The prescribed mean curvature equation with Neumann condition

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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

Transcript of The prescribed mean curvature equation with Neumann condition

Page 1: 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.

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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

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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.

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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

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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).

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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).