fr Tu.vda= f H(x)dX- fs TU.vda.

A NOTE ON THE CAPILLARY PROBLEM

BY R O B E R T F I N N

Stanford University, Stanford, CA 94305, USA

The purpose of this note is to extend the result of Theorem 3 in the preceding paper [2] to a configuration not amenable to the methods of t h a t reference.

w

Consider an open set ~ in Euclidean n-space, bounded in part b y a surface Z of class C (2) whose mean curvature H r. does not change sign; the sign of H r. is chosen to be posi- tive when the curvature vector points into ~. As in [2], we use the symbols Y~ and ~ to denote also the area and volume of these sets.

Let u(x) be a solution in ~ of ( ) = n H ( x ) , 1

div v V u W ~= 1 + IVul ~, (1)

where H(x) is prescribed in ~, and set

Tu=~Vu

(2)

in ~. Denote by v the exterior directed unit normal on Z. We observe first t h a t i[ H(x) is bounded on one side, then/or any open subset Z * c ~,, the quantity S~* Tu .,~da is uniquely defined as a limit o/ integrals over sur/aces converging in any uni/orm way to Z * / r o m within

~. To see this, it suffices to suppose H > 0 and to show the result when E* is the part of lying interior to a (small) sphere S centered on Y~, so t h a t •* and a part S * c S together bound ~ * c ~. Integrating (1) b y parts in the subregion cut off by an approximating surface F*, and passing to the limit as F*-~Z*, we find

lim

fr Tu.vda= f H(x)dX- fs TU.vda.

I'*~r.* * * *

This work was supported in par~ by AF contract F44620-71-C-0031 and NSF grant GP 16115 at Stan- ford University.

2 0 0 ROBERT F I ~

Here t h e first t e r m on t h e right exists because H > 0, t h e second exists because I T u v I < 1;

thus, ~z* Tu,~da is defined b y t h e t e r m on t h e left.(1)

W i t h this definition in m i n d we shall prove t h e following result:

THEORV.M 1. Let u(x) be a bounded solution o/ (1) in )I. I / H r ~ ( n / ( n - 1))H0>~0and H(x) < g o in )1, then Sz* T u . v d a < Z* on a n y open subset ~* ~ ~. I / I H~I < ( n / i n - 1))H 0 and H(x) >t H o in )t, then ~ T u . v d a > - Z* on a n y open subset ~,* ~ ~,.

Remark. I n [2] it was assumed, chiefly for reasons of n o t a t i o n a l simplicity, t h a t T u v defines, as a limit f r o m within )1, a function cos y almost everywhere on Z. U n d e r this additional hypothesis T h e o r e m 1 states t h a t there is no b o u n d e d solution in )1 for which cos y = 1 (resp. cos y = - 1) on a n y open subset of Y,. I t is in this sense t h a t T h e o r e m 1 completes t h e result of T h e o r e m 3 in [2].

T h e o r e m 1 has a geometrical content, which one sees b y n o t i n g t h a t H(x) is t h e m e a n c u r v a t u r e of t h e surface $ defined b y u(x), t h a t ((n - 1 ) / n ) H ~ is t h e m e a n c u r v a t u r e of the (vertical) cylindrical wall Z over Z, a n d t h a t T u . v]~ is t h e cosine of t h e angle ~ between $ a n d Z at points of contact. Thus, u n d e r t h e given hypotheses, t h e t h e o r e m restricts t h e w a y s in which $ can (in a limiting sense) be t a n g e n t to Z.

T h e o r e m 1 is best possible in a n u m b e r of ways. I t is false for u n b o u n d e d solutions, as follows f r o m a c o n s t r u c t i o n due t o S p r u c k [5], which we shall also use in t h e proof. I t is false if H(x) > ((n - 1)/n)H ~ (respectively H(x) < ((n - 1)/n)HZ), as follows f r o m t h e example of w 2.3 in [2]. If H(x) < H 0 < ( ( n - 1)/n)H ~ (respectively H(x) > H ~ > ((n - 1)/n)]H~l), t h e n there even exists ~0, 0 < ~/0<~, so t h a t there is no solution u(x) in )1 for which limx-.z T u . v >/cos ~0 (respectively limx-~r. T u . v < cos ~/0), see T h e o r e m 3 in [2]. W e show later b y example t h a t in t h e class of all b o u n d e d solutions, no such ~0 can be f o u n d u n d e r t h e present hypotheses; nevertheless, we do show (Theorem 2) t h a t if lu(x)l < M in )1, t h e n there does exist a Y0, d e p e n d i n g on M, t h u s s t r e n g t h e n i n g T h e o r e m 1 for a n y class of e q u i - b o u n d e d solutions of (1).

T h e u n d e r l y i n g heuristic c o n t e n t of T h e o r e m 1 is t h a t if t w o surfaces $1 a n d S3 are t a n g e n t at a c o m m o n p o i n t p, if t h e m e a n c u r v a t u r e v e c t o r h2 of S~ at p is either directed oppositely t o t h e corresponding v e c t o r h I of S1 or satisfies ]h21 < ] h i ], t h e n S3 c a n n o t lie on t h e side of $1 into which h 1 points. This m u c h is geometrically evident. I f I hs[ = I hl I, t h e n n o t h i n g can be said unless i n f o r m a t i o n is k n o w n on these vectors in a n entire neigh- b o r h o o d of p; however, it is still easy to give conditions t h a t lend themselves t o analysis.

A related problem for which it is less e v i d e n t w h a t h a p p e n s appears when t h e surfaces (1) We note that since I Tu. v I < 1, there holds J'n* H(x)dx < c~ whenever a solution of (1)can be defined in )1".

A NOTE ON THE CAPILLARY PROBLEM 201 ha ve a manifold F of contact t h a t serves as a b o u n d a r y for one of t h e m (say S~) so t h a t

$2 is not known to be continuable across F as a surface satisfying the hypotheses. I t is this t y p e of situation t h a t we discuss here, for the special case in which $~ is a cylinder and $2 projects simply onto the base of $1-

The problem has appeared previously, incidental to other investigations; the result of Theorem 1 was given b y Jenkins and Serrin [3, L e m m a 3] for minimal surfaces, l a t e r - - with the same method of p r o o f - - b y Spruek [5, L e m m a 4.2] for surfaces of constant mean curvature, in the case n = 2 and under a hypothesis t h a t the intersection manifold F is continuous over the base plane.

The conceptually interesting new feature of the present note is t h a t no hypothesis at all is introduced on F; in fact, the surfaces are not explicitly required to contact. Thus, assuming only t h a t u(x) is a bounded solution of (1) in T/, and t h a t Z satisfies the (neces- sary) curvature inequality, one can infer information on the angle of contact (defined in a limiting sense) between the two surfaces. This is the natural setting for the capillary problem, which is the physical motivation for this note and for the two papers appearing with it. I n t h a t problem, a fluid surface is to be determined b y the condition t h a t it makes a prescribed angle with bounding walls; no reference to b o u n d a r y curves appears in the problem's formulation.

Before proving Theorem 1, we derive a preliminary result:

LEMMA 1. Let ~ and Z be as above, with Hr. ~ ( n / ( n - 1))H o. Suppose there is a point p E Z at which H r ' ( p ) > ( n / ( n - 1))H o. Then there exist~ (locally) a surface If of constant mean curvature H [ , such that p E 1 f, Hr. (p) > H [ > (n/(n - 1)) Ho, and If A ~ = r

Proof. We adopt a coordinate frame with p at the origin, so t h a t the x, axis is normal to Z at p and points into T/, and so t h a t the other coordinate axes coincide with the prin- cipal directions on Z.

Denote b y St ... ~n-t the principal curvatures of Z at p. Thus, Z~ -1 2 j = ( n - 1 ) H ~ (p).

For a n y H0 T in the indicated range, set e = H r . ( p ) - H o T > O , ~ / t = 2 t - e . Then ~t<~t,, i = 1, ..., n - 1, and Z~ 1 ~t ^{= ( n - -} 1)H0 T. We consider the initial value problem for the equa- tion

~ , = ( n - 1)Ho T W 2 = 1 + Iv~ol ~,

i = i z i

with initial d a t a

in--2

~(X 1 . . . X n - 2 , 0 ) = ~ 1~ ~tX 2

~ (xl . . . x,-2, 0 ) = 0

~Xn-1

14- 742909 Acta mathematica 132. Imprim5 lc 18 Juin 1974

(3)

(4)

2 0 2 ROBERT FINN

'/V

i I

Fig. 1

on the hyperplane x~_l=0. The existence of a solution of (3, 4) in a neighborhood of x = {xj} = 0 is assured b y the Cauehy-Kowaleski theorem [4].

At x = 0 there holds (O]Oxj)q) ~ 0 , ] = 1 ... n - 1, and (O~/(Ox~Oxj))~ = 0 if i ~ j . Thus, for j = 1 ... n - 2 , ~j is the jth principal curvature of the surface Y: ~0(x) at x = 0 . B u t b y (3), H0 Y is the mean curvature of Y. Hence ~ - 1 is the ( n - 1 ) t h principal curvature at x = 0;

in some neighborhood of x = 0, the surface Y satisfies the requirements of the lemma.

Proo/o/ Theorem 1. I t suffices to consider the ease H ~ ~>(n/(n- 1))H o >~ 0, as the other is analogous. We suppose the theorem false, and restrict attention to t h e given open sub- set, which we again denote b y •.

If H ~= ( n / ( n - 1 ) ) H o , we identify Y, with Y and select an a r b i t r a r y point p o n this surface. Otherwise there is a point pEY~ at w h i e h . t t ' ~ ( p } > ( n / ( n - 1 ) ) H o . By L e m m a 1, there is a surface Y through la and exterior to T/, such t h a t H f i s constant and satisfies H ~ (p) > H Y > ( n / ( n - 1))H 0. L e t St(l) ) be a sphere of radius :e centered at p.and bounding, with p a r t of Y, a region ~ * with ~/* =7/~* N T/4=r (see Fig. 1). We denote the two parts of the b o u n d a r y of ~/* b y Z * ~ Z and b y F * ~ St(1)), and define u ( p ) = lira supx-,p u(x), xe~/.

We use in a basic w a y a theorem of Spruek [5, w 9]: i / e is su//iciently small, there exists a solution v(x) o/

( 1 ) W~= + , V v , ~ (5)

div ~ V v = - ( n - 1 ) H Y, 1 in ~ * , such that lim~-.r, v(x) = 0 and lim~_, y v(x) = + oo.

A N O T E ON T H E C A P I L L A R Y P R O B L E M 203 W e p r o c e e d in several steps:

(i) L e t ~ = v - c , where c is chosen so t h a t on F*, ~ <

glbx~N*

(ii) L e t ~$ ( x ) = ~ ( x + ~ 5 ) , where ~ is t h e u n i t exterior n o r m a l t o Z a t p, a n d choose sufficiently small t h a t t h e r e still holds ~$ (p) > u(p). ~$ (x) is defined in a region ~ o b t a i n e d b y displacing TI* a distance ~ along ~-

(iii) T h e f u n c t i o n w(x) =~$ (x) - u ( x ) is defined in ~ ~ ~*. D e n o t e b y ~0 t h e open com- p o n e n t in ~ h a v i n g p as a b o u n d a r y point, a n d in which w > 0 . Tl ~ is b o u n d e d b y Z ~ Z*

a n d b y a closed set F ~ ~ N ~* (see Fig. 1).

(iv) W e n o w consider t h e f o r m a l i d e n t i t y

f ~~ [div (w V~) - div (w VU) ] dx = f z,w[T~ - Tu] " ~*da

1 A 1

H e r e t h e integral o v e r ~0 is to be u n d e r s t o o d as a limit of integrals o v e r surfaces a p p r o x , i m a t i n g ~'.0 f r o m w i t h i n ~0, a n d t h e integral o v e r F ~ does n o t a p p e a r since

w=O

set,(1)

F r o m t h e construction of w(x), we find t h a t t h e i n t e g r a l on t h e left in (6) is non-nega- tive. T h e i n t e g r a n d in t h e o t h e r integral o v e r 7//~ can he e x p r e s s e d as a positive definite q u a d r a t i c f o r m in t h e c o m p o n e n t s of Vw; t h u s this integral is positive.

T o s t u d y t h e integral o v e r Z ~ we choose as a p p r o x i m a t i n g surfaces a f a m i l y Z ~ t h a t , in suitable local p a r a m e t e r s , t e n d s t o Z ~ t o g e t h e r w i t h t h e first order d e r i v a t i v e s of t h e position vector. T h e convergence will t h e n also be in area, a n d we find, since [Tu] < 1, t h a t ff t h e e l e m e n t

danc ~o

da~ Z ~

~iTu"~d(~n=~r~o(TU"~)z~da+sn

where ~n-~ 0 as n-~ ~ .

B y definition, limn_,~r ~z~Tu 9 ~ da~ = ~z~ ~ da, a n d since b y h y p o t h e s i s ~r.,Tu.

"~da

Z* a n d ~ o ~ ~ , , t h e r e follows~ using again [Tu I < 1, ~z* T u . v d a = Z ~ Thus,

lim ~

(Tu.'~)~oda=Z ~

W e h a v e b y the a b o v e construction t h a t on Z ~ [T~] ~< cos ~0 < 1 for all sufficiently large n. L e t #n be t h e m e a s u r e of t h e set on Z ~ on which ~(Tu.~)r~<cos~0. T h e i n e q u a l i t y (') The difficulties arising from possible irregularities of F ~ can be overcome by a simple approxi- mation procedure, based on the fact that I ~~ is a level set of w(x); cf. footnote (2) in [1].

204 ROBERT FINN

]Tu] < 1 now implies, with (7), that/~n-~ 0. I t follows then from 0 < w < M in ~0 t h a t

~ 0w[T~s - T u ] . ,~da< (2+ e) M/xn I :

and finally, from (6)

f *

0 < | [ T ~ - Tu]. Vw dx ~< 0.

Jn ^e This contradiction establishes the theorem.

w

The above proof yields somewhat more information than is stated in Theorem 1. To achieve the contradiction it suffices to have

c o s ~ , o = m a x T ~ . v ~ < inf 1. ~ T u . ~ d a ,

(the integral on the right being understood, as before, in a limiting sense), so t h a t the theorem will be improved as soon as a uniform bound on the left-hand term can be ob- tained. Clearly this bound depends, for given Z, only on e and on the distance 5 t h a t ~/*

can be displaced while maintaining the inequality ~ ( p ) > u ( p ) . ~ can be chosen depending only on the geometry, t h a t is, to permit the Spruck construction in a region 7~* of the desired form. Assuming the geometry fixed, the maximum permissible 5 depends only on the bound for ] u(x) ] in T/. This remark proves one of the inequalities of the following re- sult, the other one being obtainable, as before, analogously.

THEOREM

/ying lu(x)l < M there holds cOS$o~ inf~.c~(1/Z*) S~.Tu.vda. I / I H ~ I < ( n / ( n - 1 ) ) H 0 a n d H(x)>~H 0 in 711, then there exists 71,

0<71<Yt,

] U(X) ] < M , then cos ~1 ~< s u p E , c z (1/Z*) S~. T u . v da.

Remark. If, as is done in [2], it is assumed t h a t T u - v defines, as a limit from within T/, a function cos ~ almost everywhere on Z, then the result of the theorem becomes t h a t there cannot hold cos y ~> cos ~0 (resp. cos ~ ~< cos ~1) on Z.

w

The restriction ]u(x)l < M of Theorem 2 is necessary. This can be seen by considera- tion of the properties of the Spruck surface; alternatively, we present here a more ele- m e n t a r y example.

A N O T E ON T H E C A P I L L A R Y P R O B L E M 205 L e t ~ be t h e disk x 2 + y 2 < l , a n d Z t h e arc x 2 + y 2 = l , y > 0 . T h e cylindrical surface Z in E u c l i d e a n 3-space, t h a t projects o n t o Z, has m e a n c u r v a t u r e ~ - 1 _ ~ - ~ - ~ 1 ~ . If Z is r o t a t e d a (small) angle a a b o u t t h e x-axis, in such a w a y t h a t points on t h e positive y-axis are t u r n e d a w a y f r o m t h e positive z-axis, we o b t a i n a surface Z~: u~(c, y) defined over a n d satisfying t h e e q u a t i o n

1

L e t pa be a point on Z ~ N Z. pa lies at t h e intersection of a g e n e r a t o r on Z t h r o u g h Z, a n d of t h e image g e n e r a t o r on Z ~. I t follows t h a t t h e angle y~ between Z a n d Z ~ at p~

satisfies 0 < ~ ~< ~, u n i f o r m l y on Z, a n d hence lim~_,0 ~ . Tu~](x.y) = 1 u n i f o r m l y for (x, y) E Z.

The surface Z~: u~(x, y) is however defined a n d b o u n d e d in N (although of course n o t equi- b o u n d e d in a) for all a > 0.

w

W e r e m a r k finally t h a t t h e sense of T h e o r e m s 1 a n d 2 is to give conditions u n d e r which I Tu~[ c a n n o t be close t o u n i t y on a b o u n d a r y set of significant size. T h a t t h e size of t h e b o u n d a r y set m u s t enter into t h e result is easily seen f r o m t h e example at t h e e n d of w 3.3 in [2], which shows t h a t u n d e r t h e h y p o t h e s e s of t h e t h e o r e m s a n y value of T u . v i n t h e closed interval [ - 1, 1] can be o b t a i n e d at an isolated b o u n d a r y point.

R e f e r e n c e s

[1]. CoNcus, P. & FINN, R., On a class of capillary surfaces, J. Analyse Math., 23 (1970), 65-70.

[2]. - - On capillary free surfaces in the absence of gravity. Acta Math., 132 (1974), 177-198.

[3]. JE~-xI~s, H. & SERRIN, J., Variational problems of minimal surface type, I I : boundary value problems for the minimal surface equation. Arch. Rational Mech. Anal., 21 (1965-6), 321-342.

[4]. PETROVSKY, I. G., Lectures on Partial Di]]erential Equations (translated from the Russian).

Interseience Press, :New York, 1954.

[5]. SPRUCK, J., Infinite boundary value problems for surfaces of constant mean curvature.

Arch. Rational Mech. Anal., 49 (1972-3), 1-31.

Received July 17, 1972