# Global regularity of the solutions to the capillarity problem

(1)

## S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze

e

o

### 1 (1976), p. 157-175

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to

CLAUS

### GERHARDT (**)

0. - Introduction.

Let S~ be a bounded domain of

with smooth

### boundary aS2,

and let A be the minimal surface

a

solution of the

### capillarity problem

can be looked at as a

solution U E of the

in

to the

conditions

on

where

are

and

### y = (yl, ... , yn)

is the exterior normal vector to 8Q.

and URAL’CEVA

solved this

### question partially:

In the case n = 2

### Spruck

could show the existence of a solution C

### provided

that aS2 is of class

### 04, fJ belongs

to such that 0 8 1 and and

### provided

that H has the form

methods are

two-dimensional.

the

### preparation

of this article the author was at the Université de Paris VI as fellow of the Deutsche

### (**)

Mathematisches Institut, der Universitat

### Heidelberg.

Pervenuto alla Redazione il 30 Novembre 1974 e in forma defini tiva il 18

1975.

Here and in the

we sum over

### repeated

indices from 1 to n.

(3)

A different

### by

Ural’ceva which will be valid for

dimension. She

### proved

the existence of a solution u E

under the

### assumptions

where H satisfies

and where S~ is

to be convex

is constant.

The last

are rather

### restrictive,

and it is the aim of this paper to exclude these restrictions

### by

a suitable modification of Ural’ceva’s

In the second

### apply

this result to the

### capil- larity problem

with constant volume-which is an obstacle

### problem-and

we shall show that this

### problem

has a for any p &#x3E; n.

Since the paper of Ural’ceva is written in Russian we shall

many

### proofs

of that paper almost

### literally

for the convenience of the reader.

1. - A

estimates

### for

In this section we shall assume that is a solution to the dif- ferential

### equation (0.2), (0.3). Furthermore,

let us suppose that 8Q is of class

### C2,

and that the functions

and

the conditions

and

the

### following

theorem is valid.

THEOREM 1.1. Under the

### assumptions

stated above the modulus

the

u can be

a constant

on

n, a, and on the

constant

(4)

PROOF. First of

let us extend

### fl

and y into the whole domain D such

### that fl belonging

to still satisfies

### (1.3),

and such that the vector

is

### uniformly Lipschitz

continuous in S~ and

bounded

### by

1.

These extensions are

### possible

in view of the smoothness of aS2.

Ural’ceva’s

we are

### going

to prove that the function

is

bounded in S~

### by

some constant which

on the

we have

### just

mentioned in Theorem 1.1.

### Precisely,

we shall show

that v is bounded

near aS2. The

### global

estimate then follows from well-known interior

bounds.

In order to prove the main result we need some lemmata which will be derived in the

We denote

the

of u

and

### by ~ _ (~1, ..., ~~+~)

the usual differential

on

i.e. for

we have

where v =

### (v1,

..., is the exterior normal vector to 8

Then the

Sobolev

### Imbedding

Lemma is valid:

LEMMA 1.1. For any

g E the

where

### Jen

is the n-dimensional

### Hausdorff

measure, and where the con- stant

on n and

### IH(x, u(x)) In.

PROOF OF LEMMA 1.1. This Sobolev

for functions

### equal

to

zero on all of aS2 was established in

for solutions of

### (0.2). Here,

we shall

not assume that g is

to zero on 8Q.

(5)

Denote

= dist

### (x, 8Q),

the distance function to

and let

for kEN.

Let be

Then

has

values

### equal

to zero, so that in view of the result in

in-

### equality (1.8)

is valid for gk .

If k goes to

the

to the

g, while

is estimated

The last

converges to

### (cf. [5; Appendix III]),

hence the result.

Next we need to technical lemmata. Let us denote

### by aii

then we have

LEMMA 1.2. On the

S~ we have the

### following

estimate

where the constant C2

on 2S~ and the

constant

### of fl.

(6)

PROOF OF LEMMA 1.2. Let zo be an

### arbitrary boundary point

and let us

introduce new coordinates y =

### y(x)

which are related to x

an

### orthogonal

transformation such that the

is directed

### along

the exterior normal vector at zo :

that in a

of zo the sur-

face 8Q is

### specified by

If we now differentiate the

with

to the

### operator

then we obtain at x,

since we deduce

### Thus,

we have in view of

### (0.3)

and hence the relation

### (1.19)

is also true for s = n.

On the other

### hand, combining (1.19)

and

we derive that the left side of

is bounded at zo

### by

which the assertion is

### proved.

LEMMA 1.3. In the whole domain Q the

### following pointwise

estimate is valid

where the constant

on

### IDyln,

and on a.

11 - Annali della Scuola Norm. Sup. di Pisa

(7)

PROOF OF LEMMA 1.3.

the

### proof

we have to work in the

### (n + 1) -

dimensional Euclidean space rather than in the n-dimensional one; there- fore we

a

= ...,

as

### being

defined in via the

### mapping x -* (x, 0).

Let us introduce the notation 0 for the Euclidean

norm in Rn+l

We shall consider the Hessian matrix of

evaluated at

qo =

where Xo is an

but fixed

in Q. Let

...,

be the

### eigenvectors

of that matrix and ..., be the cor-

is itself an

which is

### just

the exterior normal vector of the surface 8 at the

that the

### eigenvectors

are numbered in such a way that Zn+l =

we have 0.

the

z; ,

are

to

and we

derive

if we denote

the scalar

of the

### corresponding vectors,

where the indices run from 1 to it

### + 1,

then we obtain

To estimate

at xo, y let us observe that we may sum from 1 to n

in this

since u does not

on xn+1.

### lBforeover,

as 11. is the trace of a

### product

of matrices it is invariant under

### orthogonal

transformations of the co-

ordinate

we

in mind that

0.

From

### (1.28)

we conclude in view of

where

(8)

On the other

### hand,

we have

where now we also sum over the

### repeated indices i, j,

and k from 1 to n

### Furthermore,

I to estimate for

### ~~-)-1

we observe that for any

which does not

on there holds

and

hence

### Finally,

y if differentiate v with

### respect

to zs for s -=1= n

we obtain

where the

and-in

### Taking

the estimate

(9)

with some suitable constant c4 into

### account,

we thus deduce from

and

the relations

and

we then conclude

### Hence,

there exists a constant e3

on a,

and known

### quantities

such that in view of

### (1.29)

from which the assertion

follows.

As we are

### treating

the case of a non-convex domain and a

we

need the

### following

estimate

LEMMA 1.4. For any

### positive

E we have the estimate

where the constant

on and

### IH(x,

PROOF OF LEMMA 1.4. Let for i

..., n, we have

we deduce the

(10)

_ ~ ~ yi

this

and

### summing

over i from 1 to n

### yields

hence the result.

to now we have

which we shall

need for

certain

### expressions

that will appear in the

### following

calculations. As we mentioned at the

we are

to show

or

is

### uniformly

bounded in S~. To

### accomplish this,

let us look at the

If we

= 0

and where

h is

### large,

then we obtain in view of

and

that

we deduce from

### (1.47)

in view of the Lemmata 1.2 and

### 1.3,

and in view of the

### assumption (1.3)

where C6 is a suitable constant

any

### positive

Cl-function.

We shall use this relation with 27 =

where h is a

number

### and ~, 0~ly

a smooth function.

(11)

the notations

the relations

and

into

we derive

### Moreover,

from the Lemmata 1.1 and

and from

we conclude

the Holder

(12)

and

small

### enough

we deduce

from which we derive

a well-known

we conclude

### Now,

the boundedness of

follows

that

is

where

E Q:

&#x3E;

and

is

p.

As a first

we prove

LEMMA 1.5.

that the

Theorem 1.1 are

### satisfied.

Then we have

PROOF OF LEMMA 1.5. We insert in the

### inequal- ity (1.49)

and conclude with the

of the relations

### (1.51)-(1.54)

Thus it remains to prove

sa v dx or

n is bounded.

To

we consider the

(13)

### Choosing q

= u the result follows from the

Lemma

After

### having

established the estimate

### (1.63)

we use the relation

### (1.65)

once more, this time with q =

### 0)

and we obtain in view of the

### Hence,

we deduce

where we used the

To

the

### proof

of the boundedness of the

### integral (1.62)

we ob-

serve that

for some suitable constant a.

we have

a suitable

bounded in U.

### Together

with well-known interior

estimates

this

the

### proof

of Theorem 1.1.

2. - Existence of a solution u.

In view of the a

### priori

estimates which we have

### just

established the existence of a solution will be

a

method.

THEOREM 2.1.

that the

S~ is

class

and that H

their

that H

(14)

Then the

value

has a

solution u E

where the

a, 0 a C

is determined

the above

### quantities.

PROOF. Let z be a real number with

and consider the

value

Let T be the set

T is

not

for uo = 0

to

### it,

and we shall show that it is both open and closed.

In view of the

we obtain an a

bound of

for any 7: E T

of i

### (cf. ). Furthermore,

let us remark

that any solution uZ E

is of class

### C2~°‘(SZ)

with some fixed a, 0 ex

### 1,

such that the norm of uz in

is bounded

of 7:.

To prove

### this,

we first deduce from Theorem 1.1 that

is

bounded

### Then,

we choose a smooth vector field

such that

is

and such that

From

Theorem

### 2.2]

we conclude that the

has a solution E

### C‘~°‘(,S2)

for any 7:. Moreover in view of

and

we derive

(15)

### Hence,

we obtain from the

of the solution

we can

conclude that is

### uniformly

bounded

where the constant is determined

known

### quantities.

From the estimate

it follows

### immediately

that T is closed.

On the other

let To E T.

we consider the

value

as before. Since

### IDuTI.a depends continuously

on Ty it turns out that

But this

### yields ict

= u-r for those i’s.

the

### proof

of Theorem 2.1 is

### completed.

For our considerations in the next section it will be necessary to bound the norm of u in the function space

p &#x3E; n,

a constant which

on

### luln,

p, n, the 02-norm of

and on the

LI-norm of

### H(x, u(x)).

THEOREM 2.2. Under the

### assumptions of

Theorem 2.1 the norm

u in

n p is bounded

a constant

determined

the

### quantities

mentioned above.

PROOF. Since in the interior this result follows from the well-known

we have

### only

to prove it near the

Let r be a

of the

### boundary

and suppose that an open subset Q*

of Sz

to T is transformed into some open subset G of the

via a

such that

y,n =

In G the

assumes the form

and the

condition

### (0.3)

is transformed into

(16)

We are

### going

to prove that the norm of

= in

can

be estimated

the

mentioned in the

### theorem,

where we assume

that u - and hence i1- is of class C2.

It will be sufficient to bound the LV-norm of where k ranges from 1 to n and r from 1 to n - 1. The estimate for then follows from the

### equation.

We

know from the results of

p.

### 468]

that the norm of i1 in

### H 2,2 (G)

n with some suitable a can be estimated

be

functions in

### CI(O) vanishing

on aG -

Then we obtain from

and

in this

and

### Thus,

we deduce

where we have set

and

is the

is any function

to

on aG -

The constant

### H3 depends

on the quan- tities mentioned in the theorem.

is

### arbitrary,

which vanishes on

Then the

### inequality (2.17)

is satisfied so that we obtain

with some constant

on

and

we

### finally

deduce

(17)

Since the bilinear form

is coercive and

### non-degenerate,

and since the coefficients

are continuous

we conclude from

Theorem

that

### belongs

to and

that its norm can be estimated

a constant

on the

constant of the

### bkl7s,

and on the modulus of

### continuity

of the coefficients.

3. - The

### capillarity problem

with constant volume.

In a former paper

### 

we considered the variational

### problem

We could prove that this

### problem

has a solution r1 Leo

### (Q),

and u

also minimizes the functional

in the convex set where A is a suitable

multi-

to assume

and

and

we shall

### give

sufficient conditions which

### imply

that the variational

has a

### (unique)

solution for any finite p. It will be

### important

to remark that .H need not be

monotone in t.

we deduce from

that in the

### following

we may consider the variational

(18)

where H is

to

the

we shall

### obviously

assume that the conditions of Theorem 1.1 are

and

to

### H 2,, (Q).

But we still have to

### impose

a further conditions on y which ensures, that a solution U C of

satisfies the

condition

which is

### absolutely

necessary in order to obtain a

### priori

estimates for the

we suppose that the relation

is valid on aS2.

we have the

### following

result

THEOREM 3.1. Under the above

the variational

has a

any

p, which is

### uniquely

determined in that

### function

class.

PROOF. Let 8 be the maximal monotone

and let

### 6e

be a sequence of smooth monotone

to 6 in such a

way that

be a

### positive

constant such that

we consider the

value

### problems

We shall show that the a

### priori

estimates of Section 1 are still valid in this case, where the estimate

on and on known

### 1)

The Lemmata 1.1-1.5 are still

### valid,

but the constants

on it.

(19)

The

### only difficulty

arises in the estimate

since we

### apply

this estimate with

### ~==~’max{~~2013~0}y

we deduce that for the critical term

in

is

### positive

and can therefore be

where

on

and a.

### Hence,

we obtain

where the constant is

that is

### uniformly

bounded. But this follows from the strict

of H.

### Thus,

we deduce from Theorem 2.3 that for any

### uniformly bounded,

where we may assume without loss of

that

### H, and satisfy

the further smoothness conditions of Theorem 2.2.

To

the

of Theorem

### 2.1,

we shall show that the relation

is valid in Q.

But this estimate is an immediate consequence of the

and

set 1pe = 1p - ê

=

we

deduce from

we

### obtain q m

0 in view of the strict

### monotonicity

of H.

In the limit case a

of the

converges

to some

function U

(20)

ana

But this is an

### equivalent

formulation of the variational

### problem (3.5),

if

we restrict the variations to the convex set r1

as one

checks.

After

finished the

### present

article the author became

### acquainted

with a paper of Simon and

who

similar results.

REFERENCES

### 

E. BOMBIERI - E. GIUSTI, Local estimates

the

### gradient of non-parametric surfaces of prescribed

mean curvature, Comm. Pure

### Appl.

Math., 26 (1973),

pp. 381-394.

 P. CONCUS - R. FINN, On

in a

### gravitational field,

Acta Math., 132 (1974), pp. 207-223.

 C.

Existence,

and

behaviour

### of generalized surfaces of prescribed

mean curvature, Math. Z., 139

### (1974),

pp. 173-198.

 C. GERHARDT, On the

### capillarity problem

with constant volume, Ann. Sc. Norm.

### Sup.

Pisa, S. IV, 2 (1975), pp. 303-320.

C. GERHARDT,

### Hypersurfaces of prescribed

mean curvature over obstacles, Math.

Z., 133 (1973), pp. 169-185.

 C. GERHARDT, Existence and

### regularity of capillary surfaces,

Boll. U.M.I.,

10 (1974), pp. 317-335.

 O. A. LADYZHENSKAYA - N. N. URAL’CEVA, Linear and

### quasilinear elliptic equations,

 O. A. LADYZHENSKAYA - N. N. URAL’CEVA, Local estimates

solutions

and

Comm. Pure

### Appl.

Math., 23 (1969), pp. 677-703.

 J. H. MICHAEL - L. M. SIMON, Sobolev and mean-value

on

Rn, Comm. Pure

### Appl.

Math., 26 (1973), pp. 361-379.

M.

### SCHECHTER,

On Lp estimates and

### regularity,

I, Am. J. Math., 85 (1963),

pp. 1-13.

 L. M. SIMON - J. SPRUCK, Existence and

a

### capillary surface

with prescribed contact

### angle,

to appear.

 J. SPRUCK, On the existence

a

with

contact

Comm. Pure

### Appl.

Math., to appear.

### 

N. S. TRUDINGER, Gradient estimates and mean curvature, Math. Z., 131 (1973),

pp. 165-175.

 N. N. URAL’CEVA, Nonlinear

value

minimal

### surface

type, Proc. Steklov Inst. Math., 116

### (1971),

pp. 227-237.

 N. N. URAL’CEVA, The

the

### capillarity problem,

Vestnik

Univ. No. 19 Mat. Meh. Astronom.

### Vyp.,

4 (1973), pp. 54-64, Russian.

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