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

n),

0 < d < co, U

q=p/(p-l).

5V(g) whenever

is an open subset

O
:s CUlglq dllvll)l/q, g

€

n) X(JR

and

spt g C U

9.l1d

E h IIvll , x) 2: Then

ff(IIvll ,.) . is

d

for

IIvll

almost all

x

€

U •

a real vatued uppersemicontinuous function on U.

- 27 -

W. K . Alla r d

We have proved this theorem already ( s ee (5) of Lecture p = "",

Two) in case

p <

In case

q = 1

t he argument ne eds

co

to be mor e delicate and uses t h e Isoperimetric Inequality ; we omit it. Suppose now V satisfies the hypotheses of the Lemma. Let

D be the set of points

ek ( Ilvll ,.)

oscillation of dTJ.

a

in

U

n s pt llVIl

, restricted to

sptllVIl , do es not exceed

ek ( Ilv ll> ,.)

Owing to the uppersemicontinuity of

elementary that

D is dense in

U

, it i s

n sptllvll. Also, if a e D it

is e asy t o see that for some sufficiently small

s a ~~sfies

for Which the

r

t h e hypotheses of the Regularity Theorem.

n sptllVIl

t he var i f old

We con clude

that an open dense subset of

U

entiable subman i f ol d of

whose t ang ent map satisf ies a HBlde r

lRn

condition (locally) with exponent

i s a continuously differ-

1- kip .

How b ig can t h e exceptional set, that i s the set of points where

s ptllVIl

is nondifferentiable, be ?

It can b e arb itrarily

large as we see from the following Example.

Suppose

= lR 2

n (x: x 2

~

and M 2

= oj ,

~

M 2

lR

= lR 2

is smooth.

n (x :. x 2

Let

= f( x l )}

are smooth 1 dimensional submanifolds of

boundary.

and let

f: JR

Let

N

lR 2

n (a: a

2

= f(a

l)

r ·O}.

Cl early ,

; not e that JR2

~

wi th empty

- 28 -

w.

0

=

Ef(lIvll,a)

r

As is well known, we

-N J.(N) > O.

closed, we see that

a Ii sptllVIl ,

if

J. i f a e (sptllvll) - N , 2

nected with

if

aeN.

m~

take

Because

Obvious:Iy,

spt IIvll

N

to be tota.l1y discon-

N is tota.l1y disconnected and

N is precise:Iy the set of points where the

Ef(lIvll,.) , restricted to

density

K. Allard

sptllVIl , is discontinuous.

does not have the structure of a manifoJ.d near

the points of N. One wou1.d J.1Jte to be abJ.e to put conditions on insure the reguJ.arity almost everywhere of -V. be a difficu1.t probJ.em.

We have the following

Open Question.

V e T

:m n

,

0

<

d

< 00

BV(g) = 0

Suppose

k

e JR

n

BV that

This appears to

U is an open subset of

,

,

whenever

g e X(JR n )

and

spt g C U

and

~(lIvll ,a) 2: d for IIvll almost Let that

all

E

R = sptllvll () (a: sptllvllis smooth near IIvll(u",R)

e U • a

J.

Is it true

= 01

Of course, it is not true that

U'" R

= ~.

For example,

a;rry sum of the varifoJ.ds corresponding to a finite collection of k dimensionaJ. affine subspacee of

in the Open Question.

:m n

will satisfy the conditions

- 29 -

W. K. Allard We now make some remarks regarding the proof of the Regularity Theorem.

It is proved by constructing nonparametric approx-

imations to the surface under consideration, which approximations are close to the original surface in terms of certain geometrically defined quantities which measure the deviation of the surface from being disk-like.

One calculates the Laplacian of these nonpara-

metric approximations and uses well known a priori estimates for th~

Laplacian to obtain geometric information about the approxi-

mations and therefore about the original surfaces.

This is pos-

sible because taking the Laplacian of these approximations amounts to calculating the first variation of area with respect to certain deformations of the original surface, modulo an error which is small relative to the aforementioned geometric quantities. We conclude with another corollary to the Regularity Theorem. There is a positive number

Theorem.

1)

with the followiag property:

Suppose M is a smooth closed k-l dimensional subman d f'o.ld of

-n-l whose mean curvature vector is alw/Ws normal t o

-n-l Ji

Ji

''1hose area does not exceed

(l+rj)lsa(k) ; then Me S

and

for some

S e G(n,k) •

This assertion. is proved by applyiDg the Regularity Theorem to the cone C = (tx: 0 < t < 1 No"te that

and

11. k(C) ~ (l+rj)a(k)and that

x e M)

s!c] = 0

sufficiently small, Closure C is differentiable at

thus if

o.

1)

is

- 30 -

W. K . .Allard

REFERENCE3

L,

Allard, W. K., On the first variation of a

varifo~d,

Ann. of

Math., Vol. 95, Ma;v ~972, pp. 4~7-491. 2.

Almgren, F. J., Jr., The theory of

varifo~ds,

Mimeographed

nct; es, Princeton, 1965.

3.

Federer, H., Geometric Measure Theory, Die Grundlehren der math. Wissenshaften, Band

1969.

~53,

Springer-Verlag, New York,

CENTRO I NT ERNA Z IO NALE MA T E MATICO ESTI VO (C . I. M .E . )

F . J. AL MG REN Jr .

GEOMETRIC MEASU R E THEORY A ND ELLIPTI C VAR IA T IO NAL P RO B L E MS

Corsd

t enuto

a

Var enn a

dal

24

a go s t o

al

2

settembre

1 972

GEOMETRIC MEASURE THEORY AND ELLIPTIC VARIATIONAL PROBLEMS

by F. J. ALMGREN

Jr.

(Princeton University)

"During t he l a st three decades the subject of geometric measure theory ha s developed from a collection of isolated special results into a cohesive body of basic knowledge with an ample natural struoture of its own, an d with strong ties to many other parts of mathematics. These advances have given us deeper perception of the analytic and topological foundat ions of ge ome t ry-, and have provided new directions to the ca lculus of variations.

Reoently the methods of geometric

measure theory have led to very substantial progress in the study of qui t e gene r a l elliptic va riational problems, including the multidimensional problem of least a r ea . " "[FRl - Preface]. The se lecture notes are intended as an introduction to the collection of mathematical techniques and results known as geometrio measure theory l argely from the point of view of certain problems arising in the calculus of variations.

1 The preparation of these lecture notes was supported in part b,y a grant from the National Science Foundation, in part by funds from the Science Rese arch Counoil in oonnection with the Symposium on Global Analysis, 1971-1972, at the Universi~ of Warwick, and in part by funds from the International Atomic Energy Agency -an d the United Nations Educational, Scientific, and Cultural Organization in connection with the Summer College on Global Analysis and its Applications, 4 July - 25 August, 1972, a t the International Centre for Theoretical Physics, Trieste.

- 34 -

F. J. Almgren Jr. There are six partsl PART A.

Some phenomena of geometric variational problems.

PART B.

Geometric variational problems in a mapping setting

and

associated varifolds. PART C.

Surfaces as measures.

PART D.

A regularity theorem.

PART E.

Estimates on singular sets.

PART F.

Caratheodory1s con struction for k dime nsional measure s in Rn and the structure of sets of finite Hau s dorff me a su r e .

There a r e also twenty illustrations. Figure 1.

A disk with five han dl e s .

Figure 2.

A s impl e closed unknotted boun dary cu rve C of f inite length.

Figure 3.

The oriented surface S having boun dary C an d of l e as t area ha s infinite topological type .

Figure

4.

A di sk T with infinitely man y handles converging t o a boun da ry point.

Figure .5.

The 1 dimensional set Y conn ec ting the 3 poin t s of the boun dary B and of le a st length has an interior singularity of co dimension 1.

Figure 6.

The unique minimal partitioning configuration for t wo regi ons of prescribed volumes.

Figure

7.

A "soap bubble like" minimal partitioning configu r ation consisting of six r e al an a l y t i c surface s meeting a t 1 20

0

along four smooth ar cs which meet at equ al an gles a t t wo poi n ts . Fi gu re 8.

The Mob i u s band as a s u r f a ce of l ea st a r ea .

Fi gu r e 9.

The triple Mobius ban d as a s u r f a ce of lea s t ·a r ea .

- 35 -

F. J . Almgren Jr. Figure ·1 0.

The su r fa ce S r e t ra ct s onto i ts boundary C.

Figure 11 .

A soap film with a boun dary wire which i s not closed.

Figure 12.

The curve C of lea st l engt h is not of class 2.

Figure 13.

y

Figure 14.

A di sk with tentacles.

Figure 1 5.

The one dimensional Cantor t ype set A.

Figure 16.

Polygonal estimates on

Figure 17.

The maps

Figure 18.

The sets

X(a,r,V,s).

Fi gure 19.

The s e t s

S( z,r,u, ¥).

Figure 20.

S(z,fn_l' ~n(a)exp(~o(),o<) - S( z, in-I' ''In( a) , 0<)

x1/3•

boundary(S).

Tt n•

c; X( a , 2J n _l s , v, s },

- 36 -

F. J. Almgren Jr.

PART A

======

Suppose k ~ n are positive integers and one is given a n• reasonable surface S of dimension k in R

Assume also one is

given a suitable function n R

f :

JC

G(n,k)

~

+

R ;

here G(n,k) denotes the Grassmann manifold of all unoriented k dimensional planes through the origin in Rn

or, equivalently,

n the space of all k dimensional tangent plane directions in R ~(S)

Then one can define the integral~f-! over S by the formula

E(S)

S F(p,

~S

k

Here Tan (S,p)

~

Tank~s,p)) d~kp

G(n,k) denotes the k dimensional tangent plane

direction to S at p for p ~ S , and n Hausdorff measure over R Hausdorff measure of a

If.k

denotes k dimensional

defined in C.)(l).

The k dimensional

n k dimensional submanifold M of R

of

class I agrees with any other reasonable definition of the k area of M.

However, with the use of Hausdorff

make mathematically precise the notion of

k

measure, one can

k dimensional area on

- 37 -

F . J. Almgren Jr.

surfaces which may have essential s xngularities. Wi t h this terminology, there are several problems one might wish to study. ~~~~~=E~~~~~~

(existence).

Among all surfaces S having, say,

a prescribed boundary B (and possibly satisfying other constraints), is there one minimizing

~(S)?

g~~~~=E~~~~~J (regularity).

first problem, how nice is it? like a smooth

If there is a solution to the In particular, is it generally

n k dimensional submanifold of R ?

- 38 -

F. J. AJr.ngren Jr. ~M~~=E~~~~~~ (structure of singularities).

What kind of

singularities are possible in solutions to the first problem (if any)?

And if singularities are possible, what language

should one use to

de~cribe

them?

~~~~~M=E~~~~~~ (computation).

If there are solutions to the

first problem, how does one explicitly f ind them?

To make these problems mathematically precise, there are of course several questions to be' settled, namely,

(1)

I'fuat is a surface?

(2 )

What is the boundary of a surface?

(3)

What are reasonable conditions to put on F ?

(We already

have been assuming implicitly that F is ~k measurable, for example). Before attempting to answer these questions, it is perhaps usefUl to consider the these problems.

phenomena which arise when one studies

In the following examples, the integrand

F is

restricted to being identically 1 , i.e. for each (p,

'ff ) •

Rn.lC G(n,k),

F(p,,,,)

1

The problem of minimizing

- 39 -

F . J. Almgren Jr.

~(S)

thus becomes the problem of minimizing ~k(S) , in other

words, the problem of minimizing the

k dimensional area of S •

The various problems associated with minimizing area are sometimes called collectively Plateau's Problem in honor of the Belgian physicist, J. Plateau, of the last century who among other things studied the geometry of soap films and soap bubbles (see [GJl] i n this volume in this regard).

EXAMPLE 1.

Suppose C is a f ixed simple closed curve in R3 of

finite length and let DO say

DO

= R2 C'\

(x,y): x

2

denote a fixed 2 dimensional disk, + y2

~

1) .'

The first person to make

significant progress in a special formulation of the problem of least

2 dimensional area was the late J. Douglas who showed i n

particular: THEOREM [DJ]

Among all maps

such that

~DO

maps homeomorphically onto C , there exists a mapping of least 2 dimensional area

("2 dimensional area" as used in this theorem

is defined explicitly in B.6Y. Unfortunately, there are no known direct extensions of this result to higher dimensional disks.

- 40 -

.F: J. Almgren Jr. Now for

disk with

m

= I,

2, 3,

m handles (a

•••

let D denote a fixed 2 dimensional m

D appears in figure 1), and let Am

5

(Figure I)

the

denote the infimum of the areas of~mappings that

aDm

such

maps homeomorphically onto 0 (mappings realizing

this infimum area do not seem to be known to exist). can always "pinch out" a handle before mapping

Since one

D one has the m

obvious inequalities ?;

lim A m m

?; 0

•

For a curve like that sketched in figure 2 (a simple closed unknotted curve of finite length), W.H. Fleming has proved the (Figure 2) strict inequalities ,. lim

A > 0

m m

[FW].

The significance of these inequalities is that if one wishes to solve the problem of minimizing area among all oriented surfaces having boundary

minimum among oriented ~faces of fi~te topologicaJ 0, then one cannot find an absolut;
indeed, the list of

D 's is a complete topological classification m

of all compact orientable 2 dimensional manifolds having a

- 41 -

F. J. Almgren Jr.

circle as boundary.

On

the other hand, there is a surface S

(sketched in figure 3) which perhaps deserves to be called the (Fiigure 3) oriented surface of least area having is a

2

C as boundary.

dimensional real analytic submanifold of

mean curvature at each point and the area of

The surface

S

S

R3

S

C

~

having

0

equals

is of infinite topological type arid is

homeomorphic with the surface

T sketched in figure 4.

(Figure 4) (the

nature of this surface

T suggests why, frequently,

homological conditions in geometric measure theory are stated in terms of the Vietoris or eech theories rather than the singular theory).

EXAMPLE 2.

Suppose

2 B consists of three points in F which

of an are the vertice~ equilateral triangle with center at Then among all

1 dimensional sets

o

S which are the un ions of

non+.rivial rectifiable arcs and through which each point of is pathwise connected to each other point, the unique set (see figure 5)

2

R •

€

of least total length consists of the union

Y

B

- 42 -

F . J . Almgren Jr.

(Figure 5) o~

the three line segments connecting the points

(The proof of this is not completely trivial).

o~

B

to

0

This problem is

as naturally posed as one could ask, the integral (length) is real

~alytic,

Y is un ique.

the boundary

B is algebraic, and the solution

Nevertheless the solution has an interior singularity 2

of codimension 1 ,namely

0 eYe R

where the three line segments

meet. EXAMPLE 3.

complex dimension let

V is a complex algebraic variety of

Sup~ose

U e f:n

k

in complex

n dimensional space

be open and bounded such that

tJ(V n U)

~n , and

is suitable

(in the language of"C.3(4) , V n U is required to be a 2 k dimensional integral current in of

V, of course, gives

2n R

~

t

n

);

the complex structure

V a natural orientation so that

V n U together with that orientation becomes an oriented (#.2k, 2k)

rectifiable and

[FHl 4.2.29].

Let

1l2k measurable subset of R2n

W be any other oriented

surface (integral current) such that W = Vn U or the 2k area of W (N(W»

aw = is

2k dimensional i(Vn U).

Then either

strictly larger than

- 43 -

F . J. Almgr-en Jr.

the -2k area of

Vf"I U q~(Vf"I U)).

Furthermore, this is true

whether or not V has singularities .

In particular, if one

wishes to solve the problem of minimizing oriented area, and really achieve the least area, then one must at times admit as singularities in the solutions to Plateau's problem at least all the singularities occuring in complex algebraic varieties. EXAA~LE

4.

R. Thorn has constructed a 14 dimensional compact

real analytic manifold integral homology class

without boundary with a

M

d

7 dimensional

which cannot be represented by any

7 dimensional smooth submanifold [TR].

On the other hand every

integral homology class of any compact smooth Riemannian manifold can be represented by an oriented surface (integral current) of least area (mass) in that class.

Thorn's example thus shows that

sometimes there can be topological obstructions to surfaces of least area being free of singularities. EXM,~LE

5.

Consider the following partitioning problem. , m~

>

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 .
conc~rned

with geometric iXitegrals Several definitions

are in order.

B.3 DEFINITIONS. (1)

A funf'tion

i: rh~R+

parametric form if and only if

i

is called an integral in

has the form

(*)

above and,

- 53 -

F. J.Almgren Jr.

in adlii tion, for each

f

&

c/t i

!(f)

ep:

A function

(2)

and each diffeomorphism Ig: W~ Vi ,

IV

(feg)

n R

1<

X

Hom (R

k

,

Rn)~ R

is

called an integrand in parametric form i f and 'Only if the

~:c/'t~R defined by

function

(*)

is an integral in parametric

form. (3) integral

I: 4

A function

~ R+ is called a geometric

~ has the form

if and only if

(*)

and, in addition,

there exists a function G{n,ky ---:, R+

such that for each

'! (f) Here

=

p

f

Go

S Q

f{IV)

N{ f, p) = card f-

~;

_

F[p, Tank{f{IV) , p)] N{f,p) l

[P}

for each

p

Q

d ![kp •

f{ W)

called a geometric integrand i f and only if the function defined by

(*)

B.4 REMARK.

~

is a geometric integral.

To the best of my knowledge the ' expression

"integral in parametric form" was introduced by C.B. Morrey, Jr. to

- 54 -

F. J : Almgren Jr.

suggest that the appropriate integrals to be stuaied over

parametr~c

surfaces (surfaces which are not the graphs of functions) are integrals in parametric form [ fflCl].

This terminology has been

a source of confusion, however, since, in particular, integrals in parametric form are precisely the integrals of form

(*) which

do not depend on the parametrization of the domain. For sufficiently nice

f e

u1r it

is clear that any geometric

integral has the i nvar i an ce property characteristic of integrals in parametric form. and

In case the maps in ~ are Lipschitzian

Tank(f( W), p) is understood to mean the

(IF

Lt(W), k )

approximate tangent cone, Tank(,J/k L f(W)" k)[FID. 3.2.l6J, then any geometric integral is indeed an integral in parametric form. Also, in case class 2, range

i

is an integral in parametric form Which is of

!

Co

R+, and

i

is independent of orientation,

then C.B. I.1or r ey , Jr. has shown that

i

is a geometric integral

[l.iCl] .

Integrals in parametric form, and

~~ometric

i nt egr al s i n

par t i cul ar , are of special geo metric significance s ince t he value of the integr al depends only on the geo me try of on the particular

f( IV)

and not

parametrization whi ch produces it.

I know of no natural geometric, phy sical, or bi ol og l Cal problems

- 55 -

F. J. Almgren Jr.

involving integrals in parametriG form

whic~

are not also

geometric integrals. The

k dimensional area integrand i s an example of a

geometric i nt egr and while the Dirichlet i nt egran d is neither a geometric i nt egran d not an integrand in parametric form. One refers to a varuat i onaj, problem as :being a geometric variational nroblem in case the

~nly

integrals which arise in

the problem are geometric integrals.

The greatest initial difficulty with the study of geometric variational problems (in the mapping setting) i s that, in general, there is no straightforward way to obtain solutions by t he s o called direct met hod of the calculus of vari ations.

The term

"direct method of the calculus of variations" is u s ed loosely in a number of different contexts.

Roughly speaking the "direct

method" makes sense in a variational problem when the formulation of the problem itself guarantees the existence of a sequence (fiji

of mappings in ~, any sui tabl e convergent (in ~)

subsequence of whi ch would yi el d a s olutlon to the problem;

- 56 -

F. J. Almgren Jr.

for example, if one seeks a minimum for lim.!(f .) 1 1

inf

!

2,

one might have

The direct method of the calculus of

variations is applicable wh en sequences

[f

i

ji

as above do , .~

fact admit subsequences convergent to solutions. The main difficulties wh i ch I know about in the attempt "0 use the direct method in geometric variational problems seem best illustrated by examples. B. 6 EXAMPLES.

Suppose

k

=2

, n

=3

,

R2 fl [ (x,y): X2 + Y2 <

w

consists of all Lipschitzian mappings f: closure

for

f

£.It. !

W~ R3

such that

is, of course, the

f(x,y)

=

(x,y,O) e

R3

2 dimensional area integral.

Suppose one is given the geometric variational problem of finding h ~

/t such

that

'j (h) = inf

is, of course, such a mapping.

i.

The mapping

h(x,y)

=

(x,y,O)

However, in g ener a l , there is

- 57 -

F. J. Almgren Jr.

no reason for a minimizing sequence to admit a reasonable convergent subsequence as the following examples show: (1) in

Let

R3 with rational coordinates.

. sequence [Pl

3 Pl , P2 ' P3 ' .•. e R

,

P2

lim

with

,

... , Pi~

C

i

i

f . (VI)

for each

~

h(W)

One can construct a minimizing

j (f ) = inf ~ = 7f

example by deforming the map h out of the flat disk

be a list of the points

= 1, 2, 3,

...

,

for

above by pulling thin "+'entacles" R3

in

i

such that

as indicated in figure 14,

(Figure 14) the total area of which "tentacles" is no more than l/i. this case, each point in

(2)

Let

and define for

R3

is a limit point of the sequence

W be given the usual polar coordinates i = 1,2,3,

Clearly the sequence above problem.

(hogiJi

However,

(r,e)

the diffeomorphisms

g .(r,9) = «l/i)r + (1 - l/i)r ~

:::n

i

, e).

is a minimizing sequence for t te

- 58 -

F•. J , Almgren Jr.

for x

l im . (hog .) (x,y) ~

~

lim. (hog . ).(x,y) ~

~

=

(x,y,O)

for

x

2

2

+ y2 < 1 ,

2

+ Y

= 1

so that no suitable convergent subsequences exist.

i : ,/'f -'R+

Suppose F:

n

R

)/

'i

=

=

llJ

T f(P)

N

f ~ or"

plf(W) F[p, Tank(f(W), p)]

Now define for

and

+ R , and for each

G(n,k)~

(f)

is a geometric integral,

Ilk

L

fe W)

almost all

k n (p,Tan (f(W), p»e R

It

N(f,p) dllkp •

n

R.

p~

G(n,k) ,

note that one can write

I

(f)

IR

=

n

F0

ljJf

d

[Il k

t, N (f ,

.) ]

which equivalently -can be written,

I(f) = in which

I F R x G(n,k)

~fll

measure over

[i{k L N(f, .)]

n)/ R

is Lipschi tzian). and

F.

d[~f# [-&lk

LN(f,

.)]3

determines a unique Radon

G(n,k) (assuming, say,

W is bounded and

Note that this measure is independent of

A definition is in order.

1

f

- 59 -

F. J. Almgren Jr.

B.8 DEFINITION [AW]. n R

By a

k dimensional varifold in

n one means a Radon measure over R x

the space of all

G(n,k),

k dimensional varifolds in

is a bound.ed open set in Rk

f: W--') Rn

and

n

~kCR ) flenotes

Rn•

In case

W

is Lipschi tzian

one denfl"&es by

.i.'#

Ivil

~k(Rn)

Q

'fJf # [~k

the varifold' eorresponding to

1. N(r, .)] as in B. 7

above. B.9

REMARK.

As was noted in B.7,

One can also verify, in a reasonably straightforward manner, that for each

f ~

4

and each diffeomorphism

so that clearly the mapping by its associated varifold

g:

W ~IV ,

f ~ cit is not uniquely determined f~lwl. 11"

Of interest, however, is

the fact that spaces of Radon measures have very strong convergence properties in the weak topology. B.lO over

DEFINITION. G(n,k)

In the present context we have :

A sequence

(vJ i

of Radon measures

(i.e. a sequence of elements of

- 60 -

F . J. Almgren Jr. is said to converge weakly to a limit Radon measure . V if and only i f for each continuous function with compact support,

J

F

B.ll [ViJ i

[v. ~ ~j

F:

n)( R

+

G(n,k) - . R

dV

A sufficient condition that a seauence

PROPOSITION.

of Radon measures over

n)(' R G(n,k)

contain a subsequence

which converges weakly to some limit Radon measure

j

o

is that for each

B.12

r

<

EXAMPLES.

<

V

00

Let us return to the examples of

B.6

above which illustrated the difficulties of the attempt to use the direct method i n geometric variational problems. corr~sponding

(1)

Let

We have

examples . [fiJ i

be as in

B.6(1).

Then i t is easily

verified that (weakly) where in

h(x,y) = (x,y,O)

B.6.

disk in

Geometrically, 2 R

>t

(03

Co

as in

B.6

is a solution to the problem

h#lwi corresponds to the unit flat

R3 which "solves" the problem, but without

a particular parametrization prescribed for that disk.

- 61 -

F . J. 'Alm gr-en Jr.

(2)

Let

[gi}i

be as in B.6(2J above.

Then also

(weak ly)

h.,(L\wl tr

- 62 -

F. J. Almgren Jr.

PART C

======

been suggested by the various examples and discussion in Parts A and B, one approach to the study of geometric variational problems is based on a correspondence between suitable surfaces and measures on appropriate spaces.

Indeed, the natUral setting

for geometric problems in the calculus of variations seems to 06 that in which surfaces are regarded as intrinsically part of RL n) (in particular as measures on spaces associated with R ratt6~ than as mappings from a fixed

k dimensional manifold, even

though with this approach one is not able to use the traditiona: methods of functional analysis for showing the existence of solutions. (1)

The main reasons for doing this are the following: Mappings from a fixed compact

k dimensional manifold

cannot take into account the phenomena of the examples given in Part A.

For example, one cannot consider surfaces of infinite

topological type, surfaces having singularities not realizable by mappings like the singular curve of the triple Mobius band, or surfaces haVing boundaries defined in certain ways.

- 63 -

F. J. Almgren Jr.

(2)

Many significant results have been obtained. from the

study'of geometric variational problems in the measure theoretic setting

~n

contrast with the virtual absence of · such results in

hi gher dimensions and codimensions in the mapping setting. [AW] [AFl] [FlU. 5][FH2][GJl] (3)

It seems reasonable to hope that once the singularities

of the measure theoretic solutions are

under.stood, one will be

able to solve the mapping problem as a consequence. (4) theory

Topological methods analogous to those of Morse's

are availaole in the measure theoretic setting i n

contrast with the absence of such methods in the mapping setting[AF2]. For example, the "Condition COl of Palais and Smale CPS] is not satisfied by any geometric variational problem (no t e B. 6 ) . (5)

The techniques developed in the study of geometric

problems in a measure theoretic setting have aided in the solution of r elated problems. (a)

For example:

The basic theorems of

cl ass~cal

integral geometry

have been proved in wha t seems to be thp.ir most natural setting[BJ]. (b)

Various long standing qu es t i on s in the theory of

Lebesgue area have 'be en settled [FH3].

- 64 -

F. J . Almgren Jr.

(c)

Extens~ons

of Bernstein's

~heorem

that a globally

defined nonparametric minimal hypersurface must be a hyperplane have been proved in dimensions up to 8 and counterexamples have been shown to exist in higher dimensions [FHl 5 ·4.l8][BDG]. (d)

Proofs have been given of the regularity

almos~

everywhere of "weak" solutions to some nonlinear elliptic systems of partial differential equations, and examples exhibited which show sometimes unique solutions to such systems contain

esse~t~al

discontinuities [MC2][GM]. (6)

In the measure theoretic setting,

many important

natural geometric constructions are available, the analogues of which i n spaces of mappings seem unnatural. C.2

RE~aARK.

The geometric measure theorist routinely

works with a number of different measures and measure theoretic surfaces.

The following definitions

ar.~

terminology both are

representative of the most common "surfaces" of the geometric measure theorist and enable a careful statement to be made of sQme of the regularitr and singularity results in Parts D an d

~.

- 65 -

F. J. Almgren Jr.

C.3

DEFINITIONS AND TERMINOLOGY.

g"ppose

0

~

k

~

n

are integers. For each

(1)

n

A c:: R

the

k dimensional Hausdilrff

A, denoteq ~k(A), is the greatest lower bound of

measvre of I

all' t

such that

0 ~ t ~

countable covering

I

and for every ·

t10

It ~

0 there exists a

A with

G of

ClC.(k) [diam (5)/2l

56 G

~

t

[FlU 2J

(2)

A subset

n A c R is called (~k, k) rectifiable

if and only if irk(A) < ~ compact class

and there exists for each

k dimensional differential submanifold 1 such that Itk(A -

M() -e , .

measurable one can choose M

E

In case

(3)

measurable,

/I All

defined by requiring B c

n R , and

A

Co

so that Rn

is

ME

In case

It k ([A

-J

£

> 0

u

of

n R

cf

A is Ilk M J u · [M £

f

,oJ

AJ)< e •

(Jlk, k ) rectifiable and Ilk

~k L A denotes the measure over Rn

dl k

LA)(B)

Tank(1t k L. A , a)

approximate t angent cone to

A at

=

itk( Afl B)

denotes the

for each

("Il k

a. [ FHl 3.2.161.

, k) For It kL. A

- 66 -

F. J. Almgren Jr.

almost all

a

orientation for A

w:

~.

--t

n, R

in

Ais a:n Jtk

GO(n,k)

(a) = Ta:nk(1f k

Co)

Ta:nk(ll

n R

L..

LA, a)

L.

A , a).

a:nd ~: GO(n,k)

Eo

G (n,k).

An

A measurable function

such that for Ilk

ma:nifold of all oriented in

k

Here

a e A ,

a i -_Jst all GO(n,k)

is the Grassmann

k dimensional pla:nes through the origin

--+ G(n,k)

is the natural projection.

The pair

(A, w) is called a:n oriented (Uk, k ) rectifiable

a:nd Ilk

measurable subset of (4)

If

A

Co

n R

n. R

is (Ilk, k) rectifiable a:nd" k

measurable, the integral varifold associated with IA\ ' is the Radon measure over [~ n R

It

A Radon measure

n R

defined by

k .Jl,k ;;k Ta:n ('I L A, .) J# [" V 6

n) ¥k (R

A, denoted

L.

AJ

is called a:n integral varifol d

if a:nd only if it is a finite or convergent infinite sum of measures

[I Ai I J:i.

corresponding to (Ilk, k ) rectifiable and

~k measurable subsets ~, A2, A

3,

•••

of

n• R

The most

accessible source for the basic theorems about varifolds is to which the readers attention is strongly directed.

[A~ J

- 67 -

E . J. Atmzr-en Jr.

n A k dimensional curren~in R

(5)

is a continuous

n of linear. functional on the snace gk(R all differential forms

)1.

of d.egree

k

and class

n• R

on

o:l

If · k ;> .0

k-l dimensional current defined by setting

If E: Cek-l

Suppose the pair

here d

;

each

If''' T(

gk ,

If )

=

pL ('"

n R

(p),

here we are identifying simple unit all

If')

= T (d ey )

A is bounded.

and (A,~)

cr

The rectifiable

is given by requiring for

(p)

Go(n,k)

with the submanifold of all A

k vectors in the Grassmann vectorspace

k vectors in

F

n

as a

If' is the exterior derivative of f

associated with

T

oT(

~:

is an oriented (Ilk , k ) rectifiable

(A, Col)

and measurable subset of current

T is a

n, R then the boundary or

k dimensional current in

for each

and

A k dimensional current

"k T

in

Rn n R

of is

called rectifiable if and only if it is the finite or. convergent infinite sum of rectifiable currents associated with oriented rect ifiable and above.

Jl k

measurable subsets of

A k di mens i onal current

i nt egr al current in

n R

T

is called a

if an d only i f both

T

R11

as

k dimens ional and ~T

are

- 68 -

F. J. Almgren Jr.

rectifiable currents (for

k = 0 , there is no requirement

regarding . ~T which, of course, is not defined). The most basic theorems about integral currents in general are thp deformation theQrem [FHl 4.2.9J, the compactness theorem [FHl ~.2.17J, and the approximation theorem [FHl 4.2.20J. readers

The

attention is strongly directed to these fundamental

_:·:- ~ t s .

The compactness theorem alone is the essential ingredient in

~~~y

existence results for elliptic geometric problems by the direct method.

The necessary lower

ellipticity.

sem~continuity

One USUally takes as the "k dimensional area"

k dimension integral current

a

denoted

T

in

Rn

the mas s of

T

:" : ~

,

M(T), defined by setting

~ (T) = sup Note that

follows from the

~2T)

fT( If ): tp " 2~(T)

s>.

sup

I crt s

1.)

so that, in par-t icul.ar , the mass

C1;

can be much larger thp~ Jrk(spt T).

T

(6)

In a natural way rectifiable currents may be regarde :

as having the integers as "coefficient group".

For each int c:<:-e:"

-.

11 ~ 2 , the k dimensional flat chains mo dulo 11 in Rn defined essentially as the quot i ent subgroup of the group of

a :'d

F. J. Almgren Jr.

n R

k dimensional rectifiable currents in homomorphism

Z. ~ 7l.!( 2

7L).

induced by the

Theorems corresponding to the

deformation theorem, the compactness theorem, and the approximation theorem hold for flat chains modulo

"j

. [FHl 4.2.26].

- 70 -

F . J . Almgren Jr.

In this part we give a careful statement of a very general regularity result for solutions to elliptic geometric variational problems. D.l

DEFINITIONS.

(1)

An integrand

with ellipticity bound

c

following condition holds. k dimensional disk and of

n R

--

G (n,k) -+ R+is ca:'leJ eilbt ic

FO:

S

if and only if Suppose

D

f:

n n R --. R

follows that

dD G S) .

(2)

An integrand

n R

C

>

0

and the

is a flat

is a compact (~k, k) rectifiable su~ ~ , "

which can not be mapped into

function

c

such that

dD by any Lipschitzian f'{p )

=

p

for p

&

dD

+

F:

such that for each

p _ RN,

bound

FP:

'7f e.

G(n,k).

(it

Then

G (nk) --" R

i s called

elliptic i f and only if there is a continuous function

c(p) ; here

"'---'-

~

c : Rn -+ R+

i s elliptic with ellipticity +

G(n,k)-R,

~ ('J1') = F( p, "If )

for

- 71 -

F. J. Almgren Jr.

()

Suppose

t

0

~

CIC

1

<

R+ _

.i

~d

R+

is

monotonically nondecreasing with l'

t-(l+~)

Suppose also Let

S

Co

cl os t:d .

:a

dt

n

<.

+

x

R

be

(/tic, k) rectifiable and

n

R

G (n,k) - . . R

One says that

S

is

=' ,.e

(S f"\ [z: cr(z)

I

I> 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)


n

R

n ~ R

is Lipschi tzian,

(b) [[z: ep (z ) I z] U'P [z: (c) r = diam ([z:'

RE~~RK .

In case

I

z]

If (z ) I

Ulf

k = n-l

(z:

zJ ] () B

f

(z)

I

=

zj) <

the ellipticity of

D.l(2) above is equivalent to the uniform convexity of ,ea ch

n ps.R •

For all

y! ,

F

cf

in

pP for

k, the set of elliptic integrands

as

i n D.l(l) contains a convex neighborhood in the class 2 topology of the f:

k dimensional area integrand.

n, n R ~ R

f# F

Also for each

is elliptic if and only if

F

diffeomorph~sm

is, for

f

- 72 -

F. J. Almgren Jr.

as in D.l(2).

Finally the elliptiCity of

various Euler-Lagrange

equ8+·~ns

F implies that the

which arise are nonlinear

strongly elliptic systems of partial Q1Iferential equations, and, "in the small", the ellipticity of

F

is equivalent to

the strong ellipticity of these equations. D.3

THEOREM [AF1].

D.l(3) above and that

S

Suppose ~

(E,~)

F, S, B are as in

~,

at,

minimal with respect to B

,Then k (1) hk([s ·.- spt(ll ,\.. S)] u

[spt(ll k

It k

[spt(/l]<:

(b) spt(ll of

Rn

k

of class 1

L-

L. S) 1"\

S) U

S) -

U of

n R

(U u B) ]

=0

(2) Ther e exists an open subset (a)

l-

is a

with tangent

Holder continuously with exponent

(S u B)]

0

such that ,

k dimensional submanifold ~ planes which locally. vary oe..

The special interest of this theorem is that the hypotheses are SUfficiently weak to , apply to all the natural constrained geometric variational problems of w4ich yield rectifiable sets. 2, 5, 6, 7, 8

in Part

The

whi c ~

I

kn O\l', '

~urfaces

the 1Jol u t i o"'\s of

discussed ,i n examples

A come 1n t his category in particular.

- 73 -

F. J. Almgren Jr.

Very little is known at the present time about the structure of the singular sets of solutions to general elliptic geometric variational problems, except for their existence.

However, for

the area integrand there has been substantial progress.

The

following three theorems are representative of the present state of lrnowledge. E.l

THEOREM [FH2]. Rn

current in (1) in

Rn

Let

such that

There exists an

such that

dT =B

~B

B be an

= O.

n-2 dimensional

inte~~s:

Then

n-l dimensional integral current

T

and

~(T) = inf [~(S): S ~ n-l dimensional integral curren~ in (2)

n R

and

~ S = B3

There exists an open set U in (a)

io at most

The Hausdorff dimension of

Rn

such that

spt T ,.., (U v

spt B)

n-8.

(b) submanifold of

spt T n U is an Rn

having

n-l dimensional real analytic

0 mean curva!ure at each point.

- 74 -

F. J. P.Ur.ngren Jr.

E.2

-Let . 1.s k

THEOREM [FH2].

:t

k-l dimensional flat chain modulo 2 in

(1) n ~ R

There exists a

and let

n R

such that

B be a ~B

=0

k dimensional flat chain T modulo 2

such that

~(T) = inf

~(S):

S

is a

k dimensional flat chain Rn

modulo 2 in (2)

There exists an open set (a)

is at most

E.3

spt T " n R

U

such that,

spt T -

(U

\J

spt B)

having

R3

~here

mean curvature at each point.

0

such that exists a

such that

~

k dimensional real analytic

~

THEOREM [GJ2]. ' Let

modulo 3 in

modulo 3

Rn

U in

B)

k-2.

submanifold of

(1)

~S =

and

The Hausdorff dimension of

(b)

M0T)

n

B be a

; ;3 '-

O.

1 dimensional flat chain

Then

2 dimensional flat chain

T

aT = B and

[M(S): S is a modulo 3

2 dimensional flat chain in

R3

and

- 75 -

F . J . Almgren Jr. '

(2)

class 1

spt T (a)

A

~

(b)

C

~

R

C

=0

AU

C

~

,

1 dimensional submanifold of

R3

(d) such that

is a

A

3 having

0

2 dimensional real analytic submanifold

mean curvature at each point,

Each point

c/Y'" A

Furthermore for each

p

in

has exactly i,

A.

~

C

haa an open neighborhood

3

components, say ~, A

u (cAP 11 .

2

C)

REMARK.

See [GJl]

cI1I' A

3

•

is a Holder

continuously differentiable manifold with boundary, and

E.4

2!

with tangent lines which locally vary Holder' continuously. (c)

.Qf.

apt B

,-v

whenev e~

for a complete

classification of the interior local structure of mathematical "soap bubble" and "soap film" like surfaces, inc;"''l.ing the first mathematical verification of the century old "axioms of · Plateau".

- 76 -

-

F. J. Almgren Jr.

PART F

Caratheodory's construction for k dimensional measures in >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


= inf

The measure ~

F (\ iSI diam(S) ~

\I'

l

L:

S E: G

$(S)I GC

F~

called ~ ~

i ll

countable and A C UG}

.!!! Cllrathe 0 dory , s

,A

c a",

oonstruction ~

1

F is given b,y the formula

Severai

dis tinot ~°'lsures obtained b,y 6arathe 0 dory 's cons truction are

desoribed by the chart belowl

0<:

MEASURE

SYMBOL

k dimensional Hausdorff measure

k dimensional spherical measure k dimensional integral-~ometric measure with exponent t, 1 ~ 1;,-«00 ~ d~mensional

when t '" 1 unnamed for

Gillespie measure

1 -< t

j{k

"gk

J

k. t

lQ~

F

all nonempty subsets of Rn (equivalent!y, all nonempty closed or open subsets of Rn

j(A)

for A € F

~(k)(diam(A)/2)k

all closed balls in Rn all Eorel subsets of Rn all nopen convex subsets of R

(t (1p(G(n,k)~

.-

k t ) lit G(n,k)(L (pA)) d(n,kP k

(pQ))

t

~n,kP

) lit --J

co

.(00

k dimensional Gross measure k dimensional Caratheodory measure k dimensional integral-geometric measure with exponent 00

§k

ek

=~

J:;'

all Borel subsets of Rn all nopen convex subsets of R all Borel subsets of Rn

sup .,(k(pA) 'p e: G(h,k)

ess sup £k(pA) PEG(n,k)

"J

;-< :P

8'

(Jq

'1

(1)

::l

'-< '1

- 79 -

F. J. Almgren Jr . In the chart above,

Q -_ Rnn SI xr O!- x 3., ~_ 1

G(n,k) of

f or 3.' = 1 , ... , k an d x

j

= 0 f

1, or J' = ' k -1, ••• , nj

is the Grassmann manifold of all k dimensional linear subspaces

If, is the orthogonally invariant Radon measure over G(n,k) for

~n,k

which

dt k

Ifn, k(G(n,k)) = i , denotes the Lebesgue k dimensional measures over eleme=ts of O(n,k)

defined in the obvious manner, G(n,k)

is also identified with the spaoe of all orthogonal projections

p of Rn onto k dimensional linear subspaces of Rn qy associating to the projection p the 'linear subspace

p(Rn).

- 80 -

F . J . Almgren Jr.

F.3.

Observations about the measures resulting from Caratheodory's

construction. ~t: 0:;; t ;;

and

.0/

Suppose F is a family Qf subsAts of an,

""J

~ 0< ~ ~

00, CPJ

is the size

S

approximating measure,

t

is the result of Caratheodory1s construction from

(1)

~

SI F

on F.

n All open subsets (hence all ~ ~ ~ sUbsets) of ·R are

measurable.

To see this one first recalls:

CARATHEOIDRY I S CRITERION FOR MEASURABILITY. ~ ~

space X,

ill

l!!. ~ j!

open subsets of X ~ jA- measurable

measures

g!:!l!!

only

g jA (A U B) ~ r(A) + /,,(B)

whenever A, Be X ~ dist(A, B) > 0 [FHl 2.3. 2( 9)1 Clearly if A, BeRn, dist(A, B) > ~~(AUB) =

b , then

(f(~A) +


from which it follows that

'f(AU B) (2)

If

ill ~

4J(A) + lV(B).

=

members of F ~ Borel sets, ~

ssssx.

subset of

n R is contained in ~ ~ ~ ~ equal ~ measure and ljJ

let A C Rn be given, and choose

00

00

for each

for · each

such that AC

v

.

U

i =l

Borel

To s e e this, assume all the members of F are Borel sets,

regular measure.

2, 3,

.!!!. ~

S~

1:

and

i=l

J.

The set

S(S~) -< = Glb(A) +

b

Ih

n U S~ v=l i =l 00

T

J.

~

J.

is a Borel set containing A such that

n 00

j =l

Tl/ ' J

~(A)

=

q>~ (T&).

The set

Y = 1,

- 81 -

F. J. Almgren Jr.

is a Borel set containing A such that ~(A)

= ~(TO)'

One notes, in particular, that the several measures

def1ne~

on

the chart above are all Borel regular measures.

(3)

.5 1:!.

If ~

family F is invariant ~ Euclidean isometi'ies ~

invariant ~ ~ transformations

isomefries, ~ l.jJ

1::!.

E.!

F induced ~ Euclidean

invariant under Euclidean isometries.

the several measures defined on the chart above are all

In particuiar,

invarian~

under

Euclidean isometries.

(

4)

.fu

measures

10k ill. ,

ok

Jd

ok ,tXt'

Rk

G1k

~t' ~ (f

.

~ ~ ~ k

dimensional Lebesgue measureotk (~ precisely the orthogonal inclusion image

E.!

the standard k dimensional Lebesgue measure) ~ k dimensional

~ subspaces

E.!

Rn•

As was noted above, Lebesgue k dimensional measure

over Rk i s characterized by its Borel rsgularity, its invariance under Euclidean isometries, and the condition £.k(Rkn ~x: 0 ~ xi ~ 1

for

i = 1, ... , k}) = 1.

In view of (1), (2), (3) above it i s not difficult to show that sufficient to veritY (4) are the equ a lities .J{k(Q) = Jk(Q) = J~(Q) = tl~(Q) = ~k(Q) = 1. These equalities follow almost immediately from the definitions and the Vi tali covering theorem. [FHl 2.8.7] •

(5)

E.!

.fu

measures J{k,

J k, d~, &2~, ~

k dimensional submanifolds of Rn

~

E.! ~

Cfk

subsets .

1 ~ assign such subsets

"correct" k dimensional measure.

local.

~~~

The problem is, of course, purely n f: Rn_R is Lipschitzian, then for

One notes first that i f

and A C Rn, j{k(f(A»

~ (LiP(f»kj(k(A),

...,gk(f(A»

s

(Lip(f»kj{k(A).

- 82 -

F. J. Almgren Jr.

Since each k dimensional submanifold of an of class 1 is locally the image of an open subset of a k dimensional linear subspace of an under a class 1 diffeomorphism

g: Rn~ Rn

with Lip(g)

very close to 1, the asserted equality between J{k,

and Lip(g-l)

Jk

rea sonable definition of the "correct" value follows. addi tionally that if such that

f: Rn _

Rn

,>

Lip(f) and Lip(f-l)

~

( pE.G(n,k)

both

and any

One notes

is a bilipschi tzian diffeomorphism are very c]~se to 1, then

cfhpf(Q))t dh'n k P\ lit , ')

S J.k(pQ)t dOn , kP)l!t (pEG(n,k) is very close to 1.

The equality of l>(k,J

k,

j~, :Q~, and CJk on

k dimensional submanifolds of Rn of class 1 is now essentially a consequence of the Vitali covering theorem and the obvious behavior of these measures under homothetic maps Rn----1> Rn (L e. uniform expansions and contractions).

(6)

For each

A eRn,

where

The right hand inequality follows since each subset B of Rn is contained within a ball of diameter is that Jk

~

2 diam(B)

[2n/(n+l)]k/2j{k

(the strongest known inequality

which follows from Jung' s 1Iheorem

(FHl 2.10.41] · whi ch states than an arbitrary bounded subset B of

Rn

is contained within a unique closed ball of s mallest diameter no larger than

[2n/(n+l)j1/2 diam(B).

the definitions.

The left hand inequality follows from

The middle inequality i s a consequence of the observation

- 83 -

F . J . Almgren Jr .

r (.k ( pB(O , l ) ) t j ( pE. G(n,k)

) lit

dO' k P n,

- - - - - - - - - - - - - = --------~--

(\PE. S

,Lk(pQ)t

G(n,k) DEFINITION.

only if J(k(B)..( s u bman i f o l d

~1

~

~

n

,

A Borel s et B

in

Rn is called Qtk, k) rectifiable if and

€ >

and fo r e ach

n

(. of cla s s 1 of R

me asures j(k,

~ ~ which ~

el.(k)Ct •

dt kP)l/t

° there exists a k dimensional

such that

Jek([A - MEl V [ Mf, - A]) -<

(7)

=

.J k , J~, Q~" ~

0(k, k) rectifiable.

i mmediate consequence of

e•

( 5),

gk

~~.2£ subsets.2.!

This fact i s essentially an

(6), and the definition above.

- 84 -

F. J. Almgren Jr. F.4._ Computation of the measures .1<.1, ,81, ~~, J~, and

fj-

on ~

[Fit1 3.3.19J. With each closed 2 D ~ B(z,r) C R = C one a s s oci a t e s the family Fn(D)

~ dimensional set A of Cantor type

ci~cular disk

cons i sting of the n disks B(z + r(l - 1/n)exp«k/n)2~~), r/n) corresponding to k

=

1, 2, ••• , n.

Inductively one then constructs

of closed circular disks by the formulas G = F 4 4(B(O,l) Gn =UtFn(D): De Gn_lJ

n- UG

for n

5, 6, 7, ....

Finally one defines the Cantor type set A =

(see figure 15)

n

n=4

and seeks to estimate the measures j{l,.Jl."

~~)~~

and

9 on A.

\'1e

have the following immediate estimates: (1)

the smallest distance between the centers of distinct disks of

F (B(z,r») is n

(2)

2(1 - l/n)sin(~/n)r.

no halfplane bounded by a straight line through the center of

B(z,r) meets more th ~n n/2 + 1 members of Fn(B(z,r». (3)

for n ~ 4, Gn

consists of

f (4) (1

n:/3l

disks each with radius

n = 3l/nl

the minimum distance between distinct members of Gn equals (from

»,

2(1 - l/n) sin('\1'/n)r - 2Jn

2(1 - l/n)sin('lI'/n)(nfn) - 2Jn 2Jn (n - 1)sin(1'l/n) - I]

> 2fn[(n - 1)2/n - IJ (because

sin(x»2x/1Y

for

0 ~ x
~Jn'

- 85 -

F. J. Almgren Jr.

~

Estimate E.!l 1t leA)

,leA).

One notes that

~ leA) ~ .sl(A) ~ 2 because

for n = 4, 5, 6, 1 We seek to show that ~ 1 (A) ~ 1l' .

Es~ma te E.!l ~i(A).

o

!n

.<. 1..<. 'lr

an d choose

~ d

implies

(i)

n ~

(ii) (iii) To show

~

> 0

such that whenever

Let

n ~ 4 the condition

5,

(n-l)sin(lf/n)'7 'A , (n-l)sin(1rIn) - 1 > " [1/2 - l/(n ..l») •

~i(A) ~

'II'

it will be sufficient to show that in ca s e H is a

countable covering of A consisting of convex open sets S with

diam(S) ~

b,

then

where

S(S)

S

~l(pQ)

pEG(2.1) Q = R2n[(x,y):

In c.se

S

O~ %~ 1, Y

= B(z,r)

=

di2,lP

oj.

2 for some z € R

and some

0 <: r":oo,

then~l(pS) = 2r for e uch p € G(2,1) and, since ~2,1(G(2,1» = 1, one has

S

p€G(2,1)

t\pS) d)'2 IP ,

We define (): G(2,1)--(-1Y/2, 1'
= 2r .

by th e requirement that for each

- 86 -

F. J. Almgr-en Jr.

peG(2,1),

is the acute angle between the x axis

2

and peR ).

12 , 1 =

One notes also that

(1/"')

trf2

S.

-i/2

(lfrr) de so that cosB

de

= 2/'lr

In particular t hen, in case S .. B( z ,r), ~(S) = '!1' r. Suppose H ·i s a countable covering of A by convex open s e t s each wi th di ameter no l arger t han J • H i s finite.

Since A is compact one can assume

Also, s in ce A is compact,

E

inf ae A

s o that, choosing each member of

v

sup

S€ H

t d:i:s t( a ,

2 R - H)} > 0

su f f i cien t l y l arge tha t

2 j ~ <.

is contained in so me member of H.

G~

that each member of H cOnt ains some member of Gv i n order to estima te

~n over R2 for n

e; , one has t hat

k(S ) fo r S €

4, 5, 6 , •••

=

~ n (T) =

Also one can a ssume

•

H it is us eful to def ine mea sures

by t he formula

J n card( Gn n 1 BIB

1\

T ;, ¢1)

2 is a measure since it i s de f ine d over al l subs e ts of R and 2 R (T) ~ ~ F ~ (v) when ever Fe 2 ,F i s 'coun t ubLe , an d Tc U F).

(~n

\fn

Since

ca rd G n_l

= ~

/ _ 1-

= ( I,n / I0. l ) card n_

card Gn

1\

\ II 'f n - l

~

Gn , one has

If

-n

We now consider sever al pos s i bi lit i e s. ~

1.

S E H and S

t( S) ~ !( D)

=

'l(

~.!

f~ =

s ingle

11' 't'y( S )

~

» ?

Then

D of G".

Des an d

"Ii Y+1 t s i

as was compu te d a bove . Case 2.

S€

H.2Ei S meets a t l ea st

s mallest in teger su ch t hat

4 ~ n ~ V

~

members of

C\I .

Let n be the

and S m ~e ts a t l eas t two members

- 87 -

F. J. Al m gr en Jr. of G. (a)

,

In particular,

Jn

(since diam S ~ S

= ~

me mbe r s of Gn is l arger t han In 0 )

. (b)

,

5 (by the choice of [, above),

n ~

(c)

an d the distance between distinct

S meets a single membe r

D = B(z,fn_l)

of B n_l

(by the choice

of n }, We now define the normal retraction f:

R2~E

f(x) = x f (x )

=

for x

E

-In)

E,

z + (;n -l - f n )( x-z)/Ix-zl

and ·we also define B () S

= B(z, In-l

I¢ ,

for x €

2 R - E;

c (B) to be the center of B for BE G and, in ca se n,

one choose s

1; ( B) E f-l (B) n boundary (S), f [ S ( B)]

=

c(B)

in ca se

c (B)e- S

(see figure 16 ) .

li e now verify

for e a ch BE G n•

We c on sider two pos s i bil i ties :

Po s s ibil ity 1.

c(B ) E. S, f[~(B)1 = c (B).

....f\ 'V n+l(B n s)

Then

~ 1lj1 (B)

n

/)..In <: (n-l)sin('lt/n)!n

=(n-l ) sin('lt/n )fn_/n ( s in ce I n-l

= npn

and

fLt(B)1

-

If[s( B)]

- c(B) \

= c (B)) (1 - 1/n)s in(1I'/ n)Pn _l -\f[s(B)1 - c(B)\

wh ich is the de s i r e d ineq ua li ty . P os sibility 2.

c ( B)

if S.

I f t h i s pos s i bi li t y oc curs

B (1 S is

- 88 -

F. J:. Almgren Jr. con tained in a half plane bounded by a . fltraigb
+ l/(nt-l)]Jn

<: ((n-l)sin('ll'/n) - iJjn

b above

(by the choice of

so that condition (iii) holds)

(n-l)sin(1r/n )J n -

If[S(B)1 - c(B) 1•

Since boundary(S) is a simple closed curve the members of G which meet S n can be a r range d sequentially Bl, B2, ••• , Bm = BO so that the corresponding poin ts ~(Bl )' ••• ,S(Bm) occur in a natural order along One recalls the gene r a l f act that

boundary(S).

2 ~( S) = ~l(boundary( S» ( immedi a t e l y obvious in ca se boun dary(S) is polygonal, which suffices here}, s o t hat 2 ~( S) = ~hboundary(S}) m

~ EI~(B.) - s(B. 1)\ j=l J J-

(since

Li p(f) = 1)

~ j~ Ic(B ,) =l J

- c(B , l} \ - \fU(B ,)] - C(B ,)\ - If[;(BJ'_l)] - C(BJ'_l)\ J J J-

m

~ 2 2: [(1 - l/n}sin('l!"/n}f -1 - \f[s(B.)] - c(B ,} \ j=l

(by the estimate f a ct t hnt

n

(1)

=

2

m

Z

J

"1

above on the distance between centers and the

Bm = BO) ~ 2

J

~

j=l not-

1 (B , 1\ S )

'>,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
,

) lit .

00

(this follows from Holder's inequality

,

is nonde cnee s Ing in t for

1 ~ t ~"".

But it has been shown that

Hence one concludes 2

In particular J<.l(A) = 2, ai(A) j{l(AI1 B) = 2fn

and

forl
=1r'.

One concludes also for n

~i(A

flB) = 1rfn

~ 4 that

- 90 -

F. J. Almgren Jr . whenever

B€

G n

a partition of A.

1A

Ilfn

since the

sets

A (l B

are congruent and constitute

Now

() B: B

e

41

for some n ~

G

n

is a basis for the relative topology of A.

(j(lL A)j2

(~L A)!-rr.

=

l

(M

It follows that A contains no

Hence

, 1) rectifiable subset of positive

measure. n A Eorel set C in R is called purely (j(k, k ) unrectifiable

DEFINITION.

if and only if C contains no Borel set B which is with J{k(B)

>

~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

.<

B..(

1, •

s-ldist(x-a, V)..( lx-al..( r}

(see figure 18). We a sse r t t hat

.f2.!:M l

almost all points a of A,

Gfl(.1(} LAn X(a,lXl,v, s)a1 ,

whenever

V E 0(2,1)

0..(

~

~

lim sup (2t)-1-'<.1 (Afl X(a,t,V,s»

=

t_O+

B

<. 1.

1/(2n)

It suffice s to show that for every

sufficiently large integer n there exi s t s a positive number

t..( 2/ n_l

s u ch that l[A ( 2t)-Jt f\ X(a,t, {XI u"ln(a) = 1/( 2~).

i s not much smaller t han

01,

sn

To do this we define

S(z,r,u,~) = B( z,r)f\ tXI (x-z). u ~ r ;co s ( ~ >1 whenever

n, z € R r > 0, u E Sl, 0

~ t ~

'tr (see f igure 19) and observe

t hat

( 2fn_l)-1~1(A n S ( z'Jn_l'u,~» is close to As s ume

o
~/'Ir

B(z'fn_~) E 0n_l and n i s l arge.

in ca se

aE: A, a E B(zdn_l) E 0n_1' V= tal x."ln(a) = OJ,

11'/2, sin("") = s , 0

-
11'" , cos(t)

=

1 - 2/n , and observe

S( z'fn_l'1 n( a)exp (J: ),0() - S( z'!n_l''?n(a)'f)C X(a, 2/n_l'" (see figure 20).

One notes that if n is l arge, t hen (4fn_ls)-lXl[A () X(a, 2jn_lS, V, s)

is not much smaller than

0(/( 21's)

=

1/2 fr •

f

V, s)

is 'Small and

- 92 -

F . J .. Almgren Jr. DEFINITION. of all pairs

We denote by 8* the subset of

(a,V)

~ >~,

suoh that for all

~l(A() X(a,r,V,s»

sup

lim sup s_O+

rs

O..c.r<~

AX G(2,1) oonsisting

= 00 •

In view of the estimate above, ,1{l(A_

n

THEOREM [FHl 3.3.91.

a c s",

\al(a,V)~SJ)

O.

Let A' C Rn,b(k(A')-<

00

V€ B(2,1)

e:

p

,

G(n,k),

~

whenever

r

k

s

k

~ ~ > O. ~ J..k(pB)

x E: B

PROOF.

j{~AI () X(x,r, ker(p), s)]

'>up 0<: r
lim su~ s_O

=

~

= O.

One notes that X(p,r, ker( p), s)C p -lu(p(x), rs)

whenever

r,s

> 0; and therefore

P#~kLAI)U(p(X), rs)

lim sup

(r,s)~(O,O)

whenever

x

e:

=

00

(rs)k

We define the Radon measure )A over Rk

B.

by the formula

)'4-(8) = infh#GKkLAI)(T)1 SC T, T is a Borel setJ for eaoh

k• 8 C R

In particular lim sup

J -'> 0+

whenever

z E: p(B) .

One takes

B

that for every

p

tlk(B(z,.l') )

The theory of derivat ion of measures implies clk(pB)

= tal EO:

)4(B(z,y»

(a, ker(p»

G(2,1),

EO:

8*~

O.

in the above theorem and concfudes

- 93 -

F. J. Almgren Jr.

J.l(pA) ~ ll(pB) + .r.1(p(A--B»

~ 0 + .1(l(p(A_B»

~ Lip(p) ~hA-B) =

I t follows that

gl(A) = 0

o. and

J~(A)

0

for e ach

1

~ t ~

00 •

- 94 -

F. J. Almgren Jr.

F.5.

The structure of sets of finite ;H=a::=u:;:s::=d=o=r=ff;: measure.

As indicated

in F.l, there evolved from the stuqy of the behavior on sets of Cantor type (like the set A of F.4) of the various k dimensional measures over ~, first a pattern of structure for general sets of finite k dimensional

measure and, second, the general acceptance of Hausdorff measure J(k and integral-geometric measure n

measure~ in R •

J~

In this section

as the two most important k dimensional we summarize the principal structure

resul ts. DEFINITION.

A Borel set

and only if J{k(B) -c submanifold

Me

]I in

~

and for each

00

.e >-

~

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<,

k) rectifiable.

n If M is a k dimensional submanifold of R of class 1, then

PROOF.

locally in M, j{k L M is a Radon measure, open subsets of M can be j(k L M approximated from the inside by compact sets, and arbitrary subsets can be ~kL M approximated from the outside by open sets, etc. DEFINITION.

n A Borel set C in R is called

~ (Mk, k) unrectifiable

if and only if C contains no Borel set B which is k wi th X ( B)

> O.

THEOREM.

If A

~ ~ ~ ~ ~ Rn ~ j{k(A)'<;

Jek ~ unique ~ sets B, C B

1! (~k,

k

(M , k) rectifiable

k) rectifiable

PROOF.

E.!.

n R ~ ~

~ C ~purely

Maximize b{ k L A

over

A

00

,

~ there exist

= BUC,

Bf\C =

¢,

(~k, k) unrectifiable.

(~k, k) rect ifiable Borel subsets

- -95

~

F . J. Almgren Jr. The cen1iI'al f act related to t ile structure of purely <Jlk,k) unrectifiable sets is the following. THEOREM [FHl

~ C c If be ~ ~ ~ ~ ~k(C)':'
3.3.13'1.

.m d suppose C is purely

~k, k) unrectifiable.

~ for ~n,k ~ all

pEG(n,k), J..k(pC)

~ particular, J~(A)

= 0

for 1

=

O.

~ t ~

The example of F. 4 shows tha t

00 •

&.~( C)

need not be O.

The proof of the above theorem rests in substantial measure on a detailed analysis of the sets C n X(a,r,V,s) (recall the density estimates of such sets in F.4).

Certain properties

of these sets imply rectifiability; for example,

THI~OREM [FHl 3.3.5 J. diam(E) .:: r, p €

e: E.

&:lli

G(n,k), ~ E

whenever a

n Suppose r ::> 0, 0 <: s « 1, E C R

~

Let

X(a, r, ker(p), s) = q

q6

ker(p) E G(n, rr-k}, 1

~

~ ~ a e E,

(1)

card(p-l(p(a)))

(2)

f = p(E) X q(E) (\lJp(a), q(a)): aE E5

~ Lipschitzian

&:!:h

Lipschitz constant at ~ lis,

(3)

In

~ E is ~ ~ set, then E i s ~\ k ) rectifiable.

n In summary then, any subset of R of finite j{k measure is

~k almost

uni~uely the disjoint union of an (~k, k) rectifi able subset of Rn n• together with a purely ~k, k) unrectifiable s u bs e t of R e ach purely

Furthermore,

(~, k ) unrectif iable s u bse t of Rn pro jects to £.k mea sure

under a l mos t all orthogonal projection s

k• n R _'> R

0

- 96 -

F. J. Almgren Jr.

REFERENCES [AW]

W.K. Allard,

On the first variation of a varifold,

Ann. of Math.,

(1972).

[AF1] F.J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints [AF2] '

(in preparation).

, The theory of varifolds. calculus in the large for the integrand

A variational

k dimensional area

(multilithed notes), Princeton, 1964.

[BDG] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math 7 (1969), 243-268. [BJ]

J.E. Brothers, Integral geometry in homogeneous spaces, Trans. Amer. Math. Soc. 124 (1966), 480-517.

[DJ]

J. Douglas, Soiution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), 263-321.

[FHl] H. Federer, Geometric Measure Theory, Die Grundlehren dar mathematischen Wissenschaften in Einzeldarstellungen, Band 153, Springer-Verlag Berlin - Heidelberg - New York, 1969. [FH2]

, The

sin~lar

sets of area minimizing

rectifiable currents with codimension one and of 'area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 79 (1960), 761-771. [FH3] ________, Currents and area, Trans. Amer. Math. Soc. 98 (1961), 204-233.

- 97 -

F. J. Almgren Jr.

[FW] W.H. Fleming, An example in the problem vf ·least area, Proc. Amer. Math. Soc. 7 (1 956),

1065-l074~

[GM] E. Giusti and M. Miranda, Sulla reeolarita delle

soluzion~

.deboli di una classe a{ sistemi ellittici guasi-lineari, Arch. Rat. Mech. Anal. 31 (1968/69), 173-184. [GJl] Jean Guckenheimer, Singularities in "soap bubbles" ant: "soap films", I.C.T.P. lecture notes, 1972 •. [GJ2]

, Regularity of the singular sets of 2 dimensional area minimizing flat chains modulo 3 in B3 , Ph.D. Thesis, Princeton University, 1973.

[MC1] C.B. Mor r ey , Jr.

Mult i pl e integrals in the Calculus of

Variations, Die Grundlehren der mathematischen Wissenschafte~

in Einzeldarstellungen, Band 130, Springer-Verlag

Berlin - Heidelberg - New York, 1966. [MC2]

, Partial regularlty results for

non-lineD ~

elliptic systems, J. Math, Mech. 17 (1968), 649-670. CPS] B.S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soq •. 70 (1964), 165-172. [RE]

E.R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type (with Appendix by J.F. Adams), Acta Math 104 (1960) 1-92.

[TR]

B. Thorn, Quelques propri etes globales des varietes different iables, Comment. r,iat h .Hel v . 28 (1954), 17-86.

- 98 -

F . J. Almgren Jr.

(f)

CD LO

o

«

- 99 -

F. J . Almgren Jr.

!

.s: N Q)

L

:J

OJ

h-

-a


~

...-:>

a

-rJ

OJ

C


(1)

c

.-r:> -

:J

'+-

U OJ

4-

-u0

U

.x: e

If)

(1)

0-

E if)

«

c

0


>

L

~

u

0 co

-0

c

:J 0

-0

Figure 3 '%j

;-. ~

S (JQ '1

III

::s c... ~

The. oriented surface S having boundary C

and of least area has inf i nit.e topo\oSic.al type.

t-

o

0

- 101 -

Jr, F. J. Almgren .

U) (])

-

-0

-r (1)

L ~ {JJ

u.,

C

«r

.c

2 <0

E

+.:>

C '-

0

o.,

0 CO

-0

...0 c (l)

c

~

~

0

.-0

c

eo

.c

-rJ

C+-

~

.-

3

f-

.x: if)

.-

-U

«

0

OJ

c

a-:> L (1)

>

c 0

u

- ,10 2 -

F. J. Almgren Jr .

Fig u re

5

o

The

I dimensional

set

Y

connecting the 3 points of the

boundary B and of least l e nqt. h has an interior singularitj of codimens ion

I.

- 103 -

F ,. J. Almgren Jr.

Figure 6

The uni9ue minimal par t.i t.ioni nc

confkguration for two regions

of presc.ri bed vol umes.

v

- 104 -

F, J, Almgren Jr',

A "s oe p bubble lIke)) minimal

part i t ion i n 9 con f \' 9u ro t ion consisting of six real enc\ytic 0 surfaces meeting at 120 a lon g four smooth arcs' wh ic h meet at e.9 ua \. a \'"\g les at two poi nts .

- 105 -

F .. J . Almgren " r.

ro

(1)

s,

ea

~

(f) (CJ

(1)

'-i0 (l)

U

<0

ro Q)

L

::>

en

u,

'+s;

::>

<.f)

<0 ([)

CO

-0

s:: (G

..D tf)

J

-D

:0

~ (1)

.s:

l-

- 106 -

F. J . Almgren Jr.

y•

_ 107 -

F . J . Almgren Jr .

o

~

o

-

- ,108 -

F. J . Almgren Jr.

Fi 9ure\ \ a

Fi 9u re 11 b

A soap fi I m w'lth a boundary w'lre which is not clo se d .

- 109 -

F. J. Almgren :J r .

N

I

0

(\j (j)


G Q)

4-

O

S~

-P

OJ

0

c

LL

(/)

.-

..c -r-:>

(J)

c

Q)

-r:> U) CD

(1)

40

C o >L

--.

a

I

-l

u

ill

-C

l-

- 110 -

F. J. Almgren Jr .

Fi 9ure 13 a

I

+

-0-

+

o,

+

+

I

+

0I

Y=/G .

1.

J

+

rigure 13 b

- 111 -

F . J . Almgren Jr.

- 11 2 -

F . J . Alm gren Jr .

<:(

+><1J

cJ)


0-

::>->

~

L

0

-r=>

C ·

(C1

U

-

CO

s:: 0

(j)

C

(J)

E -0 (])

ill L

~

OJ

u,

C 0 (l)

-C

l-

:.. 113 -

F. J. Almgren Jr.

Figure \6

f

-.

Polygona \ estimates on boundary (5)

- 114 -

F. J . Almgren Jr.

f\9 ure \7 The. maps

~n'

F . J . .l\ltngr en Jr.

_ 116 -

F . J . Almgren Jr .

\u \

,,

----r

- 117 _

. .. . . : '.: : ::'

S( Z ,jn-I ,

F. J . Almgren Jr.

~~ (a)eXp(kad,cx.)_ 5(Z,/n_1 , "l~(a),(3) C

X(a, 2.fn_1 s, V, s)

Figure 20

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.LM.E.)

E.

GIUSTI

MINIMAL SURFACES WITH OBSTACLES

Corso

tenuto

a

Varenna

dal

24

agosto

al

2

settembre

1972

Ei. Giusti

Introduction We will be concerned with the problem of existence and regularity of solutions to the non-parametric Let

Pla~eau

problem -with obstacles.

0 be an open bounded set in in, and let - A be a closed set

contained in the c19sure of

0,

o.

Let

~(x)

be a function defined in

00 , and let w(x) be a function in 0 such tliat

~(x) ~~(x)

in

Roughly speaking, our problem consists of finding a function defined in

00 n A u(x~ r

0, such that:

= ~(x)

on

00,

a)

u(x)

b)

u(x) ~ w(x)

c)

the "surface" graph of u

in A, minimizes the area between all graphs of

fUnctions verifying (a) and (b). We shall refer to this problem as Problem Depending on the choice of the set

[~,A, w]

•

A, we have different situations;

we shall consider in detail the -following three cases: 1) ~ obstacles,where

A coincides with

n.

This is the simplest

situation, and the results are quite satisfactory, at least from the point of view of the existence and regularity of the solution. 2)

Discontinuous obstacles. and

oA

no

Here

A is the closure of its interior,

is a -non-void regular surface of dimension

n-l •

3) 1h1!!. obstacles, when A is a regular (n-l) -dimensional oriented surface in

n.

We note that, in the previous classification, no mention has been made ~

boundary data

and of the obstacle

~

and

V;

of the fUnctions

of course, depending on the regularity of the

W, we have various subcases.

On the

other hand, one of the f ea tures of the problem under consideration is that

- 122-

E. Giusti the regularity of the solut ion op.uends more on t he shape of the s et on the smoothness of the

functions , ~

and

W,

A t han

so that we can consider onl y

the ca se of regular data and yet obtain a complete picture of the situation. For instance, we can su ppose t ha t the bou ndary of and that

~

i

Q

~f

2 C , while

class

Problem

In this way, although not comnletely.

W

0,

dO, is a C'-surface

is Lipschitz-continuous. become R more precisely stated,

[~,A, wl

In f act, we must make precise the class of func-

tions competing 'for the minimum, and poss ibly we ha ve to extend in a suitable way the area functional in order to cover the class under ' consideration'. The following well-known example shows t hat in general one can not hope to get a continuous solut i on. Let r a dius

1/2

n be the unit disc in Let

tion ' of the pr obl em

~

=0

and

( ~,A, wl

2

E

, a nd let

W=M

A be the closed dis c of

I t is cl ea r that a co nti nuou s so lu-

must be e qua l to M in the interi or disc , an d

must be a solut i on ·of t he minimal su rface e quat ion in the annulu s , t ak i ng the values zer o on the ext erior cir cle and

M in t he int eri or .

Qn the other

hand, if, M is big enough , such a s olution cannot exist , a nd he nc e t he origina l problem has no continuous solutions, in spit e of the r e gularity of t he data. It is evident from the preceding example that one must c onsi der discontinuous functions dealing with discontinuous a nd thin obstacles. We will work in t he clas s of upper

semi~continuous

f unct i ons, a nd we

shall prove t he existence of a so lution to the problem a f te r ext endi ng t he area funct ional t o thos e funct ions. Let us remar k, however, that the class of fUnct i ons co m p e tin g to minimize t he a r ea is by no means uniquely determined, a nd t hat our approach to the problem in consideration is only one of the poss ible opt ions, t he

- 123--

E. Giusti choice between them being justified both and

~

~~steriori

by the results obtained

priori by the aAsthetical feelings of the author. The paper is divided into :roqr chapters:

1.

Smooth obstacles

2.

Discontinuous and thin obstacles; existence theory

3.

Regularity of the solution; the favorable case

4.

Regularity of the solution; the general case.

At the end of every chapter, a sectiOn is dedicated to a discussion of additional results and open problems. Although

That section also contains the references.

effort has been made to report all the material relevant to the

subject, some unintentional omissions are ineVitable, and the author can only lpologize.

1.

Smooth obstacles Let

tion in that

n be a bounded open set with C3-bbundary, let ~(x) be a C2-func-

on,

and let

~(x) ~ *(x)

in

*(x)

be a Lipschitz-continuous function in

n

such

'on .

In this chapt er , we shall prove the following theorem: Theorem 1.1. problem .QJ.l..§

lAo

Let

[~,n,*]

on

be ~ surface of 1lQn.-negative .!!!illm curvature.

Then ~

has!!. unique solution in the ~ of l!ll L1pschitz-continu -

functions. For Lipschitz-continuqus functions, the area of the graph is given by

a(v) It is not difficult to show that

=

Jn /l+ IDv 1 2

dx •

- 124 · -

E. Giusti n

Lg 2 < 1

(1.1)

.i

• i=O

For ·that, we first observe that

J(v 'di v g+g )dx J(-g.Dv+g0)dx .$. a(v)

•

=

nOn

we have

J

(-"7.Dv+1 )dx

n C~

Now tions

l is dense in L

-.gy,

g

o,y

j

J(v div g +g

o and

= a(v)

•

it follows that there exists a sequence of C=-func-

n 2 .E gi i=O Y

, with

0

YOY

<

-

converging to

1

=

)dx

J

(-g 'Dv+g

0

Y

OY

"7,

1

L

in

0

o

1

, so that

)dx -. a(v)

(1.1) fol lows at once . From (r.r), we conclude that

a(v)

respect to the uniform convergence.

is lower semi-continuous with

In fact, if v

n

is a sequence of

Lipschitz-continuous functions, uniformly convergent to a Lipschitz-continuous ·funct i on

v, we have, for every choice of the functions

J(v div g+g )dx

o

0

=

8

i:

~= 0

J(v

n

div g+8 )dx.$. lim inf G,(v )

>

0 ,

the class

lim

Let us consider now, for

R

~=

0

(v €'Lip (0)

v

= cp

on

where

=

sup I v(x)-v~Y)1 \x-y x, yEO

xh

is the Lipschitz constant of the function

v .

00

v~ 1jr

n

in

0)

- 125 -

E. Giusti For R~R o' let

un

R In

greater than some

L cP, 1jr) R(

R the class o'

Ls ' non-void.

L (cp, 1jr ) , there exists a unique minimum for R

be a minimizing sequence.

a(v) .

Let

In fact,

Then the Ascoli-Arzel1l. theorem provides u

a(v)

provides the required mini-

is lower semi-continuous, the function

u

R

R

in

L (CP, 1jr) • R

a . subsequence uniformly. convergent to a function

Since

mum. The uniqueness follows at once fran the strict convexity of the functional.

ia. u

R

IUR h .$

We obviously have

R.

It is worth noting that, i f

IU R h < R

gives the minimum in the class of all Lipschitz-continuous functions.

fact, let

v(x)

be an arbitrary function in

enough, the function

~

+ t(v-u

R)

L (cp, 1jr ) - . S

belongs to

If'

It

I

is small

L (CP, 1jr ) , and hence the R

function

has a local minimum at

t

= O.

Since

g(t)

is an absolute minimum. In particular, we have

is convex, this local minimum g(O).$ g( l )

that is,

The proof of Theorem 1.1 is then reduced to the proof of an 1! m:iQtl bound for the Lipschitz constant of lC.

u

R•

A supersolution (resp., a subsolution) for the area functional is a

Lipschitz-continuous function

w(x)

such that

a(w) .$ a(v)

,

In

- 126 -

E . Giusti We have the following ~

1.1.

Let

ul;'l

LR(cp,ljf),andill, w(x) 2:..ljf(x) Proof.

in

'01

'0.

~

the minimizing function for the

~.!!.supersolution~

Suppose ~

Suppose first that

w(x)

w(x)2:.cp(x)

Iwll.:s R .

> cp(x)

in

~

in

~~

J:!l. On

Th'en 'w( x ) 2:. uR(x)

en .

~

J:!l. o

We have

and

so that, if

is non -void, we have

and hence

But

thi~

area in

is impossible, because

u

R

is the only function minimizing the

LR(cp,ljf)

Then

A

=0 ,

and the lemma is proved in the case

w(x)

> cp(x)

general case follows at once, considering the supersolution -w( x ) +€ , and letting

E -+

0 •

q.e.d .

A similar argument proves the following two lemmas:

The E

>0 ,

- 127 -

E. Giusti 1.2.

~

~ let

that

Let

vex)

u

L

R(CP2''il2) O. Then

Then

u~l)

Let

vex) S uR(x)

u~2)

and

, respectively. u(l) < u(2)

in

Lemma 1.4.

Let

Then

(1.2)

max

.ill do. Suppose

minimize the

~

CPl S CP2

in

LR(CP1''il

do

in

and

0

l)

and

'ill S'il2

in

hav~:

u~l)

LR(CP2''il2)

vex) S cp(x)

.ill 0'.

Suppose that

--R-R- ·

Finally, we

L cP, 'iI) , R(

be the minimi,zing ftrhction -f or the -~ in

be.~ subsolution such ~

Ivl l SR.

~ 1.3:

R

and

IU~1)_U~2)\

u~2)

minimize the

Smax

(na;

Proof.

Let us denote by

u~2)+M

minimizes the area in the class

~

\CP1-q>2 1,

in

m::.x o

LR(CP1''il

l)

and

\'ill-'il2 11

M the right-hand side of (1.2).

The function

LR(CP2+M, 'il2 +M) , and hence, by

Lemma 1.3, we have in Changing the role of

rn. and

Theorem 1.2. ~

Let

u(l) R w(x)

and

u(2) we get (1.2). R'

and

vex)

0

q.e.d.

~ respectively. ~ supersolution

subsolution for the area functional such that w(x)

= vex) = cp(x)

and

w(:x;) 2:. 'iI(x)

in

o.

- 128 -

E . Giusti Let

11.

R>

LR(cp,'lr) •

Proof.

, ~ let

uR(X)

min1Inize the ~ in

Then

We must estimate

point

\tl l )

= max (Iwh, lvii,

y . belongs to

.

\uR(x) -uR(y)

I.

Let us suppose first that the

Lemmas 1.1 and 1.2 give at once the inequality

0;)

and hence

(1.3) Now suppose that neither of the points T = y-x.

and, if

x, y

belongs to

0;), and let

Let

f(x)

is a function in

be the function in

n, let

n( T)

defined by

It is clear that the functions set

n

n neT»~

in the class

uR(X) ,

~(uR''lr)

U~T)(X) and

minimize the area (in the

LR(u~T),

Fran Lemma 1.4, it follows that there exist points z"

in n

n neT)

z'

'lr(T» in

, respectively . o(n

n neT»~

and

such that

On the other hand, either

z'

or

Z'-T

belongs to

and (1 .4) we get the conclusion of t he theorem.

0;), and hence fran (1.3)

q .e.d.

- 129 -

E. Giusti Theorem 1.2 gives the required constant of u

~

priori bound for the Lipschitz

R ' provided one can construct a supersolution and a subsolu-

tion satisfying the hypotheses.

The remaining part of this chapter will be

devoted to such construction. lEo

For

t > 0 , let at

(x

€

a

d(x) < t)

rt

- (x

€

a

d(x) = t) ,

d(x) = dist (x,On)

where

C3

If the boundary of a, On , is of class

, then the following'

results are not difficult to prove : ali

There exists an

Eo > 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

<

~

, then the derivatives (in the sense of

are measures, and

the total variation in v)

f

~f

and

g

f 11+\Df!2

K

1

the iast integral being

K of the vector-valued measure

U(K)

are in •

= u(g).

then a(f)

a(f)':::'

distri~utions)

(dx,Df)

f = g Hn- l-almost everywhere,

,and

is the j-dimensional Hausdorff measure in

(H j

En .) vi)

Let

n =~

where·.r let

gl

U ~

ur

(see Figure),

is a regular hypersurface; and

~

be Lipschitz-con-

tinuous functions in ~ and let

f

and ~,

be a positive Lipschitz-

r.

continuotis function in

Let.

rgl

in ~,

-t ~

vex)

in ~,

max (gl,~}+f

in

r .

Then a(v)

=

JA+IDg ~ + 2

vii)

Let

JfdH

r

2

dx

lI

_ == n l

n =~ u ~ u r

+ J 11+1%1 2 ~

dx

(f)

o

r

JI1+ Invl2 + 2 JfdH n _l n r

as before; then

a

+ f 1~-~ldH

= a (f)

'\

for every upper semi-continuous

+

f.

a

Ci;z

(f)

n

-1

- 137 -

E; Giusti 2B.

The following lemmas will be 'very/useful in the following.

The first

is essentially the Dini theorem on uniform convergence. ~ 2.1.

K , be !!. canpact set, and let

Let

such that

f

n

~

f.

Let

g(x)

Lemma 2.2.

0

Let

be!!. sequence in 'L«(K)

.!!!.'. K such that

continuous function

~!!.

g(x) > 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
00 and

upper semi-continuous functions in
in

~

00. Suppose further that

Let

~+

l