the degree (see Theorem 2.7) we obtain d(v, Bi, v(x)) = d(v, B(x, ro), v(x)) = sgn det Vv(x).

160

LOCAL INVERTIBILITY OF SOBOLEV FUNCTIONS

Ch. 6

Claim 2. sgn det Vv(x) = sgn det Vv(x') for every x, x' E v'' (C') \ v-' (v(8Bi U A,))-

Assume that x, x' E v-' (G) \ v-' (v(8B, U Ai)). Using Claim 1 and the fact that the degree d(v, Bi, -) is constant on each C', we obtain that sgn det Vv(x) _ sgn det Vv(x'). Now we conclude the proof of (b). Let I {j E N : detVv(x) > 0a.e.x E

v''(Ci)) and J = {j E N : detVv(x) < 0 a.e. x E v-'(C')}. Set

B, :=U v-'(C')nB,, B,2 :=U v-'(C')nBi, 3EJ JE!

and

Ni:=Bi\(B, UBt)Then Bi := B, U B? U Ni and by setting D1 := U,B; , f12 := U,B,2 and E 52 WI U f12), we obtain that CN (E) = 0 and 521, S22 have the required properties.

0 Proof of Corollary 6.3 To obtain that w E W1'71 (v(D), D) we take s = xg- in Theorem 6.1. If q > N(N - 1), then w E W' N, detVw(y)

= det Ov(w(y)) >

0 a.e. y E v(D)

and so, by Corollary 5.19, we deduce that w is continuous. Hence v and w are homeomorphisms and v is an open mapping in 52' for some fY C 1 open, where

LN(52\52')=0. 6.2 Energy functionals involving variation of the domain The variational treatment of crystals with defects leads to the study of functionals of the type

E(u,v) := J W(Vu(x)(Dv(x))-')dx, n

where 52 C RN is a reference domain, W is the strain energy density, u is the elastic deformation, and v represents the slip (rearrangement) or plastic deformation, with det (Vv(x)) = 1 a.e. x E 52. The underlying kinematical model for slightly defective crystals was introduced by Davini (1986) and later developed by Davini and Parry (1989). As it turns out, matrices of the form

Du(x)(Ov(x))-' represent lattice matrices of defect-preserving deformations (neutral deformations), and, taking the viewpoint that equilibria correspond to a variational principle, Fonseca and Parry (1987) studied the structure of some kinds of generalized related variational minimizer (Young measure solutions) for the energy problems were also investigated in Fonseca and Parry 1987).

§6.2 ENERGY FUNCTIONALS INVOLVING VARIATION OF THE DOMAIN 161

The lower semicontinuity and relaxation properties of E(-, ) were addressed only under additional material symmetry assumptions on W. The existence and regularity properties for minimizers of E(-, -) were obtained by Dacorogna and Fonseca (1992). Following this work, we stress the fact that the direct meth-

ods of the calculus of variations fail to apply to this problem, as sequential is not sufficient to guarantee the existence of minimizers. Indeed, if W(F) = IFIz, it was shown by Dacorogna and Fonseca (1992) that there are no minimizers in {(u, v) E W l-00 x W1,001 u(x) _ x on i1, det (Vv(x)) = 1 a.e.} if 0 < r < N = 2, while for r > N existence is weak lower semicontinuity of

obtained for smooth (u, v) (Dacorogna and Fonseca 1992, Theorem 2.3). It is clear that if is a minimizing sequence and if IVu4,(Vvn)'1j2 is bounded in L1, then

Vun(Vvn)-1

L in L', unto = uo, det (Vvn) = 1 a.e.

and so, if some type of lower semicontinuity prevails, then

L It would remain to show that L will still have the same structure, precisely

L = Vu(Vv)-1 where uI - = uo, det (Vv) = 1 a.e. X E Q. Using the div-curl lemma it follows

that if un -- u in W 1.00 w - *and v -- v in W1,00 w - *, then Vun(Vvn)-1

Vu(Vv)-1 in L°O w-*.

Note that (6.27) is always satisfied if W is a convex function. On the other hand, formally, as det (Vv) = 1 a.e. and setting w = u(v'1), the energy becomes

40)

W(Vw(y)) dy,

which is now an energy functional involving variations of the domain. Hence, under this new formulation, quasiconvexity seems to be more appropriate than convexity (see Acerbi and Fusco 1983, Ball 1978, and Dacorogna 1987). Suppose that W is a quasiconvex function, i.e.

W(F)
0, r > s > 1, p > 1, q > N, p + NQ 11 = .1 (W > 0 if r = s = 1). If un - u in W"p(fI)N, vn v in W1.q(SZ)N, and if det (Vvn) = 1 a.e. in Sl, then

L

W(Vu(Ov)-I)dx < liminf J W(Vun(Vvn)-I)dx - n-++oo ft

Before proving Theorem 6.9 we make some remarks.

Remark 6.10 (i) It is clear that if u E WI.p, v E W1,9 and det Vv = 1 a.e., then Du(Vv)'1 E

L''. (ii)

If r > 1, then s < r is a necessary condition, as the counterexample by Murat and Tartar shows (see Ball and Murat 1984). Here r = s = 2 = NJ I = (0,1)2, W(F) =r det (F), u,, u inr HI(Sl), vn(x) = x, and det Du 2: liminf Jn

detVun. fnn

(iii) The growth condition cannot be dropped even if W is polyconvex and nonnegative. Precisely, if the relation between p, q, r, and s does not hold, the conclusion of Theorem 6.9 may be false. Indeed, using the example by

Mali (1993) with q = +oo, p < N - 1, W (F) = det F, N = r = s, we may u in W",P, u(x) _- x with vn(x) __ x, and find u,, f Idet (Vu) I > liminf fo Idet (Vun)1.

Moreover, the growth condition prescribed in Theorem 6.9 is the wellknown growth condition ensuring weak lower semicontinuity of E(u, id) in WI.p (see Acerbi and Fusco 1983 and Dacorogna 1987).

(iv) It is natural to ask whether or not these results can be extended to the < q < N, since, due to Miiller's result (1990b) and if we assume that det Vv = I a.e., then Det Vv = det Vv a.e. in IL case

§6.2 ENERGY FUNCTIONALS INVOLVING VARIATION OF THE DOMAIN 163

(v) Having obtained lower semicontinuity of the energy in Theorem 6.9, the question now amounts to showing that one can find a minimizing sequence {Dun(Vvn)-1} where {un} is bounded in W1.p and {vn} is bounded in W 1.O°. Actually, one only needs to show that there exists a sequence {f} C W 1"°°(Sl, S2) such that vn o fn is bounded in W 1-°° and

detVfn(x) = 1 =1 fn(x)

a.e. X E 11

xEOS2,

= V(un o fn) (V(vn o fn))-1. Due to the examples provided by Dacorogna and Fonseca (1992), we know that this may not be possible since the infimum of E may be zero, which prevents the existence of a minimizing sequence bounded in W 1.p x W l .Q. since

Vun(Vvn)-1

As is usual in variational problems for which existence of minimizers is not guaranteed (such as variational problems for material that changes phase or, as in our case, for slightly defective materials), rather then studying the macroscopic limit of we focus our attention on the properties of the minimizing sequences. The following may contribute to a better understanding of why the boundedness of {Vun(Vvn)-1} may not entail the boundedness of {Vun} and {Vvn}. Using Theorem 6.9 we show that we can construct a minimizing sequence of the form {VuE(Vvv)-1} with IIot4 llp = 0(E;), IIVVEIIq = 0(-a), for any a,0 > 0. Consider the `perturbed' family of variational problems EE(u, v) := J W(Vu(Vv)'1) dx + e°pIIVu1IIP + E09IIVvEIIq,

where ulan = uo, det Vv = 1 a.e., and Zw'M fn v(x) dx = 0. Using the direct method of the Calculus of Variations, Poincare's Inequality and Theorem 6.9, it follows immediately that there exists (ui,vv) E W11p X W' such that EE(u(,vC) = inf {EE(u,v) : (u,v) E W1'1' x W1,1, detVv = 1 a.e.}

.

Then, given an admissible pair (u, v), E(u,v) = lim EE(u,v)

E-o+

> lim sup EE

v,)

E-»o+

> lim sup E(uE, v.), E-.o+

> inf E. Using the same reasoning with lim E(u1, vE) and taking the infimum in (u, v) we conclude that inf E = lim E(uE, vE) E-.o+

LOCAL INVERTIBILITY OF SOBOLEV FUNCTIONS

164

Ch. 6

and

Iloucllp =

Ilovcll4 = 0(E ).

The following two lemmas will be useful in proving Theorem 6.9.

Lemma 6.11 Let W, 0 be two open sets of RN such that SY CC 12, let q > N, and let v, vn E W1.9(11)N be such that det Vv(x) = det Vvn(x) = 1 a.e. X E 12. Assume that vn v in W1,9(12)N. Then then exists a subsequence of {vn} (not nlabelled) such that for almost every xo E 12' then exist open sets D, Dn C S2'

containing xo and them exist no E N, ro = r(xo) > 0, w : B(yo, ro) -+ D, wn : B(yo,ro)

Dn, with yo = v(xo) such that for n > no,

wn o vn(x) = x a.e. X E Dn, vn o wn(y) = y for every y E B(yo, ro) and vn(Dn) = B(yo, ro),

w o v(x) = x a.e. x E D and v(xo) 96 v(x) for x E D, x 0 xo, v o w(y) = y for every y E B(yo, ro) and v(D) = B(yo, ro),

wn,w E W''. Proof Using Corollary 5.19 and the Ascoli-Arzela Theorem we obtain that, up to a subsequence, vn converges to v uniformly in 12'. By Lemma 6.5 for almost every xo E SY, there is RD > 0 such that B(xo,Ro) CC W, N(v, B(xo, Ro), y) = 1 for almost every y E C8, d(v, B(xo, Ro), y) = 1 for every y E Cp, d(v, B, y) = 1 for every y E B \ v(8B),

for every nonempty open set B C v-1(Cp) n B(xo, Ro) such that GN(v(8B)) _ 0, where Cp is the connected component of RN \v(8B(xo, Ro)) containing yo v(xo). Since v is differentiable at xo and det Vv(xo) 96 0 we may assume without loss of generality that N(v, B(xo, -o), yo) = 1. Fix 0 < e < d(yo, v(8B(xo, Ro))) and choose no E N such that Ilvn - v11 < e. Set

A , :_ {y E CR.: dist(y, v(8B(xo, Ro))) > e}. It is obvious that A, is a nonempty open set. Claim 1. d(vn, B(xo, Ro), y) exists and is equal to 1 for every y E A. and every

n>no. By Theorem 2.3, together with the fact that d(v, B(xo, Ro), y) = 1 for every

yECa,wehave d(vn, B(xo, Ro), y) = 1

(6.28)

for every y E A, and every n > no. By Lemma 6.7 there is 0 < ro < Ro such that

§6.2 ENERGY FUNCTIONALS INVOLVING VARIATION OF THE DOMAIN 165

B(yo, ro) cC AE and v-' (B(yo, ro)) n B(xo, Ro) cc B(xo, Ro).

(6.29)

Claim 2. We claim that B(yo, ro) CC Cn

(6.30)

where CL is the connected component of RN \ vn(8B(xo, Ro)) containing yo. We first prove that AE C RN \vn(8B(xo, R0)). Assume, on the contrary, that there is y E AE flvn(8B(xo, Ra)) and choose x E 8B(xo, R0) such that y = vn(x). We would then have Ivn(x) - v(x)I2 = J Y - v(x)I2 > e > Ilvn - vii, which yields a contradiction. Fix r' > ro such that B(yo,r') C AE. We have that B(yo,r') is a connected set included in RN \ vn(8B(xo, Ro)) and containing yo. We deduce

that B(yo, r') C C and B(yo, ro) CC C. Set

D:=v-'(B(yo, ro)) fl B(xo, Ro) CC 52', Dn :=vn'(B(yo, ro)) fl B(xo, Ro) CC l'.

By (6.28), (6.29), (6.30), and using arguments similar to those in the proof of Theorem 6.1, together with Corollary 6.3, we deduce that for n > no there exists

wn : B(yo,ro) - Dn and w : B(yo,ro) -+ D such that wn, w E W'' 7r (B(yo, ro)) N, and

wn o vn(x) = x a.e. X E Dn, vn o Wn(y) = y a.e. y E B(yo,ro),

w o v(x) = x a.e. X E D and v(xo) y6 v(x) for x E D, x 34 x0, v o w(y) = y a.e. y E B(yo, ro). Finally, by Lemma 6.6, we conclude that vn(Dn) = v(D) = B(yo, ro).

Remark 6.12 (i) It can easily be seen by the preceding proof that if the conclusion of Lemma

6.11 holds for r = r(xo) > 0, then it also holds for 0 < r' < r. Thus, as v is continuous on D, v(x) ,jA v(xo) for x E D and x # xo, we deduce that

lim max{Ix - xo12: x E D, v(x) E B(yo, r)} = 0. r-.0

(ii) It is possible to show that limn+oo GN(DODn) = 0. First, we prove that lim GN(D \ Dn) = 0. n-+oo Let FE := B(yoi ro -e) and O, := v-' (FE) fl D. We claim that for each fixed e there exists no = no(e) E N such that n a no implies OE C D. Indeed,

LOCAL INVERTIBILITY OF SOBOLEV FUNCTIONS

166

Ch. 6

since {vn} converges to v uniformly, there exists no - no(e) E N such that IIv - vnI1 < 2 for every n > no. If X E Oe, we obtain Ivn(x) - Yo12 S Iv(x) - Yo12 + 1v(x) - vn(x)I2 < ro

and so x E D. As UEO, = D and the sequence {O,} is nonincreasing, we have lim GN(D \ Oc)

0,

which, together with the fact that CN(D \ D,,) < GN(D \ O,) for n > no, yields limn+oo, N(D\Dn) = 0. Next, we prove that limn. +oo £N(Dn \ For

n > no. We have

{xE B(xo,Ro):r- 20 for every u,v E E, we have 2 2

Iu,VI>Iu-VI2

(7.17)

with equality if and only if u E OB or v E OR If N = 2 we define 7r1

Gl(u,v) :_ 1 1091U - vI2, G2(u, v) _ loglu, vl,

u0v,

and if N > 3 let Gl(u, v) G2(u, v) :=

-1 2(N - 2)wNlu - v12 -1

2(N - 2)wNlu, VIN -2'

u u

v' v,

where WN = LN(B). Define

G:=G1 -G2. For a E B and r > 0, we set E(a, r) = B \ B(a, r) rt B and S(a, r) = 8B(a, and we define

U:={(u,v):u,vE RN, ug6v}.

FIRST APPLICATION OF THE DEGREE THEORY TO PDES

§7.5

195

Lemma 7.37 Let F E C1(U) and let L > 0 be such that IF(u,v)12
0 such that (see Exercise 7.2) IG1(u,v)12
0 such that If(u) - f(v)12 < KIu - v12 for every u, V E B. Thus we can still apply Lemma 7.37 to eC u'9 If (u) - f (v)) to obtain e

If (u) - f (v)) du av; v fB'(u'v) 8G(u, v) 8 f (v) f 82G(u, v) du If (u) - f (v)) du - f 8v; 8v2 B B 8v,8vj

=

f BIG( v) If (u) - f (v)) du avick

If ,>(v) .

We deduce that 82x(v)

8v^

- JB

82G(u, v) If(u)8v;avi

b f(v))du+f(v), N

hence 02'(u,') avifty is continuous and (7.22) together with (7.23) yield Oz(v) = f(v)

for every vEB. Step 4. We prove that IIxIIz 0, and f E C(D x (0, +oo)). If f (x, 0) > 0, then (7.24) is said to be a positone problem, and there is an extensive literature on this subject (see Amann 1976, Ambrosetti and Hess 1980, and Hess and Kato 1980). We say that (7.24) is semi-positone if

f (x, 0) < 0 for all x E D.

(7.25)

Work on this class of problems can be found in Castro, Garner, and Shivaji (1993), and Smoller and Wasserman (1987), among others. For related applications of the degree theory to boundary value problems, we refer to Rabinowitz (1971, 1975).

We will show that a rescaling argument and the theory of Leray-Schauder topological degree will enable us to obtain positive solutions for the semi-positone problem (7.24) for certain values of A, provided f (x, ) grows linearly at infinity.

Precisely, we assume that there exists m > 0 such that 1(u) lira u-+oo U = m,

(7.26)

and we define

m where Al is the first eigenvalue of -A with zero boundary conditions. Let c1 be the corresponding eigenfunction, with v1 > 0 in D and IIVIIL2(D) = 1. As in the

DEGREE IN INFINITE-DIMENSIONAL SPACES

200

Ch. 7

previous section, we denote by F the Green operator for -A with zero Dirichlet boundary conditions; F : C(D) -i C(D) is a continuous, compact operator such that v = F(g) if and only if

Ov = g(v) V

=0

xED

xEOD'

In the case where f = f (u), we will show that A,,. is a bifurcation from infinity, i.e. there exist A,, -. \,,. and un E C(D) such that IIuIIn -, +oo and un - anF(f(Iunl)) = 0. Setting wn := unllunll-', elliptic theory and (7.26) yield that, up to a subsequence,

wn -' to in Cl"°(D) for some 0 < a < 1, where llwll = 1 and

Ow = ailwl 1W

=0

xED

xEBD'

(7.27)

By the maximum principle to > 0 and as to i4 0, we must have to = -fpl, for some -y > 0. Thus un > 0 in D for n large. Ambrosetti, Arcoya, and Buffoni (1994) used this argument to prove the theorem below, where

a(x) := limes (f (x, u) - mu) , A(x) := lira sup (f (x, u) - mu) . u-+oo

Theorem 7.40 If f E C(D x [0, +oo)) satisfies (7.25), (7.26), then them exists e > 0 such that (7.24) has positive solutions u E W2,p(D) n W02 2(D), for all p > 1, provided either (i) a > 0 (possibly +oo) in D and ,\ E [am - e, ate[ or (ii) A < 0 (possibly -co) in D and X E )A00, aoc + e). In the remainder of this section, following Ambrosetti, Arcoya, and Buffoni (1994), we prove that ,\oo is a bifurcation from infinity.

(7.28)

We remark that u is a positive solution of (7.24) if and only if u is a positive solution of u) = 0, where

t(,\, u) := u - \F(f(Jul)). We rescale 0 as

§7.6

SECOND APPLICATION OF THE DEGREE THEORY TO PDES IIzII 4 (A, T7

to

)

z

201

0

z=0.

Due to (7.26) the function W is continuous. It is clear that A is a bifurcation from infinity for (7.24) if and only if it is a bifurcation from the trivial solution

z=0forT=0.

Lemma 7.41 For every A < A there exists r = r- (A) > 0 such that 4i(µ, u) # 0 for all p E [0, AJ, I Jul I > r.

Proof Suppose, on the contrary, that there exist A' < A,,, {A :n E N), {u,,: n E N}, such that

An - A', IIunll - +oo, un = AnF(f(lunl)). Setting wn := unllunll-1, elliptic theory and (7.26) yield that, up to a subsequence,

win C" (D)

wn

for some 0 < a < 1, where IIwII = 1 and Aw = A'mlwl W

=0

XED

xEBD'

(7.29)

By virtue of the maximum principle, w > 0 and so w = rywl, for some ry > 0. Since IIwII = 1, and in particular w 34 0, we must have A'm = Al, which is in O contradiction with the hypothesis A' < A,,. The proof of the following lemma can be found in Ambrosetti and Hess (1980).

Lemma 7.42 If A > A there exists r = r+ (A) > 0 such that 4;(A, u) 96 t 0, l lul l > r.

Setting An := A - n and A,+, := A. + we choose an increasing sequence {rn} +oo and rn > max{r (A;), r+(A,+,) }. If A < A- , by Lemma such that rn 7.41 and Theorem 7.10 d (W(A, ), B (0,

n)

0 I = d (W(0, ), B = d (I, B (0, 1.

\

(O, !) '0)

')'0) (7.30)

On the other hand, by Lemma 7.42 and Theorem 7.10 and for all t E [0, 1), d

B

(0, n) , 0)

= d (W(A,+ ) - tcoi, B (01 n) +O)

202

DEGREE IN INFINITE-DIMENSIONAL SPACES

Ch. 7

We claim that

To this end, and by virtue of Theorem 7.8, it suffices to prove that the equation

T(A ,0 has no solutions on B (0, B (0, such that n)

n).

Indeed, suppose that if there is a function zn E T(,\n , zn) _ W1'

Setting un := IIznII-2zn, we have IIunII ? rn and

un - anF(f(Iunl)) =

(7.33)

Due to (7.26), we may assume that If(Iul)I 0 such that for every u, v E B, u 34 v, we have L

101(U, V)12

IG2 (u, v)i2 < - Iu FU_ vI2 '2'

IVuG1(u,v)I2 ` Iu-V1 N_1, 2

IvuG1(u,v)I2 < Iu

vI2 ,

IVuG2(u,v)I2