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To appear in Abstract and Applied Analysis
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nonparametric surfaces - variational methods
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\title
Existence and regularity of weak solutions to the prescribed

\endtitle
\title
mean curvature equation for a nonparametric surface
\endtitle 

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

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



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\lema{Abstract:}

We give conditions on the boundary data and the 
function $h$ in order to obtain multiple solutions for the
problem (1) below, using variational methods. We also study the
regularity of the solutions.

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\bigskip

\tit{Introduction}


The prescribed mean curvature equation with Dirichlet condition
for a nonparametric surface $X:\Omega \longrightarrow R^3$, 
$X(u,v)=(u,v,f(u,v))$ is the quasilinear partial differential equation

$$
\text{(1)}
\cases
(1+f_v^2)f_{uu}+(1+f_u^2)f_{vv}-2f_uf_vf_{uv}= 
2h(u,v,f)\left( 1+\left|
\nabla f\right| ^2\right) ^{\frac 32} \text{ in } \Omega &\\
f=g \quad \text{ in }\partial \Omega 
\endcases
$$
where $\Omega $ is a bounded domain in $R^2$, 
$h:\overline \Omega \times
R\longrightarrow 
R$ is continuous and $g\in H^1(\Omega )$.

We call $f\in H^1(\Omega)$ a weak solution of (1) if $f\in
g+H_0^1(\Omega )$ and for every $\varphi \in C_0^1(\Omega )$

$$\int_\Omega \left( ( 1+| \nabla f| ^2)
^{-1/2} \nabla f\nabla \varphi
+ 2h(u,v,f) \varphi\right) dudv=0 
$$

It is known that for the 
parametric Plateau's problem, weak solutions can be
obtained as critical points of a functional 
(see [2], [5], [6], [7], [9], [10]).

The nonparametric case has been studied for
$H=H(x,y)$
(and generally $H=H(x_1,...,x_n)$ for
hypersurfaces in $R^{n+1}$)
by Gilbarg, Trudinger, Simon, Serrin, among other 
authors. 
It has been proved [12] that
there exists a 
solution 
for any smooth boundary data if
the mean curvature 
 $H'$ of $\partial \Omega$ satisfies:
$$H'(x_1,...,x_n) \ge \frac n{n-1} \vert H(x_1,...,x_n)\vert$$
for any
 $(x_1,...,x_n) \in \partial \Omega$, and
 $H \in C^1(\overline \Omega,{R})$ satisfying
the inequality:
$$\vert \int_{\Omega}H\varphi \vert \le \frac {1-\epsilon}n  
\int_{\Omega}\vert D\varphi \vert$$
for any $\varphi \in C_0^1(\Omega,{R})$ and some $\epsilon > 0$. 
They also proved a non-existence result ([12], corollary 14.13):
if
$H'(x_1,...,x_n) < \frac n{n-1} \vert H(x_1,...,x_n)\vert$ 
for some $(x_1,...,x_n)$ 
and the sign of $H$ is constant, 
then for any
$\epsilon >0$ there exists 
$g \in C^\infty(\overline \Omega)$
such that
$\Vert g \Vert_\infty \le \epsilon$ 
and that 
Dirichlet's problem 
is not solvable.

We remark that the solutions
obtained
in [12] are classical. In this work 
we find weak solutions of
the problem
by variational methods.

We will prove that for prescribed $h$ there exists an associated functional
to $h$, and under some conditions on $h$ and $g$ we will find that this
functional has a global minimum in a convex subset of $H^1(\Omega )$, which
provides a weak solution of (1). 
We denote $H^1(\Omega )$ the usual
Sobolev space, [1].

\bigskip

\tit{The associated variational problem}

\medskip

Given a function $f\in C^2(\Omega )$, the generated
nonparametric surface associated to this function is the graph of $f$ in 
$R^3$, parametrized as
$X(u,v)=(u,v,f(u,v)). $

The mean curvature of this surface is

$$h(u,v,f)={1\over 2} {Ef_{vv}-2Ff_{uv}+Gf_{uu}\over
(1+f_u^2+f_v^2)^{3\over 2}} $$
where $E,F$ and $G$ are the coefficients of the first fundamental form
[3], [8].

For prescribed $h$, weak solutions of (1) can be obtained as critical 
points of a functional:

\medskip

\lema{Proposition 1}
Let $J_h:H^1(\Omega )\longrightarrow R$ be the functional defined by 
$$J_h(f)=\int\limits_\Omega \left( ( 1+| \nabla f| ^2)
^{1/2}+H(u,v,f)\right)dudv \tag2$$
where 
$H(u,v,z)=\int_0^z2h(u,v,t)dt$.
Then
(1) is the Euler Lagrange equation of (2).

\lema{Remark} If $f\in T = g + H_0^1(\Omega )$ is a critical point of $J_h$, 
then $f$ is a weak
solution of (1).


\demost{Proof} For $\varphi \in C_0^1(\Omega )$, integrating 
by parts we obtain:
$$dJ_h(f)(\varphi )=2\int_\Omega \left({1\over 2}{Ef_{vv}-2Ff_{uv}+Gf_{uu}\over 
(1+f_u^2+f_v^2)^{3\over 2}}-h(u,v,f)\right)\varphi dudv $$

\medskip

\tit{Behavior of the functional $J_h$}

In this section we will study the behavior of the functional $J_h$
restricted to $T$. 
For simplicity we write
$J_h(f)=A(f)+B(f) $, with

$$A(f)=\int_\Omega ( 1+| \nabla f| ^2)
^{1/2} dudv , \qquad \qquad B(f)=\int_\Omega H(u,v,f)dudv. $$

We'll assume that $h$ is bounded.

\lema{Lemma 2}
The functional $A:T\longrightarrow R$ is continuous and convex.

\demost{Proof}
Continuity can be proved by a simple computation.
Let $a,b\ge 0$ such that $a+b=1$. By 
Cauchy inequality, it follows that

$$\sqrt{1+| \nabla (af+bf_0)| ^2}\le a\sqrt{1+| \nabla
f|^2}+b\sqrt{1+|\nabla f_0|^2} $$
and convexity holds. 

\lema{Remark}
As $A$ is continuous and convex, then it
is weakly lower
semicontinuous in $T$.

\lema{Lemma 3}
The functional $B$ is weakly lower semicontinuous in $T$.

\demost{Proof}
Since $h$ is bounded, we have that 
$$| H(u,v,z)| \le c| z| +d $$

>From the compact immersion $H_0^1( \Omega) \hookrightarrow L^1(\Omega)$ 
and the continuity of Nemytski operator associated to $H$ in 
$L^1( \Omega)$, we conclude that $B$ is weakly lower semicontinuous in $T$
(see [4] and [11]).

\bigskip

\tit{Weak solutions as critical points of $J_h$}

\medskip

Let us assume that $g \in W^{1,\infty}$, and consider
for each $k>0$, the following subset of $T$:

$$
\overline M_k=\{ f\in T:\quad\quad \| \nabla (f-g)\|
_\infty \le k \}. 
$$

$\overline M_k$ is nonempty, closed, convex, bounded, then it is weakly compact.

\lema {Remark:} as $g \in W^{1,\infty}$, taking $p>2$ we obtain, for any $f \in \overline M_k$:
$$\Vert f-g \Vert_p \le c  \Vert \nabla (f-g) \Vert_p $$
Then, by Sobolev imbedding, $\Vert f - g \Vert_\infty \le c_1\Vert f - g \Vert_{1,p} 
\le \overline c k$ for some constant $\overline c$.
We deduce that $f \in W^{1,\infty}$ and $f(\Omega) \subset K$ for some 
fixed compact 
$K \subset R$. 
Thus, the assumption $\Vert h \Vert_\infty < \infty$ 
will not be needed.

\rm
Let $\rho $ be the slope of $J_h$ in $\overline M_k$ defined by:

$$
\rho (f_0,\overline M_k)=\sup \{dJ_h(f_0)(f_0-f);f\in \overline M_k\} $$
(see [6] and [9]), then the following result holds:

\lema{Lemma 4}
If $f_0\in \overline M_k$ verifies 

$$J_h( f_0) =\inf \{ J_h( f) :f\in \overline M_k\} $$
then $\rho (f_0,\overline M_k)=0$.

\demost {Proof}
$$dJ_h(f_0)(f-f_0)=\lim_{\varepsilon \longrightarrow 0}
{J_h(f_0+\varepsilon (f-f_0))-J_h(f_0)\over\varepsilon} = 
\lim_{\varepsilon \longrightarrow 0}{J_h((1-\varepsilon)f_0+\varepsilon f)
-J_h(f_0)\over\varepsilon }.$$ 

When $0<\varepsilon <1$ we have that $(1-\varepsilon )f_0+\varepsilon
f\in \overline M_k$, and then
$dJ_h(f_0)(f_0-f)\le 0$ for all $f\in \overline M_k $.
As $dJ_h(f_0)(f_0-f_0)=0$, we conclude that 
$\rho ( f_0,\overline M_k) =0$. 

\lema {Remark:} 

Being $J_h$ weakly semicontinuous and $\overline M_k$ a 
weakly compact subset of $T$, $J_h$ achieves a minimum $f_0$ in
$\overline M_k$. By lemma 4, $\rho(f_0,\overline M_k)=0$. 

\rm 
As in [6], if $f_0$ has zero slope, we call it a
$\rho$-critical point.
The following result gives sufficient conditions 
to assure that if $f_0$ is a $\rho$-critical point, 
then it is a
critical point of $J_h$.

\lema{Theorem 5}

Let $f_0\in \overline M_k$ such that
$\rho(f_0,\overline M_k)=0$, and assume that 
one of the following conditions
holds:

i) $dJ_h( f_0)(f_0 -g) \ge 0$

ii) $\Vert \nabla (f_0 - g) \Vert_\infty < k$ 

\noi Then $dJ_h( f_0)=0$.

\demost{Proof}

As $\rho (f_0,\overline M_k)=0$, we 
have that $dJ_h(f_0)(f_0-f)\le 0$ 
and then $dJ_h(f_0)(f_0-g)\le dJ_h(f_0)(f-g)$
for any $f\in \overline M_k$. 

We'll prove that $dJ_h(f_0)(\varphi )=0$
for any $\varphi \in C_0^1$. 
Let $\widetilde{\varphi }=
{k\varphi \over 2\| \nabla \varphi \| _\infty }$, 
then $\pm\widetilde{\varphi }+g\in \overline M_k$, 
and then
$dJ_h(f_0)(f_0-g)\le \pm dJ_h(f_0)(\widetilde{\varphi })$. 

Let us suppose that $dJ_h(f_0)(\widetilde{\varphi })\ne 0$, 
then $dJ_h(f_0)(f_0-g)<0$. 

If i) holds, we immediately get a contradiction.
On the other hand, if ii) holds, there exists $r >1$ such that $g + r(f_0-g) \in 
\overline M_k$. Then $ dJ_h(f_0)(f_0-g) \le r dJ_h(f_0)(f_0-g)$, a contradiction.

\lema {Examples}

Let us assume that 
$\int_\Omega 
\dfrac {\nabla(f-g)\nabla g}{\sqrt{1+\vert \nabla f \vert^2}}dudv \ge 0$ 
for any
$f \in \overline M_k$. 
Then condition i) of theorem 5 is fulfilled for example if:

a) $| h(u,v,z)| \le c( z- g(u,v))_{+}$
for every $(u,v)\in \Omega$, $z\in {R}^3$, for some constant  
$c$ small enough.
 
b) $\int_\Omega h(u,v,f)(f-g)dudv \ge 0$ for every $f \in \overline M_k$. 
As a particular case, 
we may take $h(u,v,z) = c(z-g(u,v))$ for any $c \ge 0$.

c) $h(u,v,z) = -c(z-g(u,v))$ for some $c > 0$ small enough.

Indeed, in all the examples the inequality $dJ_h(f)(f-g) \ge 0$ holds for any $f \in 
\overline M_k$, since
$$dJ_h(f)(f-g)=\int_\Omega \left({\nabla f\nabla ( f-g)
\over\sqrt{1+| \nabla f|^2}} 
+2h(u,v,f) (f-g)\right)dudv= 
$$

$$\int_\Omega \left({| \nabla (f-g) | ^2\over
\sqrt{1+| \nabla f| ^2}}+2h(f-g)\right) dudv+
\int_\Omega {\nabla(f-g)\nabla g\over\sqrt{1+|\nabla f|^2}}dudv \ge
$$
$$\int_\Omega \left({| \nabla (f-g) | ^2\over
\sqrt{1+| \nabla f| ^2}}+2h(f-g)\right)dudv.$$

Then the result follows immediately in example b). 
In examples a) and c), 
being $\Vert \nabla( f-g) \Vert_\infty  \le k$ we can choose 
$\widetilde{k}$ such that 
$\sqrt{1+\Vert \nabla f \Vert_\infty ^2}\le \tilde k$.
Then 
$$\int_\Omega \left( {|\nabla ( f-g) | ^2\over
\sqrt{1+|\nabla f|^2}}+2h(u,v,f)(f-g)\right)dudv
\ge \int_\Omega \left(\frac {|\nabla (f-g)| ^2}{\tilde k}
-2c(f-g) ^2\right)dudv$$
$$\ge {1\over\tilde{k}}
\|\nabla(f-g)\|_2^2-2cc_1^2\|\nabla (f-g)\|_2^2=({1\over\tilde{k}}-2cc_1^2)
\|\nabla(f-g)\|_2^2$$
where $c_1$ is the Poincar\'{e}'s constant associated to $\Omega.$

Thus,
the result holds for
$c \le \frac 1{2\tilde{k} c_1^2}$. 

\lema {Remark}
As it happens in the preceding examples, it can be proved that if 
$dJ_h(f)(f-g) \ge 0$ for any $f \in \overline M_k$, 
then $g$ is a weak solution of (1). Indeed, if $dJ_h(g) \ne 0$, 
from theorem 5 it follows that $\rho (g,\overline M_k)>0$. 
As $J_h$ achieves a minimum in every 
$\overline M_k$, we may take 
$k \ge k_n \longrightarrow 0$, and $f_n$ such that
$\rho (f_n,\overline M_{k_n}) =0$. 
As $\overline M_{k_n}\subset \overline M_k$, 
condition i) in theorem 5 holds and then $dJ_h(f_n)=0$. 
It is immediate that
$f_n \longrightarrow g $ in $W^{1,\infty}$ and then it follows easily that
$dJ_h(g)=0$. 

Furthermore, for constant $g$ we can see 
that if $dJ_h(f)(f-g) \ge 0$ for any $f \in \overline M_k$, 
then $g$ is a global minimum 
of $J_h$ in $\overline M_k$: 
let us define
$\varphi(t)=J_h(tf+(1-t)g)$, then
$\varphi'(t)=dJ_h(tf+(1-t)g)(f-g)$. 
As $0 \le dJ_h(tf+(1-t)g)(tf+(1-t)g-g) = tdJ_h(tf+(1-t)g)(f-g)$ 
it follows 
that
$J_h(f)-J_h(g)=\varphi(1)-\varphi(0)=\varphi'(c) \ge 0$. 

\medskip

\tit{Multiple Solutions}

In this section
we study the 
multiplicity of weak solutions of (1).
Let us consider:
$$\overline N_k=\{f\in \overline M_k\cap H^2:
\|\frac{\partial^2 f}
{\partial x_{i}\partial x_{j}}\|_2\le k\} $$
$\overline N_k$ is a nonempty, closed, bounded, and convex subset of 
$T$, therefore $\overline N_k$ is weakly compact.

\rm
Then we obtain 
the following theorem, which 
is a variant of the Mountain Pass Lemma:

\lema{Theorem 6}

Let $f_0\in \overline N_k$ be a local minimum of $J_h $ 
and assume that
$J_h(f_1)<J_h(f_0)$ for some $f_1\in \overline N_k$.
Let
$$
c=\inf_{\gamma \in \Gamma }\sup_{t\in [0,1]} J_h (\gamma
(t)) 
$$
where $\Gamma = \{ \gamma \in C([0,1],\overline N_k) :
\gamma (0) = f_0, \gamma (1) = f_1 \}$.
Then there exists $f\in \overline N_k$ 
such that $J_h (f)=c$ and $\rho (f,\overline N_k)=0.$

\rm
We remark
that 
$f$ is not a local minimum of 
$J_h$. 
This kind of $f$ is called an unstable
critical point.

The proof of theorem 6 follows 
from theorem 3 in [6] and 
lemmas 7-8-9 below.

\lema{Lemma 7}

The functional $J_h$ is
$C^1(\overline N_k).$

\demost{Proof}
Let $f,f_0\in \overline N_k$. Then
$$\left| dJ_h(f)(\varphi )-dJ_h(f_0)(\varphi )\right| 
\le \left\| \varphi \right\| _{H_0^1}\left( \left\| 
\frac{\nabla f}{\sqrt{
1+\left| \nabla f\right| ^2}}-
\frac{\nabla f_0}{\sqrt{1+\left| \nabla
f_0\right| ^2}}\right\| _2+
\left\| N_h(f_0)-N_h(f)\right\| _2\right) $$
where $N_h$ is the Netmyski operator 
associated to $h$. Being
$$\left\| \frac{\nabla f}{\sqrt{1+\left| \nabla f\right| ^2}}-\,\frac{\nabla
f_0}{\sqrt{1+\left| \nabla f_0\right| ^2}}\right\| _2\leq \left\| \sqrt{%
1+\left| \nabla f_0\right| ^2}\nabla f-\,\sqrt{1+\left| \nabla f\right| ^2}%
\nabla f_0\right\| _2 \le \kappa \left\| f_0-f\right\| _{H_0^1}$$
and
$N_h:L^2 \longrightarrow L^2$
continuous,
the result holds.

\lema{Lemma 8}
The slope $\rho $ is 
$H^1-$continuous.

\demost {Proof}
Let 
$f_n \in \overline N_k$ such that 
$f_n\longrightarrow f_0$ in $H_0^1$. 
For $\epsilon >0$ we take 
$g_n\in \overline N_k$ such that
$\rho (f_n, \overline N_k)-\frac \epsilon 2<dJ_h(f_n)(f_n-g_n)$.
Then
$$\rho (f_n, \overline N_k)- \rho (f_0, \overline N_k)
 \le dJ_h(f_n)(f_n-g_n)+\frac \epsilon 2-dJ_h(f_0)(f_0-g_n)$$
$$\le \left\| dJ_h(f_n)\right\| _{\left( H_0^1\right) ^{*}}\left\|
(f_n-f_0)\right\| _{H_0^1}+\left\| dJ_h(f_n)-dJ_h(f_0)\right\|_
{\left( H_0^1\right) ^{*}} \left\| (f_0-g_n)\right\| _{H_0^1}+
\frac \epsilon 2 < \epsilon$$
for $n \ge n_0$. 
Operating in the same way with
$\rho (f_0, \overline N_k)-\rho (f_n, \overline N_k)$, we conclude
that $\rho (f_n, \overline N_k) \longrightarrow \rho (f_0, \overline N_k) $.

\lema{Lemma 9 (Palais Smale condition)}

Let $\left( f_n\right) _{n \in N}\subset 
\overline N_k $ such that 
$\lim_{n\longrightarrow \infty }
\rho
(f_n, \overline N_k)=0$. 
Then
$\left( f_n\right) _{n\in N}$ 
has a
convergent subsequence in $H_0^1(\Omega )$.

\demost {Proof}
As $f_n \in 
\overline N_k $, we may suppose that
$f_n \longrightarrow f$ weakly.
Let $\Psi _n=$\ $f_n-f$. We'll 
see that $\Psi_n \longrightarrow 0$.
Indeed,
$$
dJ_h(f_n)(\Psi _n)=\int_\Omega \left( \frac {\nabla f_n}
{\sqrt{1+\left| \nabla
f_n\right| ^2}} \nabla \Psi _n
+2h(u,v,f_n) \Psi _n \right) dudv= 
$$
$$
=\int_\Omega \frac 1{\sqrt{1+\left| \nabla f_n\right| ^2}}
\left| \nabla \Psi _n\right| ^2dudv+\int_\Omega 
\frac{\nabla \Psi _n}{\sqrt{1+\left| \nabla f_n\right| ^2}}
\nabla f dudv+ $$
$$+\int_\Omega 2h(u,v,f_n) \Psi _n dudv. $$
Then for some constant $c$
$$c\left\| \nabla \Psi _n\right\| _2^2\le \rho
(f_n, \overline N_k)-\int_\Omega \frac{\nabla \Psi _n}{\sqrt{1+\left|
\nabla f_n\right| ^2}}\nabla f dudv-\int\limits_\Omega
2h(u,v,f_n) \Psi _n dudv $$
By Rellich-Kondrachov theorem 
$\Psi _n\rightarrow 0$ in $L^2(\Omega )$, and then

$$\left| \int_\Omega 2h(u,v,f_n) \Psi _n dudv\right| \le
2\left\| h\right\| _\infty \left| \Omega \right| ^{\frac 12}\left\| \Psi
_n\right\| _2\rightarrow \;0 $$
and
$$\left| 
\int_\Omega 
\frac{\nabla \Psi _n}{\sqrt{1+\left|\nabla f_n\right| ^2}} \nabla f 
dudv\right|$$ 
$$=\left| -\int_\Omega \frac{\Delta f}{\sqrt{1+\left| \nabla f_n\right|
^2}}\Psi _ndudv-\int_\Omega \Psi _n\nabla \left(
1+\left| \nabla f_n\right| ^2\right) ^{-\frac 12}\nabla f
dudv\right| $$

$$\le \left\| \Delta f\right\| _2\left\| \Psi _n\right\| _2+\left\| \nabla
f_n\right\| _\infty \left\| \nabla f\right\| _\infty \left\| D^2f_n\right\|
_2\left\| \Psi _n\right\| _2\longrightarrow 0.$$

\lema {Example}

Now we'll show with an example that problem 
(1) may have at least three $\rho$-critical points in $N_k$:

Let $g = g_0$ be a constant, and $h(u,v,z) = -c (z-g_0)$ for some constant
$c>0$. 
Then, $g_0$ is a minimum of $J_h$ in $\overline M_{k_1}$ for $k_1$ small 
enough,
and a local minimum in $M_k$ for any $k \ge k_1$.
 
Moreover, taking $\Omega = B_R$, 
$f(u,v) = g_0 + R^2 - (u^2+v^2)$, it follows that 
$$J_h(f) - J_h(g_0) = 2\pi (o(R^3) - \frac c6 R^6)$$
and taking $k=2\sqrt \pi R$ 
it holds that $f \in \overline N_k$. 
Hence, if $R$ is big enough it follows that $g_0$ 
is not a global minimum 
in $\overline N_k$.
Furthermore, we see that
the proof of lemma 4 
may be repeated in $\overline N_k$, 
and then the
minimum of $J_h$ in $\overline N_{k}$ is 
a $\rho$-critical point.
From
Theorem 6 there is a third 
$\rho$-critical point which is not 
a local minimum of $J_h$.

\tit{Regularity}

As we proved, 
problem (1)
admits
(for an apropiated $k>0$) a weak solution in a subset
$\overline{M}(k)=\left\{ f\in T\quad /\quad \left\| \nabla (f-g)\right\|
_\infty \le k\right\}.$ 

Let us consider $p>2$, and $f_0 \in W^{2,p}(\Omega) \hookrightarrow 
C^1(\overline \Omega)$
a weak solution of (1). Then 
$
L_{f_0}f_0 = 2
h(u,v,f_0)(1+ \nabla f_0^2)^{3/2}
 \text{ in } \Omega$
where for any $f \in C^1(\overline \Omega)$ 
$L_{f}:W^{2,p}\longrightarrow L^p$ is the strictly elliptic operator given by 
$$L_{f}\phi=
(1+f_v^2)\phi_{uu}+(1+f_u^2)\phi_{vv}-2f_uf_v\phi_{uv}$$

In order to prove the regularity of $f_0$, we study the 
equation (2)
$
L_{f_0}\phi = 2
h(u,v,f_0)(1+ \nabla f_0^2)^{3/2}
 \text{ in } \Omega$, $\phi = g \text { in } \partial \Omega$. 
           
\lema {Proposition 10}

Let us assume that $\partial \Omega \in C^{2,\alpha}$, 
$g \in C^{2,\alpha}$
and $h \in C^\alpha$ for some
$0 < \alpha \le 1- \dfrac 2p$. Then,
if $\phi \in W^{2,p} $ is a strong solution of (2), 
$\phi \in C^{2,\alpha}(\overline{\Omega })$.

\demost{Proof}

By Sobolev
imbedding
$\phi \in C^{1,\alpha}(\overline \Omega)$. 
Then
$L_{f_0}\phi \in C^\alpha(\overline \Omega)$ 
and 
the coefficients 
of the operator $L_{f_0}$ belong to $C^\alpha$.
By theorem 6.14 in [8], the equation 
$Lw = L_{f_0}\phi$ 
in $\Omega$, $w = g$ in $\partial \Omega$ is
uniquely solvable
in $C^{2,\alpha}(\overline \Omega)$, 
and the
result
follows from the uniqueness
in theorem 9.15 in [8].

\lema {Remark}

As a simple consequence, we obtain that $f_0 \in C^{2,\alpha}
(\overline\Omega)$, 
 by the uniqueness in $W^{2,p}$ 
 given by theorem 9.15 in [8].

\lema {Corollary 11}

Let us assume that $\partial \Omega \in C^{k+2,\alpha}$, 
$g \in C^{k+2,\alpha}$
and $h \in C^{k,\alpha}$ for some
$0 < \alpha \le 1- \dfrac 2p$. Then $f_0 \in C^{k+2,\alpha}
(\overline \Omega)$.

\demost {Proof}

It is immediate from Proposition 1 and theorem 6.19 in [8].

\medskip
\centerline{ACKNOWLEDGEMENT}

The authors thank specially 
Prof. J.P.Gossez for the careful reading of the manuscript and 
his suggestions and remarks.

\bigskip

\tit{References}

[1]  Adams R. A., Sobolev Spaces. Academic Press, 1975.

[2]  Br\'{e}zis H. and Coron J.M., Multiple solutions of
H- system and Rellich's conjeture. Comm. Pure Appl. Math. 37
(1984), 149 - 184.

[3]  Do Carmo M., Differential Geometric of Curves and Surfaces. 
Prentice-Hall, 1976.

[4]  Goldstein Costa D., T\'opicos em Analise n\~ao
Linear e Aplicac\~oes as Ecuac\~{o}es Diferenciais. I.M.PA., 1986.

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

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

[7]  Lami Dozo E. and Mariani M.C., Solutions to the
Plateau problem for the prescribed mean curvature equation via the mountain
pass lemma. Studies un Applied Mathematics 96 (1996), 351 - 358.

[8]  Osserman R., A survey of minimal surfaces, Van Nostrand
Reinhold Company, 1969.

[9] Struwe M., Plateau's Problem and the Calculus of Variations. 
Math. Notes 35, Princeton University Press. Princeton 1989.

[10] Struwe M., Non uniqueness in the Plateau problem for
surfaces of constant mean curvature, Arch. Rat. Mech. Anal. 93
(1986), 135-157.

[11] Vainberg M. M., Variational Methods in the Theory of Non
Linear Operators, Holden-Day, San Francisco, 1964.

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

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


Dpto. de Matem\'{a}tica, Fac. de
Cs. Exactas y Naturales, UBA. PAB I, Cdad. Universitaria, 1428. Capital,
Argentina.

${}^*$ CONICET


\tit{Address for correspondence:} Prof. M. C. Mariani, Dpto. de
Matem\'{a}tica, Fac. de Cs. Exactas y Naturales, UBA. PAB I, Cdad.
Universitaria, 1428. Capital, Argentina.

\tit{email:} mcmarian\@dm.uba.ar

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