ON CAPILLARY FREE SURFACES IN A GRAVITATIONAL FIELD
BY a n d PAUL CONCUS
Lawrence Berkeley Laboratory Berkeley, CA 94720, USA
ROBERT F I N N Stanford Unlversity Stanford, CA 94305, USA
W e p r e s e n t here t h e second p a r t of our s t u d y of t h e e q u a t i o n div
-~ Vu ~,~1 ~x~ \ W OxJ
for a scalar function u(x) over a n n-dimensional d o m a i n ~ with b o u n d i n g surface ~]. F o r i n f o r m a t i o n ' on physical b a c k g r o u n d a n d smoothness h y p o t h e s e s we refer t h e r e a d e r t o t h e I n t r o d u c t i o n in [7], where we s t u d y t h e case ~ i n d e p e n d e n t of u. T h e interest for t h e present work, in which ~ is p e r m i t t e d t o d e p e n d on u explicitly in certain ways, centers on t h e capillary e q u a t i o n
N u ~ div ~ V u = ~u (2)
where u # 0 is a constant, u n d e r a b o u n d a r y condition
T u . v - ~ ~. V u = cos 7 1 (3)
where v is u n i t exterior n o r m a l on Z a n d 7 is prescribed (see [7]). However, we shall dis- cuss considerably more general situations t o which our m e t h o d s apply.
w
W e impose on ~d(x; u) in (1) a single requirement:
A : F o r a n y 5 > 0 there exists M8 < ~ , such t h a t at least one of t h e conditions
AI: {~<~-1~ u<M,}
A~:
{~>~ - ~ - 1 ~ ~>-M~}
This work was Supported in p a r t b y t h e US Atomic E n e r g y Commission, a n d in p a r t b y A F contract F44620-71-C-0031 a n d bISF g r a n t GP 16115 at Stanford University.
208 P A U L C O N C U S A N D R O B E R T ~ N N
holds for all x E ~ , t h a t is, limu~r :H(x; u ) = ~ uniformly for x E ~ in the case A~, with ana- logous expressions in the other cases.
Remark. The capillary equation (2) is contained in A~ and A , if x >0, and in A a and A, if x < 0 .
T ~ O R E M 1. Suppose A 1 (resp. A~) is satisfied by ~(x; u), and let u(x) be a solution o]
(1) in an n-ball B~ o/radius 6. Then u(x) < M s + 6 (resp. u(x) > - M s - 6 ) ]or all x E B~.
COROLLARY: I[ there is a [ixed 6 > 0 such that ~ can be covered by an interior/amily {B~}, then the conclusion holds uniformly in ~ , for any solution u(x) over ~. I] ~ is arbitrary and r is distance to Z, there still holds u(x) < M r (resp. u(x) > -M~).
Proo/ o/ Theorem 1. Suppose first t h a t A 1 holds; let xEB]. Choose 6', 0<(Y <~, and let S~,, be an n-sphere (boundary of an (n + 1)-ball) of radius 6' whose center (x 0, %) lies o n t h e vertical through the center (xo, 0) of B]. If u o is sufficiently large, S~. will lie above the solution surface S: u =u(x). Let u0 be the largest value of u 0 for which S~. contacts S, and let Pl = (xl, ul) be a point of contact. Then Pl lies on the lower hemisphere of S~, (it cannot lie on the equatorial sphere since ] Vu [ < ~ at these points), and S~, shares with $ a com- mon normal at Pl. None of the normal curvatures (at Pl) of curves on S through Pl (con- sidered as positive when the curvature vector is directed into S~,) can exceed 116', for otherwise there would be points of S interior to S~., contrary to the construction. I t fol- lows t h a t the mean curvature of S at p~, as defined by the left side of (1), cannot exceed 1/6'. Thus ~(xl, ul)<l/~' , from which, b y At, Ul<~M,~. Hence u ( x ) < M , . + 6 ' in B~,. The proof is completed by letting 6'-+6.
Similarly, if A~ holds, one finds u(x) > - (Ms +6).
R~mar~8.
(i) A somewhat stronger (geometrical) theorem could have been stated. I t would have sufficed to know t h a t u ( x ) < M z (resp. u ( x ) > - M ~ ) whenever the maximum ~ of the principal normal curvatures of S satisfies x~ < 6 -1 (resp. x~ > --6-1).
(if) For the capillary equation n ~ ( x ; u)-~xu, x > 0 , we m a y take Ms=n](x6) for the case A 1 or A 2. Thus, in this case there holds }u] < n ] ( x 6 ) + 6 for any solution in B~.
(iii) We note t h a t the corollary holds without explicit hypotheses on the boundary Z of ~ , and t h a t no hypothesis is introduced on the boundary behavior of u(x). Under hypotheses A 1 and A2, it provides an a priori bound on [u(x)[ in any compact subdomain of an arbitrary ~ .
ON C A P I L L A R Y F R E E S U R F A C E S Il~ A G R A V I T A T I O N A L F I E L D 2 0 9
(iv) I f ~u ~> 0 (as in the capillary equation with u > 0 ) the result can be improved somewhat b y using as comparison surface a rotationally symmetric solution ~0(r; (~) of the equation, determined b y the condition %(~; 5) = ~ . I t is not hard to show the existence of such solutions (cf. w 2) and to estimate t h e m from above. The result u < ~(r; (~) on I x I = r then follows from the general comparison principle of w 3.6. I n the case of the capillary equation the i m p r o v e m e n t obtainable in this w a y has the order 0((~) as 5->0.
(v) I t should be noted t h a t the proof of Theorem 1 does not use the m a x i m u m prin- ciple, a n d the result holds for m a n y equations for which the m a x i m u m principle does not apply, either for a solution of (1) or for a difference of solutions. I t is this fact t h a t sug- gests t h a t a corresponding result be sought for the case of the capillary equation (2) with reversed gravitational field, u < 0. The best result of this sort we can offer is the following:
T~IEOREM 2. Suppose A a (resp. A4) is satis/ied by ~(x; u). Then i/ u(x) i8 a solution o/
(1) in a ball B~, there is a point xEB~ /or which u ( x ) ~ - M S (resp. u(x)~<Ms).
The proof is analogous to t h a t of Theorem 1, the point x being in each case the pro- jection onto the base hyperplane of a point of last contact of $ with S~. The method yields, however, no global bound throughout the domain of definition, even if (as in the capillary equation with u < 0 ) A s and A 4 hold simultaneously.
w
The extent to which a global estimate can be obtained under hypotheses such as A 3 or A 4 remains open. We have stated such estimates in w 11 of [5] a n d in w 3 of [6]; although there is evidence t h a t the statements given there have physical meaning, we have since found t h a t our demonstrations of these results are in the full generality indicated incom- plete, iThe following remarks bear on this point, a n d show t h a t a t least for n = 2, the re- sult o/Theorem 1 still holds, with Mo = n/I ~ [ 5, /or the rotationally symmetric solutions o/ (2) with ~ < O.
2.1. Consider solutions of
div ~ V u = - u 1 (4)
in n = 2 variables, t h a t are rotationally symmetric about a vertical axis. Such solutions are functions u(r) of a single variable, and satisfy the equation
-- r~--~ \ V ~ / - u . (5)
210 PAUL CONCUS AND ROBERT F I N N
We s t u d y solutions u(r) of (5) such t h a t u ( 0 ) = u 0 < 0 , Ur(0)= 0. The local existence of such a solution can be proved b y the method of Lohnstein [1 I, 12] or of J o h n s o n a n d Perko [10], although this case does not seem to have been explicitly studied in those papers.(1) We summarize here the global behavior of u(r), in its functional dependence on %. Some of these features were already known to Bashforth and Adams [2] and to W. Thomson [13], although perhaps not in m a t h e m a t i c a l rigor. We state here only the results we have est- ablished; complete details will appear in a later paper.
2.2. I], i n the initial value problem o] 2.1, u0* < 1/3, then the solution u(r) exists ]or all positive r. I t has a n i n l i n i t y o] zeros. F o r a n y two successive extrema ra, r~ o] u(r) there holds
lu(r0) l < lu(ro) l.
A s y m p t o t i c a l l y as u o ~ 0 the ]irst zero r~ is the first zero o / J o ( r ) , r 1 "" 2.405.2.3. A s u o decreases ]ram 0 to - co, there is a ]irst value u o =Uol , such that the correspond- ing ]unction u(r) cannot be continued ]or all r as a solution ol (5). The continuation is possible only i n an interval 0 < r < r ~1~, and lira u ' ( r ) = co. There holds -3-1/4 >u01 > -35i4; the
r.~r(1)
p o i n t o / i n / i n i t e slope on the solution curve is precisely the second p o i n t o/intersection of this curve with the hyperbola r u = - 1 (see Fig. 1). (We note t h a t numerical computations [4]
yield the value u01 ~ - 2 . 6 . )
2.4. H % = % 1 , the solution can be continued indefinitely as a solution of the para- metric system
d~ 1
ds u r sin v 2
du d~ = sin ~o (6)
dr
= c o s ~.
Here s i s a r c length on the solution curve, ~p is the angle measured counterclockwise from the positively directed r-axis to the t a n g e n t line. There holds everywhere - (~t/2) < ~ ~< (~/2);
the function u(s(r)) is a solution of (5) except at the single point r = r (1~.
2.5. As u 0 decreases past u01, the solution continues to exist everywhere as a para- metric solution of (6); however, the point (r cx~, u u)) moves below the hyperbola ru = - 1, and the solution continues, with decreasing r, to a second point (r ~2~, u ~2~) of infinite slope, lying above the hyperbola (see Fig. 2)(~). On this branch, u(s(r)) is a solution of the equation
(1) W e h a v e b e e n u n a b l e t o o b t a i n a c o p y of L o h n s t e i n ' s d i s s e r t a t i o n , a n d w e h a v e h a d to infer its c o n t e n t f r o m his l a t e r p a p e r s a n d f r o m t h e general r e p o r t b y B a k k e r [1].
(2) T h i s s t e p in t h e discussion is b a s e d p a r t l y o n n u m e r i c a l c o m p u t a t i o n ; we h a v e n o t y e t establish- ed t h e r e s u l t f o r m a l l y in t h e full s t r e n g t h i n d i c a t e d here.
O N C A P I L L A R Y F R E E S U R F A C E S I N A G R A V I T A T I O N A L F I E L D 211
--- ~ ' U = - - 1
Fig. 1
~ J
- - - r u = -- 1 (1)uo= -- 6
Fig. 2 Fig. 3
Nu ( rut ] = +u
(7)in which t h e sign of t h e right side is t h e reverse of t h a t in (5). A t (r (~, u(~)), the curve re- verses again, a n d continues indefinitely as a solution of (5).
2.6. As u 0 continues to decrease, t h e p r o c e d u r e repeats, le~ding t o t h e f o r m a t i o n of r e p e a t e d " b u b b l e s " (Fig. 3). All inflection points on t h e meridional curve lie b e t w e e n t h e t w o h y p e r b o l a s r u = • 1. D e n o t i n g b y bm t h e m e r i d i o n a l c u r v a t u r e referred t o t h e u-axis, we h a v e kin(u(1) ) ~ 0 according as r ( J ) u (j) ~ - 1 . T h e entire curve lies a b o v e t h e h y p e r b o l a
r u = - 2. A s y m p t o t i c a l l y for large I u0 l,
212 P A U L CONCUS AND ROBERT FINN
2 2 3
- - - - ~ r ( 1 ) ~
UO ~0 U03 :'
while for all % < - 1 0 there holds UO
2 4 2 4
< u ( ~ < % - -
71 2 ~/2"
UO Uo
2.7. There is evidence t o s u p p o r t t h e assertion t h a t as u0-~ - c~, t h e " b u b b l e " solu- tions converge, u n i f o r m l y in c o m p a c t a excluding t h e origin, to a new n o n p a r a m e t r i c solu- tion u = U(r) of (5), satisfying
1 1 1 - r 4
- - < U ( r ) < . . . . (8)
r r (1 +r4) a
for 0 < r <r(1), where r(1 ) is t h e first zero of U(r).
This n e w solution is defined for all r > 0, a n d yields a solution of (5) with a n isolated singularity at t h e origin. W e h a v e n o t y e t d e m o n s t r a t e d this convergence, n o r h a v e we shown t h e existence of U(r), b u t we hope t o r e t u r n to these m a t t e r s in a subsequent paper.
H e r e we note in passing t h a t t h e right side of (8) is precisely t h e negative of t h e m e a n cur- v a t u r e of t h e surface defined b y t h e left side of (8).
2.8. N u m e r i c a l calculations of U(r) were performed on t h e CDC 6600 c o m p u t e r using a variable-step-length f o u r t h - o r d e r A d a m s - M o u l t o n m e t h o d . T h e first t w o t e r m s of t h e formal a s y m p t o t i c expansion
U ( r ) - - ~ - ~ r +O(r ), (r~O) 1 5 3 7
r
p r o v i d e d t h e initial height a n d slope with which to begin t h e numerical integration at small values of r. T h e result b e h a v e d s t a b l y with respect to changes in t h e initial value of r a n d appears t o s u p p o r t t h e conjecture of w 2.7. T h e numerical solution for U(r) is com- pared with t h e left a n d right sides of (8) a n d with bubble solutions corresponding t o several choices of %, in Figs. 4, 5.( 1 )
2.9. The surmised existence of U(r) indicates t h a t t h e result of T h e o r e m 2 c a n n o t be e x t e n d e d to a global b o u n d for solutions t h a t are n o t r o t a t i o n a l l y symmetric, t h a t is, T h e o r e m 1 would be qualitatively incorrect u n d e r t h e h y p o t h e s e s A 3 or A 4. To see this, we consider t h e solution U(r), r 2 = ~ + ~ , a n d a ball B~ (as in T h e o r e m 2) which lies close t o b u t does n o t include t h e origin. As indicated in T h e o r e m 2, By will c o n t a i n points a t (1) We wish to thank W. H. Benson and F. C. Gey for making available their computer programs and carrying out some of the calculations.
213
1 1 - r ~ 1
r ( l + r 4 ) a/2' U ( r ) , - - r Fig. 4
J
j/'/
% = - 4 , U(r)
Fig. 5 a ON CAPILLARY FREE SURFACES IN A GRAVITATIONAL FIELD
u o= - 8 , U(r) u o= - 1 6 , U(r)
Fig. 5b Fig. 5c
214 P A U L C O N C U S A N D R O B E R T F I N N
which I U(r)] <Ma, but it can also be made to include points at which I U(r) l is as large as desired.
2.10. B y restricting attention to a wedge-shaped region formed b y two lines through the origin and meeting with angle 2~, we realize the situation studied in [5, 6]. If the solu- tion U(r) exists, then the surface it defines meets the boundary walls of the corresponding cylindrical wedge in an angle ~=~/2; thus a + ~ > g / 2 , yet the solution surface is not bounded, in apparent conflict with the result stated in w 11, (ii) of [5].
2.11. The construction of w167 2.1, 2.2 yields as corollary the nonuniqueness of the solu- tion of the capillary problem, (2, 3) in f2, when u < 0. For example, in the case ~ = z/2, the horizontal plane u = 0 yields one solution for any choice of f~; if s is the disk r < r~, where r~ is the first maximum of u(r), then u(r) yields a second solution for this domain. Calcula- tions indicating criteria for uniqueness of rotationally symmetric solutions, with u < 0 , are given in [4].
w
If information is known on the boundary behavior of u(x), then Theorems 1 and 2 can be sharpened. Write Z = Z ' + ~:0, Z' being the set of points x E Z which lie interior to ( n - 1) dimensional surface neighborhoods of class C (1). Consider a surface S defined b y a solution u(x) of (1) in ~ , such t h a t u(x) E C (1) up to Z'. The angle ? =?(x) between S and the bound- ing cylinder walls Z' over Z' is then well defined. Denote b y Z ~ the bounding cylinder wMls over y0.
THEOREM 3. Suppose A 1 is satis/ied by ~/(x; u) and suppose there is a lower hemisphere S~, lying partly (or entirely) over ~ , that does not meet Z ~ and that meets Z' (i/ at all) in angles yz satis/ying 0 <~Ts ~<~(x) at each contact point that projects onto x E ~'. Letting B~ be the pro- ]ection o/ S~ onto the hyperplane u=O, there holds u(x)<M$ + e) /or all x E B~ N f2. I / A 2 holds and i/there is an upper hemisphere/or which ~(x) ~<~s ~<~, then u(x) >1 - M ~ -(5 in B~ fl f2.
CO~OLr.A~r. I / / o r / i x e d 6 > 0 , ~ can be covered by a / a m i l y o/such balls B~, then the indicated bounds hold uni/ormly in ~ .
I t suffices to prove the theorem for the c a s e A1, as the other case is analogous. If
?s <?(x) at all contact points, the proof is formally identical to t h a t of Theorem 1; we need only note that because of the condition ?s <?(x), none of the contact points corresponding to u =u0 can lie in Z. If we are given only ~z ~<?(x), consider a concentric sphere S~,, 0 <5' <(~, and the corresponding projection B~,. One verifies easily t h a t ?s. < ? s at any points of
O N C A P I L L A R Y F R E E S U R F A C E S I N A G R A V I T A T I O N A L F I E L D 215 contact t h a t lie on a corresponding generator of Z'. Thus, the proof follows for x E B~. fl again as it did for Theorem l; the result in the general case is obtained b y letting (3 (3.
We also have:
T ~ O R E ~ 4. Suppose A a (resp. A4) is satis/ied by ~4(x; u), and that g~ satis/ies the hypothesis o/the Corollary to Theorem 3. Then in each B3 there is a point x E BE/or which u(x) >~ - M$ (resp. u(x) ~<M~).
3.1. We illustrate the above theorems b y considering, as in [7, w 3.5], a wedge region W with b o u n d a r y E defined b y
m--1
r = x s e c ~ , r2=x2+ ~ y ~ , 2<~m<~n. (9)
t = 1
Here ~] is s m o o t h except at the ( n - m) dimensional " v e r t e x " continuum E~ r = 0. I f /> (~/2) - ~, R/(1 + sin ~) >(3 >R/2, the spherical cylinder
m - 1
S~ : (x - (3) 3 + ~ y~ + U s = ( R - (3)3
j = l
will m e e t the walls of the cylinder Z' over Y / = Y~ - E ~ in an angle 7s < Y, and will lie interior to the cylinder r = R.
Suppose condition A 1 is satisfied b y :H(x; u). L e t u(x) be a solution of (1) in ]OR=
~0 A {r < R}, and suppose t h a t on the p a r t Z ' of the b o u n d a r y of this domain, the solution surface meets Z ' in an angle 7(x)>~ ( ~ / 2 ) - a. We shall show t h a t u(x) is bounded above as the vertex is approached in a n y w a y from within 1 0 .
To do so, consider first the sphere, for arbitrary {bi},
~ : ( x - ( 3 ) 2 + ~ l y ~ + ~ (yj-bj)2 + u 2= (R-(3) 2
~=1 i = m
and its p r o j e c t i o n / ~ o n t h e base space u = 0 . Clearly 3~ again meets Z ' in the angle ~s <~'.
Theorem 3 then yields immediately t h a t u(x) < M s +(3 in W f3/~.
We now observe t h a t the (bj} are arbitrary, and it follows t h a t the same result holds in W flB~, with B~ the projection of S~ on the base space y j = 0 , ?'~>m. L e t t i n g (3-+R/2, we find t h a t the bound holds uniformly in B~/2 up to the vertex. Finally, we note t h a t if R* = (2/(1 + [/5)) R, t h e n WS* can be covered b y balls of radius R/2, each of which lies in the set r ~< R and meets Z ' in an angle not exceeding y; thus the same method yields a uniform bound in ~0 n*. An analogous discussion holds under the condition A s.
We summarize the result:
216 P A U L C O N C U S A N D R O B E R T F I N N
Let ~H(x; u) satis/y A 1 (resp. A~). Let u(x) be a solution o/ (1) in ~ R and suppose the solution sur/ace meets the part Z' o/ the bounding walls in angles ? ( x ) / o r which ~ >~ (zt/2) - (resp. ~ <~ (~t/2) + zr Then, without regard to the conditions on the remainder o/the boundary, there holds u(x) <Ma/~ + R/2 (resp. u(x) > - M ~ - R/2) in ~ a * , R* = (2/(1 + VS)) R.
I n fact, a bound holds in a n y ~/~R' with R' < R. The value R* was chosen because of the simple exphcit nature of the estimate in this case.
3.2. Now suppose ~ + ~ < (zt/2) -Co for some eo > 0 at all x E Z ' . Let ~ be a sphere of ra- dius 8, with center on the hne of s y m m e t r y a t distance Q from the vertex Z ~ I f ~ meets Z ' in an angle ~<7o ~<7, there follows ~ <~ sin ~ see 70 <q. Thus, no set of these spheres of fixed radius can cover all points in the corner. The method yields only the growth estimate u(x) ~<MQ.+q' (resp. u(x)>~--MQ.--~') with ~' =~ sin ~ see ?0, for points x at distance ~>Q from Z ' , as ~ 0 . We proceed to show in a particular case t h a t this estimate is qualita- tively the best t h a t can be expected.
3.3. We consider, for ~ > 0 , a solution u(x, y ) = u ( x , YI ... Yn-1) of the equation N u = x u , defined in a region ~R bounded between parts of the spherical surface r 2 = x ~ . . . . t 2.,J1= 1 y~ = R 2, 1 the conical surface Z': r = x s e c ~r 0 < ar ~/2, 0 < x < R, and the vertex •0: r= O. L e t
~0=glbx. ~. I f ~0>~(zt/2)-x, then u(x, y) is bounded above near ~0, b y 3.1. Suppose
~0 < (~/2) - ~, a n d set k 0 = s i n ~ see V0. The function
)
~(~,y;ro)=~
- t , t = V k ~ - l + ( ~ / r ) ~(10)
is t h e n defined a n d positive in ~R- We assert t h a t for a n y R ' < R, there is a constant C, depending only on the geometry and on R' (and not on the particular solution considered), such t h a t
u(x, y) < ~ ( x , y; ~0) + C (11)
uniformly in ~ w .
The proof can be obtained, with minor modifications, from a similar result, given in our earlier p a p e r [6] for the case n = 2. We present here an alternative, more geometrical proof, modeled on the considerations of w 1 of this paper.
We suppose first 70 > 0 , and consider 9o in the range 0 <90 <Fo, 9o to be determined later. We introduce a function
and set
/(~)={2M_2V~-M-~_X2,
O<;t~<M M < . 2 < . V 2 M
~M(X, y; ~0) = l[~(x, y; ~o].
OIq C A P I L L A R Y F R E E SURFACES IN A GRAVITATIONAL FIELD 2 1 7 Thus, ~0 M is defined in t h e s u b d o m a i n ~M in which ~0 ~< t/2M. W e note t h a t on t h e spherical cap W = ~/2M t h e n o r m a l derivative (a/~v)q)M= oo.
T h e calculation of NWM in :~M a n d of T~M. V on •' is facilitated b y t h e observation t h a t t h e level surfaces of ~M are spheres t h a t m e e t Z ' in t h e c o n s t a n t angle 7o. W e find
NqJM << . U~M + U(X, y), JU(X, y)J < Cr a (12) T ~ M ' M ] Z , = COS ~0 + / A ( r ) , - - Cr 4 <lu(r) ~<0 (13) u n i f o r m l y as r t e n d s to its m i n i m u m , for all 70 in the range considered, a n d for M > M o > 0.
W e n o w choose ~0 < 7o, a n d r0< R, so t h a t for r < r o there holds cos ~50 + lu(r) > cos 7o.
Clearly, b y (13), it suffices to choose ~5 o so t h a t cos 750 > cos 7o + Cry, for r o sufficiently small t h a t this inequality is possible.
N o w set WM = ~M + C, a n d choose C t o be t h e smallest value for which eOM >/U in ~M.
F o r this choice of C, there m u s t be at least one point p E ~M at which u ( p ) = O)M(p)- T h e point p c a n n o t lie on t h e inner cap 9 = I/2M, since (~/~V)gM = c~ on this cap; similarly, since T(~M. M > COS ~'0, t h e surface 9M meets t h e cylinder walls Z ' over X' in an angle smaller t h a n 7o; t h u s p c a n n o t lie on Z ' unless it lies on t h e outer cap r = r o.
I f p is a n interior point of :K M, t h e n u(p) = ~OM(p) = (PM(P) + C, a n d since at p, t h e m e a n c u r v a t u r e of t h e surface u(x, y) c a n n o t exceed t h a t of t h e surface WM(X, y), there holds uu(p) <U?M(P)+~(P); thus, uC < 7 in this case.
I f p lies o n t h e outer cap r=ro, t h e n f r o m
u(p)=cfM(p)+C=~(p)+C
w e find C <maxr=r, [u(x, y) - 9 ( x , y)]. T h e o r e m 1 provides an a priori b o u n d for u(x, y) on the are r = r0, a n d a b o u n d for ~ on this arc is k n o w n explicitly.
I n b o t h events, C is b o u n d e d a priori i n d e p e n d e n t of M, a n d we are free to let M - ~ 0%
This yields a b o u n d of t h e f o r m
u(x, y) ~< ~(x, y; 7o) + C (14)
in a n y :~R', a n d it remains o n l y t o investigate t h e transition ~5 o-~ 7o-
Choose ro< rain (1, R} a n d sufficiently small t h a t ~5 o can be chosen to satisfy cos 7o > c o s 7o + Cry, as above. I f we choose ~5 o sufficiently close t o 7o t h a t also cos ~5o < c o s 7o+2Cr~ we note, using t h e explicit form of W(x, y; 7), t h a t there is a c o n s t a n t Co such t h a t
~0(x, y; ~5o) < ~(x, y; 7o) + Cor~/r (15)
for all points (x, y) for which r < ro, a n d with C O i n d e p e n d e n t of r o in t h e range considered.
I n particular, in t h e range r~ ~< r < ro, there holds
~(x, y; 7o) < ~0(x, y; ~o) + Cor~, a n d hence b y (14)
1 5 - 742909 A c t a m a t h e m a t i c a 132. Imprim~ lc 19 Juin 1974
218 P A U L C O N C U S A N D R O B E R T F I N N
u(x,
y) ~< ~(x, y; ~0) + C < ~(x, y;Vo) + C + Cot ~
in this range.
Using again (15), we m a y choose ~1, T0 ~<~1 <V0, such t h a t for r ~<r0 ~ there holds
~(x, y; ~1) < ~ ( x , y;
~,o)+Cor~/r
so t h a t in the range ro 4 ~< r ~<
~(x, y; Pl) < ~(x, y; to)
+Cor~
We note (a/O(cos ~))~ >0; thus
q~(x,
y; ~x) >~(x, y; ?0) a n d it follows t h a t on the spherer = r 2
u(x,
y ) - ~ ( x , y; ~1) <C+Co~
Applying the above proof of (14) in :~r:, with ~0 replaced b y ~1, we obtain
u(x,
y) < ~(x, y; :~1) + C + Co~< ~(x, y; ~0) + o + Oo (to ~ + r~)
in the range
r~<r<r~.
I t e r a t i o n of this procedure, with ~ replaced successively b y to, 4r0 s ... yields the estimate, for all r ~ ro,
u(x, y) <~(x, y; ~0) + o + ~0 co/(1 - r 0 ~) which completes the proof of (11) in the case ~040.
Finally, suppose ~0 = 0. I n this ease, q(x, y; 0) satisfies the b o u n d a r y condition exactly, t h a t is,/~(r)--0 in (13). We consider first an interior region :K~ obtained b y translating the cone Z slightly along the axis of s y m m e t r y . On the new conical wall ~ ' there holds
>~0 > 0, hence the above proof can be repeated, yielding the stated result in : ~ . Since p(r) = 0 , the estimate is in this case independent of the a m o u n t of shifting; thus we are free to let Z ' slide back to Z, a n d the assertion follows again in :~R-
3.4. We obtain from 3.3. a universal a priori bound above, for all solutions of (2) de- fined in :~R- A corresponding bound holds of course from below, and is obtained from the given one under the transformation u ~ - u , ~ - ~ g - y. Under this generality, little more can be said. However, if it is known t h a t ~ ~<?1 < (~/2) - ~ on Z ' , t h e n there holds Mso
~(x, y; 71) - C <
u(x,
y)uniformly in a n y :~R',
R'< R,
where again C depends only on the geometry a n d on R'.We do not at present have a geometrical proof of this fact, a n d we refer the reader to w 3.7, where it is obtained under weaker conditions as a consequence of a much more general result. Alternatively the analytical proof given in [6] for the case n = 2 , ~ ~-~1, can be mo- dified, using the results of w 1, to yield the assertion.
ON CAPILLARY FREE SURFACES IN A GRAVITATIONAL FIELD 219 3.5. W e collect t h e a b o v e results, t o g e t h e r w i t h others t h a t are p r o v e d analogously, in a general s t a t e m e n t . T o do so, it is c o n v e n i e n t t o introduce a function
~p(x,y;,~)=fq~(x,y;7) if k~=sin~ a s e c ~ 7 < 1,
to
if ]c2~> 1, where ~(x~ y; 7) is defined b y (10).I n t e r m s of (I)(x, y; 7), we t h e n have:
T H E O R E M 5. There is a constant C, depending only on ;~, R, R ' < R and not on the par.
ticular solution u(x, y)), such that i / u ( x , y) satisfies N u = ~ u , x > 0 , in ~ R and 70~<7(x)~<71 on Z', then
(I)(x, y; 71) - C <~ u(x, y) <<. qP(x, y; 70) § (16) in :Kn'.
W e h a v e i m m e d i a t e l y :
C O ~ O L L A R r : Under the above hypotheses, i / 7 = c o n s t . on Z', then (i) i / ~ >~ ](~/2) - 71, then - C <~ u(x,y) <~ C in ~ n . ;
(ii) i / ~ < I (~/2) - 71, then q~ (x, y ; 7) - C <~ u(x, y) ~< ~(x, y; 7) + C in ~R..
Thus, if 7 - - c ~ t h e a s y m p t o t i c b e h a v i o r of u(x, y) is characterized c o m p l e t e l y t o within a (universal) a d d i t i v e constant. More precisely, all solutions for which (~/2) - ~ ~< 7(x) ~< (~/2) + ~ are b o u n d e d in ~ n ' , while if 71 < (7~/2) - ~ or if 70 > (x/2) + ~, t h e n all solutions are u n b o u n d e d . I r r e s p e c t i v e of b o u n d a r y behavior, no solution in ~ a can grow f a s t e r in m a g n i t u d e t h a n O(r -1) a t t h e vertex.
3.6. T h e o r e m 5 can be o b t a i n e d a l t e r n a t i v e l y , a n d u n d e r w e a k e r s m o o t h n e s s h y p o - theses, as a consequence of a m a x i m u m principle, which is closely related to, b u t n o t equi- v a l e n t to, t h e result of T h e o r e m 6 in [7].
THEOREiVl 6. Let Z = Z ~ + Za + ZB be a decomposition o/ Z, such that Z~ is either a null set or o/ class C (1~ and E ~ is small in the sense introduced in w 3.5 o / [ 7 ] . Let u(x), v(x) be o/class C (~ in ~ , and suppose
(i) N u > N v at all x e ~ /or which u - v >O.
(ii)/or any approach to ~ / r o m within ~ , lim sup [ u - v ] < 0 .
(iii) on ED, T u . v <~Tv. v a l m o s t e v e r y w h e r e as a limit(1)/tom points o / ~ . Then u(x) ~<v(x) in ~ .
(1) A somewhat weaker hypothesis suffices; el. the remarks in the preceding note [9].
220 P A U L C O ~ C U S A N D R O B E R T F I N N
The proof is identical to the proof of Theorem 6 of [7] for the case Za dg Z0, as in that proof no use was made of the set for which u - v <0. We note that the conclusion of Theorem 6 of [7] for the case Z~ c Z 0 can not be expected to hold in the present situation.
3.7. We are now in position to give an independent proof of Theorem 5, without smooth- ness conditions or bounds for u(x) on Z. We suppose only t h a t Tu.v exists as a limit almost everywhere on Z'. I t suffices to prove the right-hand inequality, as the left inequality fol- lows analogously.
We choose Z ~ to be the vertex of ~R, ~ the conical walls, and ~ the set r = R'< R
in
~R.
Suppose first 0 <~0 < (:~/2) - ~. Since ~'0 ~ ( x ) , (13) implies t h a t for a n y constant C the function v(x, y) =~0(x, y; ~0) + C satisfies (iii) for sufficiently small R', for a n y ~0<?0 . From (12) we see t h a t C m a y be choosen so that, (a) v(x, y)>~u(x, y) on Z a, and (b) hrv~ <
.Nu=uu at all points where u>v, uniformly as ~0~?0 . Thus, (i) and (ii) hold, and since Z ~ can be covered b y balls of radius bounded from zero, it follows from Theorem 3 t h a t C can be chosen independent of the particular solution u(x, y) considered. Since (iv) obviously holds in this situation, Theorem 6 yields u(x, y) -~v(x, y) in ~R.. The transition :~0~?0 is effected as in w 3.3.
If ~'0= 0, then (13) holds with/~(r)-= 0 and one m a y choose ~0=~o; no transition is then needed.
If ? 0 > ( : ~ / 2 ) + a, the transformation u - ~ - u , ? - ~ : ~ - ? reduces the problem to the previous case.
Finally, if it is known only that (:~/2) - a ~< 70 <~ ~(x), then a uniform bound above for a n y solution in ~R" is obtained b y the methods of w 3.1.
3.8. Theorem 5 was verified physically in an ad hoe experiment b y Mr. Tim Coburn in the Medical School at Stanford University. Mr. Coburn constructed a wedge b y machin- ing one edge of a 1/2 inch thick rectangular piece of acrylic plastic 4" high, then placing the edge in contact with a face of a similar piece of the same plastic; the configuration was placed on a flat horizontal plastic surface, and the bottom of the wedge was then covered b y a small amount of distilled water. The results corresponding to the half angles a of ap- proximately 12 ~ and 9 ~ are shown in figure 6. Both photographs are on approximately the same scale, the scratch mark on each corresponding to a height at t h e vertex of about 7 cm.
We interpret the result with the aid of Theorem 3. Since for distilled, water a = 78 dynes/cm, we find u=~g/a=980/73 > 13; hence, choosing 8 = [/~13 a n d using the remark (ii) following the proof of Theorem 1, we obtain lu(x)l <0.8 cm. whenever a+~,~>:~/2.
This is above the observed rise height for a = 12 ~ but less than 1/10 the height observed for
ON C A P I L L A R Y F R E E S U R F A C E S I N A G R A V I T A T I O N A L F I E L D 221
Fig. 6a. ~=12 ~ Fig. 6b. ~ = 9 ~
= 9 ~ (The liquid failed to rise t h e entire height of t h e plates in t h e latter case p r e s u m a b l y because u n i f o r m c o n t a c t between t h e t w o plates could n o t be m a i n t a i n e d all t h e w a y to t h e top). W e conclude, in particular, t h a t t h e c o n t a c t angle of distilled w a t e r with acrylic plastic (under t h e given conditions) has a value between a p p r o x i m a t e l y 78 ~ a n d 81 ~
3.9. W e consider n o w t h e case ~ < 0 . The material of w 2 suggests t h a t T h e o r e m 5 can- n o t be e x p e c t e d t o hold as s t a t e d in this situation. W e show, however, t h a t a weaker f o r m of t h e result~ suggesting t h e qualitative b e h a v i o r w h e n ~ < 1 ( ~ / 2 ) - 9 ' 1 , still holds w h e n
~ < 0 .
T H E O R ~ 7. Let u(x, y) satis/y 2Vu=,~u, ~< O, in ~R. I/, on Z ' , 9'>~9'0 > ( ~ / 2 ) + (resp. 9' ~<9'1 < (~/2) - ~), then there is a sequence o/points in ~R tending to the vertex Z ~ along which u(x, y) > 2/r (res T. u(x, y) < -(,~/r)), with
> n - 1 sin ~ + cos 9'0 I n - 1 sin a - cos 9'1~
(sin \resp" X > Y !
Proo/. Suppose 9' ~> 9'0 > (~/2) + ~. Consider t h e region A~ c u t off f r o m ~ R b y t h e plane x = r . I n t e g r a t i n g t h e e q u a t i o n over A t a n d using t h e estimates T u . o > - 1 on t h e plane, T u . ~ = cos ? ~< - cos 9'0 on Z ' , we find
u ( . u(x,y) dydx< w,_l(r sec ~)n-1 (sin ~ + cos 9'0)
222 PAUL COI~CUS AND ROBERT FINN
w h e r e O)n_ 1 is t h e v o l u m e of a u n i t ( n - 1 ) - b a l l . I f there were to hold u ( x , y ) < 2 / Q for some c o n s t a n t 2 when x = ~ < r, we would h a v e
f ~ u ( x , y) d y d x > ~2 r l (g t a n ~) n- l m~_ ldQ = n _ 1 f o ~ ~Jt ( r t a n ~ ) n - 1
r
f r o m which t h e result follows.(1) T h e other case is p r o v e d analogously.
3.10. W e r e m a r k t h a t t h e derivatives of H(x) were at no time used for a n y of t h e re- sults of the first p a r t of our s t u d y [7]; t h u s all t h e results of t h a t paper a p p l y equally to a n y situation discussed in t h e present work, w h e n e v e r t h e solution u(x) is k n o w n to be bounded. I n particular t h e y a p p l y w h e n e v e r t h e h y p o t h e s e s of t h e Corollary t o T h e o r e m 1 are satisfied, as these h y p o t h e s e s yield a n a priori b o u n d on a n y possible solution in ~ . As an example, we note t h a t if u(x) is a solution of N u = g u , g > 0 , in a star-shaped do- m a i n ~ with s m o o t h b o u n d a r y E, then, in t h e n o t a t i o n of [7], (10) of [7] implies
~g ~ _
- ~ = cos 7 H~"
n
3.11. T h e criterion in t h e corollary to T h e o r e m 5 relates closely t o r e c e n t w o r k of E m - m e t [8] on t h e existence of solutions of a variational problem associated with (2, 3). F o r a d o m a i n ~R, E m m e r ' s criterion Iv I I/1 + L 2 < 1, when applied to t h e b o u n d a r y a t t h e vertex, is equivalent to t h e condition u > I ( ~ t / 2 ) - 7 1 ; thus, his criterion is a l m o s t identical to t h e one ensuring b o u n d e d n e s s of all solutions in ~ . I f n = 2 a n d ~ < ] ( ~ / 2 ) - 7 ] , T h e o r e m 5 shows there is no solution in t h e r e g u l a r i t y class considered b y E m m e r . M. Miranda has p o i n t e d o u t t h a t t h e variational expression a d m i t s no finite lower b o u n d in this case.
I t should be noted, h o w e v e r t h a t if x > 0 , solutions of (2, 3) regular in ~ a n d in t h e class s can be shown t o exist in (essentially) t h e class of d o m a i n s considered in w 3.6, whenever Z ~ = r This follows f r o m t h e m e t h o d of E m m e r in c o n j u n c t i o n with t h e results of w 1 a n d general a-priori interior estimates for t h e derivatives of t h e solution, cf. [3].
W e r e m a r k t h a t E m m e r ' s condition ~ > 0 is necessary, even for s m o o t h boundaries;
this follows f r o m Corollary 3.1 in [7].
Acknowledgment
I t is a pleasure for us t o t h a n k Mr. B e n j a m i n Swig for his kindness in m a k i n g avail- able t h e facilities of t h e public rooms at t h e F a i r m o n t H o t e l in San Francisco. T h a t corn- (1) A somewhat better estimate for 2. could have been obtained by using more carefully the bound - u ( x , y)<~/r. In view of the apparently tentative nature of the result, this improvement does not seem to us at this time to be of great value.
ON CAPILLARY FREE SURFACES I N A GRAVITATIONAL FIELD 223 fortable a n d pleasing milieu afforded us p r i v a c y a n d a n o n y m i t y t h a t were i m p o r t a n t for o u r efforts. W e e n j o y e d also t h e c o n c o m i t a n t privilege, during m o m e n t s of relaxation, of observing t h e m a n y fascinating a n d changing vignettes of e v e r y d a y life t h a t passed be- fore us in t h e l o b b y of w h a t i s certainly one of t h e world's m o r e splendid hotels.
Notes added in proo/:
1. T h e surmised (w 2.7) existence of t h e singular solution U(r) is n o w proved; details will a p p e a r in a w o r k b y these authors, n o w in preparation.
2. W e h a v e been informed t h a t t h e solution U(r) was encountered i n d e p e n d e n t l y in a c o m p u t a t i o n a l s t u d y b y C. H u h (Capillary H y d r o d y n a m i c s . . . Dissertation, Dept. of Chem.
Eng., Univ. of Minnesota, 1969).
3. W e n o t e a p a r t i c u l a r choice for t h e t w o constants C in (16). Define b y ~K ~, j = 0, 1, t h e p a r t of ~ R between t h e v e r t e x ~0 a n d t h e outer cap of a sphere, of radius ~j = m i n {~, 6 sin a[ sec 7J] }, centered on t h e axis at distance 6 < R/2 f r o m Z ~ W e m a y t h e n choose C = n ( g 6 j ) - l § in ~ J on (respectively) t h e right a n d t h e left of (16). H e r e ej(6)->0 with 8, ej = 6j if sin ~ I sec 7j I >~ 1.
References
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[2]. BASHFORTH, F., & ADAMS, J. C. A n attempt to test the theories o/ capillary action by com- paring the theoretical and measured ]orms o] drops o/ fluid. Cambridge University
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[3] BOMBIERI, E. & GIUSTI, E., Local Estimates for the Gradient of Non-Parametric Surfaces of Prescibed Mean Curvature. Comm. Pure Appl. Math., 26 {1973), 381-394.
[4]. CoNeus, P., Static menisci in a verical right circular cylinder, J. Fluid Mech., 34 (1968), 481-495.
[5]. CoNcus, P. & FINN, R., On the behavior of a capillary surface in a wedge. Proc. Nat. Acad.
Sei., 63 (1969), 292-299.
[6]. - - 0 n a class of capillary surfaces. J. Analyse Math., 23 (1970), 65-70.
[7]. - - On capillary free surfaces in the absence of gravity. Acta Math. 132 (1974), 177-198.
[8] EMMEt, M., Esistenza, unieita' e regolarita' nelle superfiei di equilibrio nei eapillari.
Annali Univ. Ferrara, Sez. V I I , 18 (1973), 79-94.
[9]. FINN, R., A note on capillary free surfaces. Acta Math., 132 (1974), 199-205.
[10]. JOHNSON, W. E. & PERKO, L. M., Interior and exterior boundary value problems from the theory of the capillary tube. Arch. Rational Mech. Anal., 29 (1968), 125-143.
[11]. LOHNSTEIN, Th., Dissertation. Berlin 1891.
[12]. - - Zur Bestimmung der Capillarit~tsconstanten; Bemerktmgen zu der Arbeit von Herrn Quincke. Ann. Phys. Chem. (Wiedemann Ann.), 53 (1894), 1062-1063.
[13]. THOMPSON, W., Capillary attraction. Nature, 505 (1886), 270-272; 290-294; 366-369.
Received July 17, 1972
Received in revised ]orm September 20, 1973
