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To appear in the Belgian Bull. Simon Stevin
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prescribed mean curvature equation - Green function
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\title
Solutions of H-systems using the Green function
\endtitle 

\centerline
{P. Amster.  M. C. Mariani and D.F. Rial}

\centerline
{Departamento de Matem\'{a}tica, FCEyN-UBA }



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\lema{Abstract:}

We find a solution to the mean curvature equation with 
Dirichlet condition using the Green representation formula.
Moreover, given $H_0$ and $g_0$ 
for which there exists a solution 
to the problem, we prove that for $H$ and $g$ in 
appropriate
neighborhoods of $H_0$ and $g_0$, there still exists a solution. 

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\bigskip

\tit{1. Introduction}

We consider the Dirichlet problem 
in a bounded smooth 
domain $\Omega \subset {\R^2}$
for a vector function 
$X: \overline{\Omega}\longrightarrow {\R^3}$ satisfying 
the prescribed mean curvature equation 
$$
\text{(1)} 
\cases
\Delta X=2H(u,v,X) X_u\land X_v \qquad \text{ in }\Omega &\\ 
X=g\ \qquad \text{ on }\partial \Omega
\endcases
$$
where $X_u$ and $X_v$ are the partial derivatives of $X$,
$\land$ denotes the exterior product in ${\R^3}$. We'll 
assume that
$H:\overline \Omega \times \R^3\longrightarrow {\R}$ 
is continuous
and that $g$ is smooth. Without loss of 
generality, we may extend $g$ to a
harmonic function in $C^1(\overline \Omega)$.

Problem (1) and the general Plateau problem have 
been studied by variational methods for constant $H$ and
$H=H(X)$
 in [BC], [H], [LDM], [S], [W], among other authors.
Topological methods are applied for the case $H=H(u,v)$ in
[AMR].

\tit {2. Existence of a solution}

We recall the Green representation 
formula for the Dirichlet problem [GT], valid for 
$X:\overline\Omega\lra \R^3$: 
$$X=\int_\Omega G \Delta X +\int_{\partial \Omega}\frac
{\partial G}
{\partial \nu}X$$
where $G:\overline\Omega\times\overline\Omega\lra R$ is defined by
$G(w_1,w_2)=N(w_1,w_2)+h(w_1,w_2)$, with $N$ the newtonian potential
and $h(w_1,\cdot )$ harmonic such that
$h(w_1,\cdot )=-N(w_1,\cdot )$ on
$\partial \Omega$. 
Let $s_1=supr_{w_2}\Vert G(\cdot ,w_2)\Vert_1$, 
$s_2=supr_{w_2}\Vert \nabla_{w_2} G(\cdot ,w_2)\Vert_1$
and $s=max \{ s_1,s_2\}$.
For $R>0$, we consider 
the compact set 
$K_R=\overline \Omega \times (g(\overline\Omega)+ RB_1)$, 
where 
$B_1$ is the closed unit ball in $\R^3$.

Then we can
prove 
the following theorem:

\lema {Theorem 1}

Let $f(R)=\frac {\Vert H|_{K_R}\Vert_\infty}R 
(\Vert \nabla g\Vert_\infty+R)^2$. 
Then, 
if $f(R)\le \frac 1s$ for some
$R >0$, problem (1) admits a solution.

\demost {Proof}

Being $g$ harmonic, $g=\int_{\partial\Omega}\frac{\partial G}
{\partial \nu}g$.
Then, if we define the operator 
$T:C^1(\overline\Omega)\lra C^1(\overline\Omega)$
given by 
$$TX(w)=g(w)+2\int_\Omega G(\cdot ,w)H(\cdot ,X)X_u\land X_v,$$
any solution of (1) may be regarded as a fixed point of $T$.

By Arzel\`a-Ascoli, $T$ is compact. Moreover,
for $\Vert X-g\Vert_{1,\infty}\le R$, we have that
$$\Vert TX-g\Vert_{1,\infty}\le s\Vert 2H(\cdot ,X)X_u\land X_v\Vert_\infty
\le sf(R)R\le R$$
for some $R>0$. Then, $T(B_R(g))\subset B_R(g)$ and
by Schauder's Fixed Point Theorem we conclude that 
$T$ has at least a fixed point.

\medskip 
As a simple consequence, 
we see that (1) admits a 
solution when $\nabla g$ is small enough. Indeed, fixing
$\overline R$ 
such that $\overline R\Vert H|_{K_{\overline R}}\Vert_\infty\ge\frac 1s$, 
and
calling 
$h=\Vert H|_{K_{\overline R}}\Vert_\infty$, we obtain:

\lema {Corollary 2}

Let us assume that
$\Vert \nabla g 
\Vert _\infty \le \frac 1{4sh}$. Then (1) admits a solution 
in $ B_R(g)$ for some $R \in (0,\overline R]$.

\demost {Proof} 

For $R\le \overline R$, we have that 
$f(R) \le \frac hR (\Vert \nabla g\Vert_\infty+R)^2$. 
Then, the hypothesis of Theorem 1 is 
fulfilled if $(\Vert \nabla g\Vert_\infty+R)^2\le \frac R{sh}$ for 
some $R \in (0,\overline R]$, and a simple computation shows that
this is equivalent to the condition 
$\Vert \nabla g 
\Vert _\infty \le \frac 1{4sh}$. 

\lema{Remark}: in particular, for 
$H=H(u,v)$ problem (1) is solvable for  
$\Vert \nabla g 
\Vert _\infty \le \frac 1{4s\| H \|_\infty}$

\rm

\tit {3. Solutions for small perturbations of $H$ and $g$}

In this section 
we'll prove under some conditions 
that if (1) is solvable for some 
$(H_0,g_0)$, then there exists a solution for any 
$(H,g)$ close enough to
$(H_0,g_0)$:

\lema {Theorem 3}

Let us assume that (1) admits a solution 
$X_0 \in W^{2,p}(\Omega,\R^3)$
for some 
$g_0 \in W^{2,p}$ 
with $p>2$
and 
$H_0=H_0(u,v)$.
Then, if 
$2\Vert H_0\nabla X_0\Vert_\infty < \sqrt {\lambda_1}$
where 
$\lambda_1$ is the first eigenvalue of $-\Delta$,
problem (1) is solvable for any
$(H, g)$ close to $(H_0,g_0)$ in $L^p\times W^{2,p}$.

\demost{Proof}

Let us consider $H$, $g$ such that
$\Vert g-g_0 \Vert_{2,p} < \delta_1$ and
$\Vert H-H_0 \Vert_{p} < \delta_2$. 

We look for a solution 
$X$
of (1), which is equivalent, taking 
$Y = X-X_0$, to find a solution of
the 
equation 
$$
\cases
\Delta Y=2H(u,v)(Y+X_0)_u\land (Y+X_0)_v -2H_0(u,v) X_{0_u}\land X_{0_v}
\qquad \text{ in }\Omega &\\
Y=g-g_0 \qquad  \text{ in }\quad \partial \Omega &
\endcases
$$
If we consider 
the operator 
$LY = \Delta Y -2 H_0 (X_{0_u}\land Y_v + Y_u \land X_{0_v})$,
last equation becomes
$$LY=2H_0Y_u\land Y_v
+ 2(H- H_0)(Y+X_0)_u\land (Y+X_0)_v$$

By lemma 4 below and the Sobolev imbedding 
$ W^{2,p}\hookrightarrow C^1(\overline \Omega)$
we may 
define a continuous
operator $T:C^1\lra C^1$, given by
$T(\overline Y) = Y$ where $Y$ is the unique solution in
$(g-g_0)+W^{2,p}\cap W_0^{1,p}$ 
of the linear problem
$$LY = 
2H_0 \overline Y_u\land \overline Y_v
+ 2(H- H_0)(\overline Y+X_0)_u\land (\overline Y+X_0)_v$$ 
Moreover, as
$$\Vert T(\overline Y)-( g-g_0)  \Vert_{2,p} \le 
c (\Vert L(T(\overline Y)) \Vert_p+\Vert L(g-g_0)\Vert_p),$$
the range of a 
bounded set is bounded with $\Vert \quad \Vert_{2,p}$, and
by the compactness of the imbedding $W^{2,p}\hookrightarrow C^1$, 
we conclude that $T$ is compact.
Furthermore, for
$\Vert \overline Y \Vert_{1,\infty} \le R$ 
we obtain:
$$\Vert T(\overline Y) \Vert_{1,\infty} \le \Vert g-g_0 \Vert_{1,\infty}
+ c_1c (\Vert L(T(\overline Y)) \Vert_p+\Vert L(g-g_0)\Vert_p)$$
$$\le k\delta_1
+c_1c \Vert 
2H_0 \overline Y_u\land \overline Y_v
+ 2(H- H_0)(\overline Y+X_0)_u\land (\overline Y+X_0)_v 
\Vert_p$$ 
$$\le k\delta_1 + c_1c(\Vert H_0 \Vert_p R^2 + 
\delta_2(\Vert \nabla X_0 \Vert_\infty+R)^2)$$
\noi for some constant $k$. 
Then, if 
$R$, $\delta_1$
and $\delta_2$ are small enough,  
we have that 
$T(B_R(0)) \subset B_R(0)$
 and 
the result follows from
Schauder's Theorem.

\tit {4. A technical lemma }

In this section we extend a classical result for a 
linear second order elliptic operator defined in
$W^{2,p}(\Omega,\R)$:

\lema {Lemma 4}

Let $L:W^{2,p}(\Omega,\R^3) \lra L^{p}(\Omega,\R^3)$ 
be the linear elliptic operator given by
$LX = \Delta X + AX_u + BX_v + CX$,
with $A,B,C \in L^{\infty}(\Omega,\R^{3\times 3})$, 
$2<p<\infty$
and 
assume 
that
$r:= \left( \frac {\Vert |A \vert ^2 + \vert B \vert ^2 
\Vert_\infty}{\lambda_1}\right)^{1/2}
 < 1$ and
$C \le \delta < \lambda_1(1-r) $, 
where
$\lambda_1$ is the first eigenvalue of $-\Delta$.

Then:

\noi  a) There exists a constant $c$ such that
$$\Vert X \Vert_{2,p} \le c \Vert LX \Vert_p$$
for every $X \in W^{2,p} \cap W_0^{1,p}(\Omega,\R^3)$.

\noi  b) The Dirichlet problem
$$
\cases
LX = f
\qquad \text{ in }\quad \Omega &\\
u = \varphi \qquad  \text{ in }\quad \partial\Omega
\endcases
$$
is uniquely solvable in $W^{2,p}(\Omega,\R^3)$ for any 
$f\in L^p$, $\varphi \in W^{2,p}(\Omega,\R^3)$.

\demost{Proof}

Let $Z_n \in W^{2,p} \cap W_0^{1,p}(\Omega,\R^3)$
be  
a sequence 
such that $\Vert LZ_n \Vert _p \lra 0$.
Then $\Vert LZ_n \Vert _2 \lra 0$, and
as 
$\int LZ_nZ_n \le -\|\nabla Z_n\|_2^2+
\Vert (|A \vert ^2 + \vert B \vert ^2 ) ^{1/2}\Vert_\infty 
\|\nabla Z_n\|_2\| Z_n\|_2
+ \int CZ_nZ_n\le
 ( r -1 + \frac {\delta}{\lambda_1}) \|\nabla Z_n\|_2^2$,
we conclude that 
$\Vert \nabla Z_n \Vert_{2} \lra 0$. 
By Poincar\'e's inequallity, we obtain
 that $\Vert Z_n \Vert_{2} \lra 0$
and hence 
$\Vert\Delta Z_n \Vert_2 \lra 0$. 
As a) holds 
for  
$\Delta$ and $1<p<\infty$ (see [GT]), 
choosing a subsequence we may assume that 
$\Vert Z_n \Vert_{2,2} \lra 0$, and by Sobolev imbedding
$\Vert Z_n \Vert_{1,p} \lra 0$. This shows that
$\Vert \Delta Z_n \Vert_{p} \lra 0$, a contradiction.

In order to prove b), we'll apply a continuation method:
let us define for
$0\le t\le 1$
the operator
$L_tX=\Delta X + t(AX_u + BX_v)$. 
We'll
show that for any 
 $f \in L^{p}(\Omega,R^3)$, 
$\varphi \in 
W^{2,p}(\Omega,R^3)$
the equation (*) 
$L_tX=f \text { in } \Omega$, $X=\varphi \text { in } 
\partial \Omega$ is solvable for any $t \in [0,1]$.
For $t=0$ this is immediate; moreover, we can see that

1) If (*) has a solution (for any $f$) for every $t \in [0,t_0]$, 
then there exists $\epsilon_0$ such that (*) has a solution 
for every $t \in [0,t_0+\epsilon_0)$: let
$t=t_0+\epsilon$, 
and
$\overline X \in W^{2,p}(\Omega,R^3)$ fixed, then
$A \overline X_u + B \overline X_v  \overline X \in L^p(\Omega,R^3)$ 
and
the equation 
$$\cases
L_{t_0}X = f - \epsilon (A \overline X_u + B \overline X_v + C
\overline X) \quad \text {in } \Omega &\\
X = \varphi \qquad \text {in } \partial\Omega &
\endcases
$$
admits a unique solution $X$. 
Thus we define an operator
$T:\overline X \lra X$, and 
for $X=T(\overline X), Y=T(\overline Y)$, 
as $X=Y$ in $\partial \Omega$ we have by a):
$$\Vert X - Y\Vert_{2,p} \le c(t_0) \Vert L_{t_0} 
(X - Y) \Vert_p = c(t_0) \epsilon \Vert A(\overline X_u - \overline Y_u)
+ B (\overline X_v - \overline Y_v)
+ C ( \overline X - \overline Y) \Vert_p$$
$$\le c(t_0) \epsilon (\Vert A \Vert_\infty \Vert \overline X_u - \overline Y_u 
\Vert_p + \Vert B \Vert_\infty \Vert \overline X_v - \overline Y_v 
\Vert_p + \Vert C \Vert_\infty \Vert \overline X - \overline Y
\Vert_p)$$
Thus
$$\Vert T(\overline X) - T(\overline Y)\Vert_{2,p} \le c(t_0)
(\Vert A \Vert_\infty+\Vert B \Vert_\infty+\Vert C \Vert_\infty)
\epsilon 
\Vert \overline X - \overline Y \Vert_{2,p}$$ 
Choosing $\epsilon$ small enough, $T$ is a contraction and then it
has a (unique) fixed point. 

2) If (*) has a solution (for any $f$) for every
$t \in [0,t_0)$,
then (*) has a solution for $t_0$: as in the previous
case,
we define $T(\overline X) = X$, 
where $X$ is the only
solution of
$$\cases
L_{t_0-\epsilon}X = f - \epsilon (A \overline X_u + B \overline X_v + C
\overline X) \quad \text { in } \Omega &\\
X = \varphi \qquad \text  { in } \partial\Omega &
\endcases
$$

Note that 
$\Vert L_t X \Vert_p \ge \Vert L_{t_0} X \Vert_p - 
\Vert (L_t - L_{t_0})X \Vert_p \ge \dfrac 1{c(t_0)}\Vert X \Vert_{2,p}
-\Vert (L_t - L_{t_0})X \Vert_p$, and 
$$\Vert (L_t - L_{t_0})X \Vert_p \le \epsilon k\Vert X \Vert_{2,p}$$
\noi for some constant $k$.
Thus, choosing $\epsilon$ and $c$ such that,
$\dfrac 1{c(t_0)} - \epsilon k = 
\dfrac 1c >0$, we see that we may consider
$c(t_0-\epsilon) \le c$
in a neighbourhood of $t_0$. As in the previous case,
$T$ is a contraction for $\epsilon$ small enough.

\lema {Remark: }

Lemma 4 holds for a general 
linear second order elliptic 
operator $L$ 
defined in
$W^{2,p}(\Omega,\R^n)$, 
considering $\lambda_1$ the 
first eigenvalue of
the second order part of $L$. 

\rm

\tit {References}

[AMR] Amster P. Mariani, M.C, Rial, D.F: Existence and uniqueness of H-System's 
solutions with Dirichlet 
conditions. 
To appear in Nonlinear Analysis, Theory, Methods, and Applications.

[BC] Brezis, H. Coron, J. M.:Multiple solutions of $H$ systems and
Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), 149-187.

[GT] Gilbarg, D. Trudinger, N. S. : Elliptic partial differential
equations of second order, Springer- Verlag (1983).

[H] Hildebrandt, S.: 
On the Plateau problem for surfaces of constant mean curvature. 
Comm. Pure Appl. Math. 23 (1970) 97-114.

[LDM] Lami Dozo, E., Mariani, M. C.: A Dirichlet problem for an H-system 
with variable H. Manuscripta Mathematica 81 (1993), 1-14. 



[S] Struwe, M.: Plateau 's problem and the calculus of variations,
Lecture Notes Princeton Univ. Press (1988).

[W] Wang Guofang: The Dirichlet problem for the equation of
prescribed mean curvature, Analyse Nonlin\'eaire 9 (1992), 643-655. 

\bigskip 

\tit{P. Amster${}^*$,  M. C. Mariani ${}^*$ and D. F. Rial}


Dpto. de Matem\'{a}tica, Fac. de
Cs. Exactas y Naturales, UBA. Pab. I, 
Ciudad Universitaria. (1428)
Buenos Aires, Argentina.

${}^*$ CONICET


\tit{Address for correspondence:} 

Prof. M. C. Mariani, 
Dpto. de Matem\'{a}tica, Fac. de Cs. Exactas y Naturales, UBA. Pab. I, 
Ciudad Universitaria. (1428) Buenos Aires, Argentina.

\bigskip
{\bf email:} pamster\@dm.uba.ar - mcmarian\@dm.uba.ar - drial\@dm.uba.ar 

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