The prescribed mean curvature equation with Neumann condition
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Transcript of The prescribed mean curvature equation with Neumann condition
Pergamon
Nonlrneor Ano/ym, Theory, Merhods & Applicalions, Vol. 22, No. 9, pp. 1147-1152, 1994 Copyright 0 1994 Elsevier Science Ltd
Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .@I
THE PRESCRIBED MEAN CURVATURE EQUATION WITH NEUMANN CONDITION
ENRIQUE LAMI Dozot$ and MARIA CRISTINA MARIANI$
t Departement de Mathematique, C.P. 214, Universitt Libre de Bruxelles, 1050 Bruxelles, Belgium; $ Instituto Argentino de Matematica, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
(Received 2 November 1992; received for publication 2 June 1993)
Key words and phrases: Mean curvature equation, Neumann problem for the H-surface system
1. INTRODUCTION
LET B = ((u, v) E R2; u2 + v2 < 1) denote the unit disc and H: R3 --, R a given function. We consider the following boundary value problem: given f: dB + R3, find X: B + R3 such that
C (1) AX = 2H(X)X, A X, in B
(NJ (2) g = f on dB
where n(u, IJ) = (u, v) is the exterior normal to aB. As in [l-6] for Dirichlet data on aB, we call (l), (2) the Neumann problem for the H-surface
system. Any surface of constant mean curvature H,, (or Ho-surface) satisfies (1) and the general aim
is a better understanding of Plateau’s problem of finding a surface with prescribed mean curvature H which is supported by a given curve in R3.
We consider boundary data f E C’(aB, R3), function H continuous and bounded, and we call X E H2(B, R3) a weak solution of (N) if
VX*Viq + 2H(X)X,,r\X;y, = 0 for every v, E C,‘(B, R3)
=f
where Tr: H’(B, R3) --* L2(aB, R3) is the usual trace operator [7]. Denoting D(Y) = $ jB (VYJ‘ the Dirichlet integral, and for < E R’,
we will study a special behavior of the functional
D,(Y) = D(Y) + f Q(Y). Y, A Y, B
near a weak solution of (N) which is not a local minimum of DH.
1147
1148 E. LAMI Dozo and M. C. MARIANI
Notations
We denote WmPp(B, R3) the usual Sobolev spaces [7] and Wme2(B, R3) = H”(B, R3). For X E H’(B, R3),
IlXIlL2@B,R”) = ( jaB In xJi)l/i and for YE L”(B, R”), 1) YI(, = ess supi Y(w)l. For example
llfalm = E”~wIMr,l
IiQiim = ywlQK)I
and
IlffW)lL = ess supIH(X(w))I WEB
Finally, o denotes the tangent to aB, and concerning Dw (resp. V) we denote
whenever this limit exists (resp. dv(X)(p)).
2. NECESSARY CONDITIONS
THEOREM 1. Let H: R3 -+ R continuous and bounded, and suppose that X E Hz@, R3) is a weak solution of (N) verifying IjH(X)Xll, < 1. Then
Proof. We have that AX = 2H(X)X,, AX, a.e. in B, then
o= [-AX + 2H(X)X, A X,] . X B
=
i
[IVX(‘+ 2H(X)X*X,r\X,] - B
a,$*X
2 2SV - 2b%Wk I& A &I - Il~ll~~~~~,~~~llfII~~~~~,~~~ B
2 2~~1 - WWF&) - ll~ll~~~~~,~~~Ilfll~~~~~,~~~~ and it follows that
Prescribed mean curvature equation 1149
3. EXISTENCE OF SOLUTIONS
THEOREM 2. Let f E C’(M, R3) and suppose that Hz R3 + R is a function satisfying the following properties:
(i) HE C'(R3)n W19m(R3) and Q E Lm(R3,R3); (ii) there exists g E W2,“(B, R3), g harmonic in B, verifying that 0 < ]]H]l,l]g](, < +$ and
ag/&zlaa = f, and a positive number c > I(Vg]], such that
ImN 5 wc2Nrl - Mm)+ for c E R3
where /2, > 0 is the first eigenvalue of -A in H,(B). Then g is a weak solution of (N) and either g is a local minimum of DH in W2’“(B, R3) fl (X E H2(B, R3); Tr(aX/&-) = f, Tr X = Tr g] or there exists a sequence (X,) in W2’“(B, R3) of distinct weak solutions of (N) with X,, + g in W2+‘(B, R3).
Proof. From (ii), H(g) = 0 on B and g is harmonic, so (N) holds trivially. Now, we choose a positive number 6r such that 6r < min(c - ]]Vg]],, (3/2]jH(],) - \]g11_]
and define
We have that MI is a nonempty, convex, closed and bounded subset of H2(B, R3), then, A4r is a weakly compact subset of H2(B, R3) and by lemma 2 in [8], DH is weakly lower semicontinuous in Ml, because IIQGOllm 5 ll~ll~llXll, 5 ll~ll,dlx - gll, + IIAJ < 3 for XE Ml.
Hence, there exists X, E Ml such that
Suppose that g is not a local minimum of DH in
W2,“(B, R3) fl 1 X E H2(B, R3); Tr $T = f, Tr X = Tr g
1 .
Then, X, # g and Ml is convex, then, if E E [0, l] we have that D&X,) I DH(XI + E(X - X,)) for all X E Ml and it follows from lemma 3 in [8] that
0 I ; DH(Xl + E(X - X,)) = c&(X,)(X - X,).
Then, d&(X,)(X, - X) I 0 and given ~1 E C,‘(B, R3) verifying
Ilvllcc + IIvvllm + C lldij~llca 5 619 dD,W,)(X, - .!?I 5 dDn(X,)(v). 1si.j 52
Hence, we deduce that either dD,(X,)(X, - g) = dD,(X,)(p) = 0 and then, from
Trg=f
for all X E Ml, Xl is a weak solution of (N) or d&(X,)(X, - g) < 0. We will prove that the second case is not possible.
1150 E. LAMI Dozo and M. C. MARLNI
First, we note that
B(X)1 5 (Wc2)(lXl - ll&+ 5 Wc2)(IXI - lgl)+ 5 Wc2)IX - gl
a.e. in B; and I)vXIJ, I IIv(X - g)Ij_ + IIVgll, I c for all X E Mi, then
j 2H(X)X, A x, * (X - g) 5 - 21x, A X,@,/c2)IX - g/2 2 - IVX12(1i/C2)IX - g12
B i
2 -dix - AIt 2 -IlVW - km .i B
Also Se Vg * V(X - g) = -jB Ag * (X - g) = 0, because X - g E &(B, R3). Finally, from lemma 3 in [8], we have that
dWX,)(X, - g) = 5
[V-X, * VW, - 8) + =fW,)Xl, A Xl, * (Xl - 811
B
=
.1 Ivw, - &?)I2 + vg * VW, - g) + =fW,)XI, AXI” * (XI - g)
B .r B s B
2 llV(Xl - &M - llV(Xl - g>ll”, = 0.
Now, we choose 6, = minlb,, +(llXI - & + llV(X, - g)ll, + Cl si,js2 Ib,W~ - g)ll,I and define
M2 = g + ~1; VI E ff’(B, R3), Tr$ = Tr o1 = 0 and IIpllm + IIv~I\~ + C IIaij~11~ I b2 . 1 si,j52 1
Then, there exists X2 E M, such that
X2 is a weak solution of (N) and X2 # g, Xi, because Xi $ M2. Hence, we can define a sequence (X,) C W2*-(B, R3) of weak solutions of (N) in
W2sm(B, R3) such that X,, --t g in W2,“(B, R3).
Remark 1. If llQllrn < 3, the condition 0 < llEill,l(gll, < z 3 is not necessary. In this case, we can define the sequence of convex subsets of H2(B, R3) as follows
M, = g + p; a, E H2(B, R3), Tr $ = 0 = Tr (D and JJpoJJ_, + JJv~~JJ,
+ C Il~ijdlm s c - lIvgllco P llsi,jr2 1
and
a2 = min c - (IVgll,, & IIxI - gllc0 + Ilv(xI - g)Ilm + C llaijvlllm lsi,js2
M2 = g + 9; P E ff2(B, R3), Trz = Tr a, = 0 and ll~llm + J)vq)l, + C Il8ijplJm I a2 . lsi,js2 1
Prescribed mean curvature equation 1151
Remark 2. Suppose that H has a sequence & converging to to as unique zeros in R3, then for f = 0, the constant function g = &, is a weak SOhtiOn Of (N), there iS a sequence gk = <k Of
weak solutions of (N) and it seems possible that g is not a local minimum of DH.
4. A UNIQUENESS TYPE THEOREM
It is well known that for a constant H E R and for zero Dirichlet boundary conditions, the only weak solution of an H-system is 0 [9]. On the same vein we have the following result.
THEOREM 3. Suppose that H is constant and that f = 0. Then any weak solution X of (N) is constant.
Proof. We extend X to R2 by a reflection, setting
Y(U, v) = i
X(u, v) in B
X(U/?, v/r*) in R2\B where u* + v* = r*.
A direct computation shows that
i
AY = 2HY,r\ Y, in B
AY= -2HY,/\Y, in R'\B.
If we now consider the conformal measure function
F(u, v) = 1 Y,l* - lY,l* - 2iY, * Y,
we have that the trace of F on aB from the inside and from the outside are equal, because so are the tangential derivatives of Y, and the normal derivatives are 0, by hypothesis. Hence, FE C’(C, C) and F is holomorphic in C\aB, then, F is holomorphic in C. But from
IF(uv ~11 5 2 .i R2 (I r,l 2 + I a*) = 4 (I r,l* + lY,12) = 4
s (1X,)* + 1X,12) < +co
RZ B B
we deduce that F = 0 and then, we have that IX,]* - IX,]* = X,, *X,, = 0 in B. Hence,
and from ax/an = 0 on dB, it follows that X is a weak solution of the problem
t
AX = 2HX,, AX, in B (Dir)
x=c in aB
with c E R3, and there is only one weak solution of (Dir): X = c [9].
REFERENCES
HILDEBRANDT S., On the Plateau problem for surfaces of constant mean curvature, Communspure appl. Math. 23, 97-l 14 (1970). HILDEBRANDT S., Randwertprobleme fur flachen mit vorgeschriebener mittlerer Krummung und Anwendugen auf die kapillaritatstheorie Teil I. Fest Vorgegebener rand, Math. Z. 112, 205-213 (1969). STRUWE M., Plateau’s problem and the calculus of variations, in Lecture Notes. Princeton University Press, Princeton. STRUWE M., Non uniqueness in the Plateau problem for surfaces of constant mean curvature, Archs ration mech. Analysis 93, 135-157 (1986).
1152 E. LAMI Dozo and M. C. MARIANI
5. BREZIS H. & CORON J. M., Multiple solutions of H-systems and Rellich’s conjecture, Communspure appl. Math. 37, 149-187 (1984).
6. WENTE H. C., A general existence theorem for surfaces of constant mean curvature, Math. Z. 120,277-288 (1971). 7. ADAMS R., Sobolev Spaces. Academic Press, New York. 8. LAMI Dozo E. & MARIANI M. C., A Dirichlet problem for an H-system with variable H, Manuscripta Math.
(to appear). 9. WENTE H. C., The differential equation AX = 2H X,, A XV with vanishing boundary values, Proc. Am. math. Sot.
50, 59-77 (1975).