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## A NNALES DE L ’I. H. P., SECTION C

o

### 5 (1991), p. 499-522

<http://www.numdam.org/item?id=AIHPC_1991__8_5_499_0>

© Gauthier-Villars, 1991, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(2)

### of prescribed Gauss curvature

J. I. E. URBAS

Centre for Mathematical Analysis, Australian National University,

GPO Box 4, Canberra 2601, Australia

Vol. 8, 5, 1991, p. 499-522. Analyse non linéaire

ABSTRACT. - We

the

### boundary regularity

of convex solutions of the

of

### prescribed

Gauss curvature in a domain Q c ~n in the case

that the

of the solution is infinite on some

open,

convex

### portion

r of Under suitable conditions on the data

we show that near r x R the

of u is a smooth

a

submanifold of

and that u

is smooth. In

u is Holder

continuous with

### exponent 1 /2

near r.

Key words : Gauss curvature, boundary regularity, free boundary, Legendre transform.

1. INTRODUCTION

In this paper we

the

### boundary regularity

of convex solutions of

the

of

### prescribed

Gauss curvature

Classification A.M.S. : 35 J 60, 35 J 65.

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire - 0294-1449 Vol. 8/91/05/499/24/\$4.40/© Gauthier-Villars

(3)

on a domain Q c in the case that

for some

open,

convex

r of aS2. The

condition

is to be

as

for

### all y

E r. This situation arises in a number of

connected with

and in

that

### ( 1. 2)

holds on a suitable

r of aS2 is a

in the

of the interior

of solutions of

in the papers

Before

our

we recall the

### problems

in which the above situation arises. Let us suppose that Q is a

convex domain

in ~n and is a

function. It is

### easily

shown that the

condition

where ffin is the measure of the unit ball in

### fR",

is necessary for the existence of a convex solution of

### (1.1).

In order to solve the Dirichlet

for

with

### arbitrary boundary

data cp of class it is known from the papers

### , , , 

that two conditions are necessary.

we

to ensure the

of an a

### priori

bound for the solution u, and in addition

in order to obtain a

estimate. As shown in

### ,

this

condition is not necessary if instead we

### impose

some restrictions on

1 ~a~~. If

but not

### ( 1. 6),

then the Dirichlet

### problem

may not be solvable in the classical sense

it is shown in

that there is a

### unique

convex solution of

### ( 1.1 )

which solves the Dirichlet

in a certain

### optimal

sense: the solution u is the infimum of all convex

of

above cp on or

### alternatively,

u is the supremum of all convex subsolutions of

below cp on

The Dirichlet

condition is

### generally

satisfied in the classical

sense on some

### parts

of aS2 and not at other

At

### points

at which it is not satisfied we

### have |Du= oo, provided

K satisfies a mild

restric-

tion near

for

### example

for some p > n. In

### fact,

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

(4)

is

### sufficient, by

an examination of the

of Lemma 3.6 of

also

Lemma

and Remark

the

### proof

of Lemma 2.2.

In the extremal case

it is shown in

### , 

that there is a convex solution of

which is

and

if

for some

p > n

above

on ao.

At

the

theorems which

### regularity

of u near r are the Holder estimates of

These are valid

much more

and their

### proof

makes no use of the

condition

### (1.2).

It is reasonable therefore to

that better

can be obtained

### by using ( 1. 2).

Our results are in fact valid for more

Gauss curvature

### equations

of the form

where v denotes the downward unit normal to the

of u

and K E

X is a

### positive

function. Here Sn denotes the lower

### hemisphere

of the n-dimensional unit

Sn~

### Rn+1.

The situations described above for

also arise for

if we

a number of

on K

Theorem

and

Theorem

### 4.10).

Our main result is the

As

### usual, BR

denotes the open ball in (~n with radius R and centre 0.

THEOREM 1. - Let SZ be a bounded domain in (~n and suppose that

some

### x0~~03A9,

which we may take to be the

I-’ = aSZ

is a

convex

aS2.

that

K E

x f~ x is a

and u E

### C2 (SZ)

is a convex solution

### of ( 1 . 8) satisfying ( 1 . 2)

and such that

and

where ~1, ~,2 are

### positive

constants. Then there is a number p E

on n,

### Ro, h,

~,1 and ~2, such that the

hold:

u E

### C°°1~2 (SZ n

and for any x, y E S2

### n BP

we have

Vol. 8, n° 5-1991.

(5)

where

on n,

~,o, ~.l and ~2.

is a

any a

and we have

where

on n,

~o, ~.2 and a.

u is of class

any a

and we have

where

on n,

### R0, r,

0, pi, 2, a and infu.

q

we have

and

some

2 and some a E

then

u is

class and we

have ~

and

where

and

on n,

po, ~1, ~,2,

a and inf u.

q

### If

If the K and r are

then

and u

are

of Theorem 1 are

with r =

### aSZ,

then we obtain

the conclusions of Theorem 1

dist

### (x, ~03A9) 03C1}

for a suitable

p > 0. We may then

the interior

result

Theorem

### 1.1,

to extend the estimates to all of

We state this

case

THEOREM 2. -

Q is a

convex domain in

K E

x R x is a

### positive function

and u is a convex solution

such that

and

### ( 1. 11) hold,

and

Then the assertions

to

### (iv) of

Theorem 1 hold with Q n

Q and r

aSZ. The

the constants

...,

on

r

should now be

### replaced by

Q.

We shall prove Theorem 1 in the

### remaining

sections of the paper. In Section 2 we shall derive some

estimates for

### graph

u near r.

We also reformulate our

as a free

### boundary problem

for a

function w, obtained

u

as the

### graph

of a

function w over a suitable subdomain of the

to

u at

### (0, u(0)).

In Section 3 we show how the information obtained in Section 2 can be used to deduce

### appropriate regularity

assertions for the

### Legendre

transform of w, from which the assertions of Theorem 1 follow.

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

(6)

As will become clear

some of the

### hypotheses

of Theorems 1 and 2 may be weakened

### slightly. First,

u may be assumed to be a

solution of

### ( 1. 8)

in the sense of Aleksandrov

also

### )

rather than

a classical solution.

### However,

under the conditions of Theorem

### 1,

u is a

classical solution on

for some p E

on n,

0

Remark

the

of Lemma

the

condition

### (1.2)

may be weakened to a measure theoretic condition

which arises

In

### fact,

it is this condition which is first shown to be satisfied in the papers

the

condition

is then deduced

### by arguments

similar to those used in the

of Lemma 2.2.

### Analogous boundary regularity questions

for solutions of the nonpara- metric least area

were studied

Simon

His results were

### improved

and also extended to the mean curvature

in situations

### corresponding

to those mentioned above for the Gauss curvature

Lau and Lin

and Lin

Our

is similar to

### [ 11 ], , 

in that we also reduce the

to a free

In our case,

the free

is not a

and

more

information is

before

we can

### apply

standard results.

It will be evident that the

### techniques

of Section 2 are also

to

certain other

of the form

for which the

condition

### ( 1. 2)

arises in a similar fashion as for the Gauss curvature

we

### impose

suitable conditions on

the function

In

fast

of

with

### respect

to Du is necessary.

in the case of

### equation ( 1. 8)

we may also allow some unboundedness or

### decay

to zero of K as x

### approaches

r. We

shall make some remarks about these extensions later. In these

we obtain a free

which in

is

or

and the

### techniques

of Section 3 do not

any further

### regularity. Nevertheless,

even in these cases it is

reasonable to

somewhat better

of

than is

### provided by

the Holder estimates of

For

of the

form

the

condition

also arises

fast

as x

even without

to Du.

we cannot

assertions for

### u Ir

such as those of

Theorem 1. For as

and Yau

have

### shown, uniformly

convex domain in ~8n

satisfies

B dist

### (x,

Vol. 8, 5-1991.

(7)

for some constants

B >

a > 1 and

n +

### 1,

then the Dirichlet

has a

### unique

convex solution u E

for any

data cp, and we

### have |Du = o0

on aS2. Similar assertions are also valid for

of the

from

where

satisfies 0 and

for all

z, x R x

where

a,

### P,

y are constants such that

and

Theorem

### 2.19).

2. ESTIMATES FOR GRAPH u

The main results of this section are a Holder estimate for the normal vector field v to

u and a strict

estimate for

u, both

of these

valid at any

### point (xo, u (xo))

with These

results hold under weaker

### regularity hypotheses

than stated in Theorem

we need

assume r to be

### uniformly

convex and of class

and the bound

is not needed.

We

with some

### preliminary

estimates for u. From

we have the

### following preliminary regularity

result.

LEMMA 2.1. - There is a number

on n, such

that u E

we have

where C

on n,

### Ro,

r and ~.°.

Remarks. - The estimate

does not

on the

condition

it uses

the

### regularity

of r and the lower bound of

and is

of any

condition. The

of

in

the

### exponent cx = 1/2 n,

but this can be increased

an iteration

described in

The

is not known at

### present,

but for our purposes any

### positive exponent

suffices.

From Lemma 2.1 we see that

### by replacing Ro by Ro/2

we may assume

from the start that ue

### (Q (1

It is also convenient to

S2

Q (~

### BRO.

This involves no loss of

### generality

and has the

that Q is convex, which makes the

of the

lemma

### technically simpler.

It is a refinement of the

used in

to show

### that Du|

= 00 on a suitable

### portion

of ~03A9.

Annales de l’lnstitut Henri Poincare - Analyse non linéaire

(8)

LEMMA 2.2. - There are numbers El, C >

on n,

r

such

each x E

x E S2 : dist

El

we have

### Proof. -

Choose coordinates so that the

xn axis

### points

in the

direction of the inner normal to aQ at 0. It suffices to prove

for

### x = (0, ~2) with ~~~1=~1(n, Ro, r,

because a similar

with

and

### ~1/2

if necessary, will

### yield

the estimate at any

of

Since r is

### uniformly

convex, there is an

ball

at 0 i. e.

aS2

### == { 0 ~

with the radius R estimated from above

a constant

on r. Let

be another

ball at

### 0,

and for

s > 0 consider the set

where

### x’ _ (xl,

..., It is clear that for ~>0

say

Eo = Eo

we have dist

=

where

=

### (0, E2),

and further- more,

=

+

for some

constants

and

on r and

Here

### (0)

denotes the open ball in

### [Rn-l

with radius R and centre 0.

Let

Then

is a

convex,

in

is enclosed

Let

and G be the

Gauss maps of

and

E

The

of G is

since E is

### smooth,

while for

is defined to be the set of outward unit normals of

of

at

### ç,

and for any set E c

.

=

+ E

and let

### G

be the Gauss map of ~. Since each

encloses

### ~o, it

is evident that if T is any

of at

### 03BE0, T

can be translated in the

direction to

a

of

at some

n

### LJ.

Thus

Vol. 8, n° 5-1991.

(9)

virtue of

and an

argument.

To

### proceed

further we need to define some set functions. Let

Since Q is convex, M is a

convex

in

As in

we define

=

a

For any set E c

we define

A result

also

asserts that if

B are

subsets of

then

### XV (A) U XV (B)

has measure zero.

### Furthermore,

if E is

a Borel set, then so are

x"

and

now with the

of the

we see from

### (2. 5)

that

because each nonhorizontal n-dimensional

T of M

an

= T

x : xn + 1=

of

for

each From

we have

### However, by (2 . 4)

the second set on the

hand side of

is

contained in

and so is

### empty,

since on r. Thus from

we obtain

or

the definition

where

### po = Du (xo)

and Xt. is defined as above with v

u.

### Next, using

the fact that u is a convex solution of

### (1 . 8),

we have

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

(10)

where h

=

+

(n +

for any

and

### estimating

the left hand side of

from

### above,

we obtain

for suitable constants

and

It follows that

for E small

say

now that

and

=

### (0, ~2),

we see that we have

since

= dist

The

of the lemma is

Remarks. -

The

### proof

of Lemma 2.2 does not

### require

u to be a

classical solution of

All we

is that u is a

### generalized

solution

in the sense that

for any Borel set EcQ.

### Furthermore,

if u is not differentiable at x, then in the statement of the lemma we need

Du

the

of any

of

u at

u

If u is a

solution of

and the

of

Theorem 1 are

### satisfied,

then u is in fact a classical solution on

for some

### positive

p > 0. For if xo E Q

then for p small

we

have

Vol. 8, 5-1991.

(11)

where po is the

of any

T of

u at

We assert that

### T n graph

u cannot contain a line

with an

on SQ x R.

### Clearly

we cannot have such a line

with an

on r x

virtue of the

condition

### ( 1. 2),

while if there were such a

L with an

on

then

the uniform

### convexity

of r there would be a

u

E L

with

for some

constant

on

and r. But

### then,

since T is also a

of

u at

### u (x 1 )),

we have

since u is convex. This contradicts

if p is small

so our

assertion is

The

of u on Q

### n Bp

now follows in a

standard way, see for

In the

### proof

of Lemma 2 . 2 the

condition

was used

### only

to assert that the set

in

is

All that is

for the

### proof

is that this set have measure zero. Thus the

condition

could be

### replaced by

the weaker measure theor-

etic condition

### (iv)

Refinements of Lemma 2. 2 are

For

if we have

where

for some

then in

of

we obtain

if

for some

we obtain

From the

### proof

of Lemma 2 . 2 and

### (iv)

it is evident that the interior

result of

Theorem

### 1.1,

is valid under the

weaker

restriction with

### q >_ (n + 1)/2

rather than q > n. One can

the interior

assertions of

### ,

Theorems 4. 8 and

and

Theorems 1 and

in the case that

vements

these lines are

even if

but we shall not

pursue these

### questions

here.

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

(12)

### (vi)

Minor modifications of the

of Lemma

in

and

### (2. 14),

lead to similar results for more

for

of the form

where

for all where

### (S~)

and cx >n is a constant.

### Arguing

as above we obtain

Further modifications

the lines of

are also

but we shall

not state these.

Since

### r E C 1,1,

the distance function d defined

### by

is of class 1 near r

say for x E

for

some

on r

Lemma

### 14.16). By replacing

Si 1 from Lemma 2. 2

### by

a smaller constant if necessary, we may assume that

03C31. We may extend p, the outer unit normal vectorfield to

to

E Q :

### d (x) 61 ~ by setting

The

u, ..., of u relative to r is defined

for all

of Q :

~

### Using

Lemmas 2.1 and 2. 2 we now obtain the

### following.

LEMMA 2 . 3. - There is a number

on n, such that

on

x E o :

El

where C

on n,

po and ~1.

From

Lemma

and its

to the case

u E

we have

for all

### n ~ x E SZ : d (x) E 1 ~,

where a > 0 is the

Lemma 2.1 and C

on n,

### Ro, r,

po and pi. The estimate

now follows

with

Lemma 2. 2.

### Using

Lemmas 2. 1 to 2.3 we can obtain a Holder estimate for the normal vectorfield to

u and a strict

estimate for

u

at any

u

with xo First of

we observe that

00 on r and the

### only supporting hyperplane

of

Vol. 8, n° 5-1991.

(13)

at a

u

### (xo))

E r x (~ is a vertical

### hyperplane tangent

to r x R.

Since M is a convex

### hypersurface,

it follows that the unit normal vector-

field to M is continuous at any

u

### (xo))

E M with xo E r. Thus the unit normal vectorfield to

u

has a continuous

extension

we also denote

to

x

### R),

and

LEMMA 2. 4. - There is a number y >

on n, such that

all

and x~03A9

where C

on n,

It

### clearly

suffices to prove

for

### xo = 0

and

for other xo it can be obtained

a similar

while the case

### is

trivial.

Choose coordinates so that the

xn axis

### points

in the direction of the inner normal to r at 0. Then v = ~, on

and in

### particular, v (o) = - en

where ..., en + 1 denote the unit coordinate vectors in

### ~n + 1.

then at x we have

for some

### by

virtue of Lemmas 2 . 2 and 2 . 3. In

we may take y =

+

### P), where P

is the constant of Lemma 2. 3.

if x is any

of

the same

### argument

as above shows that

is the

### point

of r nearest to x. It follows that

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

(14)

since v = ~. on

### r,

r is of class and

The lemma is

### proved.

Next we come to the strict

estimate.

LEMMA

### 2. 5. -

There are numbers

such that

T is the

tangent

to

u at

=

u

where xo

then

any

X =

u

E

x

we have

where II is the

onto T and

on n,

~o

### Proof. -

As usual it suffices to prove this for the other cases

obtained

a similar

### argument.

We suppose for convenience that

and let E2,

...,

denote various

constants

### depend- ing

on some or all of the

n,

~,o and ~,1.

### By

Lemma 2. 4 there is a number

such that

x

### IR)

can be written as the

### graph

of a function w

defined on a suitable subdomain U of T obtained

x

onto T. In

if we choose new

### coordinates ya

= xa for a n, yn = - xn + 1

+ 1= xn, then

Lemma 2. 4

we have

for some R > 0

on n,

~o and ~1, where

is the

of

onto

### T, and y’ = (yl, ... ,

We may also assume that

In

Lemma 2.1 we have

where a is the

### exponent given

in Lemma 2 . 1 and denotes the open ball in

### (~n -1

with radius R and centre 0.

We also

r x (~ near

as the

of a function

is

### independent

of yn, and since r E 1 is

### uniformly

convex we have

Vol. 8, n° 5-1991.

(15)

for

E

we also have

for all

D.

Let y =

E D and

be the

on the lower

of D such that

=

thus

### Using

Lemma 2. 1 we have

where and a is the

of Lemma 2.1.

then

from

and

we obtain

while

2

### C21 y~ ~°‘

we have

Thus in any case we have

for all A similar estimate for all

follows

the

of

w. The estimate

### (2 . 32)

with 8 = 2 À follows

from

so the

lemma is

Remark. - Holder

### continuity

estimates for v similar to

### (2 . 31 )

can also

be obtained if we allow some

to zero or

to

of K as

x

r. For

if we have

where

### K E Lq (Q)

for some q > n, and ~, > 0 and

### 8 E [o, 1)

are constants, then

### by , Corollary 3 .11,

we have

Annales de l’Institut Henri Poincaré - Analyse non linéaire

(16)

for all x, y eQ

where C

on n,

### Ro, r, ~,

and 8. This leads to the

estimate

On the other

as we have

the second

of

### (2. 23).

The combination of

and

### continuity

estimate for v of the form

### (2 . 31 )

with

y > 0 and C > 0 now

we have

Strict

### convexity

estimates such as

may also be

### proved

under these somewhat more

### general hypotheses.

Further refinements of these results

are also

for solutions of

### (1 . 8)

as well as for solutions of more

### general Monge-Ampere equations,

but we shall not pursue this here.

since the

### exponent

a obtained in Lemma 2.1 and its various

for other

is not

### generally sharp,

it is clear that the

### exponents

obtained in Lemmas

2.4 and

### 2. 5,

and conditions such as

are not

### optimal.

To conclude this section we make some further observations about the function w defined in the

### proof

of Lemma 2.5. First of

### all,

from

Lemma 2 . 4 and the fact that u E

we have w E

and

where

is the lower

of D. In

is

of yn, we have

### Furthermore,

it is clear that w is a convex solution of an

of the

form

where

### K

is obtained from K

the relation

Since

### (2. 34) holds,

there is no loss of

in

### assuming

that w in

fact satisfies an

of the form

x [R x

### [R") satisfying

Vol. 8, n° 5-1991.

(17)

for almost all

z,

x R x where

on ~,1 and

### 2 in ( 1. 10), ( 1. 11 )

and in addition on n and

### sup I Dw ~.

.

D

We see therefore that w is a solution of a free

and

the

### regularity

assertions of Theorem 1 are

to suitable

### regularity

assertions for the free

### boundary X

and for w near L. From Lemmas 2.4 and 2. 5 we see that for any

any

### y~D

we have the Holder

estimate

and the strict

estimate

where

and

on n and

co

on

~o and 1. In

in

we obtain

for all

and the

### convexity

of w we also obtain

for all

The free

### boundary problem

obtained above is similar to those studied

Kinderlehrer and

and in

### fact,

their results would

### imply higher regularity

of 03A3 and w near X if we

knew that

and

### WE C2 (D U L).

Of course, we do not know such

at this

but

the method of

### 

can still be used.

3. THE LEGENDRE TRANSFORM

To

further in the

of Theorem 1 we

a

used

Kinderlehrer and

and

the free

### boundary problem

for the function w obtained at the end of Section 2

its

### Legendre

transform. In our case

### however,

we may use the full

### Legendre

transform rather than a

transform as in

### .

Since is convex and w solves

the

y --~ Dw

defines a

### diffeomorphism

of D onto

an open subset D* of

In

since

is of class

1

and

convex with

to

### y’,

we may assume that

~ E

n

is a

of

onto

the esti-

mate

we

### easily

see that for any y° E E

and any E E

### (0, R/4]

we have

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

(18)

and

where zo = ~

= Dw

is as in

and

on n,

yo

the estimate

### (2 . 54)

we see that for any Yo E L

and any E E

we have

or

for all

### 6 > o,

such that a, we have

where yo

y is as in

and 60,

are

constants

on n,

~,o and In

the estimates

and

can be

### improved,

since w = Dw = on

is of class

convex with

to

The

### Legendre

transform of w is the function w* on

where

It is

verified that

and

where =

Thus

w* is

definite in

and we have

w*

w*

### (0)

= 0 and w* >_ 0 in D*

the

### convexity

of w and the fact that w

= 0. Since

holds

E

convex with

to

### y’,

we see that we have

and

for all

E ~* = Dw

c where

B are controlled

### positive

constants.

Vol. 8, n° 5-1991.

(19)

the relations

and

it is

verified that w*

solves the

x R x is

for any

E

x R x Thus

and

we have

for almost all

E

### D*

x R x for some controlled

constants

0, 1

### and 2.

It can be shown that the estimates

and

for w

### imply analogous

estimates for w* at each

of ~*. More

from

we see that

and

for

any

### sufficiently

small and any zeD* we have and

where y, b are as in

and

are controlled

### positive

constants. We shall not prove these estimates because

are not needed

to

the

The

lemma now

the

### proof

of Theorem l.

LEMMA 3 . 1-. - Let u E

### C2 (BR ) n (BR )

be a convex solution

the

where

and suppose that

x I~ x

almost all

z, f~ x

where

is a

constant.

### Suppose

also that

and

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

(20)

all

E

=

and

E

7~ are

### positive

constants. Then there is a number p E

on n,

~,

~1 ~, such that

any oc E

we have

where C

on the same

### quantities

as p and in addition on

### Proof -

If we

knew that the estimate

with C

### depending

in addition on , would be a consequence of

well known

### global

second derivative Holder estimates for

Theorem

Theorem

### 5.5.2).

So it suffices to prove

for suitable

constants ~ E

and C

on n,

### R,

. ~

Let where A > 0 is fixed so

### large

that 1 in . Choose

new coordinates for exn, and Then

can be

as the

### graph

of a convex function v

### defined

on a subset V of the

so we have

Thus the lower

of

is

### uniformly

convex and of class

virtue of

and

we

have

and

satisfies an

of the form

where satisfies

for almost all

z,

e U x [R x where

on n,

and

.

### BR

Our aim now’ is to prove that

for suitable

### positive

constants p and C. Once we have

follows

### immediately

and the lemma will be

A bound for

on au

### proved in ,- , , 

if we knew that v were of class

on

### U.

We do not know this

### yet,

so we need to make our estimates on a suitable

### approximat- ing sequence

of solutions. First we note that

Lemma

### 2. 2,

there is

Vol. 8, 5-1991.

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