Minimal Surfaces of Codimension One

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MINIMAL SURFACES OF CODIMENSION ONE

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATICSSTUDIES Notas de Matematica (95) Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas and University of Rochester

Minimal Surfaces of Codimension One Umberto MASSARI Universita di Ferrara Ferrara Italy

and

Mario MIRANDA Universita di Trento Trento ltalv

1984

NORTH-HOLLAND -AMSTERDAM. NEW YORK. OXFORD

91

@

Elsevier Science Publishers B . V., 1984

All rights reserved. No part of thispublicalion may be reproduced, stored in a retrievalsystem. or transmitted, in any form o r b y any means, electronic, mechanical, photocopying, recording or otherwise, withour the prior permission of the copyright owner.

ISBN: 0444868739

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Arnstcrdain The Netherlands Sole disrrihutors for the U .S . A . arid Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue NewYork.N.Y. 10017 U.S.A.

Library of Congmsi Cataloging In Publlcatlon D a b

Massari, Umberto, 1948Minimal surfaces of codimension one. (North-Holland mathematics studies ; 91) (Notas de matedtica ; 9 5 ) Bibliography: p. Includes index. 1. Surfaces, Minimal. I. Miranda, Mario, 193711. Title. 111. Series. IV. Series: Notas de matedtica (North-Holland Publishing Company) ; 95. W . N 8 6 no. 95 CQA6441 510s ~516.3'61 84-1520 ISBN 0-444-86873-9 (u.s. )

PRINTED IN T H E NETHERLANDS

TO Francesca Mariangela Stefania

Marc0 and Michele


PREFACE

V a r i a t i o n a l Problems c o n c e r n i n g m a n i f o l d s , one o r more d i m e n s i o n a l , immersed in a n E u c l i d e a n s p a c e are among t h e most c l a s s i c a l o n e s .

W e mean

t h a t t h e y have been c o n s i d e r e d s i n c e B e r n o u l l i ‘ s time and have n o t o b t a i n e d a g e n e r a l s a t i s f a c t o r y treatment u n t i l t h e last t h r e e decades. A t t h e s t a r t o f t h i s c e n t u r y v e r y i n t e r e s t i n g new i d e a s a b o u t

V a r i a t i o n a l Problems f o r s u r f a c e s , are c o n t a i n e d i n t h e t h e s i s o f H . Lebesgue “ I n t e g r a l e , Longueur, A i r e “ .

Of t h e same p e r i o d o f t i m e , a r e t h e

i n t e r e s t i n g p a p e r s o f L. T o n e l l i a b o u t the l e n g t h o f t h e c u r v e s .

I n the

t h i r t i e s appeared t h e r e l e v a n t series o f p a p e r s by J . Douglas a b o u t the P l a t e a u Problem, t o g e t h e r w i t h some i n t e r e s t i n g c o n t r i b u t i o n s o f L. T o n e l l i c o n c e r n i n g V a r i a t i o n a l Problems w i t h two i n d e p e n d e n t v a r i a b l e s .

The s c h o o l

o f T o n e l l i , p a r t i c u l a r l y L. C e s a r i , worked a t t h e problem o f a d e f i n i t i o n o f t h e s u r f a c e a r e a , good from t h e v a r i a t i o n a l p o i n t of view.

But i t w a s

o n l y i n t h e f i f t i e s t h a t new d e f i n i t i o n s o f s u r f a c e s were i n t r o d u c e d and used f o r a g e n e r a l t r e a t m e n t o f c l a s s i c a l v a r i a t i o n a l problems l i k e t h e i s o p e r i m e t r i c p r o p e r t y o f t h e s p h e r e and t h e P l a t e a u Problem. I n t h e new a p p r o a c h e s , l i k e R e i f e n b e r g s u r f a c e s , Federer-Fleming i n t e g r a l c u r r e n t s , D e G i o r g i perimeters and Almgren v a r i f o l d s , i d e a s from t h e Modern Algebra, G e n e r a l Measure Theory and D i s t r i b u t i o n Theory a r e used t o g e t h e r w i t h t h e c l a s s i c a l arguments from D i f f e r e n t i a l Geometry and Real V a r i a b l e F u n c t i o n s Theory. With t h i s monograph w e do n o t i n t e n d t o p r e s e n t a l l t h e new approaches. A v e r y good and comprehensive p r e s e n t a t i o n o f t h o s e i s c o n t a i n e d i n

Geometric Measure Theory by H . F e d e r e r .

What we want t o d o h e r e , i s

e n t e r i n g i n t o t h e d e t a i l s o f i d e a s and r e s u l t s a b o u t t h e codimension o n e c a s e , where t h e c l a s s i c a l methods o f D i f f e r e n t i a l Geometry, C a l c u l u s o f V a r i a t i o n s and t h e new methods of Measure Theory and P a r t i a l D i f f e r e n t i a l E q u a t i o n s , gave t h e complete answer t o q u e s t i o n s l i k e B e r n s t e i n Problem.

viii

PREFACE

W e wish t o thank Leopoldo Nachbin f o r i n v i t i n g u s t o c o n t r i b u t e o u r work t o t h e Notas de Matematica s e r i e s .

We want t o e x t e n d o u r t h a n k s t o

the p e r s o n n e l of North Holland P u b l i s h i n g Company f o r having been unfailingly cooperative.

We are g r a t e f u l t o Augusto M i c h e l e t t i , s e c r e t a r y

o f t h e I n t e r n a t i o n a l C e n t e r f o r Mathematical Research i n T r e n t o , who d i d a n i n v a l u a b l e t y p i n g work.

October 1983

U.

Massari - M. Miranda

CONTENTS

.......................................................... I n t r o d u c t i o n .....................................................

Preface

vii xiii

Chapter one DIFFERENTIAL PROPERTIES OF SURFACES

.............................. ..................................... o f the g r a d i e n t .....................

1.1. Analytic r e p r e s e n t a t i o n o f s u r f aces 1.1.1. Unitary v ect o r f i e l d s 1.1.2.

Orthogonal p r o j e c t i o n

2

1.1.3.

C o o r d i n a t e s o f v e c t o r f i e l d s and g r a d i e n t s

3

1.1.5.

Surface curvatures

................ .................................. ........................................

3

Integrable vector f i e l d s

6

.............................. .........

7

1.2.1.

E q u a t i o n o f s u r f a c e s w i t h c o n s t a n t mean c u r v a t u r e

1.2.2.

Laplace o p e r a t o r

..........................................

..................................................... S u r f a c e area ..............................................

1.3. S u r f a c e a r e a 1.3.1.

............................... 1.3.3. Second v a r i a t i o n of t h e area .............................. An i s o p e r i m e t r i c i n e q u a l i t y ...................................... 1.3.2.

F i r s t variation of the area

9

9 11 14 16

...............................

20

1.4.2.

An isoperimetric i n e q u a l i t y

1.4.3.

The monotonic b e h a v i o r o f a r e a o f minimal s u r f a c e s

........

.................................................... ...............

A d i f f e r e n t i a l i n e q u a l i t y f o r minimal cones

.............. S l o p e o f minimal g r a p h s .......................................... 1 . 6 . 1 . Area estimate f o r minimal g r a p h s .......................... 1 . 6 . 2 . S l o p e e s t i m a t e f o r graphs ................................. 1.5.2.

8

16

A Sobolev t y p e i n e q u a l i t y

1.5.1.

7

.................................

1.4.1.

1.5. Minimal cones

1.6.

1

1.1.4.

1.2. S u r f a c e s of c o n s t a n t mean c u r v a t u r e

1.4.

1

F l a t n e s s o f s t a t i o n a r y s i x d i m e n s i o n a l cones

21 22 22 25 26 27 27

CONTENTS

X

.................. 34 ...... 35 ......................................... 36

1 . 1 . B e r n s t e i n theorem f o r f i v e d i m e n s i o n a l s u r f a c e s 1.7.1.

I n t e g r a l e s t i m a t e f o r c u r v a t u r e s o f minimal s u r f a c e s

1.7.2.

B e r n s t e i n theorem

C h a p t e r two SETS O F FINITE PERIMETER AND MINIMAL BOUNDARIES

......................................... 2.1.1. Radon measures ............................................ 2 . 1 . 2 . P e r i m e t e r of Lebesgue measurable s e t s ..................... The i s o p e r i m e t r i c i n e q u a l i t y .....................................

2.1. S e t s of f i n i t e perimeter

2.2.

2.3. 2.4.

2.5.

A f i r s t e s t i m a t e o f t h e measure o f a s e t by i t s p e r i m e t e r

2.2.2.

The i s o p e r i m e t r i c p r o p e r t y o f t h e b a l l s

................................................. Minimal b o u n d a r i e s ............................................... 2.4.1. Minimal b o u n d a r i e s ........................................ Reduced boundary

2.1. 2.8.

64

77

E x i s t e n c e and smoothness o f minimal b o u n d a r i e s

93 100 100

............

................................. ......................................... harmonic s u r f a c e s ............................. lemma .........................................

104

Smoothness of minimal b o u n d a r i e s

105

2.5.1.

Some i n e q u a l i t i e s

105

2.5.2.

Minimal and

109

2.5.3.

De G i o r g i ' s

117

........................... 129 S i n g u l a r p o i n t s o f minimal s u r f a c e s .............................. 136 131 2 . 6 . 1 . Tangent minimal cones ..................................... 2.6.2. S i n g u l a r minimal cones .................................... 140 2 . 6 . 3 . R e g u l a r i t y of s i x d i m e n s i o n a l minimal b o u n d a r i e s .......... 141 141 2 . 6 . 4 . Dimension o f s i n g u l a r s e t s ................................ simons cone ...................................................... 146 B e r n s t e i n problem ................................................ 150 2.5.4.

2.6.

43

. 79 ................... 86

2.2.1.

2.4.2.

43

C o n t i n u i t y of t h e t a n g e n t p l a n e

Chapter t h r e e THE D I R I C H L E T PROBLEM FOR THE M I N I M A L SURFACE EQUATION

............................... ..............................................

3 . 1 . The H i l b e r t - Haar e x i s t e n c e t h e o r y

152

3.2. D i r i c h l e t p r i n c i p l e

164

3 . 3 . Smoothness o f H i l b e r t

- Haar

s o l u t i o n s , t h e lgth

H i l b e r t problem

..

166

xi

CONTENTS

3.4.

The D i r i c h l e t problem f o r t h e minimal s u r f a c e e q u a t i o n . C a s e of s t r i c t l y convex domains

3.5.

..........................................

178

The D i r i c h l e t problem f o r t h e minimal s u r f a c e e q u a t i o n . G e n e r a l

case

.............................................................

181

..............................................

190

3.6.

Interior regularity

3.7.

Boundary b e h a v i o r o f v a r i a t i o n a l s o l u t i o n s

194

3.8.

The c a p i l l a r y problem

197

....................... ............................................ Chapter f o u r UNBOUNDED SOLUTIONS

................................

4.1.

Generalized s o l u t i o n s . Definition

4.2.

Compactness o f g e n e r a l i z e d s o l u t i o n s

4.3.

D i r i c h l e t problem w i t h i n f i n i t e d a t a

221

4.4.

Dirichlet

227

4.5.

Removable

............................. problem on unbounded s e t s .............................. s i n g u l a r i t i e s ..........................................

.........................................................

232

.......................................................

233

...................................................

241

..................................................

243

Appendix

References

Analytic index

L i s t o f symbols

.............................

217 220

228


INTRODUCTION

T h i s monograph i s composed o f f o u r c h a p t e r s and one appendix.

In

c h a p t e r one w e f i r s t l y show how t o d e s c r i b e a s u r f a c e by means o f i t s normal v e c t o r s .

W e g i v e t h e n formulas f o r t h e mean and t o t a l c u r v a t u r e s

and an e x p r e s s i o n f o r t h e Laplace o p e r a t o r o f t h e s u r f a c e by u s i n g t h e normal v e c t o r s and t h e d i f f e r e n t i a t i o n s i n t h e e u c l i d e a n s p a c e . Formulas f o r t h e f i r s t and second v a r i a t i o n s o f t h e area are g i v e n and an i s o p e r i m e t r i c t y p e i n e q u a l i t y i s e s t a b l i s h e d . With t h e a i d o f t h e s e new t o o l s t h e p r o o f s o f t h r e e w e l l known r e s u l t s a r e r e w r i t t e n : t h e f l a t n e s s of s i x dimensional s t a t i o n a r y cones, f i r s t l y proven by J . Simons; t h e s l o p e e s t i m a t e f o r minimal g r a p h s , by Bombieri-De Giorgi-Miranda and s u c c e s s i v e l y s i m p l i f i e d by N . T r u d i n g e r ; B e r n s t e i n Theorem f o r f i v e d i m e n s i o n a l s u r f a c e s , f o l l o w i n g Schon-Simon-Yau. C h a p t e r two s t a r t s w i t h r e c a l l i n g some u s e f u l f a c t s a b o u t e x t e r i o r and Radon measures.

D e G i o r g i ’ s t h e o r y of p e r i m e t e r s i s t h e n i l l u s t r a t e d , w i t h

i t s h i g h p o i n t s : t h e i s o p e r i m e t r i c p r o p e r t y o f t h e b a l l s and t h e r e g u l a r i t y

r e s u l t f o r minimal b o u n d a r i e s . minimal boundary,

The e s t i m a t e o f t h e s i n g u l a r s e t o f a

f i r s t l y g i v e n by H . F e d e r e r , i s shown.

The e x i s t e n c e o f

seven d i m e n s i o n a l s i n g u l a r minimal cones and t h e p r o o f of B e r n s t e i n Theorem f o r s e v e n d i m e n s i o n a l s u r f a c e s conclude t h e c h a p t e r . C h a p t e r t h r e e d e a l s w i t h t h e lgth H i l b e r t problem and t h e D i r i c h l e t problem f o r t h e minimal s u r f a c e e q u a t i o n .

A t t h e end o f t h e c h a p t e r t h e

c a p i l l a r y problem i s s o l v e d i n t h e c a s e o f a v e r t i c a l tube. The l a s t c h a p t e r d e a l s w i t h unbounded minimal s u r f a c e s , whose e x i s t e n c e i s e s t a b l i s h e d through a compactness argument.


CHAPTER ONE

DIFFERENTIAL PROPERTIES OF SURFACES

I n t h i s f i r s t p a r t w e s t u d y t h o s e p r o p e r t i e s o f s u r f a c e s o f codimension one i n e u c l i d e a n s p a c e s , which c a n be d e r i v e d from i n f o r m a t i o n s a b o u t t h e i r curvatures.

To t h i s p u r p o s e w e i n t r o d u c e a n a n a l y t i c d e f i n i t i o n o f s u r -

f a c e s , which y i e l d s a d i f f e r e n t i a l c a l c u l u s o n them a s s i m p l e a s t h a t i n euclidean spaces.

We e s t a b l i s h t h e n r e l a t i o n s h i p s between c u r v a t u r e s and

f i r s t and second v a r i a t i o n s o f a r e a . a l s o proved.

A basic isoperimetric inequality is

S p e c i a l remarks a r e made a b o u t c o n i c a l s u r f a c e s and s u r f a c e s

which a r e g l o b a l l y t h e graph o f a r e a l f u n c t i o n .

F i n a l l y w e prove t h a t

s u r f a c e s which a r e t h e graph o f a r e a l f u n c t i o n d e f i n e d o n

R5

and have

z e r o mean c u r v a t u r e must be f l a t ( B e r n s t e i n t h e o r e m ) . 1.1

ANALYTIC REPRESENTATION OF SURFACES

Here w e i n t r o d u c e a way of d e f i n i n g s u r f a c e s by t h e i r normal v e c t o r s , which v i e l d s a n a t u r a l d i f f e r e n t i a l c a l c u l u s on them.

I n p a r t i c u l a r we

g i v e a n a n a l y t i c e x p r e s s i o n f o r c u r v a t u r e s , which makes c a l c u l a t i o n s a b o u t those very easy. 1.1.1

U N I T A R Y VECTOR FIELDS

We shall w r i t e

E

n+ 1

t o indicate the

n + 1 dimensional e u c l i d e a n

d i m e n s i o n a l r e a l H i l b e r t s p a c e which w i l l b e r e p r e n+ 1 s e n t e d i n t h e c a l c u l a t i o n s by R , t h e ( n + l ) - t h power o f t h e r e a l s e t

space, i . e . t h e

n+l

Throughout t h i s monograph t h e e u c l i d e a n s p a c e w i l l b e o f dimension 3 , a t n+ 1 , which l o c a l l y c o i n c i d e s w i t h t h e l e a s t . A surface i s a subset of E graph o f a r e a l f u n c t i o n d e f i n e d o v e r a n open s e t o f

Rn.

A l l functions

c o n s i d e r e d are assumed t o b e L i p s c h i t z r e g u l a r a t l e a s t , u n l e s s e x p l i c i t a s s u m p t i o n s a r e made.

A boundary p o i n t of a s u r f a c e i s a l i m i t o f i t s

p o i n t s s e t , which d o e s n o t b e l o n g t o t h e s u r f a c e .

For r e a s o n s which w i l l

be clearer s h o r t l y , i n s t e a d o f c o n s i d e r i n g a s i n g l e s u r f a c e w e s h a l l d e a l

2


w i t h f a m i l i e s o f s u r f a c e s f i l l i n g up a g i v e n open s e t o f

,

En+'

by which

w e mean t h a t each p o i n t o f t h e open s e t b e l o n g s t o a unique s u r f a c e o f t h e f a m i l y and no boundary p o i n t s o f t h e s u r f a c e s b e l o n g t o t h e open s e t . I f a l l s u r f a c e s o f t h e f a m i l y have t a n g e n t p l a n e s a t a l l t h e i r p o i n t s t h e n , i n s t e a d o f looking a t t h e s u r f a c e s w e can c o n s i d e r t h e s e t of a l l t a n g e n t I n o t h e r words w e s h a l l a s s o n+ 1 c i a t e t o a f a m i l y o f s u r f a c e s f i l l i n g up t h e open s e t R C E a map

p l a n e s t o g e t h e r with t h e i r c o n t a c t p o i n t s .

where

i s t h e normal v e c t o r a t

v(x)

Obviously n o t a l l maps

V :

fi

R .

f a c e s o f a f a m i l y f i l l i n g up

t o t h e s u r f a c e through

x

.

r e p r e s e n t t h e normal v e c t o r s t o s u r -

Sn

-t

x

The f i r s t problem w e want t o s o l v e i s t o

c h a r a c t e r i z e such maps. 1.1.2

ORTHOGONAL PROJECTION OF THE GRADIENT

The g r a d i e n t o f a of

,

En+'

i s a map

df de

- (x)

where

=

C1

Df :

R

function f : -t R n+l E d e f i n e d by

l i m f(x+te)-f(x) t t+O

Df

where

R

i s a n open s e t

-f

For a g i v e n v e c t o r f i e l d of the gradient

,

V

e

and

:

orthogonal to

S?

v

.

satisfies

we can consider t h e p r o j e c t i o n

Sn

+

E Sn

max

T h i s new o p e r a t o r i s d e n o t e d by

6f and d e f i n e d by

Clearly

and i f

6

fl

s a t i s f i e s the i d e n t i t y

,

f2

a r e two f u n c t i o n s w i t h

f

1

(x) = f 2 ( x ) V x E

S

,

where

3


S

i s a surface orthogonal t o

1.1.3

,

V

6fl(x) = 6f2(x)

then

.

VxES

COORDINATES OF VECTOR FIELDS AND GRADIENTS Chosen a b a s i s

e . . e , = E . , is 1 3 11

vi

gradients

0

= ~ ' e . 1

and

D

6 ,

1

=

1

E

for

i = j

d dei

, 6.

~

1

n+ 1

for

= e

i

,

,

i

e.

i.e.

E

E

with r e s p e c t t o t h e b a s i s

n+ 1

-6 =

{ei)

and

and w r i t t e n

1

D.-V. 1

1

.

If

V

h= 1

the coordinates of the vector f i e l d

v* , D* , 6;

n+ 1

v

h

D

h

'

and t h e

is

{e;}

are t h e c o r r e s p o n d i n g c o o r d i n a t e s o f

1 1 w e have t h e f o l l o w i n g r e l a t i o n s h i p s

v.

=

'

1.1.4

6

and

D . = eD :

fii

1

a n o t h e r b a s i s and

V, D ,

,

v . , Di,

we s h a l l c a l l

,...,n + l

Cei'1=1,2 for i f j

C(ei.e*)v*

1 1

1

,

Di

, 6i

= C(ei-e*)D*

I

1

3

= C(e

j

i

-e*)6* l

l

.

INTEGRABLE VECTOR FIELDS

v

W e say

:

R

+

S

n

i s integrable i f i t is t h e f i e l d of orthogonal

R

v e c t o r s t o a f a m i l y o f s u r f a c e s f i l l i n g up

.

W e wish t o g i v e now a n

a n a l y t i c c h a r a c t e r i z a t i o n o f i n t e g r a b l e v e c t o r f i e l d s o f class

C1

.

We

( 6 . v . ) , whose symmetry i s i n d e p e n d e n t 1 1 In f a c t the identity

c a n a s s o c i a t e t o them t h e m a t r i x of t h e choice of

{ei}

.

shows t h a t t h e symmetry o f

(6;; V;)

( 6 i ~ i ) i s e q u i v a l e n t t o t h e symmetry o f

.

We can now answer t h e q u e s t i o n whether a

C1

v

vector f i e l d

is

i n t e g r a b l e o r n o t by t h e f o l l o w i n g THEOREM 1 .

the matrix PROOF.

A

C1

(6.v.) 1 1

vector f i e l d

v

:

R

+

Sn

i s i n t e g r a b l e i f and o n l y i f

i s symmetric.

L e t u s check f i r s t t h e " o n l y i f " .

The symmetry o f

( 6 . v . ) being a

l o c a l p r o p e r t y we c a n assume t h e f a m i l y o f s u r f a c e s f i l l i n g up

1 7

R

given a s

4


t h e l e v e l sets o f a

C2

v

The v e c t o r f i e l d

function

g :

f2

i s t h e n d e f i n e d by

+

R

with

Dg(x)

f

0

w ( x ) = Dg(x) / D g ( x )1-l

Vx

E Q

.

and t h e

( 6 . v . ) can be w r i t t e n a s

matrix

1 1

which i s o b v i o u s l y symmetric. A s f a r a s t h e o t h e r i m p l i c a t i o n i s concerned w e w i l l prove t h e l o c a l

e x i s t e n c e o f t h e s u r f a c e through a f i x e d p o i n t , i t s e x t e n s i o n up t o t h e

R

boundary of If

E Q

x

function

b e i n g a s t a n d a r d P a r t i a l D i f f e r e n t i a l E q u a t i o n s argument.

i s a f i x e d p o i n t assume t h a t

f

: A

i n

x+ 1 y i e i + f ( y ) e

i=l

L e t u s choose

a

,

R

-f

n+ 1

where

A C Rn

Iy E A 1

E (O,+m)

= v(x) and look f o r a n+ 1 i s o p e n , such t h a t i t s graph

e

i s the required surface.

such t h a t

n

X ~ . e . + t e ~> +0 ~ v) / y \ < a , V l t l < a . We need t h e n 1 1 i=l t o prove t h e e x i s t e n c e o f b E ( 0 , a ) and f : A = { y E R n / I y / < b } + R

~ , + ~ ( y , t=)

W,+~(X+

satisfying

where

A , ( y , t ) = -vi(y,t)

The s o l u t i o n

f

I

vn+l

(y,t)

of ( 2 ) c a n be o b t a i n e d through s u c c e s s i v e i n t e g r a y1 , y 2 ,..., i s the solution of the i n t e g r a l equation

t i o n s of t h e d i f f e r e n t i a l system w i t h r e s p e c t t o t h e v a r i a b l e s yn

.

I n o t h e r words

f(yl,O)

h e r e and i n t h e n e x t l i n e s t h e 0 ' s have d i f f e r e n t d i m e n s i o n s Similarly

f(y1,y2,0)

i s the solution of

5


and so o n . The f u n c t i o n s

A.

1

being continuous w i t h t h e i r f i r s t d e r i v a t i v e s , a

c l a s s i c a l e x i s t e n c e argument l e a d s t o t h e e x i s t e n c e of f

:

{YE R n l l y l < b} + R

bE (0,a)

and

s a t i s f y i n g a l l i n t e g r a l e q u a t i o n s w e have i n d i c a t e d .

A s a consequence o f t h a t

f

f(0) = O

s a t i s f i e s the i n i t i a l condition

and t h e d i f f e r e n t i a l i d e n t i t i e s

up t o

which a r e l e s s t h a n ( 2 ) .

To f i l l t h e gap we u s e a g a i n t h e i n t e g r a l

e q u a t i o n s and t h e symmetry of t h e m a t r i x consequence o f t h e symmetry o f

(6.v , ) 1 1

(D.A.+AiDn+lAj)

1 3

1

1

1

A.=~.A.+u. E n+l I 1 I 1 h=l

which i s a

:

n+ 1 D.A.+A,D

,

n+ 1

v

By d i f f e r e n t i a t i n g ( 4 ) w i t h r e s p e c t t o

D A.+Ai6n+lAj+AiVn+l h h i

y1

w e obtain

E h= 1

V D A

h h j

=


6

the equation ( 8 ) , seen as a linear integral equation for D f 1

By a repeated use of this argument one can verify that

f

is satisfied

solves (21. q.e.d.

1.1.5

SURFACE CURVATURES

From now on the vector fields v CL

and the matrix

open sets

R

of

’

( 6 . V ,)

fi

:

symmetric.

-f

Sn

are supposed to be of class

In other words we shall consider

n+ll E filled up with

C3

surfaces. What is crucial for

the sequel is to give an expression of geometric invariants of the surfaces in terms of

(6. v , )

and i,j

1 7

v .

It is easy to see that the quantities

CiGivi

, i .e. the trace and the squared norm of the matrix

( 6 . v , ) , are independent of the basis

{ei}

1 3

and equal to the sum of

principal curvatures and the sum of the squares of principal curvatures. In fact f o r a fixed el,e2,..., e

x E fi

let us choose

the surface through

en+,=v(x)

, whose

f

such that the function

x , satisfies D.D.f(O)= O 1

and the

graph represents locally

7

Under these conditions the principal curvatures of

i+j, i,j=l,2, ...,n S

at

x

.

are given by

SURFACES OF CONSTANT MEAN CURVATURE

{ - D i D i f ( 0 ) } i = l ,2

6 , V , (x)= - D . D

and

,..., n

6n+lwi (x) = 6 i. vn + l ( x )=O ( i = 1 , 2 , .

.

. f (0)( i ,j = 1 , 2 , . , , n )

1 1

1 1

,

. ., n , n + l ) .

What w e c l a i m e d i s t h e n o b v i o u s . 1.2

SURFACES OF CONSTANT MEAN CURVATURE W e w r i t e down, by means of L a p l a c e o p e r a t o r , t h e d i f f e r e n t i a l

c h a r a c t e r i z a t i o n of s u r f a c e s of c o n s t a n t mean c u r v a t u r e . EQUATION OF SURFACES WITH CONSTANT MEAN CURVATURE

1.2.1

The s u r f a c e s h a v e c o n s t a n t mean c u r v a t u r e if a n d o n l y i f

or

,

= 0

E6,(6,Vh)

k = 1 , 2 ,...,n +

h

T h e s e e q u a t i o n s become more i n t e r e s t i n g i f w e e v a l u a t e t h e c o m m u t a t o r A kh =6

6,6,

6 -6 6

.

To d o t h i s l e t u s c a l c u l a e t h e n o n symmetric p a r t of

:

=symm-EV 6 V D. = h k i i i

,

where w e have u s e d t h e i d e n t i t y

0 = 6,l

=

EfikV: i

=

2LVihkVi i

The c o m m u t a t o r w e w e r e l o o k i n g f o r is t h e n g i v e n by

8


The c o n s t a n t mean c u r v a t u r e c o n d i t i o n s become

(k=1,2,

...,n+l)

or

2

~ E h G h \ ) + cv=O

,

where

h

c

-27

2

2

(AiVh)

.

i,h

1.2.2

LAPLACE OPERATOR n+ 1 II 6,6, , which a p p e a r s i n t h e d i f f e r e n t i a l c h a r a c t e r i The o p e r a t o r h=l z a t i o n o f s u r f a c e s w i t h c o n s t a n t mean c u r v a t u r e , d o e s n o t depend on t h e choice of t h e b a s i s

The v a l u e o f choosing

6,6,$ h

i$

h e,+l=v(x)

x

is a

(X) =

, in

fact

E6h6ha t a f i x e d p o i n t

a t the point So if

{ei}

,

R

c a n b e c a l c u l a t e d by

then

.

C2

2

x E

DhDh h= 1

function w e obtain

4 (Y.f (Y))

where

f

is t h e f u n c t i o n whose

graph g i v e s t h e s u r f a c e through

The formula we have found p r o v e s t h a t o p e r a t o r on t h e s u r f a c e .

,

LFhfih

x

c o i n c i d e s w i t h t h e Laplace h W e w i l l l o o k a t t h i s f a c t from a d i f f e r e n t p o i n t

o f view i n t h e n e x t s e c t i o n .

.

9

SURFACE AREA

1.3

SURFACE AREA We d e r i v e t h e formulas f o r f i r s t and second v a r i a t i o n o f s u r f a c e a r e a .

1.3.1

SURFACE AREA Area o f t h e s u r f a c e

n

i s i t s Hausdorff

S

d i m e n s i o n a l measure,

which c o i n c i d e s l o c a l l y w i t h t h e c l a s s i c a l measure o f g r a p h s o f f u n c t i o n s ,

i.e. i f H (S)

,

S=graph f

where

= / , j m 2dy

j A a ( y , f ( y )) Hn(S)
O

%*

t h e s u b f a m i l y of

t h e r e e x i s t a n open s e t

%

AZJB

c1

.

In

containing and a c l o s e d

with

I t i s e a s i l y seen t h a t

%* c o n t a i n s a l l bounded open s e t s , t h e

complements o f a l l i t s elements and t h e i r f i n i t e u n i o n s .

%* i s a

,

a - a l g e b r a i s reduced t o c h e c k i n g t h a t

i n c r e a s i n g sequence o f e l e m e n t s o f

%* .

L e t u s c o n s i d e r , i n f i r s t p l a c e , t h e case when

Uh Bh E 8*

The p r o o f t h a t

if

{Bh)

is an

SETS OF F I N I T E PERIMETER AND MINIMAL BOUNDARIES

48

For a given

, choose Ch C Bh C A h

E>O

a(Ah-Ch)
0

3 B1 3 X 1 , B2 3 X 2

with

56

SETS OF FINITE PERIMETER AND M I N I M A L BOUNDARIES

B2EB

BIE$,

and

.

dist(B1, B2) > O

About t h e l o c a l f i n i t e n e s s o f

aB '

a. < a .

i t f o l l o w s d i r e c t l y from t h e i n e q u a l i t y

a-

I t o n l y remains t o check, f o r a l l

a. ( X ) < B

To t h i s p u r p o s e w e may assume B(X-B)

=

0

{Ah} =

o

P

.

a ( A ) O

choose

,

A 3 X - B h

a

B

,

and p i c k up a

+a

For

We have t h e n

A ~ = ) 0)

X c E

(A )

h

c

0

AhEA

since

such t h a t

BE $

with

A E A

,

B(Ah) < h - l

c1 ( A )

P

h

and

A x B

and

B hfl Ah)

c1 (

=

.

Hence w e g e t

t h e r e e x i s t s then

h

with

We have so proved t h a t f o r a l l

t h e r e e x i s t s an open s e t

X c E

AUA 2 X

h

n+l with

with

a (x)< + m and a l l B

a (AUAh) B

0 ,

which i s

exactly (13). q.e.d.

I n t h e p r o o f of

( 1 3 ) we have s e e n t h a t

Xh+

,

B(Xh)

which i s what one u s u a l l y means by a b s o l u t e c o n t i n u i t y o f

to

B .

$0

* a B( X h)

ag

4.0

,

with r e s p e c t

The fundamental r e s u l t w e are g o i n g t o e s t a b l i s h i s a R e p r e s e n t a t i o n Theorem f o r

aB i n terms of

D : E

as

n+1 -f

[O,

+a]

B

.

To t h i s p u r p o s e l e t u s d e f i n e

57

SETS OF FINITE PERIMETER

By

BO(x) w e i n t en d t h e cl o s ed b a l l o f r a d i u s

p , centered a t

x , i.e

We s h a l l prove t h e f o l l o w i n g RADON-NIKODYM-VITAL1 THEOREM.

The Sets

{X E E

n+ 1

\D(X) =

+m}

and

1 and function

is

D

6-

and

13

B

-measurable

he

(*).

Moreover

For p r o v i n g Radon-Nikodym-Vitali Theorem w e o u g h t t o g e t t h r o u g h some c o n s i d e r a t i o n s a b o u t t h e geometry o f

En+'

.

To t h a t p u r p o s e l e t p u t ,

f o l l o w i n g H . F e d e r e r L291: DEFINITION.

f

A family

T-controlled,

where

o f p r o p e r , c l o s e d b a l l s of

T E (1,

,

E

n+l

if

The f o l l o w i n g P r o p o s i t i o n e s t a b l i s h e s a r e l a t i o n s h i p between =n+ 1 f a m i l i e s and g e n e r a l f a m i l i e s o f b a l l s o f PROPOSITION 3 .

If

equibounded r a d i i ,

(*)

The s e t s

f

i s s a i d t o be

i s a family of proper closed b a l l s of

T-controlled

E

that is

{ x l D ( x ) > t} are

B-

and

a B -measurable

V t

.

n+ 1

with

58

S E T S OF F I N I T E PERIMETER AND MINIMAL BOUNDARIES

for all

PROOF.

T € (1,

+a)

, there exists

Let us consider all

a

T-controlled subfamily

-t-controlled families H c f

with

P;cf

satisfying the

following weaker implication

A

family containing only one ball

{Bs(b) 1

chosen in order to satisfy

obviously satisfies (15). Moreover the set of all families like the inclusion order, satisfies the hypothesis of Zorn Lemma. count on the existence of maximal families H

.

H

with

So we can

It is easily seen that any

maximal family satisfies (14). q.e.d. A remarkable property of

T-controlled families of balls is given by

the following proposition: PROPOSITION 4.

If P;

is a

-t-controlled family of proper closed balls

with equibounded radii, then there exist families K c P ;

of disjoint balls,

such that

PROOF.

For the proof of implication (17) we follow the same pattern as for

the proof of (14). We weaken ( 1 7 ) by requiring

(18)

Br(a)EG-K*

3 Bs(b)EK

with

l a - b \ L r + s and

"Br (a) does not intersect the balls of K " .

s>r

or

SETS OF FINITE PERIMETER

F a m i l i e s o f one s i n g l e b a l l

{Bs(b)} satisfying

t o o t h e h y p o t h e s i s o f Zorn Lemma i s f u l f i l l e d .

59

( 1 6 ) s a t i s f y (18) a n d h e r e It i s again true t h a t a

maximal f a m i l y s a t i s f y i n g ( 1 8 ) a c t u a l l y s a t i s f i e s ( 1 7 ) . q.e.d.

I f i: i s a - r - c o n t r o l l e d , f a m i l y of p r o p e r c l o s e d FUNDAMENTAL LEMMA. n+ 1 b a l l s of E , i f T = 1 . 1 and i f t h e r e e x i s t s B - ( a ) €1; w i t h L

V Bs ( b )

T

+1

2 ( l l n + l - gn+l)

than PROOF

of

+ s,

Bs(b)

Bs(b)

EL

and

,

= 0.2

Put

where w e have p u t

,

with

x

t h e n t h e number o f e l e m e n t s o f

’-‘

K = __

O < Ib-a( < K r

with Bt(c)

.

E i: ,

and e v a l u a t e , i n f i r s t p l a c e , t h e number

.

For two d i f f e r e n t b a l l s of t h i s t y p e ,

/c-al(/b-al

a + -t c - a l

i: i s l e s s

,

w e must have

.

(b-a) I b - at Then t h e number of t h o s e b a l l s c a n n o t b e g r e a t e r t h a n

(1 + ‘ / 2 P + l -

=

(1 - rl/an+l

-

I l n + l - gn+l

( V 2 ) n+l A s i m i l a r argument can be r e p e a t e d f o r t h o s e b a l l s

Ib - at , K r

.

Bs(b)

EL

s.t.

I n t h i s case

Hence t h e number o f t h e s e o t h e r b a l l s d o e s n o t p a s s l l n + l - 9n+1

,

either. q.e.d.

60


Two o t h e r Lemmas a r e a d i r e c t consequence o f t h e Fundamental Lemma. LEMMA 1.

F i s a f a m i l y o f p r o p e r c l o s e d b a l l s w i t h equibounded r a d i i ,

If

K,, K2,

then there e x i s t

F ,

n+l

N < 2(11

with

-

...,

KN ,

+1

and

9n+1

)

f a m i l i e s of d i s j o i n t b a l l s contained i n

{a}/Br(a) EF jcN BEKj

PROOF.

Chosen

L e t choose

T = 1.1

Klc6

,

1

.

6 c F be t h e s u b f a m i l y of P r o p o s i t i o n 3 .

let

6- K,

a s i n P r o p o s i t i o n 4 and r e p e a t t h i s c h o i c e i n

and so on, till we g e t b - K n+l 2 ( 11

reaches the value

1

-K

2

- 9n+1)

-

... - K N = $ ,

+1

which must happen b e f o r e

N

b e c a u s e o f t h e fundamental l e m m a . q.e.d.

p :X

+

y

Given a non n e g a t i v e Radon measure

LEMMA 2.

(0,

+m),

s u p { p ( x ) ( x € X j< + m

and a bounded s e t

there e x i s t

Y CX

and

6 :Y

X

+

with

( 0 , + ")

such t h a t

PROOF.

L e t u s a p p l y Lemma 1 t o t h e f a m i l y

(

lBp(xi)(~i)

li

.

=

For a t

g i v e n by t h a t Lemma 1 , w e must have

l e a s t one o f t h e s u b f a m i l i e s

where

F

1

is t h a t special

Kj

.

We have t h e n

By r e p e a t i n g t h e same argument f o r t h e s e t

x- UB i

(xi) ( x i )

with t h e

61

SETS O F FINITE PERIMETER

following modification of t h e function

C(x) (min

1

p

p(x),distc,UB

. Phi)

,

(xi))!

we o b t a i n a n o t h e r f i n i t e f a m i l y o f d i s j o i n t b a l l s

VxEX-UB i

lk

/BbiXkr(xk)

such t h a t

By r e p e a t i n g i n d e f i n i t e l y t h i s argument w e g e t t h e p r o o f o f o u r s t a t e m e n t . q.e.d.

PROOF OF THE DIFFERENTIATION THEOREM L e t u s check f i r s t

For t h a t it i s s u f f i c i e n t t o prove

(20)

X c {x/D(x) =

Lemma 2 w i l l g i v e u s ,

+m)

v

,

X

a E (0,

bounded

+m)

,

o(X) = 0

=)

.

a sequence

closed b a l l s , with

W e have t h e n

Being

a

a r b i t r a r y , we o b t a i n

The same argument a p p l i e d t o

R(X) = 0

,

t h a t is ( 2 0 ) .

X C { x l D ( x ) > a)

,

proves

62


A c t u a l l y t h e argument must be a p p l i e d t o a l l open sets

ADX

.

One

obtains, i n f i r s t place,

from where one a r r i v e s t o ( 2 1 ) . Hence w e have t h e f o l l o w i n g i m p l i c a t i o n

By t h e combined a p p l i c a t i o n o f ( 2 1 ) and ( 2 2 ) w e o b t a i n

W e have so proved t h a t

To check t h a t

{ x \ D ( x )> t)

is

6-

and

a -measurable w e o n l y have t o

8

prove t h a t ( c f r . t h e argument i n t r o d u c e d f o r p r o v i n g P r o p o s i t i o n 1 ) and

O h - l

,

where

h

and

ITh(x)dx = 1

, we

66


we g e t

I,

div$(x)dx(liminf h

I

/ D q h ( x )/ d x

A

,

if

$ E

which p r o v e s ( 5 ) . To prove ( 6 ) l e t u s f i r s t o b s e r v e t h a t

Then, from

and t h e p o s i t i o n

- T ~ * x , w~e g e t

$

W e have t h e n

I

/D$h/dx(PA(X)

,

V h>

E - ~

,

A-E

that is:

limsuph

I

ID$hIdxO

P(X)

.

The

is very simple,

as we will show in first place. In the one dimensional case the P(X) < m * 1 either H1(X) or H1(E - X ) is finite, is an obviously false implication. The analysis of the one dimensional case, together with a remark about the slicing of higher dimensional sets, will easily imply the following estimate

(3)

1

min Hn+l (x),H,+~ (E~+'- x)

1

n+ 1 -

(~(x)

.

, v ~ > o, V X ~ E ~ + '

( 3 ) is an improvement of the inequality proven in 1.4.2 because it holds

for all measurable sets and because the coefficient of which is strictly less than

c * (n)

.

P(X)

n+ 1 -

is 1,

We will successively improve ( 3 ) by

showing that the balls maximize the ratio

among all bounded measurable sets, which will imply the validity of the isoperimetric inequality for all bounded sets with the constant c(n)

.

The extension of this to all measurable sets will follow quite easily from that.

THE ISOPERIMETRIC INEQUALITY

2.2.1

79

A FIRST ESTIMATE OF THE MEASURE OF A SET BY ITS PERIMETER

ONE DIMENSIONAL SETS W I T H FINITE PERIMETER

An e a s y computation shows t h a t t h e h a l f l i n e s o f

E1

have p e r i m e t e r

e q u a l t o one and t h e f i n i t e i n t e r v a l s have perimeter e q u a l t o two. generally a l l subsets of

EL

More

whose b o u n d a r i e s have a f i n i t e number o f

p o i n t s , t h a t i s t h e s e t s which are f i n i t e u n i o n s o f i n t e r v a l s and h a l f

l i n e s , have p e r i m e t e r e q u a l t o t h e number o f t h e i r boundary p o i n t s .

What

i s e v e n more i n t e r e s t i n g i s t h e v a l i d i t y o f t h e i n v e r s e i m p l i c a t i o n ; c o n t a i n e d i n t h e f o l l o w i n g lemma: LEMMA.

c a r d ( 3 X ) (P(X)

PROOF.

If

f o r a l l measurable s e t s

to

x.

ax

are d i s t i n c t p o i n t s o f

{xi}i5k

then t h e r e e x i s t p o i n t s

,

{yilick

X cE1

{zi},,,

.

and i f

with

yi

9, and

= Th*XX

z.

,

as c l o s e

as w e w i s h , s . t .

lim

h

$ (y.) = 0 h

i

,

li\

$

h

(2.)

i

= 1

V i c k

We have t h e n

W e can d e s c r i b e t h e whole s i t u a t i o n w i t h t h e f o l l o w i n g s t a t e m e n t . THEOREM,

1.

The v a l u e

The f u n c t i o n 0

i s t a k e n on

half lines, the value

2

1 P : m(E ) -+ 10, + a 7 h a s i n t e g e r values o r 4 and E 1 o n l y , t h e v a l u e 1 i s t a k e n o n

+a.

i s t a k e n on f i n i t e i n t e r v a l s or t h e i r comple-

ments, a l l o t h e r p o s s i b l e i n t e g e r values are t a k e n f o r t h e u n i o n s of i n t e r v a l s and h a l f l i n e s . SLICING OF MEASURABLE SETS OF I t i s w e l l known t h a t i f

almost a l l

zEES

t h e set

En+'

XcE

n+l

= E

k

X E

s

i s measurable t h e n , f o r

SETS OF FINITE PERIMETER AND MINIMAL BOUNDARIES

80

is measurable and the function of

H

z

,

Hk(XZ)

is measurable and satisfies

nt

One may ask whether a similar relationship exists between the function

,

P(Xz)

which we expect to be measurable, and the number

Examples provided by the elementary geometry in

E2

P(X) or

. E3

,

show

that

where the strict inequality occurs quite often. P(XZ)dz represents in general, ES we need to give a new notion of "partial perimeter". In order to see what the integral

,f

DEFINITION OF PARTIAL PERIMETER. n+1 XCE

For

In case A P (X) < k,A

=

9,

measurable and

n+ 1 E we shall write

ACE

Pk

n+ 1

open, we put

instead of

P n+l * k,E is equivalent to say that the linear functionals

are bounded with respect to the norm

rnaxl@(x)

1 .

Then by Riesz theorem,

X

P (X) < k,A

I1

"i (i

k

+ a

is equivalent to the existence of

, such that

k

finite Radon measures


a r e u n i q u e l y d e t e r m i n e d and t h e t o t a l v a r i a t i o n on

/liii< -

'he

t h e v e c t o r measure

P

81

krA

a

1

=

= sup/!

(X)

coincides with

yi[izk

-4

7

1

i = l

/El.

1

(X) P k,A

,

of

A

i n o t h e r words

d i s j o i n t and c o n t a i n e d i n A

'j

A s we have remarked f o r t h e t o t a l p e r i m e t e r s

P(X)

and

PA(x)

,

!

we

have f o r t h e p a r t i a l p e r i m e t e r s too t h e r e l a t i o n s h i p s

(5)

$h = T h * x x

where

.

W e can prove now t h e f o l l o w i n g theorem

2.

THEOREM,

,

zEES

XcE

n+l

= E

k

X E

s

i s measurable,

then the f u n c t i o n P(Xz)

i s measurable and i t s i n t e g r a l s a t i s f i e s

If

PROOF.

If

A C E

n+ 1

i s open, we have

W e r e s t r i c t ourselves t o t h e proof o f

(71, the proof o f ( 8 ) being

similar. For a l m o s t a l l

H -almost k have t h e n

z E ES

t h e sequence o f f u n c t i o n s

{$h(y, z ) I h

everywhere t o t h e c h a r a c t e r i s t i c f u n c t i o n of the s e t

converges Xz

,

we

,

82


S o , a f t e r r e c a l l i n g t h e d e f i n i t i o n o f upper i n t e g r a l

w e have

-

i

\ 4

P ( X z ) d z c \ s ( l i m i n f h Ek

i = l

E

ES

By F a t o u Lemma,

As t o t h e i n e q u a l i t y i n t h e o p p o s i t e d i r e c t i o n , w e have

for almost a l l

z EES

and a l l

i n t e g r a t i n g ( 1 0 ) w i t h respect t o P(Xz)

,

4E z

[

C

1

,

k

n+l (E

(9),

I $ ( x ) 1'1

.

By

w e g e t , f o r t h e lower i n t e g r a l of

t h e f o l l o w i n g estimate

By t a k i n g t h e supremum i n ( 1 1 ) w i t h r e s p e c t t o

P(Xz)

with

)]

$

, we

get

(12) and a w e l l known p r o p e r t y of lower and upper i n t e g r a l s , imply

i s measurable and i t s i n t e g r a l i s e q u a l t o

Pk(X)

. q.e.d.


A FIRST ESTIMATE FOR THE MEASURE OF A SET OF We shall prove, by induction on PROPOSITION.

For all integers

n

En+'

83

BY ITS PERIMETER

, the following ,

and all measurable sets X

n, 1

we have

PROOF.

Assume first n = 1

.

For the Theorem, 2 with

k = s

have

Then, if

Since

(13)

X Y

, we have

P(X) < + m P(X ) < + Q I Y

, for almost all ~ E E ' ,

P(Xz)
O }

Y

5

1,'

P(X )dy(P(X) Y

O

would imply P(xz) > O

.

, for almost all z E E1 ,

which is impossible, we must have either

,

=

1

, we


84

~

,~o r H ~ I Y I =X 01 ~

{ = Ey1 I l= o~

~

.

= 0

Assuming

we would have, r e c a l l i n g ( 1 3 ) ,

(15) t oget her w i t h

min{Hl{zlX

H1{z/XZ = E 1 1 = 0

(17)

= E

1

I ,

H1{z\XZ =

011

= 0

gives

,

therefore

From (16) and ( 1 8 ) w e g e t

I n c a s e we had

(15')

Hl{yIXy =

01

=

,

0

i n s t e a d of ( 1 5 ) ,

w e would have g o t

H2(E

n+ 1

-X) cP(X)

2

.

Then o u r s t a t e m e n t i s proven f o r

n = 1

.

Assume now o u r s t a t e m e n t v a l i d f o r s u b s e t s of

En

where

f i x e d , l e t u s see how i t s v a l i d i t y can be extended t o s u b s e t s Assuming

k = n

,

s = 1

i n Theorem 2 , w e have

n> I X

of

is n+ 1 E


since the

X Y

85

are one-dimensional sets, we have

So, by the induction hypothesis, we must have

(19) and (22), together with

P (X)
0

,

one h a s

The i s o p e r i m e t r i c i n e q u a l i t y a p p l i e d t o t h e set

XnB

P '

for

p E ( 0 , Po) ,

g i v e s then

I t i s q u i t e obvious t h a t

o n l y on

n

.

c(n)

d e n o t e s d i f f e r e n t r e a l numbers, depending

95

REDUCED BOUNDARY

I f w e d e n o t e now w i t h

y(p)

t h e monotone f u n c t i o n o f

= H

n

(aB

P

nx)

,

H ~ + ~ ( x ~, B ~ )

p

w e have f o r a l m o s t a l l p o s i t i v e

y'(p)

P

t h e n from (8) one g e t s

n

,

< c(n)y'(p)

This, together with the f a c t t h a t

for almost a l l

y(p) 2 0

for

p E (0, po)

p>O

,

implies

that is

The same i n e q u a l i t y c a n be e s t a b l i s h e d f o r t h e complementary s e t E ~ + '- x

,

i.e.

The i s o p e r i m e t r i c p r o p e r t y o f t h e b a l l i m p l i e s t h e f o l l o w i n g t h r e e relations

1 (13)

Hn+l(B

) =

( n + 1) T h e r e f o r e we g e t

n

1 n+ 1

w

n+ 1

n+ 1 n

__.

SETS OF F I N I T E PERIMETER AND MINIMAL BOUNDARIES

96

Therefore (14) implies t h a t : 5

This inequality, together with the isoperimetric inequality,

( 1 0 ' ) and

( l o " ) , gives

f o r some

c(n) > O

.

L e t u s d e f i n e now

and p u t

W e have

then (7) w i l l give us

ja(P)((Ba)(c(n)Bn

,

T h e r e f o r e t h e f a m i l y of s e t s

B E (0, 1)

for

cx

(P) } P < Po

, p

E ( 0 , po)

.

i s compact w i t h r e s p e c t t o t h e

1

L ( B )-convergence.

1 There will e x i s t sequence

p $0 h

and sets

Xo

such t h a t

97

REDUCED BOUNDARY

X

(Oh) -+

,

Xo

h

1

i n the

L (B )-convergence.

1

We w i l l v e r i f y t h a t n e c e s s a r i l y

Xo = { x E B

1

I X n+l < o }

.

~1 (O)

I n f a c t , l e t us denote with

t h e v e c t o r measure

.

- D

W e have,

xXO

a € (0,

f o r almost a l l

,

1)

b u t , from ( 3 ) we o b t a i n

a ( P h (Ba ) ) (18)

...,

( 0 , 0,

-f

1 (Bu)

0 , 1)

l a ( o )I

Then, c o n s i d e r i n g t h e f a c t t h a t (16) i m p l i e s

a(') la")

(Ba)

I (Ba)

.. .,

= (0, 0,

0, 1)

,

(Ba) > 0

f o r almost a l l

, we

have

o t ( 0 , 1)

This implies

Then

"0

I

B1

x

o n l y depends on

0

From ( 1 0 ' ) and ( 1 0 " ) one h a s

(20)

{x E En+l

1

0

of b a l l s covering

and such t h a t any s u b f a m i l y o f

y(n) + 1

, Y

if

,

i s bounded, one c a n

Y

with

b a l l s h a s a n empty i n t e r s e c t i o n .

T h e r e f o r e one h a s

where

Yu

Letting

i s t h e open s e t 0 J- 0

n+ 1

.

Idist(x, Y) O

the existence

w i t h t h e same p r o p e r t i e s a s

lal-null set

Yo

.

T h e r e f o r e one

100


To prove t h e o p p o s i t e i n e q u a l i t y , l e t u s o b s e r v e t h a t

s e e n a s union o f a n

a*Xnn

l a l - n u l l s e t and a sequence o f compact s e t s

c a n be {Chlh

,

such t h a t t h e l i m i t

i s uniform f o r

x ECh

.

T h e r e f o r e f o r each

h

the

is a c o n t i n u o u s f u n c t i o n o r t h o g o n a l

VI

‘h

t,o Ch

i t s e l f , t h a t is

i s contained i n a For a c l a s s i c a l r e s u l t by Whitney, one o b t a i n s Ch 1 C -surface. From t h i s f a c t and from (241, one e a s i l y g e t s

This, together with

( a ( c a * x n n - uch) h

=

o

,

implies

q.e.d.

2.4

M I N I M A L BOUNDARIES

We s h a l l p r e s e n t now a t h e o r y o f En+1

,

n

d i m e n s i o n a l minimal s u r f a c e s i n

t h a t c o n t a i n s some g e n e r a l r e s u l t s a b o u t t h e P l a t e a u Problem.

T h i s t h e o r y was s t a r t e d by E. D e G i o r g i i n 1960.

2.4.1

M I N I M A L BOUNDARIES

W e w i l l s a y t h a t a measurable s e t

XCE

n+l

h a s m i n i m a l boundary i n

101

MINIMAL BOUNDARIES

R

t h e open s e t

whenever

i f , for a l l

A EA

i s measurable w i t h

Y

with

A cc

(Y - X) U ( X

R , we

have

.

-Y)CCA

A f i r s t i n t e r e s t i n g remark c o n c e r n s t h e r e l a t i o n s h i p between minimal

b o u n d a r i e s and s o l u t i o n s o f t h e minimal s u r f a c e e q u a t i o n , t h a t i s r e a l functions of

r e a l v a r i a b l e s whose g r a p h s are s u r f a c e s o f z e r o mean

n

curvature. Let

A

b e a n open s e t o f

Rn

and

f :A+R

a classical s o l u t i o n o f

t h e minimal s u r f a c e e q u a t i o n , i . e .

n+ 1

Consider t h e set

X c E

have c o o r d i n a t e s

(y, z )

We w i l l see t h a t

X

whose p o i n t s , w i t h r e s p e c t t o a f i x e d b a s i s , satisfying

h a s minimal boundary i n t h e open s e t

t h o s e p o i n t s having c o o r d i n a t e s

(y, z )

The e q u a t i o n (1) i m p l i e s , f o r a l l

with Y E A 1 g E Co ( A ) ,

.

I n o t h e r words, t h e f i r s t d e r i v a t i v e o f t h e f u n c t i o n

r(t) i s zero a t

t

=

=

0

I,

r/l + IgradCf ( y ) + t g ( y ) l

.

Since

r(t),r(o) , v then, i n p a r t i c u l a r ,

t

r

I 2 dy

i s a convex f u n c t i o n , w e have

E En+'

of


102

X = ( y , z) ( y € A , z < f (y)3 is € A , z ER} , w i t h r e s p e c t t o a l l o t h e r s e t s

We have so checked t h a t t h e boundary of

fl

minimal i n Y = {(y,

= { ( y , z ) Iy

z ) I Y E A , z < f ( y ) + g ( y ) l , f o r any

We must see now what happens when coinciding with

,

Y

.

i s any o t h e r measurable s e t ,

e x c e p t f o r a compact r e g i o n of

X

Given such a

Y

1 gECO(A)

l e t us c o n s i d e r t h e new s e t

2

.

Y*

defined as

where

Being

(Y-X)

contained i n

c cfl

U(X-Y)

A

.

,

g

i s bounded and h a s compact s u p p o r t

U n f o r t u n a t e l y t h e r e i s no r e a s o n why

g

should be of

b e t t e r q u a l i t y t h a n measurable. By t h e same argument used t o prove t h e i s o p e r i m e t r i c i n e q u a l i t y , see t h e Lemma i n 2 . 2 . 4 , Y

to

Y*

w e have t h a t t h e p e r i m e t e r d e c r e a s e s a s we p a s s from

.

I t remains t h e n t o check t h a t

U n f o r t u n a t e l y w e are n o t e n t i t l e d t o use t h e formula

due t o t h e f a c t t h a t

g

d o e s n o t need t o have c o n t i n u o u s f i r s t

derivatives.

W e s h a l l t h e n c o n s i d e r a smoothing sequence

funct i ons of

n

v a r i a b l e s , and p u t

{ T ~ } of

MINIMAL BOUNDARIES

Y,(Y)

= [(f+g) *T,](Y)

=

j(f+g)(n)Th(y-n)dn

We get the following inequality for (Yh)

PR(Y*) )Pa

,v

103

E >

h > E-l

0

.

:

,

-E

where Yh

=

{(y,

Consider now any

< PR(Y*)

.

2)

$ E

lY € A ,

1

[Co

(a-

Then PR

.

Z C-'

and for h >

E-'

.

-E

But we have also

then

(y)

12

dy

5 PR(Y*) , v

E> 0

-E

Therefore

q.e.d.

104


2.4.2

EXISTENCE AND SMOOTHNESS OF MINIMAL BOUNDARIES A s i t w a s remarked by D e G i o r g i i n h i s o r i g i n a l p a p e r on minimal

b o u n d a r i e s , t h e i r e x i s t e n c e is a q u i t e t r i v i a l f a c t .

In addition to the

g r a p h s of s o l u t i o n s t o t h e minimal s u r f a c e e q u a t i o n , we have t h e f o l l o w i n g general EXISTENCE THEOREM For any g i v e n s e t

n ,

there e x i s t s a set

x -R

= X-R

0

The e x i s t e n c e o f

XcE Xo

and

Xo

n+ 1

with

P(X) < + m

and any open bounded s e t

such t h a t

P(X ) = i n f { P ( Y ) \ Y - R = X - n }

0

.

i s a n e a s y consequence o f t h e compactness o f

In

sets w i t h equibounded p e r i m e t e r s and t h e lower s e m i c o n t i n u i t y theorem. f a c t , l e t u s c o n s i d e r a sequence of measurable s e t s

with

There e x i s t s a n i n c r e a s i n g sequence of i n t e g e r s set

Xo

,

j(s)

and a measurable

such t h a t

2 Xo

X j (s)

We have t h e n

X

0

-R

i n the L

=

x-R

1 -convergence. loc

and

Then

W e are n o t g o i n g t o d i s c u s s h e r e how t h e s e t

Xo

l o o k s Like i n

What i s now o u r g r e a t e s t i n t e r e s t i s t o s t a t e and p r o v e t h e r e m a r k a b l e

R .

105

SMOOTHNESS OF MINIMAL BOUNDARIES

c o n t i n u i t y r e s u l t , proved by D e G i o r g i f o r g e n e r a l minimal b o u n d a r i e s . DE G I O R G I THEOREM

x

If

h a s minimal boundary i n t h e open s e t

e x i s t s a n open s u b s e t

axnR,

R0cR

R c E

n+ 1

,

then t h e r e

such t h a t

i s an a n a l y t i c s u r f a c e ,

By a r e s u l t o f J . Simons on minimal cones (19681, one g e t s

n-n,=@

up t o

n = 6

.

J . Simons h i m s e l f i n d i c a t e d t h a t t h e cone

c o u l d c a u s e some t r o u b l e i n t h e e x t e n s i o n o f h i s r e s u l t . Giorgi-Giusti

Bombieri-De

proved t h a t Simons' cone w a s a minimal o n e .

So a q u i t e unexpected phenomenon w a s m e t i n

E8 : the e x i s t e n c e of

s i n g u l a r minimal b o u n d a r i e s . H.

F e d e r e r improved t h e g e n e r a l estimate o f t h e s i n g u l a r p a r t of

minimal b o u n d a r i e s , g i v e n by D e G i o r g i :

(R-R

H

0

)

=

0

,

Vreal

s>n-7

.

A l l t h e s e r e s u l t s w i l l be p r e s e n t e d i n t h e r e s t of t h i s c h a p t e r .

2.5

SMOOTHNESS OF MINIMAL BOUNDARIES I n t h i s s e c t i o n we s h a l l p r o v e a c o n t i n u i t y r e s u l t f o r quasi-minimal

b o u n d a r i e s , which w i l l imply DE G I O R G I ' s THEOREM.

2.5.1

SOME INEQUALITIES

Let for

xER

x

be a s e t w i t h f i n i t e p e r i m e t e r i n t h e open s e t

and

p < d i s t ( x , afi)

, put

fi of

E

n+l

.

,

106


where

P x = 0

If

w e s h a l l w r i t e simply

p)

$(X,

,

p

i s t h e closed b a l l of r a d i u s

B (x)

centered a t

i n s t e a d of

$ , which one c o u l d c a l l t h e e x c e s s o f

function

. .

p, 0)

$(X,

in

X

x

B (x)

P

,

The

w i l l play

a c e n t r a l r o l e i n t h e proof of t h e c o n t i n u i t y r e s u l t . L e t us s t a r t w i t h p r o v i n g two i n e q u a l i t i e s :

PROPOSITION, 1 : L e t =

IxEEn+l/lx1 0

,

(A,)-functions.

We must have also lim. sup Dw.1 = 0 3 3 Aj

j

such that

116


Therefore t h e

n-dimensional b a l l

A , V j> jE j On t h e o t h e r hand, s i n c e a L .

must be c o n t a i n e d i n

I

So, by d e n o t i n g w i t h

q

of radius

B:

. i s r e g u l a r , w e have

t h e mean v a l u e of

j

Dw

over

j

BZ

and o b s e r v i n g

t h a t the set function

i s non d e c r e a s i n g , one g e t s

5 Bj

.

F i n a l l y , if

u.

l

i s t h e harmonic f u n c t i o n i n

B*

I

equal t o

one has

j (W B:

Recalling Proposition 2

we obtain for

Observe now t h a t i f

fa2

U =

2

6 E (0, 1)

p - ( IyI - E )

2

,

w e have

w. 1

on

aBz

,


117

therefore

If

6

i s such t h a t

6r =

0

w e g e t , from o u r p r e v i o u s remarks,

E

which, f o r

small, contradicts the self-evident i n e q u a l i t y

q.e.d. DE GIORGI'S LEMMA

2.5.3

A fundamental s t e p towards t h e p r o o f of t h e smoothness of minimal

b o u n d a r i e s i s t h e Lemma t h a t we a r e g o i n g t o p r o v e i n t h i s s e c t i o n . W e s h a l l say t h a t a set R ~ E ~ + , i' s

K

and

h

X

,

(K, X)-minimal i n

w i t h f i n i t e p e r i m e t e r i n a n open s e t

R

if

are supposed t o be p o s i t i v e r e a l numbers.

Obviously t h e minimal b o u n d a r i e s a r e and

A20

,

More g e n e r a l l y i f t h e boundary of p > n +1

,

(K, A)-minimal f o r a l l

K)O

since they s a t i s f y

then

X

is

LP

(R)

P

X

h a s mean c u r v a t u r e

H

E LP(R)

with

118


We have t h e f o l l o w i n g

1.

THEOREM,

PROOF.

If

X

L e t us p u t

is a

X

P

From t h e d e f i n i t i o n o f

set i n

( K , Xl-minimal

= X - B

$

P

(x)

and

X*

P

fi ,

= X U B

P

then

(x)

.

we have

One g e t s ( 2 ) by a d d i n g up t h e s e t w o i n e q u a l i t i e s and r e c a l l i n g ( 1 ) . To p r o v e ( 3 ) , r e c a l l i n g P r o p o s i t i o n , 1 o f 2 . 5 . 1 for

xE

a*xnQ ,

r 40

one o b t a i n s

x, t )

xEaXnn

x h c a * x n ~ with

one o b t a i n s

x E a*X

which i s e x a c t l y ( 3 ) , f o r For

and l e t t i n g

nR

.

it i s s u f f i c i e n t t o r e c a l l t h a t t h e r e e x i s t s h x -+ x h

.

q.e.d. L e t us i n d i c a t e with

T(t)

-t ( n + 1 ) !w

I

n t1

t h e r e a l f u n c t i o n o f one r e a l v a r i a b l e

SMOOTHNESS OF M I N I M A L BOUNDARIES

and f o r

€ 1 0

,

x EE

n+ 1

l e t us put

For a given measurable s e t

One e a s i l y v e r i f i e s t h a t

119

w i t h f i n i t e perimeter, l e t u s i n d i c a t e

X

fE

i s e x t r e m e l y smooth and s a t i s f i e s

where

Moreover, f o r

x EE

n+ 1

and

2 E < f (x) < 1 - E 2 E

one h a s , f o r s m a l l

E

,

Namely, b e i n g

were i t t r u e t h a t

one would g e t

This l a s t t e r m ' i s obviously g r e a t e r than would c o n t r a d i c t t h e assumption

E

small e x c l u d e s t h e p o s s i b i l i t y

f

E

1 - E~

( x ) < 1 - E~

.

,

for s m a l l

E

Similarly

E2 < f E ( x )

t h e n one and

120


L e t u s prove now t h e f o l l o w i n g p r o p o s i t i o n :

PROPOSITION, 1 .

if

(4)

is

X

/DX,/

t/ x E E

then

PROOF.

V n z l , K>O, A > 0

g : (0, 6)

a function

Let

n+ 1

,

( 0 , +")

3

there exists

(K, A)-minimal i n a n open s e t

( B ~ -) D ~ + ~ X , ( B )

s }

;

3

S a r d ' Theorem, t h e r e e x i s t s

s j

E

(EL

j'

1 - Ef) 1

w e have

and

is a

aL. IIBt

3

surface.

C1

Being

for If

xEL.-X.

1

x E X .- L i

3

j

and

xx,(x)- f . ( x ) ) l - s

3

w e have

Recalling t h a t

we e a s i l y get (10) and

Being a l s o

I

j

.

from t h e coarea formula and

such t h a t , f o r

L , = A , (s, )

3

3

1


the hypothesis ( 7 ) ,

( 8 ) imply ( 9 ) .

127

The (11) follows from the following

inequality

The (12) follows from ( 1 4 ) and (15).

~€2,. nBt , one has

Finally, if

so, for large

j

3

,

E.

3

< 6(n,

A,

A)

and

1x1 < 1 - 2E

1/2n j

, where 6

is the

constant of Proposition 1. Then

which implies ( 1 3 ) q.e.d. DE GIORGI'S LEMMA

, K>O , A >

, a € (0, 1) there exists a real number U = U(n, K, A, a) > O such that: if x is (K, X)-minimal in the open set ficEn+l , and if x€fi , 0 € (0,01 , p E (0, min{dist(x, a i l ) , 02/A}) For

n>O

0

satisfy

then

Before starting the proof of this Lemma, let us observe that

128


rl* = rja

A/2

satisfies

Then t h e Lemma can b e a p p l i e d r e p e a t e d l y t o g e t

and, a f t e r

j

iterations,

PROOF OF DE G I O R G I ' S LEb4MA.

Suppose, by c o n t r a d i c t i o n , t h a t t h e r e e x i s t

a n d , f o r a g i v e n sequence of p o s i t i v e numbers assume t h e e x i s t e n c e of a sequence

sets

with

Q. 3

X

E ,

of sets,

j

3

such t h a t c , E l l 2 < + a , 1 3 (K, h)-minimal i n open

and s a t i s f y i n g

.

x. €a and O < p . < m i n { d i s t ( x . an.), E 2/X . } 1 1 1 3' 3 3 L e t u s d e n o t e now w i t h T . t h e i s o m e t r y of En+' 1

i n t o i t s e l f , such

that

i s a vector p a r a l l e l t o

by

e

n+l +

Let

S

j

t h e omotethie of

En+1

defined

129


L e t u s denote wi t h

F

the set

j

(S.OT.1 (X.)

I

l

l

.

One e a s i l y sees t h a t

Since

t h e sequence PROPOSITION 2 .

F.

I

s a t i s f i e s t h e hypotheses o f

L e t us use t h a t Proposition w ith

be c o n v e n i e n t l y d e t e r m i n e d )

,

t E (0, 1 ) -

and

By u s i n g P r o p o s i t i o n 3 of 2.5.2

6

= as

rg , (a

E ( 0 , 1)

.

tB

f o r t h e sequence

( s

L.

, we

in

fi ,

7

get

But w e have a l s o

From P r o p o s i t i o n 2 we d e r i v e

t h a t i s impossible i f

2.5.4

s

>

n ~

+ X/2 n+2

.

CONTINUITY OF THE TANGENT PLANE W e s h a l l p r o v e now t h a t i f

r e d u c e d boundary i n

fi i s a

X

is

(K,

X)-minimal

s u r f a c e of c l a s s

g i v e a f i r s t e s t i m a t e f o r t h e dimension o f

C

.

then i t s

We shall also

will

130


.

( a x - a*x) nn If

LEMMA.

has f i n i t e perimeter i n

X

and

A c B c n

are Bore1 s e t s

with

then

L e t us d e n o t e w i t h

PROOF.

u, v

the vectors

W e have

Moreover, t h e measure

b e i n g non n e g a t i v e , w e have

t h a t implies e a s i l y ( 1 ) . q.e.d. PROPOSITION 1.

,

xEaXnn some PROOF.

nE

If

X

is a

i f there exists

(0,

a]

and

Let us p u t

(K, X)-minimal s e t i n

a € ( 0 , 1)

O < p<min{dist(x,

such t h a t

an),

r)2’h}

~

c

n+ 1 E

and

w(X, x, p ) c r ) pn then

x€a*X

,

.

for


and p r o v e t h a t

131

v (1) i s a Cauchy-sequence.

From Theorem 1 of 2 . 5 . 3 ,

if

w e have

So, f o r

j > j,

and

k>0

r e c a l l i n g ( 1 ) of t h i s s e c t i o n and ( 3 ) o f 2 . 5 . 3 ,

we get

I f w e d e n o t e now w i t h

v

the

,

1im.V") 7

we get

and, b e i n g

we g e t IVI =

1

.

Finally, l e t

t E (0, Po) be s u c h t h a t

with

Po = p a

j0

,

and

jzj,

132


.

aj+lp(tO

P

then

.

I f t h e second p o s s i b i l i t y w e r e t o o c c u r , t h e whole segment should belong t o

ax

Ex, x + t d

and t h a t would imply ( s e e Theorem 1 i n 2 . 5 . 3 )

then

which i s o b v i o u s l y i m p o s s i b l e f o r s m a l l I n a s i m i l a r way one t a k e s care o f

p

.

t < O

.

Having o n l y used t h e c o n d i t i o n

v.a,o

,

and t h e s t r o n g e r c o n d i t i o n

implying

the l i p s c h i t z character of

aXnB

e a s i l y follows. q.e.d.

THEOREM 2 . a

C

If

X

is a

set i n

n

Q}

The smoothness o f

=

o

,

then

a*Xnn

.

a*Xnn

e a s i l y f o l l o w s from P r o p o s i t i o n 2 and

Theorem 1 , it remains t o be checked t h e claim a b o u t that

n+ 1

Q c E

s u r f a c e and H ~ { ( ~ x a*x) -

PROOF.

( K , x)-minimal

ax - a*X

.

W e know

is

136


so, f o r

E

>0

nEcQ,

t h e r e e x i s t s a n open s e t

containing

( a x - a*x) n R

and such t h a t

For a l l

Z E

( a x - a*x) n R ,

From Lemma 1 o f 2 . 1 . 1 families

K,,

K2,

let

the family

.. . ,

I-$

,

p, >

o

be such t h a t

!Bpz(z)[

s t i l l covering

contains

N

( a x - a*x) n R

d i s j o i n t sub-

.

W e have t h e n

and a l s o

p,

Since

can be chosen s m a l l e r t h a n a f i x e d

6>

O

,

t h i s proves

that is

q.e.d.

2.6

SINGULAR POINTS OF MINIMAL SURFACES I n t h i s s e c t i o n w e s h a l l a n a l y z e t h e s i n g u l a r p o i n t s o f t h e boundary

of

(K, X)-minirnal s e t s . F i r s t l y w e w i l l see t h a t a t each p o i n t of t h e boundary o f a

minimal s e t t h e r e e x i s t t a n g e n t c o n e s , t h a t are minimal s u r f a c e s .

(K, 1)We w i l l

o b s e r v e a l s o t h a t , as a consequence of D e G i o r g i ' s Lemma, i f a t a n g e n t cone

i s f l a t t h e n t h e p o i n t of tangency i s a r e g u l a r one f o r t h e

(K, X)-minirnal

137

SINGULAR POINTS OF MINIMAL SURFACES

set.

This will reduce the proof of the non-existence of singular

(K,

1)-

minimal sets to the non-existence of singular minimal cones. En+l We will observe then that if a minimal cone in is singular along one of its generating half-lines, then there exists a singular minimal cone in

.

En

Therefore, for the proof of flatness of singular

minimal cones one can assume the cone to be regular outside its vertex. At this point we will present the famous result by J. Simons: "There are no singular minimal cones in Ef". This obviously means also that there E 2 (obvious), E3

are no singular minimal cones in J. Almgren) in

E

5

6

,

E

,E

(W.H. Fleming), E4 (F.

.

7

We will finally give a quite simple proof of the fact that the cone

which is obviously singular, is a minimal one.

This fact, which stopped

the "non-existence conjecture" was firstly proved by E. Bombieri-E. De Giorgi-E. Giusti. We will close the section with a sharp estimate for the dimension of (K, h)-minimal sets, due to H. Federer.

singular parts of

2.6.1

TANGENT MINIMAL CONES Let us denote with

S for P '

p>O

,

n+ 1 E

the homotethie of

defined

by

s (x) = xp-1 P If

x

SD(X)

is a subset of

.

n+ 1 E

,

we will write

It is obvious that

X

P

to indicate the set

Let us prove now the following proposition PROPOSITION. If

x

is a

then the family of sets

ph

+0

C

is a minimal cone and

and

xh + C ,

OE~*C

(K,

X)-minimal set in

{Xp}p>o

in the

oEa*x .

L1

loc

ficEn+l

contains sequences

(En'')-convergence.

xh

and = X

h'

0

ax nn

, with

,

138


(2) of 2.5.3

PROOF.

implies

p $ 0

T h e r e f o r e , from t h e Compactness Theorem, t h e r e e x i s t s C

,

and a s e t

h

such t h a t

xh

=

x 'h + c ,

in

Lloc

(En+')

.

W e c a n a l s o assume t h a t

I DXXh I

(2)

for a l l positive

a convenient n u l l set

( 2 ) i s supposed t o be s a t i s f i e d as soon as

-r

with

For a f i x e d t E ( 0 , + m ) n+ 1 C in E - B t , assuming

r .

phtn-7

,

K>O

, x > O , QcEn+' , X

n>6

,

we would have an

(K, X)-minimal in

fi , with

Then, Proposition 3 would imply the existence of a minimal cone En+'

n>6 ,

C

in

, with H (ac- a*c) >

Blowing up

C

o

at a point

xo E aC -

{O)

one would obtain a minimal cylinder

H ~ ( -~ a*D) D > Then the cross section

D

, where

with

.

o A

of D

,

that is a minimal cone in

En , would


146

satisfy

Hs-1

can- a*/\) > o .

T h i s means t h a t i f o u r s t a t e m e n t were f a l s e i n En

E n'l

it would b e so i n

. Our s t a t e m e n t i s t h e n t r u e f o r a l l

n

. q.e.d.

2.7

SINONS CONE

J. Simons i n "Minimal V a r i e t i e s on Riemannian M a n i f o l d s " , i n d i c a t e d t h e cone

a s a p o s s i b l e minimal s i n g u l a r cone i n E . Bombieri, E . D e G i o r g i and E .

E

0

.

G i u s t i proved t h a t it w a s so

We s h a l l p r e s e n t h e r e a s i m p l i f i e d p r o o f o f t h i s f a c t . L e t u s c o n s i d e r , more g e n e r a l l y , t h e cone

and t h e f u n c t i o n f : E2k

-+

R

d e f i n e d by 4

f ( x , Y) = 1x1 - IyI

4

.

L e t u s c a l c u l a t e t h e v a l u e o f t h e minimal s u r f a c e o p e r a t o r

i.e.

Mf(x, y ) = i

where

M

for

f

,

SIMONS CONE

Since

-a-f a xi

2

-

4 1 x 1 xi

af

I

aYi =

-41yl

2 yi

we have

We g e t f o r

Mf(x, y ) =

Mf(x, y )

the following i d e n t i t y

( 4 k + 8 ) ( 1 x I 2 - l ~ 1 ~ ) ( 1 + 1 6 1 ~ 1 ~ + 1 +6 1[ 2y *1 1~6)( 1 y I 8 - lxI8) fl

Being

w e have

+ 161x16 + 1 6 1 ~61 3

147


148

t h e region 1x1 > I Y I and a s u p e r s o l u t i o n i n t h e r e g i o n

1x1 < Iy

The same i s t r u e f o r all f u n c t i o n s f (xr Y )

P

=

D

-1

f(pxr

PY) =

F o r any f i x e d non n e g a t i v e f u n c t i o n consider the function

F(t)

i s convex and I

then, i f

p>O

and

F'(G))G

i.e.

k14

,

,

w e get

which i m p l i e s

P

3

(Ix

$ECA

SIMONS CONE

1/1 + I g r a d ( f

P

-4)

I2

149

dxdy?

I n o t h e r words t h e measure of t h e graph of t h e f u n c t i o n and

k24

,

f

P '

for

p > 0

i n c r e a s e s if w e bump i t downwards o v e r t h e r e g i o n

{ ( x , Y ) 11x1 >

lYll

*

The same t h i n g happens for upwards bumps o v e r t h e r e g i o n

An argument s i m i l a r t o t h e one used t o prove t h e Lemma of p . 0 7 , a s s u r e s t h a t t h e bumps do n o t need t o be d e s c r i b e d by a r e g u l a r f u n c t i o n like

@

.

I n o t h e r words, w e c a n s a y t h a t t h e measure o f t h e boundary o f

t h e sets

xP

= { ( x , Y,

2)

i n c r e a s e s i f w e s u b s t r a c t from them any bounded s e t c o n t a i n e d i n t h e r e g i o n

{ ( x , y , z)lIxI > I Y l ,

z>ol

I

o r i f we add t o them any bounded s e t c o n t a i n e d i n t h e r e g i o n {(x, y, Letting

p*

+a

Z ) ~ l X l < I Y l ,

z O

x

-+

c

.

w e w i l l have

we have

BERNSTEIN PROBLEM

T h e r e f o r e , s e e a l s o P r o p o s i t i o n 1 of 2 . 5 . 1 , Then

C

must be f l a t .

C

151

i s a minimal cone i n

E

7

This implies

and a l s o

Then

X

must be a c o n e , i . e .

f

st

must b e a c o n s t a n t o r a 1

degree

polynomial. q.e.d. I n t h e n e x t c h a p t e r w e w i l l see, f o l l o w i n g a remark by E . D e G i o r g i , t h a t t h e B e r n s t e i n Theorem i s v a l i d i n

R'

.

CHAPTER THREE

THE DIRICHLET PROBLEM FOR THE M I N I M A L SURFACE EQUATION

W e s h a l l c o n s i d e r now t h e D i r i c h l e t problem f o r t h e minimal s u r f a c e equation ( m . s . e . ) ,

t h a t is t h e problem o f p r o v i n g the e x i s t e n c e of a

continuous functi o n :

nu a ~R -f

such t h a t

gradu

,

= o

2 where

fi i s a g i v e n open s e t of

Rn

$

and

:

an

-f

R

i s a given continuous

function. I n t h i s c h a p t e r w e s h a l l r e s t r i c t o u r s e l v e s t o t h e case

R

bounded,

i n c h a p t e r 4 w e w i l l c o n s i d e r t h e case of unbounded domains. Being t h e s e t of known s o l u t i o n s t o t h e m . s . e .

v e r y poor and t h e

e q u a t i o n non l i n e a r , t o s o l v e o u r problem w e do n o t have much c h o i c e s b u t c o n s i d e r i n g the d i r e c t method of t h e C a l c u l u s o f V a r i a t i o n s . I n f i r s t p l a c e w e s h a l l p r e s e n t t h e H i l b e r t - H a a r E x i s t e n c e Theory f o r

a c l a s s o f v a r i a t i o n a l problems i n c l u d i n g t h e area problem. T h i s w i l l g i v e u s t h e s o l u t i o n of t h e D i r i c h l e t problem f o r t h e minimal s u r f a c e e q u a t i o n under p a r t i c u l a r c o n d i t i o n s f o r

R

and

@.

W e s h a l l then p r e s e n t a

s p e c i a l method, t h a t w i l l y i e l d t o t h e s o l u t i o n of t h e problem under v e r y g e n e r a l hypotheses.

3.1

THE H I L B E R T - H A A R EXISTENCE THEORY Let

THE HILBERT-HAAR EXISTENCE THEORY

153

be a convex f u n c t i o n . For any L i p s c h i t z f u n c t i o n

u

and any open bounded s e t

fi , w e c a n

consider the i n t e g r a l

f(u)

=

l-2

.

F(gradu)dx

A q u i t e useful r e s u l t is the following

SEMICONTINUITY THEOREM.

If

tu

1

i s a sequence o f e q u i - L i p s c h i t z

f u n c t i o n s , pointwise converging t o t h e f u n c t i o n

PROOF.

u

then

Since

L(p) = a + b * p , a E R , b € R n , L ( p ) l F ( p ) V p E R

w e have a l s o , i f w e p u t f o r

A>

0

L(p) = a + b - p , a E R , bERn,

F = sup F

x>o

A

Ibl < A ,

L(p)(F(p) V p € R n

-

I t i s a l s o obvious t h a t

FA i s convex and A - L i p s c h i t z .

I f we put

w e have

f o r any L i p s c h i t z f u n c t i o n c o n t i n u i t y of

FA, t h a t i s

u.

It i s t h e n s u f f i c i e n t t o check t h e serni-

I

154

THE D I R I C H L E T PROBLEM FOR THE MINIMAL SURFACE EQUATION

To t h i s p u r p o s e o b s e r v e t h a t , f o r any f i x e d

f i n i t e s e t o f d i s j o i n t cubes

FA( u )

%FA({

{Qk]

contained

g r a d u dx) m i s Q k

A > 0 , E > 0 there i n n, such t h a t

u,

exists a

+E ,

Qk where

(jQ

jQ

dx =

k

dx)(misQk)

-'

.

k

Being

= lim

gradudx =

j

gradu.dx j I Qk 7

Qk where

\J

is t h e e x t e r i o r normal o n

aQk, we o b t a i n

k

Being

convex, we have

FA

~ ~g r a(d u j dx) f ~5jQFA(gradu.)dx 7 k

k

(gradu.)dx = 1 i m i n f . F ( u . ) 7 3 1 3

.

We f i n a l l y o b t a i n

and, being

E > O

arbitrary,

q.e.d.

155


EXISTENCE THEOREM FOR THE M I N I M U M L e t u s d e n o t e by

U

t h a t i s a s e t of functions

a set of equilipschitz functions defined on u

Rn,

satisfying

I t i s known a s A r z e l d Theorem t h e f o l l o w i n g compactness r e s u l t : f o r any

{ u j j C!J

sequence

t h e r e e x i s t s a p o i n t w i s e c o n v e r g e n t subsequence.

By means o f A r z e l d Theorem w e a r e a b l e t o e s t a b l i s h a v e r y simple e x i s t e n c e theorem f o r t h e minimum: THEOREM 1.

u

If

i s a s e t of e q u i l i p s c h i t z f u n c t i o n s c l o s e d w i t h r e s p e c t

t o t h e p o i n t w i s e convergence, i f

i s l o w e r semicontinuous w i t h r e s p e c t t o t h e same convergence, t h e n t h e r e exists

uOEU

PROOF.

Let

assume

/uj(s)

s.t.

{u.} c U 3

Is

c{ujj

satisfy

satisfying

We g e t

F(u0) 5 l i m i n f f ( u j ( s ) )

=

inf

Ell

f(u)

. q.e.d.

A s p e c i a l case, which i s o f i n t e r e s t f o r us, i s t h e f o l l o w i n g : a bounded open s e t of

Rn

and

6

is a K-lipschitz function,

Q is

11 i s t h e

156


u

s e t of a l l K - l i p s c h i t z f u n c t i o n s

F

i s a convex f u n c t i o n d e f i n e d on

F(u)

=

I,

F(gradu)dx

satisfying

Rn

and

.

Thanks t o t h e S e m i c o n t i n u i t y and A r z e l a Theorems w e have t h e f o l l o w i n g THEOREM 2 .

is

If

F:Rn

+

ncRn

i s open and bounded, i f

$I

: Rn

-+

R

is K-lipschitz,

u0

i s convex, t h e n t h e r e e x i s t s a K - l i p s c h i t z f u n c t i o n

R

satisfying

uojabl

=

$Ias2,

j n f ( g r a d u0 ) d x 5

u

for a l l K-lipschitz functions A much more i n t e r e s t i n g

F(graduo)dx
LI,,

u

,

= LI

convex t h e r e e x i s t s a v e c t o r

if

F

is differentiable a t

1

b. A

we obtain

F(gradu)dx2

A

Being

b = gradL

and

UI

=

LI aA , w e have

grad(u-L)dx = 0

therefore ( 2 ) yields t o

A

where t h e i n e q u a l i t y i s s t r i c t u n l e s s

g r a d u ( x ) = gradL(x1,

a.e. i n A

open and non empty.

B E R ~ ,s a t i s f y i n g

By i n t e g r a t i n g t h e i n e q u a l i t y (1) o v e r

(2)

A

I

,

157


158

which i s i m p o s s i b l e because

uIA >

L I ~ and

ulaA = LlaA

*

q.e.d. The remark a b o u t s t r i c t l y convex f u n c t i o n a l s h a s i n t e r e s t i n g i m p l i c a t i o n s f o r t h e minimizing f u n c t i o n s

uo

g i v e n by Theorem 2 : i f

s t r i c t l y convex, t h e n f o r e a c h l i n e a r f u n c t i o n

L

F

is

satisfying

w e must have

I n p a r t i c u l a r w e have t h e f o l l o w i n g THEOREM 3 (MAXIMUM PRINCIPLE).

$

: Rn -+ R

i s K - L i p s c h i t z and

F-minimizing f u n c t i o n satisfying

max u

R uan PROOF.

$Ian

ulaR =

uo

,

= m u;o

If

ncRn

F : Rn

-+

R

i s open and bounded, i f i s s t r i c t l y convex, t h e n t h e

i n the class of a l l

K-Lipschitz f u n c t i o n s

u

i s unique and

= max$

aa

.

The u n i q u e n e s s o f t h e minimizing f u n c t i o n i s a n e a s y consequence of

*

t h e s t r i c t c o n v e x i t y of F , were i n f a c t u a n o t h e r minimizing f u n c t i o n , u* + uo u* + uo d i f f e r e n t from u ___ would b e K - L i p s c h i t z and ___ 0' 2 on t h e o t h e r hand

which i s a c o n t r a d i c t i o n . A s t o t h e maximum p r o p e r t y ,

function

L ( x ) = max$

aa

max$

an

2

u

0

IR

it i s s u f f i c i e n t t o consider t h e constant

and o b s e r v e t h a t i t must s a t i s f y

q.e.d.

159


-uo

Observe t h a t

F

minimizes

o f s t r i c t l y convex f u n c t i o n a l s , i f LlaR

w i t h r e s p e c t t o t h e boundary v a l u e

t h e n a minimum p r i n c i p l e i s v a l i d too.

-$,

5

$Ian

then

r

More g e n e r a l l y , i n t h e case

i s a l i n e a r f u n c t i o n such t h a t

L

.

L I R 5 uOIR

What w e have remarked up t o now, becomes even more i n t e r e s t i n g under and

$

:

convex and t h e r e e x i s t s

E

>0

the following conditions f o r ( C . 1)

fl i s

Ibl = 1

(C.2)

$

where

5

i s a c o n v e n i e n t p o i n t between

M = max i,j

L-(Y)

max

IDiDj$

(5)I

.

5EnuaR

L$ (y)z

Moreover, i f y e p u t

w e have

and

y.

and t h e Taylor Formula imply:

The two l i n e a r f u n c t i o n s

satisfy

V x E

c2 .

(C.1) and ( 4 ) imply:

where

and

such t h a t

i s o f class

(C.2)

0

~ + ( y ) v, y E aR

.

x

an

3 b E Rn

with

160


(7)

IgradL

+ILK *,

.

IgradL-I(K*

REMARK ABOUT THE DIFFERENTIAL RATIOS OF THE M I N I M I Z I N G FUNCTIONS

I f c o n d i t i o n s (C.1) and ( C . 2 )

a r e s a t i s f i e d by

s t r i c t l y convex, t h e n any minimizing f u n c t i o n s c h i t z functions equal t o

where

4

aR,

on

uo must s a t i s f y

R

and

$, i f

F

is

i n a c l a s s o f K-Lip-

i s g i v e n by ( 6 ) .

K*

T h i s remark becomes v e r y i n t e r e s t i n g a f t e r t h e proof of t h e f o l l o w i n g MAXIMUM PRINCIPLE FOR THE DIFFERENTIAL RATIO (JOHN von NEUMANN THEOREM)

If

functions

u

4

is open and bounded, i f

RcRn

i s s t r i c t l y convex, i f with

uo

ulaR = luo(x) -

SUP

xEawm

minimizes

$Ian

,

U0(Y)

I

[x-yI

f

:aR+R

i s K-Lipschitz

and

F

i n t h e class o f a l l K - l i p s c h i t z

then

= sup

x

I uo ( x ) - uo ( Y ) I ~ a I x - ~y l

Y EQ

YEQ X f Y

PROOF.

Assume

z ERn

such t h a t

RZ

+

@

, where RZ

= {x+zIx

€a} .

Put

W e o b v i o u s l y have

On t h e o t h e r hand, t h e f u n c t i o n s integral

u (x+ z ) 0

and

uo(x) f C

minimize t h e

THE H I L B E R T - H A A R

161

EXISTENCE THEORY

in the classes of all K-lipschitz functions assuming their own values on

a(nflnZ). over R.

That is an obvious consequence of the minimum property of

u

0

Were it

different from

u (x) + C 0

in

Rz nR,

we would have

=

u (x)+ C 0

and

For the same reasons, if we put

we would have,

But. since

we would obtain, by adding ( 9 ) and (10),

which contradicts the minimum property for at least one of the functions UO+C,

u (XfZ). 0

162

THE DIRICHLET PROBLEM FOR THE MINIMAL SURFACE EQUATION

W e must have t h e n

w + ( y ) ~ u o ( y+)C

,v

Y

E

,~

~

~

Q

t h a t is

I n o t h e r words, w e have o b t a i n e d t h e f o l l o w i n g

u ( x + 2) - u o ( x ) 0

nQz)

x E a(R

Since

implies

x

x + z EaR

E ~ Ror

T h i s , t o g e t h e r w i t h t h e a r b i t r a r i n e s s of

1

.

V y EnZn R

w e have,

z , implies the v a l i d i t y of the

Maximum P r i n c i p l e f o r t h e D i f f e r e n t i a l R a t i o . q.e.d.

We a r e a b l e now t o p r o v e t h e f o l l o w i n g e x i s t e n c e r e s u l t THEOREM 4 .

(C.l),

if

If

$

ficRn

:Rn+R

i s open, bounded, convex and s a t i s f i e s c o n d i t i o n i s of c l a s s

and

C2

t h e n t h e r e e x i s t s a unique L i p s c h i t z f u n c t i o n

f o r a l l Lipschitz functions PROOF.

on

Let

and

u

4

In fact, i f

a r e equal t o

@

on

an

on

aR

functions equal t o

+ t ( u - u 0)

:Rn+R

through (6). The minimizing f u n c t i o n

i s any L i p s c h i t z f u n c t i o n w i t h

0

0

with

with

a l l K-Lipschitz

u

u

be any p o s i t i v e r e a l number g r e a t e r t h a n t h e

K

4 u

i s s t r i c t l y convex,

F : Rn+R

uo

K*, depending

i n t h e class o f

has t h e required property. UI

and, s i n c e

aQ u

=

0

$ 1 an , a l l is

functions

K*-Lipschitz,

163


,

K*

- graduO(X)

dt/

v(x)dx = 0

w e g e t a n i n t e g r a l e q u a t i o n of t h e f o l l o w i n g k i n d :

where

ahk ( x )

and

=

\:DhDkF(gradu0

(x)

is the differential quotient

w(x)

(x+ sa)

+t

- g r a d u O(x)

I)

dt

- uo(x)

u ( x + sa) 0 S

Being

o f class

F

t h e matrix

where

Tl,

C

2

,

s t r i c t l y convex and

uo

Lipschitz continuous,

(ahk) s a t i s f i e s

T2

a r e c o n v e n i e n t numbers:

O < T 1-< T 2
0

ahk

a r e Lebesgue measurable

3 CE €13

such t h a t

167

168


AE(O,l)

where

n,

depends on

The L i p s c h i t z r e g u l a r i t y o f

REMARK.

t h e Theorem.

'r2

and

T1

only.

i s n o t necessary f o r t h e proof o f

w

The method o f p r o o f works under t h e f o l l o w i n g weaker

a s sumpti o n s 2

W E L (R)

,

D ~ W E

L

= 4(T2/T1)

(R)

.

h = 1 , 2 ,... , n

for

As a consequence o f ( 1 9 ) and ( 2 0 ) , i f we p u t

PROOF OF DE G I O R G I THEOREM.

Y

2

,

w e obtain

y En,

for a l l

O < p < p < d i s t ( y , aR)

1 2 We have d e n o t e d w i t h

A(s)

and

s ER

.

CxERlw(x) > s )

the s e t

and w e s h a l l

o f t e n w r i t e simply

W e s h a l l prove ( 2 1 ) .

The i n e q u a l i t y ( 2 2 ) can b e d e r i v e d from t h e

application of (21) t o the function For t h e p r o o f o f v =

where

Q

C(W-S)

-w

.

(21) l e t u s u s e f i r s t ( 1 9 ) w i t h

U O l n2

i s a L i p s c h i t z f u n c t i o n equal to zero on

an.

W e obtain

SMOOTHNESS OF HILBERT-HAAR SOLUTIONS, THE 1 9 t h HILBERT PROBLEM

169

Applying t h e Schwarz i n e q u a l i t y t o t h e b i l i n e a r form

tf:

h,k=l

ahkDhwDkq

'

w e have 1/2 h , = I ahkDhqDkql

.

T h e r e f o r e , from ( 2 3 ) one o b t a i n s

kGl

'

ahkDhwDkw)q2dx

2\ A(s)

r/

ahkDhwDkw/1/ 2 h , -1

Then, by a p p l y i n g t h e Schwarz-Holder i n e q u a l i t y t o t h e l a s t i n t e g r a l , one has

a ' A ( s ) ~ l~ h g 1

1

DwDwdx(2 hk h k !jA(s).',%1

t h a t implies

Recalling (201, we obtain

D wD wdx/1/2

a hk


170

we get

which is exactly (21). The inequalities (21) and (22) have interesting consequences, if used together with the following isoperimetric-type relationship:

valid for all functions w

if

To prove (24), consider first the identity

measA(t, p)dt

’ A ( s ,p) Recalling the formulas (ll),

.

(12), (15) of p. 95 and p . 96, and the

inequalities

we obtain

for some convenient 6(n) E ( 0 ,

+a)

SMOOTHNESS OF HILBERT-HAAR

SOLUTIONS, THE 1 9 t h HILBERT PROBLEM

We g e t t h e n

which, i n t e g r a t e d o v e r t h e i n t e r v a l

(s,

Applying t h i s i n e q u a l i t y t o t h e f u n c t i o n

+a)

,

gives

[(w- s) UOl2

,

we obtain

Being

w e o b t a i n , w i t h t h e use o f Schwarz-Holder i n e q u a l i t y ,

t h e n , by t a k i n g t h e s q u a r e , w e o b t a i n ( 2 4 ) w i t h

2

B ( n ) = 46 ( n )

The j o i n t u s e of LEMMA.

then, i f

For

w

.

(21) and ( 2 4 ) allows t h e p r o o f of t h e f o l l o w i n g lemma.

U c ( 0 , 1)

, put

s a t i s f i e s ( 2 1 ) and

measA(I+uc, p-up)

PROOF.

Pnt

=

m e a s A ( s , p)

0

.

5 Bpn ,

we o b t a i n

171

172


h'

= p-up+2

-h up

From ( 2 4 ) and t h e f a c t t h a t

,

sh = s t u c - 2

0(;un,

-h

Uc

,v

integer

h>O

.

we obtain

From ( 2 1 ) , w e o b t a i n

Combining t h e s e l a s t two i n e q u a l i t i e s , one g e t s

t h a t implies the following t w o i n e q u a l i t i e s

Recalling the defining properties of proves t h e following

8

and

c

,

by i n d u c t i o n on

h

one

SMOOTHNESS OF H I L B E R T - H A A R SOLUTIONS, THE 1 9 t h HILBERT PROBLEM

6 , i.e.

Observe t h a t t h e f i r s t c o n d i t i o n on

e 2

173

1 (1-a)” 2 n

--w

and ( 2 7 ) t a k e n a s i n d u c t i o n h y p o t h e s i s ,

imply

t h a t i s t h e c o n d i t i o n f o r a p p l y i n g ( 2 4 ) t o estimate t h e i n t e g r a l

Being

A(S

+UC,

P - Up)

cA(sh,

Ph)

r

Vh10

t

w e must have

measA(s +uc, p -up)

5

epn4-nh

m e a s ~ ( s + u c p, - u p )

=

0

I

v

h/O

I

then

. q.e.d.

REMARK.

The Lemma j u s t p r o v e n , i m p l i e s t h e local boundedness o f weak

solutions

w

t o ( 1 9 ) , even though t h e y a r e n o t supposed t o b e L i p s c h i t z

continuous. What w e are g o i n g t o p r o v e now i s a l o c a l estimate of t h e Holder norm o f weak s o l u t i o n s t o ( 1 9 ) , depending o n l y o n n , y , t h e d i s t a n c e from t h e 2 boundary of R ‘and t h e L (Rl-norm of t h e s o l u t i o n s t h e m s e l v e s . To t h i s purpose, f o r a f i x e d

y ER

and

p 1

F o r any f i x e d

n

T . (x) = j T(jx)

3 and s a t i s f i e s

j

where

m

7 ECo(Rn)

i s non n e g a t i v e , v a n i s h e s

t h e sequence o f f u n c t i o n s

i s e q u a l l y bounded t o g e t h e r w i t h t h e i r g r a d i e n t s on a n y f i x e d compact s e t

188

K


such t h a t

dist(K,

aR) >

j

-1

.

T h e r e f o r e , f o r any f i x e d

t h e r e e x i s t s a n i n c r e a s i n g sequence o f i n t e g e r s

i s uniformly c o n v e r g e n t o v e r

K

h(s)

j

and

K

,

such t h a t

.

Through a c l a s s i c a l d i a g o n a l p r o c e s s w e o b t a i n t h e e x i s t e n c e of a n i n c r e a s i n g sequence of i n t e g e r s

h(s)

,

such t h a t , f o r any

j

,

the

sequence

converges u n i f o r m l y o v e r any f i x e d compact

K

.

Once o b s e r v e d t h a t

and

we can conclude by c l a i m i n g t h a t

J

K Ivh(s)

- v ~ ( ~I d, x)< E

V K C C fi V E > 0 3 B

,

for

s,

s' > a

with

. q.e.d.

REMARK.

For any g i v e n

e x i s t s a function t h i s purpose see

+EL

(an) ,

FEHlfl(Rn-fi)

C 361, by

one p r o v e s t h a t , for any

A

1

if

an

is locally Lipschitz, there

whose v a l u e s on

aR

are t h e

4 .

To

t h e same arguments u s e d i n p . 71 and f o l . , , i f v* = v i n fi , v* = F i n Rn -R:

vEBV(n)

s i m i l a r i d e n t i t y h o l d s i f , i n s t e a d of c o n s i d e r i n g t h e t o t a l

189

GENERAL CASE

v a r i a t i o n o f t h e g r a d i e n t s , one c o n s i d e r s t h e i n t e g r a l

d e f i n e d , f o r any open s e t

B

The announced formula i s , i f

,

as

,

B 2 3

W e a r e now a b l e t o s t a t e and prove a n e x i s t e n c e r e s u l t , f o r t h e

g e n e r a l v a r i a t i o n a l area problem: For any g i v e n open s e t

THEOREM 6.

RcRn

,

r e g u l a r boundary, and f o r any g i v e n f u n c t i o n function

PROOF.

vOEBV(R)

minimizing t h e f u n c t i o n a l

For a f i x e d bounded open s e t

$

with values

bounded and w i t h L i p s c h i t z 1 $ E L (aR) , there exists a

, put

ai-2

on

v* = v

a

B ~ = I

in

and f o r a f i x e d and

v* = F

in

F€BV(Rn-E) B-fi

.

From

(38) w e o b t a i n

$(v) = j

B

/

m

d

x

-

-

m

d

x

B-R

G

Minimizing

I, /

i

i n t h e c l a s s of c l o s e d under t h e

in

d

*‘

BV(R)

i s t h e same t h a n minimizing

X

v EBV(B)

equal to

F

in

B-R

S i n c e t h i s class i s

L1 ( B ) -convergence and t h e f u n c t i o n a l

190


lower semicontinuous, t h e a p p l i c a t i o n o f P r o p o s i t i o n 3 t o a minimizing sequence

EV*~I=BV(B)

w i l l g i v e u s a minimizing

j

x

d

w

d

v*o

for

,

B and t h e n a minimizing

v

v*

=

0

I

for

on

c

. q.e.d.

3.6

I N T E R I O R REGULARITY

W e s h a l l prove now t h a t t h e minimizing f u n c t i o n s , p r o v i d e d by Theorem

6 are a n a l y t i c i n

.

Q

The f i r s t s t e p towards t h e p r o o f of t h i s f a c t i s remarking t h a t t h e

set

h a s minimal boundary i n

5 2.4.1.

.

Q X R

For t h e p r o o f o f t h a t see C h a p t e r 2 ,

By r e c a l l i n g what w e proved i n C h a p t e r 2 a b o u t t h e smoothness of

minimal b o u n d a r i e s , w e c a n s a y t h a t t h e p o r t i o n o f t h e boundary of contained i n

,

n X R

i s an a n a l y t i c

singular set, closed i n

( n - 1 ) - d i m e n s i o n a l measure i s z e r o .

Another i m p o r t a n t f a c t i s t h a t t h e t a n g e n t p l a n e t o regular points contained i n

* , z*)

(x

aE

at aE

* , z *)

(x

,

aEn (nxR)

aE =

I (x,

The r e p r e s e n t a t i o n

2)

Ix, = f 3

z = f(x)

* , z* ) , b e i n g

(x

of

n

(XI,.

I n f a c t , assume

and t h e t a n g e n t p l a n e t o

t o be v e r t i c a l .

f

a t all its

aE

c a n n e v e r be v e r t i c a l .

can be r e p r e s e n t e d , i n a neighborhood o f

of an a n a l y t i c f u n ct i o n

plane a t

Q X R

t o be a r e g u l a r p o i n t f o r

,

n-dimensional m a n i f o l d , e x c e p t f o r a

, whose

Q X R

E

* , z* )

(x

as t h e g r a p h

variables, that is

. ., x j - l ,

xj+l,.

. ., X n ,

2)

1

.

i s e x c l u d e d by t h e f a c t t h a t t h e t a n g e n t

v e r t i c a l c a n n o t be r e p r e s e n t e d by a n e q u a t i o n

like = a . (x-x*)

z-z*

Since the s e t

E

.

i s t h e subgraph o f t h e f u n c t i o n

vo(x)

, we

must have

INTERIOR REGULARITY

* , z * ) , we

A t the point

(x

*

x l , - ~ - , x j - l fx j + l , . . . f x

aZ

2

since the function

Now,

must have

*

”(*

*,,

.*)

az ”(

XI,

therefore

f

,

f

... t x j-1’

s a t i s f i e s a strongly e l l i p t i c equation, t h e minimal s u r f a c e

z

w e c a n conclude w i t h t h e f o l l o w i n g i d e n t i t y

Xj+l’..

i s i n d e p e n d e n t of

continuation, implies t h a t

.

= 0

o b t a i n e d by d i f f e r e n t i a t i n g , w i t h r e s p e c t to e q u a t i o n s a t i s f i e d by

191

.,Xn,

,

z

2)

,

f 0

and t h a t , e x t e n d e d by a n a l y t i c

aE fl ( n x R)

must c o n t a i n a v e r t i c a l c y l i n d e r .

T h i s f a c t c o n t r a d i c t s t h e f a c t t h a t t h e measure o f

a E n (fix R )

is finite.

What w e have j u s t remarked p r o v e s t h a t t h e a n a l y t i c f u n c t i o n t h a t aEn (QxR)

represents

than t h e f u n c t i o n

xER

vo

for

aE ll

(aX R) .

in

R

,

vo

nearby i t s r e g u l a r p o i n t s , cannot be anything else itself.

We have a l s o t h a t t o e a c h s i n g u l a r p o i n t

i t must c o r r e s p o n d

Therefore the

a singular point

(x, z )

( n - 1) -measure o f t h e s i n g u l a r p o i n t s f o r

Another consequence o f t h e f i n i t e n e s s o f t h e measure of must be l o c a l l y bounded.

0 t h e r e must e x i s t a p o s i t i v e number

Were t h i s f a l s e f o r a f i x e d

{x.}cK 3

2.)

j‘

.

0

E

K

,

J

EaE

(KXR)

,

aEn (QXR)

T h a t i s , f o r any compact s e t

K c c

fi

y , such t h a t

we c o u l d f i n d a s e q u e n c e of p o i n t s

and a sequence of r e a l numbers

(X

aR

v

must be z e r o .

is that v

where

for

(zj)

, with

]z.-zh]>2E 3

,V

j

,

V h

,

i s a f i x e d p o s i t i v e number l e s s t h a n t h e d i s t a n c e o f

K

from

From what w e s a w i n C h a p t e r 2 , c f r . p . 118 Theorem 1 , w e would have

192


Hn(aEn (RXR))

0

let

such t h a t

s a t i s f y t h e minimal

w e have

193

INTERIOR REGULARITY

$

L e t us write these t w o identities for

Since

u -vo

=

0

on

aB-U

B

0

'h

,

r

=

u0 - v o

and i n t e g r a t e by p a r t s .

w e have

r

Dun

Dv.

r

Since

E J. 0

letting

we obtain

J

D(uO-vO) B-X

r

Dv,

Taking t h e d i f f e r e n c e , we g e t Du

D(u - V o ) 1B - X 0

*

d--

0

d

DvO

w

The l a s t i n t e g r a n d b e i n g non n e g a t i v e , and s t r i c t l y p o s i t i v e i f Duo(x)

Since

Dvo(x)

,

u

w e can conclude

and no connected components of

aB-X

0 = vo On c o m p l e t e l y b o r d e r e d by

X

uo

B-X

=

vo

in

,

w e c a n conclude w i t h s a y i n g

.

B-X

c a n be

194


c a n be extended as a r e g u l a r s o l u t i o n t o vo w a s any b a l l c o n t a i n e d i n , w e d i d prove t h a t

I n o t h e r words t h e f u n c t i o n t h e whole v

0

B

.

Since

B

.

fi

is analytic i n

q.e.d.

3.7

BOUNDARY BEHAVIOR OF VARIATIONAL SOLUTIONS

The f u n c t i o n s

vo

,

minimizing

@

do n o t need t o assume t h e v a l u e s

on

an ,

i n o t h e r words t h e i n t e g r a l

might be d i f f e r e n t from z e r o . By u s i n g t h e

r-convergence t e c h n i q u e s , D e G i o r g i and Modica

showed t h e e x i s t e n c e o f a n

L'(aB)-function

c o n t i n u i t y on t h e boundary

aB

-3

and

3

with values i n

[-1,

11

t h e values

,

w i t h no p o i n t s o f

of t h e u n i t b a l l

f o r which a

.

,

@

Such

vo

&minimizing

B

of

Rn

function

,

taking only

vo

never t a k e s t h e v a l u e s o f

exists

@

,

obviously. D e Giorgi-Modica remark p r o v e s t h a t , i n t h e case o f a d i s c o n t i n u o u s

datum

@

,

one h a s no chances t o g e t

fi

assumes a b o u t t h e domain

vo = $

on

,

no matter what one

.

So w e s h a l l r e s t r i c t o u r s e l v e s t o t h e case o f c o n t i n u o u s d a t a .

I n t h i s case, a s w e w i l l see, t h e r e l a t i o n

w i l l depend o n a g e o m e t r i c a l c o n d i t i o n f o r

fi

.

To have a n i d e a of what i t may happen, c o n s i d e r t h e f o l l o w i n g example:

n

= I x E R 2 ) l < 1x1 < 2 1

@(x) =

$ ( ( X I )

,

0 , for

1x1 = 1

M,

1x1 = 2

=

for

,

BOUNDARY BEHAVIOR OF VARIATIONAL SOLUTIONS

where

195

.

MER

Any minimizing f u n c t i o n

vo

w i l l depend o n l y o n

1x1

,

a s one c a n

e a s i l y check, t h e r e f o r e i t must be o f t h e f o l l o w i n g t y p e

where

a E C 1 , +") The jump

and e q u a l t o

a

vo M

for

1x1

bER

.

1

/ v (1) - v o ( 2 ) o f such f u n c t i o n s i s maximum f o r a = 1 0 cosh-l2 . S o , i f / M I > c o s h - ' 2 , we have no c h a n c e s t o f i n d and v ( 2 ) = M . What one can a c t u a l l y p r o v e f o r 0 -minimizing vo i s u n i q u e and t a k e s t h e v a l u e M -1 The v a l u e 0 f o r 1x1 = 1 i s t a k e n i f /Ml(cosh 2 .

v (1) = 0 0

with

any

,

is that the =

2

.

W e a r e g o i n g now t o w r i t e down a G e o m e t r i c a l C o n d i t i o n ( G . C . )

R ,

for

sufficient t o assure

f o r a l l continuous

4

.

Such a c o n d i t i o n w i l l b e s a t i s f i e d by a l l bounded

convex domains, and by a l l bounded r e g u l a r domains c o n s i d e r e d by J e n k i n s -

,

Serrin

t h a t i s t h o s e domains whose boundary mean c u r v a t u r e d o e s n o t

change s i g n . The L i p s c h i t z r e g u l a r i t y o f functional Recall

,

aR ,

t h a t w e d i d assume f o r w r i t i n g t h e

i s s u f f i c i e n t t o formulate o u r Geometrical Condition.

t h a t a l l convex domains have L i p s c h i t z r e g u l a r b o u n d a r i e s .

GEOMETRICAL C O N D I T I O N ( G . C . ) F o r each

x € X

there exists a b a l l

B

,

centered a t

The G . C . e x p r e s s e s t h e f a c t t h a t t h e measure o f minimum w i t h r e s p e c t t o m o d i f i c a t i o n s o f

an

aR

x

,

such t h a t

is a local

towards t h e e x t e r i o r of

The G . C. i s o b v i o u s l y s a t i s f i e d by convex domains, and it i s a l s o q u i t e e a s y t o check t h a t , i n t h e c a s e of r e g u l a r domains, t h e G . C . i s e q u i v a l e n t t o J e n k i n s - S e r r i n c o n d i t i o n a b o u t t h e mean c u r v a t u r e (see Chapter 2 , p.

101 and p.

102).

W e a r e now a b l e t o s t a t e and prove t h e f o l l o w i n g theorem.

R

196


THEOREM 7.

If

RcRn

satisfies the

aR is Lipschitz regular and

is bounded, if

G. C., then for any continuous

there exists a unique continuous

R and satisfying

analytic in

In order to prove Theorem 7, we have to go back to minimal boundaries and prove the following lemma LEMMA 3.

If A

is an open set of

k R

regular and satisfies the G. C., if Y A , if

x E aAn aY

then A

and

PROOF.

Since

A3Y

and

Y

and

Y

k12

,

if

aA

is Lipschitz

is a measurable set contained in

has minimal boundary in a neighborhood of

must coincide nearby aA

,

x

.

is Lipschitz regular and satisfies the

x E aAfI aY

in a neighborhood of

,

x

cannot be a singular point for

x ,aY

x ,

G.

C., since

aY

.

Therefore

can be represented as the graph of a solution

f of the minimal surface equation.

In the same neighborhood aA

represented as the graph of a Lipschitz subsolution g

can be

of the minimal

surface equation. We must also have

and

g

=

f

in one interior point.

The strong maximum principle for the

minimal surface equation can be then applied to get g - f

. q.e.d.

REMARK.

Since

aA

satisfies the G. C., in order to prove Lemma 3 it is

sufficient to know that the minimal boundary property for

Y

is satisfied

197

THE CAPILLARY PROBLEM

f o r modifications of PROOF OF THEOREM 7 .

that

contained i n

Y

Let

be a

vo

R

is analytic i n

vo

A

.

($-minimizing f u n c t i o n .

We a l r e a d y know

and s a t i s f i e s t h e n t h e minimal s u r f a c e

equation. I t o n l y remains t o v e r i f y t h a t

Assuming t h i s r e l a t i o n f a se f o r some {yh)CR

x E aR

,

t h e r e s h o u l d b e a sequence

with h

+ a # $(x) To t h i s p u r p o s e r e c a l l t h a t

vo

The minimizing p r o p e r t y of

.

must be bounded. vo

and t h e G. C . ,

s a t i s f i e d by

QXR

,

imply t h a t t h e s e t

must have minimal boundary i n a neighborhood o f belong to

aE

should o b t a i n

.

W e c a n t h e n u s e Lemma 3 f o r

E = RXR

so

vo

must b e c o n t i n u o u s up t o

The u n i q u e n e s s o f

vo

aR

E

(x, a)

,

and

and

A = Q X R

i n a neighborhood o f

o b v i o u s l y i m p o s s i b l e f o r t h e d e f i n i t i o n of

(x, a)

,

must

(x, a)

.

Y = E

We

but t h a t is

.

and t a k e t h e v a l u e s

@

on it.

i s now a n o b v i o u s consequence of t h e Maximum

Principle. q.e.d.

3.8

THE CAPILLARY PROBLEM As a n a p p l i c a t i o n of t h e r e s u l t s p r e s e n t e d above, we s h a l l s t u d y t h e

c a p i l l a r y problem, t h a t i s t h e problem c o n s i s t i n g i n t h e d e t e r m i n a t i o n o f a l i q u i d f r e e s u r f a c e i n t e r f a c e as t h e r e s u l t a n t of s u r f a c e f o r c e s , g r a v i t y f o r c e s , and boundary a d h e s i o n .

W e d i s c u s s t h e s e phenomena n o t o n l y i n t h e

two-dimensional case, which i s e n c o u n t e r e d p h y s i c a l l y , b u t a l s o i n a n a r b i t r a r y number

n

o f dimensions, because of t h e g e n e r a l m a t h e m a t i c a l

198


i n t e r e s t of t h e t e c h n i q u e s . Our a t t e n t i o n w i l l remain bounded t o t h e non-parametric t h e f r e e s u r f a c e admits a r e p r e s e n t a t i o n

S2

where

u

=

u(x)

is assumed t o be a n open bounded s e t o f

,

case, i n which

x = (xl,.

Rn

. ., x n ) E R ,

.

The t o t a l energy o f t h e system i s g i v e n by t h e sum o f t h r e e terms

a l , g , p are p o s i t i v e c o n s t a n t and

where

positive or negative.

a2 i s a c o n s t a n t which may be

We c a n o b v i o u s l y d r o p one o f t h e p o s i t i v e c o n s t a n t s

and w r i t e t h e "mathematical energy" i n t h e f o l l o w i n g way

F(u)

+ IcI u2dx + a \ aR udH n-1

d-dx

=

R

JR k E (0,

with

+m)

and

a€

(-a, + m )

'

.

The f o l l o w i n g g e n e r a l p r o p e r t y c a n be e s t a b l i s h e d : THEOREM 8 . 2

A function

c (R)n c1 ( R ) - f u n c t i o n s

U

i s a minimum f o r

= 2 k u ,

where

i s t h e e x t e r i o r normal t o

V

PROOF. f o r any

u

If

in

-L

aQ .

F , t h e n , i n f i r s t p l a c e w e have t h a t ,

i s a minimum f o r

F(u + tv)

a t t a i n s i t s minimum f o r

i n t h e class o f a l l

R

v EC; (Q) , t h e f u n c t i o n t

F

i f and o n l y i f

t = 0

.

Therefore

199


1

d dt

- F ( u + t v ) t=O

=

\

Du

Dv

n

I n t e g r a t i n g by p a r t s , w e o b t a i n

t h a t i s equivalent t o

- 2 k u = 0 ,

I f now

vEc

2

(R) n C l ( 2 ) , Du

Dv

dx

.

fi

in

through t h e same argument, w e have

+

2ki uvdx

+

aJ

n

vdH

n- 1

.

= 0

an

I n t e g r a t i n g by p a r t s t h e f i r s t addend, w e g e t

Having a l r e a d y proven ( 3 9 ) , we g e t

t h a t is equivalent to ( 4 0 ) . V i c e v e r s a , m u l t i p l y i n g ( 4 0 ) and ( 3 9 ) by any i n t e g r a t i n g (40) over

an

and ( 3 9 ) o v e r

R ,

vEC

2

1-

(a) n C (R) ,

a d d i n g up t h e two i d e n t i t i e s

and a p p l y i n g t h e d i v e r g e n c e theorem, w e o b t a i n

Du * DV

uvdx+ a

R that is

J an

vdH

n-1

= 0

,

200


d dt F ( u + t v )

F(u+ tv)

Being

1

= o

.

t=O

t

a convex f u n c t i o n of

< F(u+ tv)

t=O -

t

i n p a r t i c u l a r , for

F(U)

i.e.

F(u)

5

=

1

F ( ~ + V )

, we ,

get

,

V t

I

, v

V E C

2

m) ncl(E)

i s the least value for

F in

,

C2(R) n c l ( G )

. q.e.d.

REMARK.

Theorem 8 shows t h a t t h e e x i s t e n c e of a minimum f o r

t h a t the constant

must s a t i s f y ( 4 0 1 , t h e n

a

One can d i r e c t l y show t h a t

Assuming

a < -1

,R

=

{ x E R 2 ( 1x1 < 13

F ( u A ) = T + 2 T ( l - A - 2 ) A + k T r r h 2 ( 1 - ( 1 - )')'-A Obviously, w e have

,

c o n s i d e r for

+ZvaA

.

A>

1

F

implies

201


If the domain R a

,

has a corner, then the restriction on the constant

in order to guarantee the existence of a minimum f o r

F , is even

stricter. Consider in fact the following special case:

where

e

is the Neper number.

Consider the following sequence of functions -1 -(xllgxl)

u.(x) I

=

$(xl)

,

-1

x1 > j

for

=

j(lgj)-l

We have, for large

,

for

x 1-< j-l

.

j ,

1

cosa/2 e

F ( u . ) = j-2tga 3

+

2x1tgw.4-dxl

j-l

1 + lgxl =

___ 2 2 x lg x

1

and using the inequality

+

1

~-{xl2ex~ < 1)

Being

$'(xl)

,

1

'

202

THE

DIRICHLET PROBLEM FOR THE MINIMAL SURFACE EQUATION

we get co sa/ 2 e

-'

F(u.) 5 const+2tga I

dx

1

+

2aJl+t92LI

J

cosa/2e

-dx -=

1

1lgX 1

j-l

By t h e same argument, one c o u l d g e t

if

tga-a

7. l+tga0

given

,

aER

n c Rn

and

afi , f o r a l l functions 2

is w e l l defined, f o r a

bounded, w i t h L i p s c h i t z r e g u l a r boundary

.

u E BV(R)

The v a l u e

f w

may o c c u r b e c a u s e o f

.

kJnu dx

the t e r m

F

The f o l l o w i n g theorem i s i m p o r t a n t f o r t h e s t u d y o f THEOREM 9.

all

where

and

If

uEBV(R)

L

RcRn

and

E

aR

i s bounded and

>0

i s Lipschitz-regular,

then f o r

w e have

is a Lipschitz constant f o r the functions representing

L e t us denote with

PROOF.

{I$ .I a p a r t i t i o n of u n i t y , i . e . 3

with

3

:

an ,

i s a c o n v e n i e n t number.

CE

I$, (x)2

F

0

t/ j , x

and t h e s e t

i s f i n i t e f o r e a c h compact s e t Moreover, assume that

KcRn

.

{($j}cC:(Rn)

204


when non empty, c a n b e r e p r e s e n t e d by a n

( n - 1)

L - L i p s c h i t z f u n c t i o n of

variables. W e o b v i o u s l y have

where t h e sum i s a c t u a l l y f i n i t e . Put

u. 3

where

=

$. 1

@ .u and assume 3

:A

R

+

,

n- 1

Ac R

i s open and

is

+j

p r e c i s e l y , w e assume t h e e x i s t e n c e of a p o s i t i v e r e a l number integer

hcn

,

More

L-Lipschitz.

6

and a n

with

Du.ldx

I

+

!

lujIdHn-l

St

where

and

St =

I

\-@j(G)

=

-t

By i n t e g r a t i n g ( 4 5 ) w i t h r e s p e c t t o

t

xlxEA,

1

. over the i n t e r v a l

(0,

6)

, we

get


Adding up t h e ( 4 6 ) w i t h r e s p e c t t o

cE

where

6

= C . m a x J ~ G . 1+

1

3

~6-l

i s chosen l e s s t h a n

E

j

, we

205

obtain (44) with

,

. q.e.d.

W e can prove now t h e f o l l o w i n g theorem: THEOREM 10.

f

then

is bounded from below i n

BV(R)

From ( 4 4 ) , w e have

PROOF.

where

If

c

i s a c o n v e n i e n t r e a l number.

Therefore

2 2

> (1 - /a/m)\ / m d

R

x

-

measR

+

2 2 2k meas,

a c

q.e.d.

A s e m i c o n t i n u i t y r e s u l t c a n be e s t a b l i s h e d t h r o u g h t h e use of

Theorem 9: THEOREM 11.

If

.

206


Then

Letting

j

+ 00

, we

obtain


Being

E >

207

arbitrary, we obtain

0

F(u) - F ( u . ) 1

1

5

.

0

q.e.d.

W e a r e now a b l e t o p r o v e t h e

EXISTENCE THEOREM.

u E BV(R)

then there e x i s t s

PROOF.

If

b e a minimizing sequence f o r

{uj}CBV(R)

Let

with

F .

From what w e

have s e e n i n t h e p r o o f of Theorem 1 0 , w e o b t a i n

I,JI...?'

5

dx +

(47)

2 1 a c F(u.)+- 2k m e a d 3 min

1:

-,

< const

1- I a l m I-

W e s h a l l see t h a t ( 4 7 ) i m p l i e s t h e compactness of 1 L (R)-convergence.

{uj}

t o the

To t h i s p u r p o s e , p u t

-u . = 1

u.

3

o ,

,

in

in

R R ~ - R

and

I

(x, z ) ERn+llxERn, z < ~ . ( x )

I

If

B

i s a b a l l containing

nu aR

.

.

i n its interior, we obtain

with respect


208

T h e r e f o r e , from theorem 9, w e have

with

c*

i n d e p e n d e n t of

.

j

There e x i s t s t h e n a n i n c r e a s i n g sequence o f i n t e g e r s

j(s)

such t h a t

Since

To prove t h e convergence of

in

verify that

uniformly with r e s p e c t t o

s

and

s'

.

1 L (R)

, it

is s u f f i c i e n t to


209

T h a t i s t r u e because

where

A.

3

(0)

= {xEn

These r e l a t i o n s imply

which i s what w e needed. q.e.d.

Let

UNIQUENESS THEOREM.

u1

satisfying

Then

u1=u2 PROOF.

.

Consider t h e function

u + u 2

1 u=-------

2

we have

and

u

2

b e two elements of

BV(n)

210

THE D I R I C H L E T PROBLEM FOR THE M I N I M A L SURFACE EQUATION

and t h i s l a s t i n e q u a l i t y i s s t r c t if

u1

= minf

$:

u2

.

Therefore

,

so-

F(u)

= minF

Then we must have

t h a t implies

u1 = u2

. q.e.d.

SMOOTHNESS OF THE M I N I M I Z I N G FUNCTION Assume negative.

a € (-1, 0)

,

so t h a t t h e minimizing f u n c t i o n

t E (0,

+”)

,

From t h e i n e q u a l i t y

w e derive

must be non

We s t a y w i t h t h e assumption

L e t us prove, i n f i r s t p l a c e , t h a t

fixed

u

put

u

i s bounded.

To t h i s p u r p o s e , f o r a


The i s o p e r i m e t r i c i n e q u a l i t y a p p l i e d t o t h e s e t

gives

By u s in g (441, we o b t a i n

then

Ther ef o r e, i f

we o b t a i n

2kt- la/c > 1- l

a

/

m

,

i.e. i f

211

212


By r e c a l l i n g (441,

t> t

for

we get

where

5

If

1 1

= c(n) I + l + C + r n

t >to and

IaIJ7-z

1-

Hn+l ( E t )

>0

(to, t)

By i n t e g r a t i n g over

,

w e have

, we

obtain

1

1

65

0 -

then

1 t't

0

+

(n+l);(\;dx)=

I n o t h e r words, w e r e

w e would have

Hn+l ( E t ) = 0

that is

,

.

( n + l ) c ( i,udx);;;r

,

213


q.e.d.

SMOOTHNESS OF THE REDUCED BOUNDARY OF

E = {(x, z ) / x E n , zO

0, i f

z(0

h(z) =

we can w r i t e

Therefore, i f

F

i s any measurable s e t with

FAE = ( E - F ) U ( F - E ) C c R x R

by u s i n g t h e extended minimum p r o p e r t y of

In particular, i f

B

P

, E

i s a b a l l contained i n

, we

get

fix R

and

FAECC

B

P ' we

get

We can t h e n use t h e r e g u l a r i t y theorem proven i n c h a p t e r 2 ( s e e p . 1 3 5 ) , and c l a i m t h a t

214


is a

C1r1'4

n-dimensional manifold with

Let us denote with

Go

the projection of

a*E

n (R X R)

onto

R , we shall

firstly prove that

To this purpose, it is sufficient to check that v

a * n~ (QX R) .

at all points of

n+ 1

> O

E

can be represented as

Assume by contradiction, that

V n+l(xO, u(xo)) = 0

for some

xo E R

.

We can assume also that nearby

where

v

is a

C1'1/4-function of

The minimum property of

E

( x o , u(xo))

n

variables.

and classical results about Clra-regular and must satisfy a differential

minimizing functions, imply that v E C2 equation like

(48)

div

(

DV

n

(I+

IDV(

2

I & . , -D,VD.V

2 3/2 (1 + IDv( )

'

D.D.v = 2ky 1 1

Putting

w

=

D v

and

a. ,(p) = 13

( I + IPI we must have

wC0

and

w

=

0

at the point

2 3/2

,

)

yo

corresponding to

.


(xo, u ( x o ) ) But

w

215

.

must a l s o s a t i s f y a d i f f e r e n t i a l e q u a t i o n d e r i v e d from ( 4 8 ) , t h a t

is

n

\

In particular a t

yo

,

w e have

This i s impossible because t h e matrix

{a. . ( D V ( Y ~ ) 1 ) 17

is positive,

and t h e h e s s i a n m a t r i x

i s non p o s i t i v e , b e i n g

yo

a maximum p o i n t f o r

R0

Let us f i n a l l y v e r i f y t h a t contained i n

with

For any i n t e g e r

h

in

R ,

let

lBih)l

j(Nh

=

R

.

Assume

w

. BR

t o be a b a l l

be a f i n i t e number o f b a l l s c o n t a i n e d

such t h a t

Nh 1 [diamBih)]n'l j=l Denote w i t h

{$h]

< h-'

a sequence o f

. C2(n)

f u n c t i o n s such t h a t :


216

0 5 4 <sup u = M hR

C o n s i d e r t h e D i r i c h l e t Problem

div

(

Dv

4 L p

v = +h

If

R < -

n- 1 2Mk

,

,

on

)

= 2kv

aBR

,

in

B R

'

.

such a problem h a s a u n i q u e s o l u t i o n

see J . S e r r i n C811.

Such a s o l u t i o n h a s v a l u e s i n t h e i n t e r v a l

as one can e a s i l y check through t h e Maximum P r i n c i p l e .

LO, MI

,

T h e r e f o r e , from t h e

g r a d i e n t estimate (see p . 27 and f o l . ) , w e g e t

W e can t h e n assume t h e sequence ball B R '

B

P '

for

p < R

.

The l i m i t

{vh} v

t o be u n i f o r m l y c o n v e r g e n t i n any

of

{vh}

s o l u t i o n of t h e same e q u a t i o n s a t i s f i e d by t h e

W e c a n conclude t h e n (see s e c t i o n 3 . 6 ) w i t h

v = u

i.e.

i s a smooth f u n c t i o n i n

u

,

is regular i n

BR

.

vh

and e q u a l t o u o n

CHAPTER FOUR

UNBOUNDED SOLUTIONS

I n t h i s c h a p t e r we i n t r o d u c e a g e n e r a l i z e d d e f i n i t i o n of s o l u t i o n s t o t h e minimal s u r f a c e e q u a t i o n , w i t h t h e a i m o f u n i f y i n g t h e s t u d y o f minimal W e have s e e n t h a t t h e s e s u r f a c e s c a n b e q u i t e

s u r f a c e s o f codimension one.

d i f f e r e n t from e a c h o t h e r , a n a l y t i c l i k e t h e g r a p h s o f s o l u t i o n s t o t h e minimal s u r f a c e e q u a t i o n o r s i n g u l a r l i k e the minimal c o n e s . I n t h e f i r s t s e c t i o n we s h a l l s t u d y t h e s t r u c t u r e of g e n e r a l i z e d s o l u t i o n s and w i l l f i n d t h a t t h e y c a n be d e s c r i b e d i n t e r m s o f c l a s s i c a l s o l u t i o n s and minimal c y l i n d e r s .

I n t h e second s e c t i o n w e s h a l l p r o v e a I n the third section we

compactness p r o p e r t y f o r g e n e r a l i z e d s o l u t i o n s .

prove a n a p p r o x i m a t i o n theorem f o r s i n g u l a r minimal s u r f a c e s i n terms o f We s h a l l t h e n a p p l y t h e s e g e n e r a l r e s u l t s t o t h e

smooth minimal s u r f a c e s .

s t u d y of D i r i c h l e t Problem w i t h i n f i n i t e d a t a o n unbounded domains.

We

s h a l l conclude t h e c h a p t e r w i t h some remarks a b o u t B e r n s t e i n Problem and removable s i n g u l a r i t i e s .

4.1

GENERALIZED SOLUTIONS. DEFINITION

W e s h a l l c a l l g e n e r a l i z e d s o l u t i o n t o t h e minimal s u r f a c e e q u a t i o n ,

i n t h e open s e t

R

of

Rn

,

a Lebesgue-measurable f u n c t i o n

f : R+C-..,+..I such t h a t , t h e s e t

h a s minimal boundary i n If

f

RxR.

i s a g e n e r a l i z e d s o l u t i o n and

everywhere, t h e n

g

i t s e l f i s a generalized solution.

case, w e i d e n t i f y t h e t w o s o l u t i o n s as t h e same s o l u t i o n .

i s equal to

g

f

and

g

,

f

almost

I f t h i s is the

t h a t i s w e c o n s i d e r them

We s h a l l u s e t h i s p o s s i b i l i t y o f changing t h e v a l u e s

218

UNBOUNDED SOLUTIONS

o f a g e n e r a l i z e d s o l u t i o n on a s e t o f measure z e r o , t o o b t a i n c o n t i n u o u s g e n e r a l i z e d s o l u t i o n s a t a l l p o i n t s where t h a t i s p o s s i b l e . L e t u s r e c a l l some p r o p e r t i e s o f t h e normal v e c t o r t o a minimal

surface. aE

v

If

and i f

i s t h e e x t e r i o r normal u n i t v e c t o r t o a minimal boundary i s t h e subgraph o f a f u n c t i o n

E

(x,z) E

a t a l l points

,

f

t h e n w e have

a *E n ( R X R ) .

The i n e q u a l i t y ( 1 ) i s o f g r e a t i n t e r e s t i f t a k e n t o g e t h e r w i t h t h e e q u a t i o n s a t i s f i e d by

V

n+ 1

on

a*En

From ( 1 ) and ( 2 ) Bombieri-Giusti

,

Theorem 6 ) : i f

contained i n

R

B

i s an

, with

( a x R)

d e r i v e d t h e f o l l o w i n g consequences (see

, where

p < y ( n ) d i s t ( B , a f i xR )

n

p o s i t i v e r e a l number depending o n t h e dimension either

(3)

V

n+ 1

(x,z) = 0

v

,

( x , z ) E B n a*E

E

V

n+ 1

.

>0

with t h e b a l l

B

is a vertical

I n t h e second case

i s e i t h e r empty o r t h e graph o f a L i p s c h i t z f u n c t i o n o v e r a n open Rn.

Such a f u n c t i o n i s a c t u a l l y a n a l y t i c and a s o l u t i o n t o t h e

minimal s u r f a c e e q u a t i o n . itself.

n a*E

aEnB

c y l i n d r i c a l s u r f a c e , with p o s s i b l e s i n g u l a r p o i n t s .

set of

inf B

I n t h e f i r s t case, t h e i n t e r s e c t i o n o f t h e s e t

aEnB

is a

Y(n)

o n l y , then

or

i s made o f v e r t i c a l s t r a i g h t segments, t h e r e f o r e

,

p

(n+l)-dimensional b a l l of radius

This function coincides with the function

Observe t h a t i f t h e f i r s t case o c c u r s f o r one ball

same happens f o r a l l b a l l s o b t a i n e d by t r a n s l a t i n g p r e c i s e l y , f o r an

n-dimensional b a l l

p O

(4)

,

meas(B ( x ) - P ) = 0

*

A f i n a l remark a b o u t t h e case when

i s t h e following: i f

belong t o

y E r - G

,

,

xEP

E n

( r X

is not a vertical cylinder,

R)

V zER

then

Therefore e i t h e r

aE.

( y , z ) E;

the point

V z

,

or

(y,z)

(y,z)

BE V

cannot z.

In the

f i r s t c a s e we have

lirn f(x)

(6)

=

+

,

m

X+Y

i n t h e second case, w e have

.

l i m f(x) = -

(71

X+Y

What w e have remarked a b o u t t h e b e h a v i o r o f a g e n e r a l i z e d s o l u t i o n

r

over b a l l s

statement: i f

of radius f

p 0

and i f

already

In addition t o that, i f there exists a

r+

i s of c l a s s

a

and

For a f i x e d

d(x) = dist(x,aR).

i n a neighborhood o f

C3

p

t h e n one c o u l d f i n d p o s i t i v e numbers

where

,

T h i s c o n d i t i o n f o l l o w s from a remark by R. F i n n

normal.

mentioned i n s e c t i o n 3.5. with

,

k>O

xo r+ x . 0 '

with

and

6k>0

,

such t h a t

a c l a s s i c a l argument u s i n g b a r r i e r f u n c t i o n s , p r o v e s t h a t

u ( x ) L ~ - MV, x E R f l B

P

(where Being

M

depends o n

R

and

a

(X )

O

-Rk

only).

arbitrary this leads t o a contradiction.

k

So w e s h a l l assume t h a t

r+

r-

and

have z e r o mean c u r v a t u r e , i . e .

t h e i r measures a r e minimal under l o c a l d e f o r m a t i o n s . For s a k e of s i m p l i c i t y , assume f i r s t t h a t

$ (x)

Denote by and e q u a l t o

h for

h

t h e f u n c t i o n d e f i n e d as e q u a l t o

$(XI

for

x Er

xEr,.

For what w e showed i n s e c t i o n s 3.5 and 3 . 6 , t h e r e e x i s t s a u n i q u e u h E C L (R)

,

s o l u t i o n o f t h e minimal s u r f a c e e q u a t i o n i n

The sequence

{uh3

, with

a s a consequence o f t h e Maximum

i s non-decreasing,

Principle, there e x i s t s then

l i m u (x) = u ( x ) h h Denoting w i t h

the set

P

,

V xER

.

{xEn/u(x)=

+m}

proved: LEMMA.

The s e t

P

minimizes t h e f u n c t i o n a l

,

t h e f o l l o w i n g Lemma c a n be

223

D I R I C H L E T PROBLEM W I T H INFINITE DATA

PROOF.

If

i s an open b a l l c o n t a i n i n g

B

,

and

A c B

i s an open s e t

w i t h r e g u l a r boundary, such t h a t A n a n

r , ~ , - ~ ( a ~ n a =n o)

=

6 E H1 " ( A )

denote with

,

h The s e t

,

Eh

, @

an e x t e n s i o n o f

xEB-

and w i t h

(AUn)

d e f i n e d by

minimizes t h e p e r i m e t e r i n t h e class o f a l l s e t s

,

MCRn+'

M - n x R =

E

h

.

- n x R

Therefore, t h e l i m i t set

where for

;(x)

= u(x)

xEB-AUR

,

for

xER

,

u ( x ) = $(x)

for

xEA-n

and

u(x)

= +m

minimizes t h e p e r i m e t e r i n t h e c l a s s of a l l s e t s

MCRn+',

.

M - n x R = E - n x R

Such a minimum p r o p e r t y remains v a l i d f o r t h e t r a n s l a t e d s e t s

Since, a s

where

X

Q = P

+ +m

,

U (B - A

Therefore t h e s e t F c B ,

Un) (I

,

the set

Q

X

R

h a s t h e same minimum p r o p e r t y .

minimizes t h e p e r i m e t e r i n t h e c l a s s of a l l sets

F-fi2Q-Q

,

224

UNBOUNDED SOLUTIONS

which i s e q u i v a l e n t t o s a y t h a t

P

F,.

minimizes

q.e.d. We can prove t h e f o l l o w i n g theorem THEOREM 1 .

Assume

r-

Assume

r

@ :

-t

=

p

and t h e mean c u r v a t u r e of

t o be c o n t i n u o u s and non n e g a t i v e .

R

t h e unique minimum f o r t h e f u n c t i o n a l

F+ ,

r+

equal t o zero.

I f t h e empty s e t i s

then t h e r e e x i s t s a s o l u t i o n

o f t h e problem ( 1 2 ) - ( 1 5 ) . PROOF.

Put

=

$,(XI

rh

=

an-Ah

W e have, (19)

' 'h,h

'h,k"h,k+l

= 'h

'

'h+l,k-

Minimal Surfaces of Codimension One

Minimal surfaces of codimension one

Minimal surfaces