A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
C LAUS G ERHARDT
On the capillarity problem with constant volume
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 2, n
o2 (1975), p. 303-320
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Capillarity
CLAUS GERHARDT
(**)
0. - Introduction.
In this paper we shall discuss
capillary problems
which arisephysically
when the
equilibrium
surface of aliquid
of fixed volume in acylinder
isanalysed.
The surface u is determinedby
theprinciple
of virtual work which leads to the variationalproblem
where S~ is the cross-section of the
cylinder,
x is the(nonnegative) capillarity constant,
theobstacle 1p represents
the bottom of thecylinder,
and V is theprescribed
volume. The is the cosine of theangle
be-tween the free surface and the
bounding cylinder walls,
isabsolutely
,
bounded
by
1.The
solvability
of the variationalproblem depends
onestimating
theboundary integral by
aQ
where
Q,
is theboundary strip fx
E D:dist(x, el
and c, a constantdepending
on 8 and 8Q.(*) During
thepreparation
of this article the author was at the Universite de Paris VI as a fellow of the DeutscheForschungsgemeinschaft.
(**)
Mathematishes I nstitut der Universitdt, Bonn,Wegelerstr.
10, D - 5300 B onn . Pervenuto alia Redazione il 18agosto
1974.The
simplest
and mostfar-reaching
condition which we shallimpose
onaSZ,
in order that an estimate of this kind
holds,
is an interiorsphere
condi-tion
(ISC):
DEFINITION 0.1. S~ is said to
satisfy
an ISC of radius1~,
if for anyboundary point
xo E êJQ there exists a ball B of radius .R such that B c Q and Xo EB.
REMARK 0.1. An
equivalent
statement is to say that every interiorpoint
x E Q is contained in ball B of radius .R which lies
entirely
in Q.The main theorem which we shall prove is the
following
THEOREM 0.1..Let Q be a bounded domain
of Rn, n >- 2,
withLipschitz .boundary
aSZsatisfying
an ISCof
radiusR,
and let andbe
given functions.
Then thefollowing
results are valid(i)
The variationalproblem (0.1)
has aprovided that fl
is boundedby
Moreover,
u also solves the variationalproblem
where A is a suitable
Lagrange multiplier.
(ii) If
1p issupposed
to beof
classH2~p(S2),
n p ,00, then u has thesame
degree of
smoothnesslocally,
i. e.(iii)
In the case that x isstrictly positive
the solution isuniquely
deter-mined in
BV(Q)
and thepreceding
results are validfor
anywith ~8 ~ ~ 1,
and there exists apositive
number V*such that
provided
that1. - A
priori
boundsfor I u 1.
In the
following,
we shall consider aslightly
moregeneral
variationalproblem
than thepreceding
one: Letand j :
begiven
functions such thatwhere a is some
positive constant, and j
satisfies aCaratheodory condition,
i. e. it is measurable in x
(with respect
to the(n -1 )-dimensional
Hausdorffmeasure on
9D)
and continuous in the second variable.Furthermore,
weassume that for
Jen-1 - a.e. j(x,)
is a strictcontraction,
i. e.where a is
independent of x,
thatis convex,
The functional I is contained in this
general setting taking .H(x, t) = x ~ t
and
j(x, t) _ -~(x) ~t. Furthermore,
let us remark thatThen,
we consider the functionalcp E
L1( oQ),
also satisfies the conditionsimposed
onj.
Under the
preceding assumptions
onQ, H, and j
we can proveLEMMA 1.1. Let u be a solution
of
the variationalproblem
Then u can be estimated
by
where the constant C1
depends
on(p>n),
a, n, andon a constant co which will be
defined
in thefollowing.
Here,
we denoteby 11 - ll,,, q? 1,
the norm inBefore
proving
thelemma,
let us mention a result which has been derived in[7;
Remark2].
LEMMA 1. 2. Let Q be a bounded domain
o f Rn,
n ->-2,
withLipschitz boundary satisfying
an ISCof
radius R. Then thefollowing
estimate is validwhere
S~~
==fx
E D:dist(x, 8},
c,depends
on 8,R,
and3D,
and E is anypositive
number less than orequal
toPROOF OF LEMMA 1.1. Let k be a
positive
numbergreater
thann
and set Uk =
min (u, k).
Then Ukbelongs
to.K2
and from the minimumproperty
of ~c weget
Hence, using
the notationA(k)
=u(z) > k}
andsupposing
for a mo--ment u to be
smooth,
we obtainwhere
I
denotes theLebesgue
measure in ltn ofA(k).
In view of the condition
(1.2)
theboundary integral
can be estimatedby
or,
taking
Lemma 1.2 intoaccount, by
where we have set
Furthermore, observing
thatin view of the
monotonicity
ofH(x, ~ ),
we deduce from(1.10)
and(1.12)
the
inequality
which will also be valid for
using
anapproximation argument
(cf. [7;
LemmaA4]).
To estimate the
integral
we use the Hölderinequality,
where we denote
by
n* theconjugate exponent, lfn* = I - l fn.
Thus, using
the Holderinequality again,
we obtain from( 1.14 )
Now, applying
the Sobolevimbedding
theorem andusing
the fact thatwe derive from
(1.16)
for where
l~o
and C2depend
on a, co , and knownquantities.
Hence,
we concludeor
finally
From a lemma due to
Stampacchia [13;
Lemma4.1]
we now deduceThough u
isobviously
bounded from below it would be worth toget
thesharper
estimate(1.7),
forby
this we had also derived a bound for solutions to the freeproblem
setting formally
_ - oo .In order to obtain the lower bound we insert ~ck = max
(u,
-k)
in(1.9).
The
proof
of Lemma 1.1 can then becompleted by
similar considerationsas above.
2. - Existence and
regularity
of solutions to the variationalproblem.
Generally,
y the functional J does not have a minimum in the convexset unless we
impose
somegrowth
condition on H.However,
we canprove a rather abstract existence theorem which will be very useful in the
following.
THEOREM 2.1. Let Q and J be as in Lemma
1.1,
where we may assumethat j is only a contraction,
i.e.Let be convex and closed with
respect
to convergence inL1(,Q).
Furthermore,
let v,, be aminimizing
sequencefor
the variationalproblem
such that
and
uniformly
in c. Then asubsequence of
the Ve’S converges inL1(Q)
to somef unction u E B V (,S2 )
which minimizes J.PROOF. The theorem has more or less been demonstrated in
[7; Ap- pendix II],
but for convenience we shallrepeat
the rather shortproof.
From
[12;
TheoremXVI],
the Sobolevimbedding theorem,
and[11;
Theorem2.1.3]
we conclude from(2.4)
that the we’s areprecompact
in any
Ll(,Q), Hence,
let us suppose forsimplicity
thatve - u in Assume
by
contradiction thatJ(u)
isstrictly greater
thanlim
J(Ve). Then,
there exists apositive constant y
and a number Eo such thatIn view of
(1.8)
and(2.1)
we have the estimatewhere Qd
is aboundary strip
ofwidth 6,
and 6 issufficiently
small.Thus,
we deduceIf 8 tends to zero, then we obtain in view of the lower
semicontinuity
of theintegrals
on the left side of(2.7) (cf. [8;
formula(64)]
To
complete
theproof,
we let 6 converge to zero whichgives
the con-tradiction.
The interior
regularity
of solutions to the variationalproblem (1.6)
follows from the theorem below which has been
proved
in[8;
Theorem 6-and Lemma
4].
THEOREM 2.2. Let w be a
locally
bounded solution inBV(Q) of
the varia-tional
problem
where .~ E X
R) satisfies alllat >
0. Then w islocally Lipschitz
in Qprovided that 1p
E and we have the interiorgradient
estimate-Furthermore, if
we assume 1p to beof
classH2oV(Q),
n p .00, then wbelongs
to
c Precisely,
we have the estimatewhere A is the minimal
surface operator
indivergence form.
3. - Existence of a
Lagrange multiplier.
In this section we shall show the existence of a real number A such that the variational
problem
has a solution
UA E K2
such thatwhere the volume V is
prescribed. Thas, ui
also solvesTHEOREM 3.1.
Suppose
thatQ,
1jJ,H and j satisfy
the conditions stated in Section 1.Then,
the variationalproblem
has a solution u E r1
for
anyprescribed
volumeu also solves the variational
problem
where 2 is a suitable
unique Lagrange multiplier.
Themapping
are continuous and
nondecreasing
resp.nonincreasing. I’urthermore,
issupposed
to beof
classH2~~(S~),
then usatisfies
PROOF. For E > 0 set
Similarly,
we define Je andThen,
forarbitrary
p E R we shall demonstrate thefollowing
lemma.LEMMA 3.1. Let E, 0 E
1,
begiven. Then,
under thepreceding
assump-tions,
the variationalproblem
has a
unique
solution - > such that the estimatesand
21 - Annali della Scuola Norm. Sup. di Pisa
are
valid,
where thepositive
constants areindependent from 8 and
fl, andlill
denotes theLebesgue
measureof
S2.PROOF OF 3.1. In order to prove the existence of a solution to
(3.11 )
let v, be a
minimizing
sequence of the variationalproblem. Then, taking
the estimates
and
into account we deduce from Lemma 1.2
where we have set Co =
CRf2.
Thus,
we conclude that the sequenceis
uniformly
bounded.Hence,
the existence of a solution ue,, follows from Theorem 2.1.Moreover,
since the functionalJ~,,~
isstrictly
convex thesolution is
unique.
TheLipschitz regularity
and the boundedness ofare consequence of the Theorem 2.2 and Lemma 1.1.
To derive the estimate
(3.12),
we observe that theinequality (3.16)
is satisfiedby too;
hence the result.On the other
hand, r~ > o,
0 begiven. Then,
~a
+ 3q belongs
toK2
and we obtain from the minimumproperty
of u,,,,Therefore,
wefinally
conclude that theinequality
is valid.
Now,
the estimate(3.13)
follows frominserting
1 in thepreceding inequality.
Let us remark that we needed the
assumption only
for this estimate.Thus,
if we define for fixed swe deduce from
(3.12)
and(3.13)
and
The existence of a
Lagrange multiplier
now follows from the fact that Vdepends continuously
on u.3.2. Let the
assumptions of
Lemma 3.1 besatisfied. Then, for fixed
s, themapping
is continuous.
PROOF OF LEMMA 3.2.
Let p,
be aconvergent
sequence,and let u,,, resp. be the solutions to the variational
problem (3.11) }
with the
respective
functionals andJe,po. Then,
the form a mini-mizing
sequence for the variationalproblem
such that the
integrals
are
uniformly
bounded(cf.
formula(3.11)).
The rest of theproof
now followsimmediately
in view of the Theorem 2.1 and theuniqueness
of the solution.Thus,
for fixed c and V there exists aparameter Âe
such that the solu- tion of the variationalproblem
satisfies
Hence,
we obtainMoreover,
from Lemma 3.1 and Lemma 1.1 we deduce that~e
andare
uniformly
bounded for fixed V.Hence,
theintegrals
are
uniformly
bounded.Thus, letting
s go to zero, asubsequence
of the converges to some real number Â. Therespective
solutions then form aminimizing
sequence for the variationalproblems
Hence,
we conclude from Theorem 2.1 that asubsequence
of theconverges in to some function
uÂEBV(Q)
which solves both varia- tionalproblems. Furthermore,
the solution of the variationalproblem (3.32)
is
uniquely
determined in the class since the difference of two solu- tions must be aconstant,
which has to be zero in view of the side conditions.Thus,
the firstpart
of Theorem 3.1 isproved.
It remains to prove the
properties
of themappings h,
andh2,
since theinterior
regularity
of u follows from the estimates forue.ï.s.
First of
all,
let us observe that bothmappings
arecontinuous,
y whichfollows from the fact that
they
arecompact
and the solution ~c as well as theLagrange multiplier
 areuniquely
determined.The
monotonicity
ofhI
andh2
will be a consequence of thefollowing
lemma
LEMAIA 3.3. Let ut and u~* be solutions
of
the variationalproblem (3.33)
with
respect
to the dataÂ, 1p, j
and2*, 1p*, j*, where,
in contrast to condition(1.2 y j
resp.j*
are notforced
to be strict contractions.They
areonly supposed
to beuniformly Lipschitz
in t.Moreover,
we assume that at least oneof
the solu-tions uÂ* is
unique. Then,
obtainprovided
that the relationsand
are
valid,
andwhere, furthermore,
the -j~‘(x, - )
issupposed
tobe
nondecreasing
a. e. in3D,
which canformally
be written asSuppose
the lemma to be valid.Then,
the solution U8 of theperturbed problem
where we have
replaced H by Ht(x, t)
=H(x, t) -f- ~ ~ t,
isunique.
Further-more, the solution coincides with the one of the variational
problem
if  is
equal
to theLagrange multiplier Âs.
For0,
let the functiondescribe the
dependece
between V and1,,
and defineand
similarly.
Then,
we deduce thath2.e
isnonincreasing
for has thisproperty;
hence, nondecreasing.
In the limit case, 8 the functions and
h2.e
converge to the func- tionsh,
andh2
which can be seenby using
acompactness argument
and theuniqueness
in of the solution to the variationalproblem (3.32).
Thus,
tocomplete
theproof
of the Theorem3.1,
we havemerely
todemonstrate Lemma 3.3.
PROOF oF LEMMA 3.3.
Suppose
that ui is theunique
solution.Then,
we have
and
Combining
these relations andusing
the fact thator
equivalently
which can
easily
be checkeddistinguishing
the cases and >in view of
(3.37),
we deduce from(3.43)
thathence the result.
REMARK 3.1. Let
j(x, t)
-it
-I
andj*(x, t)
-it
-gg*(x) I
with91*
ELI(aSZ). Then,
the condition(3.37)
is satisfiedprovided
thatfor we have
4. - The case where H satisfies a certain
growth
condition.In this section we shall assume that besides the
preceding
conditions H satisfies the relationsThen,
we can prove thefollowing generalization
of Theorem 3.1.THEOREM 4.1.
Suppose
that Hsatisfies
thegrowth
conditions(4.1 )
and(4.2).
Then,
the resultsof
Theorem 3.1 remain validif
wereplace
the !condition(1.2 ) by
the moregeneral assumption
Moreover,
there exists apositive
number V* such that a solution u EH1’I(Q) of
the variationalproblem
satisfies
provided
thatPROOF. Let us remark that the solution u of the variational
problem (3.4)
is
absolutely
bounded in terms of a,Â,
and Tr(cf.
Lemma1.1),
whereas121
is estimated in terms of V
(cf.
Lemma3.1 ) . Thus,
to prove the firstpart
ofTheorem
4.1,
we haveonly
to show thatlul
is boundedindependently
of a,using
anapproximation argument
afterwards(cf.
Theorem2 .1 ) .
LEMMA 4.1.
Suppose
that Hsatisfies
the conditions(4.1 )
and(4.2),
andlet
uEH101(Q)
be a solutionof
the variationalproblem (3.5). Then,
u is ab-solutely
boundedby
some constant m, whichonly depends
onH, R, Â,
n, and sup max(y, 0).
92
PROOF OF LEMMA 4.1. We shall
only
show the existence of an upperbound,
since the lower bound could be establishedby
similar considerations.First of
all,
let us assume thatThen, u belongs
to and is a solution of the variationalinequality
where A is the minimal surface
operator
and Q’ anycompact
subdomainof Q. From
(4.7) }
and theregularity
of u weimmediately
deduceNow,
letB~
be a ball of radius R such that and letB Ro
be a concentric ball of radius.Ro,
where we assumethat the center of the balls lies in the
origin.
Let30
be the lowerhemisphere
Then,
we have I)
andFurthermore,
let M be apositive
constant such thatand
Then,
satisfies theinequality
Combining
the relations(4.8)
and(4.13)
we obtainOn the other
hand,
we know thatIflul
isuniformly
bounded inThus
we deduce
Since we have
Ro
still at ourdisposal,
we choosel~o
near 1~ such thatwhere v is the outward normal vector to
aBR.
*Partial
integration
in(4.14)
then leads to the desired resultin view of
(4.16)
and thestrong monotonicity
of theoperator A + H(x, )- Hence,
we obtainor
finally
As Q satisfies an ISC of radius
R,
it can becompletely
coveredby
balls.of radius R. Hence the estimate
(4.19)
holdsuniformly
in S~.If 1p
ismerely Lipschitz,
weapproximate 1p by
smooth functions Let us be therespective
solutions of(3.5)
whichsatisfy
the estimate(4.19).
Then,
since the solution u isunique.,
the convergeuniformly
oncompact
subdomains of Q
to u,
hence the result.REMARK 4.1. Concus and Finn
[2]
have been the first who used the ISC toget
a bound for the solution to thecapillarity problem.
In order to prove the second
part
of Theorem4.1,
let us observe that thefree problem
has a solution uo e n as follows from the
preceding
considera-tions
(we
mayformally set 1p == - oo), provided
that H satisfies thegrowth
conditions. Let be
sufficiently large
such thatThen we conclude from
(3.6)
that we may chooseREMARK 4.2. The
preceding
results(except
those in the firstpart
ofTheorem
4.1)
are alsovalid,
if S~ ismerely supposed
to be a boundedLipschitz domain, provided that j
satisfies the natural restrictionwhere
and L is a bound for the
Lipschitz
constants of theboundary representa-
tions of aS2
(cf. [7]).
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of capillary surfaces,
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in agravitational field,
Acta Math., 132 (1974), pp. 207-223.[3] P. CONCUS - R. FINN, On
capillary free surfaces
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diequilibrio
neicapil-
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