Geometric Measure Theory and Minimal Surfaces (C.I.M.E. Summer Schools, 61)

E. Bombieri ( E d.)

Geometric Measure Theory and Minimal Surfaces Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, August 24 - September 2, 1972

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-10969-0 e-ISBN: 978-3-642-10970-6 DOI:10.1007/978-3-642-10970-6 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1973 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CENTRO INTERNAZIONALE MA TEMATICO ESTIVO (C.I.M.E.) 3 0 CicIo - Varenna :- dal 24 agosto al 2. settembre

1972

GEOMETRIC MEASURE THEORY AND MINIMAL SURFACES Coordinatore : Prof. E. BOMBIERI

W. K. ALLARD: F . J . ALMGREN JR . : E

GIUSTI:

J . GU CKENHEIMER: D . KI!'{DERLEHRER :

M.

MIRANDA:

L . C. PICCININI :

On the first variation of area and generalize(Y) = y,

Whenever A

y

E

€

C:mn and a

closed cone with vertex

€

0

k

k "Dd W n M = W n q>(E ) •

C1;;sur~ A , we let in

En

Tan(A,a)

be the

consisting of those vectors v

- 7 -

W. K,. Allard

in

JR n

such that

~,x2' ••• e A - (a)

lvi-Iv.

/xi-al-l{xi-a)

lim

v=0

such that either

or

F 0 and, there x. = a and

v

lim

i-+oo

We let

J.

are points

Nor{A,a) = (w: v·w < 0

i-+oo

for all

v e Tan{A,a») •

Evident~,

Tan(M,a) e G{n,k) for each ure on IMI (A)

a eM.

Let

and

14k

if M is as above,

Nor(M,a) e G{n,n-k)

be the k dimensional Hausdorff meas-

JR n ; we set

= H k(x:

and observe that Clear~,

(x,Tan{M,x» IMI

A C JRn x G{n,k) ,

e A) ,

is a Bo:l"el regular measure on

IMI e Y (JRn) k

bounded open subset of

i f and o~ if

JR n

]R

n x G(n,k).

M intersects l/EVery

in a set of finite k dimensional area.

Fran the change of variables formula of advanced calculus we have that (6)

F#IMI

= IF(M) I

for arry diffeomorphism F

This motivates the definition of F#. sional submanifold of class 1 in (€,h,K)

En,

is a local defonnation of

Suppose

of

JRn.

M is a k dimen-

IMI e Yk(E

n)

and

JRn; using (5) and (6) we see

that

(1)

8I M/(ho )

=

~t#k[ht{M n

K)]lt=o.

We now suppose that (8)

M is a smooth k dimensional submanifold of

En With

boundary B. By this we mean that M is

~

k dimensional submanifold of class

00

- 8 -

W_ K. Allard

in

E

n

B = (Closure M) - M , and that for each b E B

,that

there are smooth f'unctions



e(x )

for

IIv J. .1I almost; all

x

€

n

K of

:JR

,

JR n,

i = 1,2 , .••.

Then

Eh IIv lI ,x)

~ e(x)

for

IIvlI

aJlnos t all

x

€

JR n .

I t i s b eyond "(;u,=, s cop e of thes e l e ctures t o gi ve a complet e proof of t he s e theorems .

In the next lecture, howev er , we wi ll

deri ve all the ge omet r i c i ngredient s of their pr oof s .

- 16 -

w.

K. Allard

Lecture Three Suppose For each

V

a € :ffi n

~a.t(X) a

a,

v(t)

€

\OR n)

and

and each

=

= IIvllex:

i

x:a

115vII

is!j. Radon measure on

t €:ffi

we set

if

Ix-al ~ t ,

if

t

Ix-al ~ t}

:R n •

< Ix-al ~ a, V(t)

= 5V(Sa. t)

We have the basic relation about Change of mass in concentric bails:

(1)

s-ltx

. (s) a,V

r-I£

a, v(r)

exp

s

d7 a,V(t)

r

tna,v(t)

s

13 a,V(t) dt tn a, V(t)

J

expJ

r

whenever distance (a,sptllvll) < r < s < In proving (1) we suppose

a = 0

and write a,~ ,7

a ·V' t3 n' 7 V' respective~. For each e: > 0 0, a.,v 0, fUnction f e:: :ffi n ~:ffi in such a way that fe:(x)

~

Ix]

Ixl grad fe:(x) ~ x Let

", e

C ~(:ffi)

and let

unifo~

q>(t ) =

as

00

for

we choose a smooth

e: ~ 0 •

t.; ",( T)dT,

t € :ffi •

For each

- 17 -

W. K. Allard

e > 0 ,let

ge(x) = cp(±:e(X))X'

x

JR

€

n

" Note that

ge

€

X"(R

and that

so that Dge(x)"S = cp'(f'e(X)) grad f'e(x)"X 1

- cp'(f'e(x)) grad f'e(X)"S (x) + kcp(f'e(x)) " Integrating with respect to V and letting

J cp(t)~(t)

=

J ~'(t)t

e ~ 0 , we have that

dl(t) -

- J cp'(t)dr(t)

+k

J cp(t)da(t)

Integrating by parts in this last expression, we see that

1 *(t)~(t)dt =-1 +

¥(t)t dl(t) +

1 *(t)dr(t)

+k

1 ¥(t)a(t)dt

so that, in the sense of' distribution theory, ~(t)dt = -

tdl(t) + dr(t) + ~(t)dt

tdl(t) - ~(t)dt

daft~ _ ~ dt a t t

We integrate:f'rom r

to

=-

~(t)dt +

= _ ~(t~d) it:X t

s

dy(t)

+ dr(t) "

1iiTtT'

to obtain (1)"

n)

- 1[; -

w.

K. Al.rar-d

From (1) we draw two basic corollaries: () .k () r -k l): a, V r ::: s l): a, V s exp

(2)

E a tt Jrs ~V V.", a, oW

dt

whenever distance (a,sptIlVID < r < s < (3)

if r

C

€

Vk(JR

n),

varies, then

8C = 0 x

€

S

and for

00

r-1txo,C(r) is constant as

C almost all

(x,S)

Both these statements follow almost immediately from (1). draw some consequences of (2).

(4)

~(lIvll ,a)

€

JR

whenever

We first

The firs+. is that lim sup 118VIIJB ta,r)

r+o~

This is an immediate consequence of (2).


0

eIl8v. II E ( a . , r ) < IIV.IIE(a.,r) , ~

a

~

-

~

~

0


Here

~

1

C is a

for

[v]

con~ tant

Suppose almos t all

€

'¥k(JRn) ,

x e JRn.

depending only on

The proof is as follows. s =

V

IT

a

Suppose €

IIvll(E n)
log A. • MJB a , t a s The inequality (6) now follows from the covering lemma of Besicovitch in the form given by [FE 2 .8.14].

- 20 -

W. K. Allard

An immediate corollary of (6) is that

-H k(M) (k-1)/k ~ c[J

(7)

IH(x) IdHkx +

M

wheID"er M,B,H

H k-1(B)]

are as in (8) of Lecture One

and ftk(M)
0

k .:.!-lIv,-" n ...>..{=x::........;:;..ef...>..(,,-,lIv..... 1I =,x:L..)_. (.;;,;.,a,=r

o.

Them among all disjointed regions m i

there regions for which

Let

for each

i

, are

- 44 -

F. J . Almgren Jr .

Ii

n-l(Q i=l

attains a minimum value?

d'\) This problem always admits solutions

and general arguments (see D.)

[AF1]

show that except for a

(possibly empty ) compact singular set of

.I;n-l

zero~

measure,

differentiable is a Holde~ continuously~n-l dimensional submanifold of

n• R

The particular form of the problem above further

implies the real analyt.ici t y of the regular part of For

i

of

n, R of course)

= 1

volume.

u. ~

;)A.

~

the unique solution to this problem (up to isometries

For

i = 2,

is a standard n

=3

n ball of the prescribed

the unique solution (sketched in

figure 6, also sketched "blown apart")

consists of three

(Figure 6) spherical pieces meeting along a circle (in this case the circle is the compact singular set of zero~ measure referred to above and in

D.).

For 1

3 , n

=3

the solution

seems to be that

sketched in figure 7 ; note that six pieces of real analytic (Figure 7) surface meet tangentially at

120

0

along four smooth arcs

which in turn meet at two vertices tangentially as the central cone over the vertices of a regular tetrahedron (see [GJl~

- 45 -

F. J. Almgren Jr.

E~~LE

6.

A Mobius band like

su~ace

(sketched in figure 8)

(Figure 8) occurs as a soap film for a wire bent in the shape of the boundary shown,

an~

also occurs as a surface of least area among

all mathematical surfaces spanning such a boundary in the sense of homology with coefficients in the integers modulo 2 (see E.2). Similarly a triple Mobius band like surface (sketched in figure

9) oocurs as a soap film for a wire bent in the shape (Figure 9)

of the boundary shown, and also ocours as a surface of least area among all mathematical surfaces which span such a boundary in the sense of homology with coefficients in the integers modulo) (see E.)). Finally a surface

S

like that sketched in figure 10

(Figure 10) (like a Mobius band on the left joined to a triple Mobius band on the right by a thin ribbon of surface, having as boundary a single simple closed unknotted curve)

C

occurs both .as a soap

film and as a mathematical minimal surface.

However, J.F. Adams

- 46 -

F . J . Almgren Jr.

has pointed out the existence of a continuous retraction (S , 0)

(0 , 0) of

~

S

onto the boundary C [RE Appendix]

so that, in no way in the sense of algebraic topology, does S "span"

O.

bound~y

of

In particular, if one wishes to regard S

C as the

then one must use alternative definitions of

boundary of a surface than those of algebraic topology

(s~e,

in

particular, the variational formulation in [AW]).

EXAMPLE 7.

Suppose one bends a wire into the shape of an t~at

overhand knot as sketched in figure lla (noteAthe two ends of the wire are free) .

Typically when such a wire is dipped in (Figure 11)

soapy water a film such as that sketched in figure lIb forms -even though the wire is not closed:

Such a film does admit a

mathematical approximation, but only with a "boundary" of substantial positive thickness.

Indeed one can prove by

tangent cone arguments (see the nice discussion of such cones in [GJl])

that such a mathematical surface is

impossible over an infinitely thin boundary of class

3.

The

significance of this, among other things, is that if one wishes

- 47 -

F. J. Almgren Jr.

to construct a theo ry of min i mal surfaces which in particular include s th e phenomena sugge st ed by s oap f i lms , t he n one mu st at t i mes

a~~it

EXAMPLE. 8 . among all

bound a rie s of

Suppose

B

L l

U

L u 2

C and

C'

of leas t t ot al l ength ,

L 3

L l

[(x ,y) : y

3-1/2(2 + x)

-2

L 2

[( x, y ) : y

3-1/2(2 - x )

l/2

L 3 and

lying i n R2~ [(x,y) ; X2 + y2 ~ 1)

B are pathwi s e connected t o each other t here a r e

exactly two di stinct s ets C l

S

C'

Then

un ions of nontrivial rect ifiable arcs··through whi ch

the poi nt s of

wher e

posi t ive thicknes s .

2 [(- 2 , 0) , (2 , 0)] c:: R .

=

1 dim ens i onal se t s

whi ch are

subst ~~tial

~

x

~

_l/2]

! x ~

2) ,

J,

1/2 2 [( x, y) : y = (1 - x )1/2 , _3-1/2 ~ x ~ 3 is th e image of

( s ee f igure 12).

C under reflect ion a cro s s the x axis

Thi s problem i s naturall y pose d , t h e i nt egral ( Figure 12)

~ length)

i s r eal analyt ic, th e boundary

[( x,y): x 2 + y2< lJ

B and the obs tacle

are algebr a ic , and t here are exac tly two

solutions (a t rivi al modifi cation

m~{ e s

the s olu ti on un ique).

Nevertheless each solut ion cu rve , although a

1 dimensional

- 48 -

F. J. Almgren Jr.

2 R

submanifold of

of class 1, is no t a submani f ol d of class 2.

The tangent l ines of

hf!.wever , CiAdo vary in a Lipschitzian mann er ,

C and

hence a f orti ori Holder cont inuously (see EXAMPLE 9.

Suppose

B =

[(0,-1), (0,1)).

2 1 dimen s ional set C in R of

B .i s , of course,

(see figure l3a).

C

D.3~

of least length connecting the points

= R 2n

Now let

Th en the unique

(x, y):

x

f: R2~ R2

= ° , -1 ~

y ~ 13

be the algebraic

(Figure 13) di f f eomor phi sm gi v en by integrand f

F: R

2

x

G(n , l)--to

to giv e a new integrand

that t he

uniq~e

= (x

f(x, y)

G

+

h)

= f#

1 di mensional set

Y3 , y ). tr~~forms

F

-1

~

x

~

(see figure l3b).

implies the ellipticity of y

= xl / 3

G

g

D of least feB)

D i s also t he graph of the f unct i on 1

naturally under

with t he obvious propert y

among those se ts connec ting t he points of Not e that

Th e length

is y

integral D

= f(C)

= xl / 3

for

The ellipticity (D.l(1)(2)) of

F

and it i s clear that the ·functi.on

is t h e unique natural solut ion t o the real analyt ic

elliptic Eul er-Lagrange differential equ a t ion a ssociated with and the standard (x, y) coordi na t es for

2 R

(see D. 2 ) .

This

G

- 49 -

F. J. Almgren Jr.

fUnction is not even of class 1, however (although it is real analytic except for a compact singular set of zero

L1

measure -

a representative. conclusion for such problems (see C.l(5)(d»).

- 50 -

F. J : Almgren Jr.

PART B

D.l

X~~~~~~g~~1=~~g£~~m~=~~=~=m~g~~~~=~~~~~~~: Suppose

one is given a suitable open set

./'t an

W in

appropriate space of mappings

with fl~ W prescribed. reasonable function

Rk•

We will denote by

f: closure

W-+ Rn , perhaps

We will suppose also we are given a

~: ./t, ~ R

In case

k

o ,

n

=1

, then

~can be regarded as, say, the space of class 1 mappings ~:

R -+R , and t h e basic problem which led to the different ial

calculus was t hat of finding a point where j value.

assumes its maxi mum

Equivalently one could seek those points where ][ takes

its minimum value, or, more generally, one could seek critical points of

~ ~

vanishes.

In case

, i.e. points at which the first derivative

~~ ~

k! 1 , n ~ 1 , then, heuristically at least,

the basic problem of the calculus of variations (in this context) is that of finding points (actually mappings a t which (most commonly)

i

assumes its minimum value, or, more

gen er al l y , the critical points of variation

d~

n) f: closure W ~ R

i ,

i.e. wh er e the first

vanishes ident icall y ( in pr a ct i ce the definition

- 51 -

F . J. Almgren Jr .

of first variation varies considerab1y from problem to problem). In a nUmber of ways, as the above phraseology suggests, there are analogies between calculus .and the calculus of

variatio~s.

The

ordinary calculus has been extended to differential .manifolds in various fashions, and it is sometimes useful to regard certain problems· i n the calculus of variations in. the language of the ordinary calculus, but extended to manifolds having infinite dimensions.

These manifolds of infinite dimension typically are

"modelled on" Hilbert spaces or Banach spaces.

Unfortunately,

such infinite dimensional manifolds so far have not played a significant role in the geometric variational problems with which we are mainly

conce~ed

The functions

here.

i:.It ~ R which

have received the greatest

mathematical attention in higher dimensions are integrals of the form (*)

I

(f) =

Here

ep:

with

i .

Vi

K

S

xeW

f(x, f(x), Df(x))

~

dX2 ...

d~

' f,"

/to

Rn ~ Hom(Rk, Rn) --+ R+ is the integrand associated

For example, the Dirichlet integrand .

'f D is given by

- 52 -

F. J. Almgren Jr.

Another example is the

k dimensional area integrand

'A which

is defined by setting

CPA (x, f(x), Df(x» here AkDf(X) :

1\ Rk

= 11 /\

Df( x) 1\

~ 1\ k Rn and

Il-\Df(X) 1/

the square root of the sum of the squares of all the of the n by n

is equal to k by k

minors

Jacobian matrix ..li!-(x) ~~ 2

ax1

~f (x)

if(x) ~

1

1

~ f (x) ch 2

Jf (x) aX k

2

2

...2.!

~f (x) c)x

cl~

2

(x)

~~(x)

~2

As

~ndicated,

weare primarily

i

corresponding to geometric integrands . 0

z])

[1

B Co

Rn ,..,

S

sucn that

~

~ (r)] F ( [S" (z: ep

+

be

minimal with respect :;\:;

)

(F

i f and. only if there exists

~

is elliptic and of class 3.

F:

(z)

I z)])

whenever (a)

Rn and the stru~ture

REFERENCES.

of sets of finite Hausdorff measure.

This part is based on and intended as a partial introduction

to sections 2.10.1, 2.10.2, 2.10.5, 2.10.6, 3.3.1, 3.3.2, 3.3.5, 3.3.13, and 3.3.19 F.l.

of [FH1] • The problems of

~

dimensional area.

A basic mathematical

objective in the study of k dimensional area in Rn has been the creation of a theory of measure and integration over k dimensional sets in Rn•

it

is now generally accepted that the cornerstone to such a theory lies in the creation and study of suitable k dimensional measures in Rn•

For such

measures to be geometrically viable they should be Borel regular (i.e. Borel sets are measurable and every set is contained>in a Bote1 set of equal measure) and invariant under Euclidean isometries.

For

k=n the

n n n dimensional Lebesgue measureL over R is characterized by these two conditions together with the requirement .(n { For

k < n

Xl

0 ~'t~ 1

for i = 1, ••• , nJ '" 1.

there are a number of distinct k dimensional measures (some

of which are discussed in F.2) which are Borel regular, invariant under Euclidean isometries, and assign to subsets of k dimensional submanifo1ds of class 1 the "correct number".

Much of geometric measure theory during

the first half of this century consisted of detailed studies of peculiar Cantor type sets on which the various k dimensional measures disagree. such set occurs as an example in F.4. n,

A second bas ic objective of the study of k dimensional area in R

of course intimately tied to the first, is the understanding of the

One

- 77 -

li' ,

J . Almgren Jr.

geometric struoture imposed on a set by the requirement that it have fini te k dimensional area.

The stud3' of the pathology mentioned above

has led to a general pattern of structure (in large measure due to A. S. Besicovitoh originally, and later H. Federer) the central results of whioh are discussed in F.5.

F.2

n• Caratheodo;r's construction for k dimens ional measures ·in ·R

The ingredients of Caratheodory's construction arel n• (a) a family of subsets F of R

If F is such a family and

0 -c

~

~

one sets F,s (b) a function

SI F-~tl 0 ~ t ~

Corresponding to eaoh ~

0.( 6 ~

aD].

~ ~

00

eq.

b

approximating measure

ql,t

given by the formula

,4'not- l(S)

(from the de f i n i t i ons ) .?, 2? '¥Yot-l (S).

J

- 89 -

F. J. Almgren Jr. Hence ~ 5(8)

8€. H .

=

1IIf1V.+l (A.)

= ? ~ard(G~+l)

(since

= I/f v+ l

d11(

>

as observed in estimate (3) above).

It foll~s that (){.l A) = 'tt' •

•

y

Computation

~ j{l(A), ,gl(A), ~ ~~(A). One notes that for convex

2, sets 8 C R

(

.

SIt l.. (p8)

pe: G(2,1)

dX2 It

is nondecreasing in t for 1 ~ t ~ since for

1 ~ s

~l(A)

Computation of

J;(A). Uan --4S 1

"1n(X)

tha t j(

X€

0a , x e; n

~k, k) unrectifiable.

and

'11 : l n

1

k) rectifiable

O.

In particular, then, A is purely

whenever

~k,

B(z,

=

First one defines for n

(x-z)/lx-zl

Jn-1)

€" Gn-1

(see figure 17).

We as se rt

almos t all points a of A have the property tha t .

d

.

~~.!!!

for ~ integer

Y

?;

5.

Sl

To prove this it is sufficient to show for

each closed proper subarc J of Sl,

Jt l(A To see this we set K

for each n> \I.

and

~ 5,

n

(\

[)v la: '1n(a) f

KjI = G~

U

DE K

Since

J})

and

F (D)n tB: n-l

O.

n

"1

n

(B)

n

(Sl - J)

I ¢}

- 91 '-

F. J. Almgren Jr.

for 11 > Il , one finds that the set

An (\

~al~n(a)¢ J}C

n;>11

n

n'?~

(AnUKn )

~ 1 measure O.

ha s

DEFINITION.

For a ERn, 0..( r ~oo , V €

lsi

X(a,r,V,s) = Rnn

G(n, n-k), 0

.
-

~

l!

k) rectifiable

if

0 there exists a k damenaLonaj,

n of class I of R such that

J.{k([A - Mg lLJ rMf. - AJ) < THEOREM.

~k,

is called

A,

]I

e,

C Rn ~ ~ ~ ~ ~ (j{k, k) rectifiable,

AVB, Allll, A - B

~~

sets which

~ (Ji O.

THEOREM.

If A

~ ~ ~ ~ ~ Rn ~ j{k(A)' 0, 0 0

exists a unique

y

The function

d(x)

in

such that, for every x On such that 2

is , of class

in

~o'

there

Ix-yl

d(x)

C

in ~o and

( )

n-L

\grad d(x)

I

=1 .

We have n-L

i Y " -~(x) = "~ l-kk (y)d(x) ~ ~ ki(y) , i=l i i=l

where

y

is the point corresponding to

(i = 1, .•• ,n-l) IF.

~,

in

On, and

ki(y)

are the principal curvatures of 00 at

An upper barrier (relative tp

in some

x

C\'

and

1/1)

> -

1 +

is a function

y. J.L(x) , defined

0 < E < 'Eo' such that

J.L ~ 1/1

in ~ '

J.L

max i~1 + max 11/11 <Xl

a

- 130 -

E. Giusti

(Here and in the following, we sum over repeated indices .) In an analuguUs way, we define a lower barrier as a function

A(X)

defined in 0e ' such that

A = cP

on

00

j

A

S -l~; Icp I on f e '

e.( A) 2: 0

We note

tha~

e.(A)

in ~ •

is, except for a positive factor, the Euler operator

relative to the area functional.

00 , and i l l t

~.!!.

,t he ~ curvature.Q!

Lipschitz-continuous function in O .

Suppose that

00 k llQn.-negative. Then there llill upper and lower

barriers.

E!:Q2!. We shall construct only an upper barrier . The lower barrier can be found by means of the same argument. First, we extend cp will be denoted again by

2 to a C - fu nct i on defined in 0

cp .

f(s)

2

is a C -function in f( 0)

(l.~)

1f'(s)

=0

this function

We will consider a barrier of the form

~(x)

where

j

= cp(x)

+ f(d(x)) ,

(O,e] such that

f(e) 2: 1 + 2 max Icpl +

o

2: 1 + ICP~l +

Itl l

=~

m::.x It I = ~ o

f "(s)

< 0 in (O,e]

- 13 1 -

E. G iusti

In this way , condit ions ( ~l) and (~2) a r e sa t isfi ed. .i nst a nce , that Let

y

~(x )

2:.

Let us prove, for

>jr ( x )

be 't he point on

00

su ch t ha t

d(x)

=

\x-y I

.

We have

a nd hen ce

On t he ot her hand,

so t hat

~( x ) ~

>jr ( x ) •

Finally, we have

d d + 2~cp d 1 + + f ,2 (i4J-(jl xix j xi x j xi xi + f ' ( 2.6::p cp d -2cp cp d ;(1+ Icp 12)~ _d cp cp ) xi xi xix j xi x j x xix j xi x j Since

00

ha s non- negative mean curvature , we hav e from (a ) tha t

3

~

is

non -pos itive, and therefore

where

c

2

depends on the C -norms of cp

l

and

d .

I f we t ake f(s) with A

~

c

l

' we ha ve

e(~ )

1 =A log

(Bs+ l)

SO, and we must only verify (1 . 5 ) .

Thi s i s

- 132 -

E. Giusti

done easily if we choose A

= oC l

and

lG.

An upper barrier is actually a supersolution in 0e: ; in fact, let

~(x)

be a Lipschitz-continuous function with support in

~

, and consider

the function

We have 0:' ( 0)

If

~ ~

0 , we have

the function

il.

0:'(0) ~ 0 , and, - taking into accou} the convexity of

o:(t) , we get 0:(0)

so that

~(x)

.s 0:(1)

is a supersolution in 0e: •

If we put mi n

w(x)

=

j

(~,M)

in

Oe:

M

with M = max (max I~I, max

of

°.

en

o

!w\) ,

we have the desired supersolution in all

- 133 -

E. Giusti In fact, let that

v 2. wand

vex)

be a Lipschitz-continuous function .in

0, such

supp (v-w) CO.

If we observe that

G(v) 2.G(min (v, M}) , it r emains to ·show only that G(min (v,M}) 2.G(w)

For that, let us remember that in

rE

port of the function

is in

in

~

, and hence

min (v,M}-w w(x)

we ha ve ~

is a supersolution i n

~(x)

2.M+l , so that the sup-

But

~(x)

is a supersolution

0 •

In an exactly similar way, we can construct a subsolution so' that

~eo-

rem 1.1 is completely proved. l H.

The existence of a solution for t he problem with smoot h obstacles ha s

been proved independently by Giaquinta an d Pepe Stampacchia (19]

.

also Miranda [21]).

[4J

and by Lewy and

We have followed es s ent ially t he methods of [4] (see Lewy and Stampa cchia work with general monot one opera-

tors, including the minimal surface operator, in the spirit of the variat ional inequalities.

Both in (4 ] and [19], one ca n f i nd the following regula rit y

result: I f the obstacle ~

1jr(x)

be l ongs to the Sobol ev space

is true for the s ol ution

u(x) •

continuously dif f er ent iabl e , then

u( x)

; , p( O)J n < p < + 00

In particula r, i f belongs to

,

the

1jr(x ) .ie.. t wice

el, a, f or every a < 1

This result is almost the best possibl e, since one ca n expe ct a t most to get a el,l-solution.

This poi nt , however, is not settled i n general; for

see Kinderlehrer

(11)

Th ,~

n

=2

cons truction of b,lrriers of section IF follows the lines

of the paper of Jenkins end Serrin

OQJ

a ) ' a ) an d a ) ffir.Y be found in Serrin 2 3 l The conclusion of Theorem 1.1 holds

; a proof of properti es

(24) •

- 134 -

E . Giusti if only one supposes that the boundary data some ex> 0

( s ee Giusti

[7J).

1

C'

belongs to

~(x)

The obstacle pr Dblem ha s been

ex , for '

consider~d

for

ot he r fu nctionals, in the framework of variational inequalities. For that, see Li ons and St ampa cc hia Stampacchia

[2.], and Lewy and

(20], Brezi s and Stampacchia

[18]

We di d not mention here the problem of the c ont a ct set ; i .e ., t he problem of the topological st ructure and of t he r egularity of the set in whi ch the s olut ion touches t he obsta cl e .

I n general , t his pr oblem i s completely

open; i n the 2 -dimensional case , some interesting r esults ca n be f ound in the papers of Lewy an d Stampac chi a

(18)

and of Kin derlehr er

(14] .

The

f i rst pape r deals with the obstacle problem fo r t he Dir i chlet integral and ze ro boundary data; if (x

€

0 : u( x) > 1jr(x ) )

is r eal analytic , then analytic curve.

1jr( x )

is strict l y concave , t hen the set

i s t opologically a n annulus . ~he

bounda ry of the set

(x

I f in add it ion €

0 : u (x)

1jr(x )

= 1jr( x ) )

Similar resul ts are proved by Kinderl ehrer for

i s an

minimal

sur f aces with obst a cles. For t he pa rametric probl em with obstacles, M. Mir anda t ha t every minimal surface wi th a Cl -obstac le in

[221 pr oves

En is of cla s s

l C

in a

neighborhood of t he obst acle . Finally, f or 2- dimens i onal su r faces i n Tomi

2.

[26] , Hildebrandt and Kau l

E3 , we ment i on the papers of

(9) , and Hildebra ndt

[8] .

Discontinuous a nd thin obst ac l e s j existence theory We shal l now consider the obstac le problem in which the i nequal i t y

u(x )

~

1jr( x)

is requi r ed t o be sat i s fi ed only on a clos ed s et

in, an d different from ,

O.

A conta ined

- 135 -

E . Giusti

is convenient to reduce this problem to an equivalent one , in

I~

which the set A does not enter explicitly .

If we define the new obstacle X € A ,

~(x )

'?i-A ,

X €

the inequality u 2:.

n

in

\jI

The new fUncti on

\jI

is identical to

the cons ideration of the problem and boundary datum

2A.

Let

n,

an d, since

A is

I n this way, we are naturally led to

[~,O,\jI]

with upper semi -continuous obs tacle

En , an d let ~(K)

upp er semi-continuous fUnctions i n K. ~(K)

in A •

~

K be a compact set in

bel ongs to

1jr

is now defined in all of

closed, it is upper semi - con tinuous.

\jI

u -2:.

be the cl a s s of all

It is known that a fUnction

f(x)

if and only if it is the pointwise limit of a decr ea s i ng

sequence of Lipschitz -continuous fUnctions (or, what is the same, of C~-fUnc -. t i ons ) ; f

k

~

f

in short, if there exists a sequence of

k

in

CO,l(K)

such that

.

For a function in

k~ ~

It is worth noting that

, we define the area functional :

~(K)

G(f) =. inf (lim inf

f

f

G(f )

~ Il+ \Dfk I2 dx; f~

K

€

CO, l ;

f

k

~

f) •

agrees with the derini tion given in lA when

is Lipschitz -continuous; this follows from the semi - continuity re sul t of

1A , observing t hat, for continuous

f , the convergence

f

n

I f

uniform conv er g0nce . We have t he following results: i) ii)

a(f)

a

coincides with the Lebesgue area for cont i nuous

is a convex functional .

f .

implies

- 136 -

E. Giusn iii)

If

fj

U(K)

is a sequence in

and

f j I f , then

f

€

U(K)

and

a(f) _ ~ lim inf u(f j) . CD iv)

j-t

If

u(~)

of

f


f(x) • ~ there exists

{f n}

such ~ g(x) > fn(X}

no

be!!. bounded open

~ in Bn with' 0' -boundary, ~ let

z'(x)

be the Lipschitz-continuous solution of the problem

w(x)

be!!. function in

(2.1) Proof.

0°,1(0) a(z)

+

J Iw-cpldHn-

00

[cp,O,V] •

Let

Then 1 •

Let ~(x)

and let

such ~ w ~ V

-< a(w)

for every

w = w + (z-w)~. k

= max'{O,

l-kd(x)} ,

We have

' On the other hand, lim

J~ID(Z-w)ldx =O,

k-+a> 0

a nd lim

f Iz-wl

k-+a> 0

so that

1~ldx = lim

(2.1) follows at once.

k-+a>

q.e.d.

k

f

~/ k

Iz-wldx =

J Iz-wldHn_l '

00

- 138 -

E . Giusti

2C. i§..

Theorem 2.1.

Let

•

0

be ~ bounded open set in

En

J

~c3-surface with .!!.Q!1.':'negative ~ curvature. Let g.

u(x)

and

v(x)

be attainable solutions of 'the problems

[f,n,g] , ·r espect i vel y .

~n'

u

*n

and

and

~ ~ f

Suppose that

f n, gn be two

v, respectively.

ca~ples

and * ~ g .

of approxi mat i ng sequenc es,

We can suppose tha t

From Lemma 2 .1, i t follows t hat, for every

f

n

> f

and

n, there exists a

k

n

such t hat

f or every

k > k

n

J

so that Lemma 1.3 gives in

Passing t o t he limit as

2E.

k

~

00

,

an d t her e a s

n

~oo

J

n

we get t he conclusion .

In a similar way, one can pr ove the following result :

q .e.d .

- 140 -

E . G iusti (1jr )

Pronos i tion 2 .1 . fu nct ions in ~

't hat

n

the nr oblem

0 ~

2F.

do ,

a nd

cP

and

respectively , s uch t ha t

*n I '.jr , and let

[cpn,O)*n]

t o the problem

be tw o s e que nc e s of

n

Then

un

~

*n .s n

on

semi -c ont i nuous

do .

Sup-

be the a t ta i na bl e solut i on to

un I u ) and

u

i s the atta inable s olut i on

[cp,O,*J .

The problem of the uniquenes s of the solution is compl etely op en .

Of

c ou rse ) it follows from 2A(v ) t hat every function in ~(O) , differen t f r om a s olut i on only i n a s et of z ero ( n - l ) - dimens i ona l mea sure) i s i t s elf a solu tion.

I t would be int er e sting to prove that these a re the only poss ibl e sol -

u tions) a nd it would be very surprising to s how t hat this i s not the ca s e . Let us note that ) if

cP

and

* are co nti nuous )

i t f ollows by

Lemma s 2 .1 and 1 .4 t ha t the a t tai na bl e solution is continuous . this is the only cont inuous solution ( s ee [ 4

1 a nd ( 19J;

a l t hou gh constructed

in different ways ) the two solut ions a re a ct ual l y the same) . on t his su bj ect is Gi u s t i

[ 5] .

quite

In this ca s e)

The liter atu r e

scarce ; we have followed he re the method of

To my knowledge ) t his is the only ge ne ral ex istence result

fo r non -parametr ic discontinuou s a nd th i n obstacles .

J. C. C. Nitsche

( 2 3] ,

who f i rst considered the problem) proved the existence of a continuou s s olu tion i n the 2 -dimen s ional case , when the obstacl e is gi ve n on a s t ra i ght line, a nd

0

i s symmetr i cal with r e spect t o i t.

Recently, Ki nderlehrer

[1 3J

r emov ed t he assumpt ion of symmetry and p roved a s imilar r esult for gene ral va r iat i ona l inequal ities . ~he

For t he pa rametric case , De Giorgi

existence of minimal su rfaces with t hin ob st a cl e s.

[3J

ha s proved

- 141 -

E . Giusti

3.

Regularity of

~

attainable ,solution; the. favorable

~

We shall show in this chapter that, under suitable assumptions, the (attainable) solutions to the Plateau problem with discontinuous and thin obstacles are Lipschitz-continuous in 0

3A. Let us start with a particular case.

Let

*

be an open set, with elosure

(~,O~~,*]

contained in 0, and consider the problem that

~

Let uS 'suppose

2 C and that the mean curvature of ~

is of class

is non-negative.

In this case . it is possible to construct a Lipschitz-continuous subsolution

",(x) . in

~,' tak-

ing the prescribed values on

dol

(see Section

W l

be the function

and let

u(x) If

in

0-01

w(x)

1).

*(x) Let

in

O-~

in

~"

be the solution to the problem

[~,O'*l]

is a Lipschitz-continuous function such that

and w(x)

= ~(x)

in

do, we have

a(u)

.s a(w)

.

In fact, since

",(x)

Now let

be a sequence, whose first element is

Wj(x)

w(x) ~ W(x)

is a subsolution, one has

Wl(x) , ~onotonically

- 142 -

E. Giusti converging to the function

~(x)

From the preceding remark, we conclude that solution of all the problems

is the (unique)

[~,O'~j]' and hence it is the attainable solu-

[~,O~,~]

tion of problem 3B.

u(x)

Let us consider now the general case of discontinuous obstacles.

It is

clear from the preceding discussion that the solution will be Lipschitz-continuous if only we can construct a subsolution ~(x)

= *(x)

dA n a , and that

in

in

~(x)

~(x) ~ ~(x)

a-A, such that

on that part of

00 which

does not belong to A . It is clear that the possibility of such construction will depend on the mean curvature of Theorem 3 .1.

dA . To be precise, we have the following theorem:

Suppose that there exists a n ~ set

B, whose boundary

2m

is ~ C3-manifold of .!l2!2-negative ~ curvature, such that

a-A Then the problem ~ontinuous

?roof.

anB.

[~,A,~] ,with ~ and

~ of class C2, has a Llpschitz-,

attainable solution.

Extend

~(x) toa C2-function in En, which we again call ~(x) ,

in such a way that to a function in

~(x) ~ ~(x)

En, such that

on

dB

n a = dA n a.

~(x) ~ *(x)

in

Next, extend

*(x )

dB. Now we can use the

procedure of Theorem 1.3 in order to construct a subsolution

~(x)

in

B,

- 143 -

E. Giusti

su ch that role of ~

= ~( x )

A( X) and

~

in

dB, and

w a r e cha nged in this case)

is the boundary datum .)

in .B

A(~) ~~(x ) ~

The f unction . A( X)

is now the obstacle , a nd

provides the r equ i red

q.e .d .

sub s ol ut i on .

The same idea works i n the case of thin obstacle s .

3C.

( Not e that the

Suppos e ,.

is gi ven

as the null set of a C3- funct i on ~ (x )

(x

A

suc h that (x

€

I D~ 1 ~ 0

0 : ~ (x ) > 0

Theo r em 3 .2 .

0

Denote by

on A ( resp. ,

€

~( x) =

0+

o) ,

( resp.,

0 _)

the s et

< C)} . We have the fol lowing r e su lt :

~(x )

Supnose t hat the re exist two open sets

B+

~.£.

C3- bOunda ry hav i ng non-nega tive mean curvature , su ch that

0+

the nroblem u~

[ ~ ,A, W],

with

cp

and

W of class

§..

= 0 n B+

. Then

Lips chitz -c ontin-

att a i na ble soluti on . It i s wor t h noting that t he sets

has zero mean curvature , and t his becau s e ti on, to both

3D.

2

C , ha s

B_ , with

B+ a nd

B

can exist only i f

A

A bel ongs, with different or i ent a-

dB+ and dB

'rh o i de a of "filling th ,; hole" by means of

8.

s u bs ol u t i on

b~ en introduced by Giusti (5J a n d , inde pendently, by Kinderl ehrer [13] • Wi t h ~±~~ the s a me method stampacchia and has

Vi gnoli

[25]

prove the a na l og ou s of Theorem 3.1 for solutions

to non-coercive variational inequalities. I f t h ef unctions

cf

and

continuous , then the solut ion Lipschitz -continuous, then exponent .

t

u(x )

a~~

if

~

ane

is continuous , an d i f

~

and

ware

i s Hold er -c ontinuous , with some pos i tive

In a ddition, if t he set

mea n curvature ,

i n Theprems 3.1 and 3.2 are

u(x )

~

B in Theorem 3 .1 has bounda r y of po sitive are Holder -continuous

with exp onent

a,

- 144 -

E. Gi usti then the solution is Holder-cuntinuous

withex~unent

0/2

(for details,

se~

Giusti [6 ·]). Another interesting problem is the nature of the contact set; Le., the set where

u(x) = ' ~(x) •

;ional case when the set cave/ and

Here, the only results known are in the 2-dimen-

A is a straight line.

If t he obstacle

.~

is con-

\jI(xo) , the conclusion is trivial.

, and let

) , so that

B be.§i subset of

x

q.e.d.

0

It may be worth noting that no assumption is made on the set

B.

In

Jarticular, for discontinuous and thin obstacles, we can conclude that t he restriction of

u(x)

same is true for

to the set A, u A

is continuous in A , provided the

\jIA

4D. We shall now consider the problem of the continuity of the solution u(x)

when x

approaches a point

x

o

in

We need the following

lew~a,

which is a simple generalization of Theorem 1.3.

~ 4.1. Let and let

eo

0

be.§i bounded open set with boundary

g(x)

3 ,

C

do, Suppose that the mean curvature of

1§. .!!2!!.-negative in .§i neighborhood of the support of

exists .§i subsolution

do of class

vex) , taking the values

g(x)

in

g.

00

.Th!:E. there

- 147 -

E. Giusti Remark.

It -is easily seen that the conclus ion of the preceding lemma holds .

supp "g

In fact, let "' \

Q ~ One~ .

~hat

define

4E.

vex)

=0

i s a C3-surface in a neighborhood

On

if~we only suppose that

We can appl y t he preceding lemma to ' \ ' and then

in a~l

Let us consider first the case of discontinuous obstacles, and let us

suppose that ·the boundary of A in that a point

dA

X ' €

o

na

Theorem 4.4 .

dA n a

,

Let

cp

and

~~~

Proof .

x

o '

Then

from

u(x)

seen from

a-A

B R

= ",(x)~(x)

ing the value

at

~~

'" 1£ dA

cp

na

continuous in

>

0

• g(x)

as a point of pos itive mean cur· x

o

is positive. X

o

be

~

in

= B(xo,R)

On,

~.9f.. class

2

C

neippborhood of

~

and

'" > 0

"'l ( x )

X

o .

a.

in

By continu-

OA n BR

such that every point in

B such that R,

d(a-A) •

point in

in ~ neighbor-

0

Let

S ~(x) S 1

~( x )

in

From Lemma 4.1, we get a subsolution v(x) in

We say

poi nt of positive mean curvature. Sup-

is

be a

B 2 , and let R/ in

a-A,

tak-

Let

u(x)

be

The function

"'( X)

in A,

= \ v(x)

i s upper semi-continuous in

a,

the solution to the problem

[CP,O' '''l] •

the problem

is a C3-surface.

as a point of positive mea n curvature.

C~-function with support in

g(x)

na ,

a-A

d( a -A)

a-A

~

We can suppose that

ity, there exists a ball

dA

'" be continuous, and let

pose that the restriction ·of of

a,

is seen from

vature if the mean curvature of

~.

a

be an open subset of

Q of

with C3-boundary, such

in

a-A

and 'is continuous in

B 2• R/

It is easily seen that

u(x)

[cp,A,If] • The conclusion then follows fran Theorem 4.3.

solves

- 148 -

E. Giusti A similar result holds for thin obstacles: Theorem 4.5.

Suppose that

l!!Y* . - Let

X

~

k continous. and that 1jr(x)

point in A that

k

~

0+ • ~ the restriction of

curvature from

4F.

be

o

q>

as

u(x)

~

is

~ C2 -function

-poi nt of positive mean

k continuous at

to 0+

All the results in this s~ction are proved by Giusti 'l 6] .

The!!. 'p:r>i or i

estimate (4.1) is a generalization of the result of Bombieri, De Giorgi and Miranda

[1] .

Since the Fl'"rnearance of that paner, many authors have dis-

cussed the existehce of ~priori estimates. for the gradient of solutions to non-linear non-coercive elliptic differential equations. of Ladyzenskaja and Ural'ceva of the

~

[15]

We mention here the articles

and--Trudinger

(27).

A very short proof

priori estimate for minimal SUL-J.i:1
=

(r/2 )'K 1 ( 0 (SnB( p, r ) ) )

~ (r/2)( d/dr )( ~~ 2(S()B(p, r»).

- 160 -

J. Guckenheimer

Letting mer) = 11. 2(SnB(p,r)), we see that (d/dr)(log(m(r))'?O which implies m(~)/r2 increases monotonically as r increases. The analogous relation for soap-bubble-like surfaces is that eKrm(r)/r2 is monotonically increasing. This fundamental result can be viewed in another way. Let Sr be the surface obtained by translating SnB(p,r) so that p is at thee orifin an~,tihen expanding this lutface by see i~ure"t a factor of l/r~ we ave tha oS lies on the unit sphere 2 2 r and}t (Sr) = m(r)/r. The above monotonicity relation thus states thattl 2(Sr) decreases monotonically as r decreases towards zero; since it is always nonegative, it must have a limit, and this limit ~y be recognized as precisely

~2(l~~S, p )-'11; where~2(t"t2LS,p) is the density of S at p (AF2]. If we now choose a sequence of radii r i decreasing to zero, the resulting surfaces Sri might behave wildly as surfaces (see (AF2]). However, as measures, since the surfaces Sri lie in a bounded region of R3 and have bounded areas, some SUbsequence must converge. Any limit to any such subsequence is defined to be a tangent ~ to p at p; we see that monotonicity implies immediately the existence of tangent cones at every point p in S. We note in passing that if C is a tangent cone to S at p, then {A.2(C) =~2(V-2LS,p)·n. Two properties of any such tangent cone C are that C is a cone (that is, its support consists of rays from the origin to a boundary on the unit sphere), and that C is area minimizing. These properties in turn imply that oC consists of geodesic segments intersecting 3 at a time at equal (120 0 ) angles. It happens that one can completely classify all the boundaries that satisfy these latter two conditions.

- 101

J. Gu ckenheimer

THEOREM [GJ2]. ]E to orthogonal rotations, there ~ precisely eight I-dimensional configurations Qn the unit sphere which ~ composed of segments of great circles intersecting precisely 3 at .!!: time at 12'0 0 angles; these ~ (1) ~ single great circle, (2) 3 half great circles (at 120 0 to each other), (3) 6 great circle segments forming the ~-skeleton of .!!: regular spherical tetrahedron, (4) 12 segments forming the ~-skeleton of .~ spherical cube, (5) 9 segments forming the ~-skeleton of ~ certain spherical prism over ~ regular triangle, (6) 15 segments forming the ~-skeleton of ~ certain spherical prism. over ~ regular pentagon, (7) 30 segments forming the one-skeleton of ~ regular spherical dodecahedron, and (8) 24 segments forming 2 quadrilaterals and 8 pentagons. Furthermore, only the cones over (1), (2) and (3) ~ ~ minimizing, and hence, ~ to orthogonal rotations, these ~ the only possible tangent cones to soap-film-like and soap-bubble-like surfaces. The cone over any great circle is just a disk; choosing a specific such great circle, we denote this cone by D. The cone over any configuration as (2) above is three half disks at 1200; choosing a specific such configuration we denote the cone by Y. The cone over any configuration as in (3) consists of 6 wedges meeting at equal angles at the origin; again, the cone over some specific such configuration is denoted by T. We note that D, Y, and T all have different areas and hence the density at any point of S determines its tangent cones there up to rotations. The cones Y and Tare sketched in figure 5. Knowing the poss ible tangent cones gives a great amount of information on the local structure of the surface; however, we can prove much more.

- 162 -

J. Guckenheimer

THEOREM [GJ2). Let §. be [the interior of) ~ soap-filmlike or soap-bubble-like surface. Then: (1) There exist unique tangent cones at every point of S [this says that for any sequence· r. decreasing to zero as ~ above, the measures corresponding to the surfaces Sri converge; this is not a trivial result.]; (2) R(S)~ { p : tangent cone to S at p is ~ rotation of D } is ~ two dimensional analytic submanifold of R3, each connected component of which has constant mean curvature, and ~ 2(R(S)) = -\-\ 2(8) [AFl);-(3) cJy(Sh lP: tangent cones to S at p is some rotation of y '} is ~ I-dimensional Holder continuously differentiable submanifold of R3, and for each PCqy(S) there exists ~ neighbood N of p such that S~N consists of the union of three 2-dimensional Holder-continuously differentiably manifolds with boundary meeting tangentially at 120 0 angles, the boundary of each being precisely N 0~y(S) [this is in part a statement of how smoothly the analytic part of S meets the singularity O-y (S ) ] ; (4) 0, ~ ) 0 and ~

+0

t\\lc:rl=

~(~,~':\-'1/I~I)

FIGURE

J..

K \ ( S')

I

Z

s ~

o+he-r\l,)\5e..)

10, t 1 aI

set of coincidence

its boundary

r = (xl'~'X3) : x3 =u(z), z =:l<J. +i~ E aI}, curve of separation. It is already known that (cf. [5], [6])

' r is a Jordan curve and

(1)

([7]):

We shall discuss a new result Theorem 1.

If

,

i s analytic and strictly concave, then

r is an

analytic Jordan curve (as a function of its arc length parameter). The first item of information we collect about the problem is that since u E

Cl(O)

and

u-

t ~0

in 0,

in I. Let us observe now that some condition about the mean curvature of ,

is necessary to conclude anytrdng at all about 01 • For suppose that

D·V

D. . J

.1

J1 + \Dv1 2

Now Dju =Djv

in

= 0 in

Bn I

Be 0,

implies that

B

a ball,

B

n Ii ¢

a..e ,

Djku =Dj k,

in

Bn I •

Hence D. J

Hence, since equa t i on in

Dju

.11 + IDul~

= 0

u E H2 , Q(0) (0 - I) U (B n I )

a.e ,

u

'in

(0-1) U (B

n

I)

is a solution of the minimal surface Wit! l

U

= Xi

i n B n I.

Hen ce

u

=v

in

- 177 -

D . Kinderlehrer

B,

independently of aI n B,

nature of the supposed

01

that the mean curvature of

in particular, independently of the

We shall see that it is necessary to know ,

is not zero.

How does one shqw that a curve is analytic?

One seeks to extend a

conformal representation of the minimal surface

Let us -briefly review this old concept.

common tangent plane to

0 E r.

Now choose a neighborhood

r'

connected subarc

c

r,

0 E r'

X:G ... UC S,

.

o

and X(t) E r,

X2 -= X2 t t2 l

and X • X =0 t t 2 l

is the

Uc S whose boundary contains a

Then there exists a 1 : I mapping

G={ltl O}

-l

-'1"2

° for

.t i)l +,~ 1

.

+.~

t >0.

+.;

where we choose a branch of ,;-

In this formula we choose the + sign

implying that r

ep2 1m -, > (jll

(I)

° for

Itj, lepll, lep2 1 small.

Hence in (12), the left hand side

(LHS)

becomes

so that

where

Ho

is the mean curvature of M at

function Fl (t,(jll,(jl2,(jl)

analytic for

0

Hence there exists a

t E G and JI~j-P small so that

~~ =Fl ( t ,'Pl , :P2 '~3 ) ' FlCa,a,a,o)

=

I~ (-L *.ihgjf~ +2(gJ-'lgl-V2~ o

- 186 -

D. Kinderlehrer

Similarly, there exist F2 and F which have the same dependence on 3 Mean curvature of M. The system

th ~

~j(t) = Fj(t'~1'~2'~3) ~.(O) J

=~ =0 J

has a unique solution which is ana,lytic in Gn{ltl<e}

and continuous

in 'On (It I

n

n

a-E

then

n-1, for some

01. >

0,

sa-

nn

is

and (12)

is valid. The proof of Massari's result requires some lemmas. LBMMA 1: sfies (13) with

f\


(x) dx E

(27)

= (

)~m

x "x 1-1

f E(x)dHn_ 1

'

t

and also, by changing variables, 1 n t -

(28)

J

Dtf (x)dx

B

E

t

=/

[x I = 1

'fE(tX)dHn _ 1·

Using (26) , (28) and (14) we d iscover the new inequality

t,_J

whi ch can be written, by using the other notations , in the form

n

(30)

~ +

'1

(x) dBn-l

] t"P,

Btn ~* E

Ixl '-

p/2

6~En(B

In-II

J

PI

~

t =P2 n

dBn _1 •

-B

It-

PI

n

{ [t

1

)

lIAIXI ! B

t

-

n

Bn_ 1 I

dx)

dt

~. E A Btl]

J

t

=9 2 +

t

= P1

- 202 -

M.

Another modification "can be" obtaiu.ed by considering

Miranda

t

Ie term

(31)

dH

and, with an integration by parts, we obtain

j

B

1 n I x 1 - 1D'I' Elxll

dx = [

f

1 -n t i D fE(x) B

p 2-B P 1

I dx

J

n-

1) dt,

t= P2

t

=P

1

t

(32)

dx) dt .

R emembering 03) and comparing

E

E UB

with the set

easily see that

~

nW ,n -l n

+

f

IAlx11 dx •

Bt

(34)

dx) dt

.

we

+

- 203 -

M . - Miranaa The for-mulae (34) and (29) provide us a new for-mula, which will be very useful, in the sequel. An interesting consequence of Lemma I can be obtained when

o c d-E n n .

In this case, from De Giorgi's results about the r eda-

ced boundary,

lim

(35)

p~o

PI

Letting

o( p1-n

tend to

J B

0

f~t-n 11

in (29) we obtain

ID 'fElxl[ dx - "n-l +(n-l)

0

p

B

Alx)

I dx)

dt .

t

The formula (36) can be extended to the case where

0 E

V* E n n

by an approximation argument . We take this opportunity to remark that

dl/fE

the set

means that id where t ion,

~

E

n f2

is the support in

x E

n -

is equal to

if necessary, of

E

il'E 1

n of the gradient of

If E

. This

then there exists a neighborhood of or

0

x

almost everywhere . By a modif'ica-

in a set of measure zero we can suppose that

(37)

Henceforth we will assume that Caccioppoli sets satisfy (37). Therefore we can state the LEMMA 2 : If

E

is a - Caccioppoli subset of

the condition (13) and if

f
n,

with

by using

the Halder inequality

(38)

Therefore 1 1--

wn P

II All

p

L (0)

f

1-~

P

The relation (39) implies that

JPit f

IAI.II

-ri

o

tends uniformly to

0

B

f

as

dx) dt

t

tends to

O.

With additional technical but non trivial calculations, the inequa lities just proved may be employed to demonstrate

IA(x)

(40)

with

l~

r

H

(Bo 0,

0(: 0

A(x)




'UP{ f"

(fdivG + GOI dx

vx )

The formula (55) has to be considered as a definition of

f. '11 n

+

I Df(x) I ·2

dx

- 211 -

M. Miranda

for

1 f € L (r2),

rivatives in

+

00

but it is an identity in the case where

L 1(r2}. The quantity

in which case we set

f

has first de-

defined by the formula (55') can be

1\ (f) = +

00

•

In the case

in '11

ID'I 2

+

dx

< +

00

there exists (for the references see the paper of Santi) a function T € f (56)

L 1(

~

with the property

(2)

lim

P-n

p-l-O

,

J

I Tf(x)

r2nB

p

- f(y)

I

dy = 0,

H

n-

1-a. e.

on

(X)

. So we can calculate

but since

T

f

is uniquely determined by

f

and . n

we write

instead of

In this way

"(f) is well defined. To prove the existence of the 1 minimum for A. in L (n) we apply direct methods, say we consider 1 a sequence f E L (r2) such that h (57)

lim ho+oo

"(fh)

inf

fA(f}

1:

- 212 -

M.

Miranda

Let us observe that

A (0) = mis n

which implies that the sequence Let us consider a ball ded over

B

B

as a function in

" (f ~~

n

+1

Ig I

on

dH n_ 1
0

n

E ~ R

. f{ P (G,A ) + meas(G()A) ) = ~n 0e: ( E,A e: '

(8)

Fo~

-- E

G~

r-:

II

A, G e: G'l n

and o(E,A) ' = lim e:+O

(9)

0

e:

(E,A)

It is easy to prove that

0

(De Giorgi's measure) is actu-

ally an exterior measure, for it is enough to use corollary 6. We can anyhow remark that this measure is somehow · constructed .

following Caratheodory's construction class

G'

n

• In fact, since the

is stable under union and since for any sequence

of Borel sets

(.)

(.)

See Almgren's lectures.

- 228 -

L. C. Piccinihi

(00

P

t' U

i=1

meas r U B inA) l i= 1 Bi ,AJ+ e: )