Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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We designate  as the relative adhesion coefficient of

the liquid–gas–solid configuration. We assume that

the cylinder walls are of homogeneous material, so

that  will be constant. In general,  is a difference of

factors that apply on the walls at the two interfaces,

with the liquid and with the external medium.

3. Gravitational energy. The work done in

lifting an amount of liquid h against the

gravity field from the base level to a height h in a

vertical tube of small section  is ghh . Thus,

the work done in filling that tube up to the

surface height u is (gu

=2),andthetotal

gravitational energy is

g



dx ½12

4. Volume constraint. In the configuration con-

sidered the volume is to be unvaried during

admissible deformations; we take account of the

constraint by introducing a Lagrange parameter ,

and an additional ‘‘energy’’ term

¼ 



u dx ½13

According to the principle of virtual work, the

sum E of the above energies must remain unvaried

in any deformation that respects all mechanical

constraints other than the volume constraint. We

choose a deformation u !u þ ", with  smooth in

the closure of , which determines a functional E(").

From E

(0) = 0 follows



r 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þru

þ u þ ðÞ

;

 



 ds ¼ 0 ½14

from which







div Tu þ u þ ðÞ





 Tu  ðÞds ¼ 0 ½15

with Tu ru=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þru

, and with  the unit

exterior normal on . Choosing first  to have

compact support in , the boundary term vanishes,

and the ‘‘fundamental lemma’’ of the calculus of

variations yields

div Tu ¼ u þ ;  ¼ g= ½16

throughout . Thus, the area integral in [15]

vanishes for any . We are therefore free to choose

 as we wish on the boundary, and the fundamental

lemma now yields   Tu =  on . We now note

that for any liquid surface u(x, y) there holds

  Tu ¼ cos  ½17

on , where  is the angle between the cylinder wall

and the surface S, measured within the liquid. Since

 is assumed to be constant, that is so also for .Itis

a physical constant: the contact angle, that must be

measured in an independent experiment, and cannot

be prescribed in advance or calculated within the

scope of the theory.

The constant , original ly introduced as a general

proportionality constant, is now characterized as

 = cos . We thus see that a physical surface of the

form envisaged is possible only if 1    1.

Physically, one expects that if <1 the liquid

will separate from the walls, while, if >1, the

liquid will spread over the walls as a thin film.

Equation [16]

and boundary condition [17]

provide a nonlinear second-order equation that is

elliptic for any function u(x, y), and also a non-

linear transversality condition on the boundary, for

determining the surface interface S. The expression

div Tu is exactly twice the mean curvature of the

surface S.If 6¼ 0then can be eliminated by

addition of a constant to u. The problem [16]–[17]

for the fluid in a vertical cylindrical capillary tube

of general section becomes thus a geometrical one:

to find a surface whose mean curvature is a

prescribed function of position in space, and

which meets the cylindrical boundary walls in a

prescribed angle .

In the absence of gravity, [16] takes the form

div Tu ¼ 2H ½18

for a surface of constant mean curvature H. The

constant H is determined by integrating [18] over ,

and using [17]:

2H ¼



cos 



½19

where 

and 

denote the respective perimeter

and area, and thus H is independent of volume.

From the known uniqueness up to an additive

constant of the solutions of [18], [17] it fol lows

that the shape of the solution surface is indepen-

dent of volume. That result holds also for [16], [17]

in view of the possibility to eli minate  from the

equation by addition of a constant, and the

uniqueness of the solutions of the resulting

equation.

Equations [16]–[17] or [18]–[17] are appropriate

for determining capillary surfaces that are graphs

Capillary Surfaces 435

u(x, y) over a base domain . More generally, any

surface S in 3-space satisfies the equation

x ¼ 2HN ½20

where H is its scalar mean curvature and N is a unit

normal vector on S. Here  is the ‘‘intrinsic

Laplacian’’ in the metric of S. This is the appropriate

relation to be applied in situations for which the

physical surface folds over itself and cannot be

expressed globally as a graph. The formal simplicity

of [20] is deceptive; the challenges arising from the

nonlinearity in the equation can be formidable, and

very little general theory is as yet availab le.

Dual Interpretation of : Distinction between

Fluids and Solids

We have already remarked the duality in connection

with eqn [10] above. It can be made explicit with a

simple experiment proposed by Dupre´. One makes a

rigid frame with a sliding bar of length l,asin

Figure 6, and dips the frame into soap solution. On

lifting the frame from the solution the opening will

be filled with a soap film, and one finds a force

F = 2l on the bar, directed orthogonal to the bar

(the factor 2 appears since the film has two sides).

The work done in sliding the bar a distance x is

F = 2lx, which can also be written F = 2A

with A an element of area. In this sense, the two

interpretations of  are formally equivalent, for

fluid/fluid interfaces.

The equivalence cannot be extended to solid/fluid

interfaces. Consider a rigid spherical ball of generic

material and radius R, freely floating in an infinite

liquid bath in a gravity-free environment, see

Figure 7a. It can be shown that the unique

symmetric solution to the problem is a horizontal

surface, as in the figure. A variational procedure as

above shows that if e

, e

are the interfacial

energy densities associated with the three interfaces,

then

cos  ¼

 e

½21

in formal analogy with the Young relation [6]. But

, e

cannot be interpreted as interfacia l forces

whose net tangential component cancels that of e

To do so would lead to a net downward force 

the ball (see Figure 7b), contradicting the supposed

equilibrium state.

Mathematical and Physical Predictions:

Experiments

In the following sections, we study the kinds of

behavior imposed on a surface S by the requirement

that it appear as solution of one of the indicated

equations and boundary conditions. Some of these

properties are quite surprising in the context of

classically expected behavior of solutions of equa-

tions of mathematical physics. The mathematical

predictions were, however, corroborated in certain

cases experimentally, as we disc uss below.

Uniqueness and Nonuniqueness

We begin by considering uniqueness questions. We

start with a semi-infini te capillary tube, closed at the

bottom, to be partially filled with a prescribe d

volume of (incompressible) liquid making contact

angle  on the container walls (Figure 8a). If   0,

any solution is uniquely determined. That is a quite

general theorem, valid for a wide class of domains 

including all piecewise smooth domains (at the

corners of which data of the form [17] cannot be

prescribed); formally, data can be omitted on any

boundary set of linear Hausdorff measure zero. In

this resul t, no growth conditions need be imposed

near the boundary (note that such a statement

would be false for solutions of the Laplace equation

under Dirichlet boundary conditions).

Next we consider a sessile liquid drop on a

horizontal plat e (Figure 8b). Again the solution is

uniquely determined by the volume and by ,

although the known proof differs greatly from that

of the other case.

We now consider a smooth deformation of the

base plane, depending on a parameter t, which

carries it into the cylinder; that can be done in such

a way that the supporting surface is at all times

‘‘bowl-shaped,’’ as in Figure 8c. Since the bowl

formation tends to restrict the possible deformations

Figure 6 Dupre

apparatus for exhibiting surface tension.

(a)

(b)

Figure 7 (a) Floating spherical ball; presumed ‘‘Young’’ forces.

(b) Normal and vertical components of Young forces; contra-

diction to presumed equilibrium.

436 Capillary Surfaces

of the fluid consistent with smooth contact with the

supporting rigid surface, one might expect that

the corresponding capillary surface S(t), arising

from the identical fluid mass, will for each t be

uniquely determined.

That is however not true, even for symmetric

configurations. We can see that from the configuration

of Figure 8d, consisting of a vertical circular cylinder

whose base is a 45



cone. We assume a contact angle

 = 45



and adjust the radius so that a horizontal

surface lying just below the cylinder/cone juncture

provides the prescribed volume. This is a formal

solution surface. Now fill the configuration with a

larger volume, so that the contact line will lie above the

juncture. The upper surface will no longer be flat, in

view of the 45



contact angle, and takes an appearance

as indicated in the figure. Finally, we decrease the fluid

volume, keeping all other parameters unchanged. As

noted above, the upper surface moves rigidly down-

ward, and it is clear that if the original surface is close

enough to the juncture line, then the prescribed volume

will be attained before the contact line reaches the

juncture. Thus, uniqueness fails.

In this construction as just described, the bounding

surface is not smooth; however, one sees easily that

the procedure continues to work if the edge and

vertex are smoothed locally. In fact, one can carry the

procedure to a striking conclusion; by appropriate

smoothing, one can construct a bounding surface

admitting an entire continuum of distinct solution

interfaces, all with the same contact angle and

enclosing the same fluid volume (Gulliver and

Hildebrandt; Finn). This can be done for any gravity

field. Figure 9 illustrates seven members of the family

of interfaces, in the particular case  = 0.

The question immediately arises as to which if

any of the continuum of surfaces will be seen in

an experiment. In fact, it can be proved that none

of the indicated surfaces is mechanically stable

(Finn, Concus and Finn, Wente). Since the indicated

family includes all symmetric surfaces that are

stationary for the energy functional, we find that

any stable stationary c onfiguration must be asym-

metric. Thus, we have obtained an example of

symmetry breaking, in which all conditions of the

problem are symmetric, but for which all physically

acceptable solution s are asym metric.

These results were subjected to computational test

by M Callahan using the Surface Evolver software,

to experimental test by M Weislogel in a drop

tower, and to experimental test by S Lucid in the

Mir Space Station. The results of the latter experi-

ment are compared in Figure 10 with the computer

calculations. In both cases, both a local minimizer

(potato chip) and a presumed global minimizer

(spoon) were observed.

The seven surface interfaces indicated in Figure 9

all provide the same sum of surface and wetting

energy, and bound the same volume of fluid. They

all satisfy an eqn [18] with constant H,in

accordance with hypotheses of incompressibility

and vanishing gravity. Thus, formally, all configura-

tions have ident ical mechanical energy. The surfaces

Figure 9 Seven spherical capillary interfaces in an ‘‘exotic’’

container of homogeneous material in zero gravity. All interfaces

bound the same volume and have the same sum of free surface

and wetting energies. If all pressures above the interfaces are the

same, then the pressures below them successively increase as the

curvature vectors of the vertical sections change from upwardly to

downwardly directed. Reproduced from Mathematics Intelligences

24(3) 2002 21–33 with permission from Springer-Verlag Heidelberg.

45°

(d)

45°

(a)

(b) (c)

Figure 8 Support configurations: (a) capillary tube, general

section; (b) horizontal plate; (c) convex surface appearing during

deformation of horizontal plate to capillary tube; and (d)

Nonuniqueness of configuration appearing during convex defor-

mation. Reproduced from Mathematics Intelligencer 24(3) 2002

21–33 with permission from Springer-Verlag Heidelberg.

Capillary Surfaces 437

are all spherical caps; however, the radii R of the

caps vary considerably. According to Y1 above, the

pressure change across each interface is p = 2=R.

Since one may assume the outer region to be a

vacuum with zero pressure for all caps, we find that

the pressures within the fluids vary greatly among

the configurations. One would thus expect that

work is done within the fluid in passing from one

configuration to another, a circumstance we have

excluded by hypothesis when determining the

family. From this point of view, the (customary)

hypothesis of incompressibility that was used in

determining the family is put into significant ques-

tion; we examine this point in some detail in the

section ‘‘Compr essibili ty.’’

Discontinuous Dependence I

Capillary surfaces can exh ibit striking discontinuous

dependence on the defining data. As initial example,

we consider the behavior of a solution of [18]–[17]

at a protruding corner point P of the domain  of

definition. For simplicity, we assume the corner

bounded locally by straight segments , meeting in an

opening angle 2<, thus forming locally a wedge

domain. In anticipation of material to follow, we

assume contact angles 

and 

on the respective

sides, 0  

, 

 . One can show that a necessary

condition for a solution surface over a domain 



in Figure 11 to have a continuous normal vector up

to P is that the data point (

, 

) lie in the closure of

the rectangle R of Figure 12. (This figure includes

also additional material anticipating the section

‘‘Drop s in wedg es’’).

For data points interior to R, this criterion also

suffices for the existence of at least one such solution

surface, for any prescribed H; such surfaces can in

fact be produced explicitly as spherical caps (planes

if H = 0). It remains to discuss what can occur with

data arising from the remaining four subregions of

the square.

If (

, 

) 2 D



, then there is no solution to

[18]–[17] in any neighborhood of the corner point

P. On the other hand, an explicit solution for any

H > 0canbefoundasalowersphericalcapon

the segment 

þ 

=   2 that se parates D

from R (see Figure 13, which indicates the

equatorial circle). Correspondingly, if H < 0then

an explicit solution can be found on the separation

line between D



and R.Thus,thereisa

Spoon (left)

Potato chip

Rotationally

symmetric

Figure 10 Symmetry breaking in exotic container, g = 0. Below:

calculated presumed global minimizer (spoon) and local minimizer

(potato chip). Above: experiment on Mir: symmetric insertion of fluid

(center); spoon (left); potato chip (right). This is a grayscale version

of a color figure reproduced from Journal of Fluid Mechanics,224:

383–94, (1991) with permission of Cambridge University Press.

Figure 11 Wedge domain. Reproduced from Finn R ‘‘Capillary

Surface Interfaces’’ in Notices of AMS 46 No.7 (1999) with

permission of the American Mathematical Society.

–

2α

(D)

)

(No graph)

(Continuous)

Figure 12 Domain R of data yielding continuous normal to

capillary surface in wedge of opening 2a < p. The symbols D

and I are clarified in the section ‘‘Behavior at a corner point.’’

Reproduced from ‘‘Capillary Wedges Revisited’’ in SIAM J. Math.

Anal. 27 No.1 (1996) 56–69 with permission from SIAM.

438 Capillary Surfaces

discontinuous change in behavior in crossing from

R to eit her of the D

regions.

This behavior was put to experimental test by

W Masica, who considered the case 0 <

= 

<=2 near the crossing point  = 

with D

, for

which  þ 

= =2. He partially filled a regular

hexagonal cylinder of acrylic plastic, successively

with two different liq uids, making respective contact

angles greater or less than 

with the plastic. For

each liquid, Masica then allowed the cylinder to fall

in a 132 m drop tower. Figure 14 compares the two

configurations after about 5 s of free fall. In the case

>

he obtained the spherical-cap solution,

which in this case covers the entire base domain 

and appears as an explicit solution of [18]–[17].

When <

, the liquid rose to the top of the

cylinder near the edges, filling out the edges over the

corner points. The surface interface S does not cover

, but instead folds back over itself, doubly covering

a portion of . Thus, a physical surface appears as it

must, but it is not a solution of [18] over .

Discontinuous Dependence II

About 1970, M Miranda raised informally the

question, whether a capillary tube Z

, whose section



lies strictly interior to a section 

of a tube Z

will raise liquid from an infinite reservoir in a

downward directed gravity field to a higher level

over 

than will Z

over that subdomain of its

section. That is true if both cylinders are circular,

and in the intervening years its correctness was

established in a number of other cases of particular

interest.

Finn and Kosmodem’yanskii, Jr. showed, how-

ever, by example that the assertion fails in a large

range of cases, and in fact can fail with arbitrarily

large height differences, uniformly over 

. Beyond

that, the con struction exhibits a strikingly discontin-

uous change of behavior, under perturbations of a

disk as inner domain. Perhaps more remarkably, the

assertion can hold with the inner domain a disk, but

with discontinuous reversal of behavior as the disk is

perturbed to neighboring disks. That was shown in a

form of the example given later by Finn, and

illustrated in Figure 15. Here the outer domain 

is polygonal , with sides that extend to be tangent to

a unit disk 

, as indicated. The angle  is to be

chosen so that 0  =2    

min

, where 

min

is the

smallest of the interior vertex half-angles of 

.In

view of the assumed infinite fluid reservoir, there is

no volume constraint, and the governing equation

[16] takes the form

div Tu ¼ u;¼ g= > 0 ½22

Taking at first the inner domain to be 

,itcan

be shown that for the corresponding solutions u

and u

of [22], the re hol ds u

> u

over 

for

2α

Figure 13 Construction of solution as lower hemisphere; g

= p  2a, H > 0. Reproduced from ‘‘Capillary Wedges Revis-

ited’’ in SIAM J. Math. Anal. 27 No.1 (1996) 56–69 with

permission from SIAM.

(a) (b)

Figure 14 Liquid in hexagonal cylinder, during free fall in drop

tower: (a) a þ g > p=2; (b) a þ g < p=2.

1 + ε

Figure 15 Discontinuous reversal of limiting height behavior. All

sides of the polygonal domain 

are tangent to the unit disk 

For the corresponding solution heights u

in 

, u

in the disk 

of radius 1 þ e,andu

in 

, there holds u

 u

< 0, for any

downward gravity. But lim

 !0

 u

) = þ1, for any e > 0.

Capillary Surfaces 439

any >0, and thus the Miranda question has a

positive answer for that configuration. But if we

replace 

by a concentric disk 

 

of radius

1 þ ",wefind

inf



x; ðÞsup



x;ðÞ

()



1 þ "

cos 



1  sin !

cos 

þ 1 þ "ðÞ

1  sin 

cos 

½23

where ! = arccos(cos = sin ), and u

is the solution

of [22], [17] in 

. Since  does not appear on the

right side of [23], there follows in particular that for

any ">0, there holds

lim

!0

inf



x;ðÞsup



x;ðÞ

()

¼1 ½24

In particular, a negative answer to Miranda’s

question appears for all gravity sufficiently small.

But as observed above, a positive answer occurs in



, for any positive gravity. Thus, the limiting

behavior as  !0 changes discon tinuously, as " !0.

We find that the two limiting procedures cannot be

interchanged: for any x 2 

, we obtain

lim

"!0

lim

!0

x;ðÞu

x;ðÞ



¼þ1:

lim

!0

lim

"!0

x;ðÞu

x;ðÞ



 const:<0

½25

Existence Questions I

For the general equation [20] there is an established

literature on existence of surfaces containing a

prescribed space curve. There is very little literature

relating to the capillarity boundary condition that

the solution surface S meet a prescribed ‘‘support’’

surface W in a prescribed an gle . The existence of

at least one such surface interior to a prescribed

sufficiently smooth closed space domain was proved

by Almgren, and then Taylor proved smoothness at

the contact curve. These are abstract theorems that

are basic for the theory but in general do not

provide specific information in particular cases of

interest.

Special interest atta ches to the nonparametric

cases [16] or [18] with boundary condition [17],

especially in view of the discontinuous behavior

properties described above. These cases were studied

in depth by a number of authors, with results that

put the above examples into some perspective.

M Emmer proved the existence of a unique

solution of [16]–[17] for any compact  having

Lipschitz boundary with Lipschitz constant L such

that

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ L

cos <1  " for some ">0. Finn and

Gerhardt (F and G) extended this condition, and

showed in particular that solutions exist in general

in piecewise smooth . This result contrasts with the

zero-gravity case [18] discussed in the section

‘‘Exi stence questions II,’’ for which solutions fail to

exist when

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ L

cos >1 at a protruding corner

(see the section ‘‘Discon tinuous dep endence I’’).

However, in the cases

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ L

cos >1 studied

by F and G the solution u(x) is necessarily

unbounded in the corner. This condition is equiva-

lent to <  =2jjat the corner. Concus and Finn

showed that if     =2

in a neighborhood 



of a corner with rectilinear sides, as indicated in

Figure 11, then the solution u(x)satisfies

ux;ðÞ



þ  ½26

independent of ,  in the range considered. Here it

is assumed that [16] is normalized so that  = 0;

when  6¼ 0 this can always be achieved by adding a

constant to u. On the other hand, if < =2

then

ux;ðÞ

cos # 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

 sin

kr

½27

where k = sin = cos  and # is polar angle relative

to a bisector at the vertex; hence u becomes

unbounded as O(1=r). Thus, the behavior changes

discontinuously as the configuration for which

 =   =2

is crossed.

This prediction was corroborated by T Coburn in

a ‘‘kitchen sink’’ experiment in the Medical School

at Stanford University. Coburn formed a wedge

using two sheets of acrylic plastic, resting on a glass

plate, and inserted a drop of distilled water at the

base of the wedge. Initially, the wedge was opened

sufficiently that  þ   =2, and he obtained the

configuration of Figure 16a, with the maximum

height slightly lower than that indicated by [26].By

closing down the angle slightly, the liquid rose to

over ten times that height, as shown in Figure 16b.

This experiment was later repeated by Weislogel

under laboratory conditions; it incidentally estab-

lishes the contact angle of water and acrylic plastic

in the Earth’s atmosphere as 80



 2



The indicated procedure provides in general a

very accurate way to measure contact angles, when

the angle is not far from =2. For  near zero or  in

the Earth’s gravity field, the disco ntinuity is con-

fined to a microscopic neighborhood of the vertex,

and can be difficult to observe. This technical

difficulty was addressed by Fischer and Finn, who

introduced ‘‘canonical proboscis’’ domains, the

theory of which was further developed by Finn and

440 Capillary Surfaces

Leise and by Finn and Marek. For such domains the

change in behavior is not strictly discontinuous, but

it is nearly so, and it extends over large portions of

the cylinder section, so that it is easily observable.

Concus, Finn, and Weislogel conducted space

experiments, demonstrating the feasibility of the

method as a means for measuring contact angles in

general ranges.

In [26]–[27] no growth conditions at the corner

are imposed; the estim ates hold for every solution

defined in 



and assuming the prescribed data on

the side walls, with no data prescribed at the vertex.

The formula [27] is the initial term of a formal

asymptotic expansion of the solution, in powers of r.

Miersemann obtained the complete expansion,

asymptotic to every order, when <  =2

.He

obtained somewhat less complete information in the

bounded case [26].

Chen, Finn, and Miersemann provided a form of

[27] that is applicable for any data (

, 

) on the

respective sides of the wedge, that arise from the D



regions of Figure 12. Lancaster and Siegel and

independently Chen, Finn, and Miersemann showed

that if 2  

þ 

   2, then every solution

is bounded at the vertex. This result holds also for

the zero gravity eqn [18].

In the case of [18], Concus and Finn showed that

in the D



regions no solution exists, regardless of H.

Again, this result holds without growth conditions.

From these considerations and from remarks in

the section ‘‘Disconti nuous depend ence I’’ follo ws

that for data in D



, all solutions eith er of [18] or of

[16] are bounded but have discontinuous derivatives

at the vertex P. Extrapolating from the behavior of

particular computed solutions, Concus and Finn

conjectured that all solutions of [18] or of [16] that

arise from data in D



are discontinuous at P.A

number of attempts to prove or to disprove this

conjecture have till now been unsu ccessful.

An existence theorem for [16]–[17] alternative to

that of Emmer was obtained independently by

Ural’tseva, using a very different approach. This

procedure yielded smoothness estimates up to the

boundary, but required a hypothesis of boundary

smoothness, so that the result does not mesh with the

discontinuous dependence behavior as does that of

Emmer. Later versions of the existence result, again

under boundary smoothness requirements, were given

by Gerhardt, Spruck, and Simon and Spruck. In the

procedure introduced by Emmer, the boundary trace is

shown to exist only in a very weak sense (which,

however, suffices for a uniqueness proof). The later

work can be adapted to show that the Emmer

solutions are smooth on the smooth parts of @.

None of the above procedures provides existence for

the zero gravity case [18]. As we shall see in the

following section, that is not an accident of the

methods, but reflects subtle properties of the equations.

Existence Questions II

We consider here the zero-gravity case [18] , over a

domain  bounded by a piecewise smooth curve ,

under the boundary condition [17]. Integrating [18]

over  and using [17], we find 2H 

=

cos . Let





 , 



=\ @



, =\ @



. The same proce-

dure over 



, using that Tu

< 1 for any u(x, y),

leads to the bound

;½> 0 ½28

where  is defined by

;½

 



cos  þ 2H 



½29

The inequality [28] must hold for any choice of





 . This provides a necessary condition for

existence of a solution to [18]–[17] in . E Giusti

showed that when 



is interpreted in a generalized

sense as a Caccioppoli set, the condition [28]

becomes also sufficient for existence.

It is easy to give specific examples of convex

analytic domains , in which subdomains 



can be

found such that [28] fails. Thus, the general

existence results for [16] do not carry over to [18],

regardless of local domain smoothness. Neverthe-

less, in many cases of interest (e.g., a circular disk or

an ellipse that is not too eccentric), solutions of

[18]–[17] do exist for any  and are well behaved.

Finn investigated the condition [28] in general by

showing the existence of a system of arcs {}  

(a) (b)

Figure 16 Distilled water in wedges formed by acrylic plastic

plates; g > 0. (a) a þ g > p=2; (b) a þ g < p=2. Reproduced

from P Concus and R Finn, ‘‘On Capillary Free Surfaces in a

Gravitational Field’’ in Acta Math 132 (1974) 207–223 with

permission of Institut Mittag-Loeffler.

Capillary Surfaces 441

that minimize . All such arcs are circular of radius

1/2H, and meet  either at smooth points in an

angle , or else at a reentrant corner point in an

angle 



 , measured on the side of  opposite to

that into which the curvature vector points

(Figure 17). All minimizing configuration s are

bounded by arcs of that form, although not all

such configurations minimize. In a typical situation

one will encounter only a finite number of such arcs,

in which case only a finite number of cases need be

examined. If  > 0 in each such case, then a

solution of [18]–[17] exists for the given  and .

It may occur that no such arcs exist; we then observe

that since [;; ] =[; ] = 0,  cannot become

nonpositive for any 



  unless a minimizing 

can be found in , contradicting the assumed

nonexistence of minimizers. Thus, the criterion is

then vacuously satisf ied, and we conclude that a

solution of [18]–[17] exists.

One has, of course, to ask what happens

physically in cases for which [; ]  0 for some

 as above. The possible modes of behavior were

studied in particular cases by Tam and later by

deLazzer, Langbein, Dreyer, and Rath; Finn and

Neel characterized the general case. Formally, the

fluid rises to infinity throughout domains 



of the

form indicated, but with H replaced by a value



< H; on the opposite side of the circular arcs ,

the fluid is asymptotic to the vertical cylinders over

. In a physical situation, the fluid will rise to the

top of the container in a nearly cylindrical region

adjacent to a portion of the container walls,

approximating the indicated beh avior and partially

wetting the top of the container. One sees that

behavior in Figure 14b, in which the fluid fills out

regions adjacent to the corners. An analogous

configuration would still be observed if the corners

were smoothed locally. If insufficient fluid is

available, a portion of the base  could become

unwetted.

Behavior at a Corner Point

Lancaster and Siegel (L and S) studied the behavior of

the limits (which they designate by Ru) of bounded

solutions of [16] or of [18] along radial segments

tending to a corner point P of a domain . These limits

can exhibit remarkable idiosyncratic behavior. For

simplicity of exposition, we restrict ourselves here to

rectilinear boundary segments at P, and assume

constant boundary angles 

, 

6¼ 0,  on the two

sides. L and S prove first that the limits Ru exist and

vary continuously with direction of approach; then

they show the existence of ‘‘fan’’ regions of directions

adjacent to those of the sides, in which the limits are

constant independent of direction, see Figure 18. They

obtain that if the opening angle 2 at P satisfies 2<

, then for data in the rectangle R of Figure 12 the fans

overlap (see Figure 18a), so that the solution is

necessarily continuous at P. For data in D

,the

solution decreases from the 

side 

to the 

side 

(‘‘D’’ behavior), subject to the Concus–Finn conjecture

(see the section ‘‘Existence questions I’’), with the

reverse behavior (‘‘I’’) in D



. Concus and Finn showed

that if 2<then in D



there is no bounded solution

of [16]–[17] or [18]–[17] as a graph. For [16]–[17],

unbounded solutions do however exist for such data

(see the section ‘‘Existence questions I’’).

∗

≥γ

Figure 17 Extremal configuration for the functional .

(a)

′

2α

A′

B ′

2α

(b)

(c)

Figure 18 (a) Fan domains APA

and BPB

of constant limiting

values; 2a < p so that the fans overlap when data are in R. (b)

2a > p; case 1. Fans APA

and BPB

of constant radial limits

appear. Limiting value changes strictly monotonically as

approach direction changes from A

P to B

P.(c)2a > p; case 2.

In addition to the two fans adjacent to the sides of the

wedge, a half plane of constant radial limits appears.

442 Capillary Surfaces

If 2>, then the fans do not overlap, and

in fact continuity at P cannot in general be

expected. Outside the indicated fan regions adja-

cent to the wedge sides, the limit values e ither

change strictly monotonically with angle of

approach, as in Figure 18b,orelsetheydoso

except for approaches within a third, central fan,

which covers a full half-space, and interior to

which the limiting values again remain constant,

see Figure 18c. L and S give an example under

which t hat behavior actually occurs. Remarkably,

intheexampletheprescribeddataarethesameon

both boundary segments. The solution is never-

theless discontinuous at P,withanintervalin

which the radial limit increases, another interval in

which it decreases, two fans of constant limit

adjacent to the sides, and a f an of breadth  in-

between.

General conditions for continuity at a reentrant

corner (2>) have not yet been established. L and

S give a sufficient condition, depending on a

hypothesis of symmetry. Since no such hypothesis

is needed when 2<, one might at first expect it

to be superfluous. However, Shi and Finn showed

that by introducing an asymmetric domain perturba-

tion that in an asymptotic sense can be arbitrarily

small, the solution can be made discontinuous at P.

That can be done without affecting any other

hypotheses of the L and S theorem.

In as yet unpublished work, D Shi characterized

all possible behaviors at a reentrant corner, subject

to the validity of the Concus–Finn conjecture at a

protruding corner. If   0 then all solutions of [16]

or of [18] in a neighborhood of P in  are bounded

at P. The further behavior depends on the particular

data, and is indicated in Figure 19. Note the analogy

with Figure 12, although the interpretations in the

figures differ in detail. Here the symbol I denotes

strictly increasing from the side 

to 

, except on

the fan regions of constant limits; ID denotes

constancy on a fan adjacent to 

, then strictly

increasing, then constancy on a fan of opening ,

then strictly decreasing, then constancy on a fan

adjacent to 

. D and DI are defined analogously.

All cases can be realized in particular configurations.

Drops in Wedges

Closely related to the material just discussed is the

question of the possible configurations of a con-

nected drop of liquid placed into a wedge formed by

intersecting plates of possibly differing materials, in

the absence of gravity. Thus, one has distinct

contact angles 

, 

on the two plates. Finn and

McCuan showed that if (

, 

) 2 R then the only

possibility is that the drop surface S is part of a

sphere. For data in D



, no such drop can exist,

barring exotically singular behavior at the vertex

points where the edge of the wedge mee ts S.

For data in D



the situation is less clear. Concus,

Finn, and McCuan (CFM) showed that local

behavior exhibiting such data is indeed possible;

however, they conjectured that such behavior

cannot occur for simple drops. In conjunction with

the above results, they were led to the conjecture

that the free surface S of any liquid drop in a planar

wedge, that meets the wedge in exactly two vertices

and the wedge faces in constant contact angles



, 

, is necessarily spherical. Here it is supposed

only that 0  

, 

 .

The behavior of a drop of prescribed volume, as

the data move from the midpoint of R to the D

regions along parallels to the sides of R, is displayed

in Figure 20. As one moves into the D



regions, the

drop detaches from one side of the wedge and

becomes a spherical cap resting on a single planar

surface, in accord with the above conjecture. As D



is approached, the liquid becomes a drop of very

large radius that fills out a long thin region in the

wedge, and disappears to infinity as the boundary of

R is crossed. However, as D

is entered, the

configuration transforms smoothly into a spherical

liquid bridge, connecting the two faces of the wedge

without contacting the wedge line.

Stability Questions

A number of authors, for example, Langbein, Vogel,

Finn and Vogel, Steen, and Zhou, have studied the

Continuous, (I), (D)

(D)

–

(I )

(DI

), (D), (I )

(ID), (D), (I

)

2(π – α)

2(π

– α)

2α – π

2α

– π

Figure 19 p < 2a < 2p. Possible modes of behavior. Repro-

duced with permission from the Pacific Journal of Mathematics.

Capillary Surfaces 443

stability of liquid drops trapped between parallel

plates, forming an annular liquid bridge joining the

plates under the capillarity boundary condition of

prescribed contact angles 

, 

on the respective

plates. These studies consi der the effects of dis-

turbances within the fluid, assuming the plates are

rigid and perfectly parallel. CFM show that from the

point of view of physical prediction, the results of

these studies may be open to some question.

Specifically, they show that unless the drop is

initially of spherical form, then infinitesimal tilting

of one of the plates always results in a discontinuous

transition of the drop form. Depending on the

particular data, the transition can be to a spherical

drop; however, it can also occur that the tilting

causes the entire fluid to disappear to infinity in the

wedge.

CFM proved that if a connected liquid mass with

spherical outer surface S cuts off areas W

, W

from plates 

, 

which it meets in angles 

, 

,as

in Figure 20, then





cos 

þ S

3 V

½30

where Sjjdenotes area of the spherical free surface

interface, V

the enclosed volume, and R the radius.

An immediate consequence is that the mechanical

energy E of the configuration is

E ¼

3 Vjj

½31

where  is surface tension. Using this result, they

show that if a spherical liquid mass meets two

wedge faces in angles 

, 

in the absence of

gravity, then the configuration has smaller mechan-

ical energy than does any connected liquid mass of

the same volume that meets only one of the faces in

the contac t angle for that face. In turn, the drop on a

single face has smaller energy than does a spherical

ball of the same volume that meets no face. Note

that in all zero-gravity cases for which stability

relative to plate tilting can be expected, the liquid

mass must be spherical.

Compressibility

Until very recently, all literature on capillarity was

based on a hypothesis that the body of the fluid

is incompressible. Indeed, from the point of view

of macroscopic mechanical measurements, most

liquids are nearly incompressible. But all liquids are

also to some extent compressible, and this property

was even conceptually essential in our characteriza-

tion in the section ‘‘Gauss’ contribution: the energy

method’’ of the surface energy, even for the nomin-

ally incompressible case. It is as yet unclear to what

extent the compressibility properties of the bulk

liquid will influence the physical predictions of the

theory. In this connection, see the remarks at the end

of the section ‘ ‘Uniqueness and nonuniqueness.’ ’

The Equations I

Finn derived two possible equations extending [16]

and [17], arising from different modelings. Both

characterize equilibrium points as stationary points

for the mechanical energy, and both are based on a

hypothesized pressure–dens ity relation  = 

(p  p

). The first equation takes account of

the change in density with height, arising from

(b) Center point

(b) In R, near D

–

(a) In R, near D

–

(A) (B)

Figure 20 (A) Drop configurations in wedge with opening

angle 2a = 50



, for three data positions on the line g

= g

(a) g = 70



(in R, near D

); (b) g = 90



(in R, near D



); (c)

g = 110



(in D



). The first two cases yield edge blobs, the third a

spherical tube that does not contact the edge line. (B) Drop

configurations in a wedge of opening angle 2a = 50



, for three

data choices in R, on the line g

= p  g

= g ; (a) g = 70



(near

); (b) g = 90



(center of R); (c) g = 35



(near D



). As D



entered, original boundary conditions can no longer be satisfied

by spherical drop, but configuration changes smoothly into drop

on single plane, with prescribed data for that plane. Reproduced

with permission from Concus P, Finn R and McCuan J (2001)

Liquid bridges, edge blobs, and Scherk-type capilliary surfaces.

Indiana University Mathematics Journal 50: 411–441.

444 Capillary Surfaces