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To appear in Revista de la Union Matematica Argentina
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nonparametric surfaces - variational calculus
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\bigskip


\centerline{\bf SOLUTIONS TO THE MEAN CURVATURE EQUATION FOR} 

\centerline{\bf NONPARAMETRIC SURFACES BY FIXED POINT METHODS}

\bigskip

\centerline{ \bf P. Amster, J.P. Borgna, M.C. Mariani and D.F. Rial}
\smallskip
\centerline{FCEyN - Universidad de Buenos Aires}

\bigskip
 
{\bf ABSTRACT}
 

We study the existence of solutions for the equation of prescribed mean
curvature when the surface is the graph of $u:\overline{\Omega }\longrightarrow 
R$, with mean curvature $H(x,y,u(x,y))$.

We give conditions on the boundary data in order to obtain at least one 
solution for
the quasilinear Dirichlet problem (1) below, with $H$ a given continuous 
function. 

\bigskip 

{\bf INTRODUCTION}

We consider the quasilinear Dirichlet problem in a bounded domain $\Omega
\subset R^2$ with $\partial \Omega \in C^2$ 
$$
\text{(1)}
\cases
(1+u_y^2)u_{xx}+(1+u_x^2)u_{yy}-2u_xu_yu_{xy}=2H(x,y,u)\left( 1+\left|
\nabla u\right| ^2\right) ^{\frac 32} \qquad \text{ in }\quad \Omega &\\
u(x,y)=\varphi (x,y) \quad \text{on }\partial \Omega 
\endcases
$$

where $H:\overline \Omega \times [\epsilon ,\epsilon ] \longrightarrow R$ 
is continuous for some $\epsilon > 0 $ and $\varphi
\in W^{2,p}(\Omega) $ is the boundary data.

The problem above is the mean curvature equation 
for nonparametric surfaces
which 
has been studied
in general
for
hypersurfaces in $R^{n+1}$
by Gilbarg, Trudinger, Simon, Serrin, 
D\'\i az, Saa and Thiel
among other 
authors. 
For $H$ independent of $u$
it has been proved [GT] 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 $\epsilon > 0$. 
The sharpness
of the geometric condition on the curvature of $\partial \Omega$
is shown by a non-existence result ([GT], 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
$\Vert g \Vert_\infty \le \epsilon$ 
for which the
Dirichlet's problem 
is not solvable.

On the other hand, D\'\i az, Saa and Thiel [DST] studied the 
general quasilinear elliptic equation 
$div(Q(\vert \nabla u \vert) \nabla u) + f(u) = g(x_1,...,x_n)$
in $R^n$ under Dirichlet and Neumann conditions. They 
studied existence and
uniqueness of the problem for nonincreasing $f$
by finding apriori bounds 
for $\nabla u$. The case $Q(r)=(1+r^2)^{-1/2}$ corresponds to 
the mean curvature equation (1), and the condition on 
$f$ becomes: $h'(u) \ge 0$.

In the present paper we study the problem by topological methods,
obtaining a solution under some restrictions on 
$\Vert H 
\Vert_\infty$ and $\Vert \varphi 
\Vert_{2,p}$ but avoiding the conditions on the curvature of 
$\partial\Omega$. The condition 
$\frac{\partial h}{\partial u} \ge 0$ will not be necessary either.

The general Plateau problem and the Dirichlet asociated problem, have 
been
studied in [AMR],[BC],[H],[LD-M],[MR]
[S1],[S2],[WG], etc.

The quasilinear operator associated to problem (1) is strictly 
elliptic 
since its eigenvalues are $\lambda =1$ and 
$\Lambda =1+\vert p \vert ^2$, where $%
p=(u_x,u_y)$ (see [GT] chapter 10).

\smallskip

The main result is the following theorem

\smallskip 

\lema {Theorem 1}

Let $p>2$ and assume that $\Vert \varphi \Vert_{2,p}$ and 
$\Vert H 
\Vert_{L^\infty(\overline \Omega \times [\epsilon ,\epsilon ])}
$ 
are small enough with respect to $\vert\Omega\vert$, the 
Sobolev's 
constant and the apriori bounds for $\Delta$
in $\Omega$. Then there exists at least 
one solution $u\in W^{2,p}(\Omega )$ of (1).

\rm
\bigskip

{\bf SOLUTIONS BY FIXED POINT METHODS}

First we note that $u$ is a solution of (1), if and only if 
$w=u-\varphi $ 
is a solution of the following equation: 
$$
\text{(2)}
\cases
(1+(w_y+\varphi _y)^2)w_{xx}+(1+(w_x+\varphi _x)^2)w_{yy}-2(w_x+\varphi _x)
(w_y+\varphi _y)w_{xy} & \\ 
\qquad \quad =2H(x,y,w+\varphi )\left( 1+\left| \nabla (w+\varphi )\right| ^2\right)
^{\frac 32}-(1+(w_y+\varphi _y)^2)\varphi _{xx} &\\ 
\qquad \qquad -(1+(w_x+\varphi _x)^2)\varphi _{yy}+2(w_x+\varphi
_x)(w_y+\varphi _y)\varphi _{xy} \qquad \text{in }\Omega  &\\ 
w=0  \qquad  \text{on }\partial \Omega 
\endcases
$$

For each $\overline v\in C^1(\overline \Omega )$ such that $\Vert \overline v + 
\varphi \Vert_{\infty} \le \epsilon$ we consider the elliptic linear Dirichlet
problem associated to equation (2)

$$
\text{(3)}
\cases
{L}_{\overline v}\left( v\right) = {F}(\overline v) \qquad \text{in }\Omega
\\ v = 0 \qquad \qquad \qquad \text{on }\partial \Omega 
\endcases
$$

where 
$$L_{\overline v}\left( v\right) = (1+(\overline v_y+\varphi
_y)^2)v_{xx}+(1+(\overline v_x+\varphi _x)^2)v_{yy}-2(\overline v_x+\varphi _x)(\overline
v_y+\varphi _y)v_{xy}$$ 
and 
$$F(\overline v)= 2H(x,y,\overline v+\varphi
)\left( 1+\left| \nabla (\overline v+\varphi )\right| ^2\right) ^{\frac
32}-(1+(\overline v_y+\varphi _y)^2)\varphi _{xx}-(1+(\overline v_x+\varphi
_x)^2)\varphi _{yy}$$
$$+2(\overline v_x+\varphi _x)(\overline v_y+\varphi _y)\varphi _{xy}$$

The linear equation (3) has a unique solution 
$v\in W^{2,p}(\Omega) 
\cap W_0^{1,p}(\Omega )$ (see [GT], theorem 9.15). 
Thus, if we consider the
Sobolev imbedding $W^{2,p}\hookrightarrow C^1$
with imbedding constant $k$ (i.e.
$\left\| u\right\| _{1,\infty }\leq k\left\| u\right\| _{2,p}$), 
we may
define an operator $T:\overline {B_{\epsilon}(- \varphi)} \subset 
C^1(\overline \Omega )\longrightarrow C^1(\overline \Omega )$ given by
$T(\overline v) =v$ if $v$ is the solution of (3) for $\overline v$.

We'll see that the operator $T$ has at least one fixed point in $%
C^1$, and this will give a solution of the original problem (1).

Our main tool will be the Schauder fixed point theorem 
(see [GT] theorem 11.1 and corollary 11.2). 
We prove first the following lemma and proposition.

\smallskip 

\lema {Lemma 2}

There exists a constant $C$ (depending only on $\left| \Omega \right|$, $p$) 
and $R>0$ such that if

$$
\left\| \overline v + \varphi \right\| _{1,\infty }\le R
$$
then for every $w\in W^{2,p}(\Omega ) \cap W_0^{1,p}(\Omega)$ 

$$\left\| w\right\| _{2,p}\leq C\left\|  L_{\overline v}\left(
w\right) \right\| _p $$                   

{\bf Proof}
\rm

We can write $${L}_{\overline v}(w)=\Delta w+S_{\overline v}(w) $$
where $S_{\overline v}(w)=(\overline v_y+\varphi
_y)^2w_{xx}+(\overline v_x+\varphi _x)^2w_{yy}-2(\overline v_x+\varphi _x)(\overline
v_y+\varphi _y)w_{xy}.$

The operator $\Delta $ satisfies the hypotheses of [GT], lemma 9.17,
then there exists a constant $C_1$ (independent of $w$) such that 

$$\left\| w\right\| _{2,p}\leq C_1\left\| \Delta w\right\| _p $$

for all $w\in W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$. Then
$$
\left\| {L}_{\overline v}\left( w\right) \right\| _p\geq \left\| \Delta
w\right\| _p-\left\| S_{\overline v}\left( w\right) \right\| _p\geq \frac
1{C_1}\left\| w\right\| _{2,p}-\left\| S_{\overline v}\left( w\right) \right\| _p 
$$
and being
$$
\left\| S_{\overline v}\left( w\right) \right\| _p\leq 4\left\| \overline v+\varphi
\right\| _{1,\infty }^2\left\| w\right\| _{2,p} 
$$
we obtain 
$$\left\| {L}_{\overline v}\left( w\right) \right\| _p\geq \left(
\frac 1{C_1}-4\left\| \overline v+\varphi \right\| _{1,\infty }^2\right) \left\|
w\right\| _{2,p} $$

The second member of the last inequality is positive if 
$\left\| \overline v+\varphi \right\|
_{1,\infty }\leq R<\frac 1{2\sqrt{C_1}},$ 
and setting $C=\dfrac{C_1}{%
1-4C_1R^2}$ the lemma holds. 

\smallskip\ 

In the following proposition we'll find $0<R < 
\frac 1{2\sqrt{C_1}},\epsilon$ 
such that $T(\overline {B_R(-\varphi )}) \subset \overline {B_R(-\varphi )}
\subset C^1(\overline \Omega ).$

\lema {Proposition 3}

Let $p>2$ and assume that $\Vert \varphi \Vert_{2,p}$ and 
$\Vert H 
\Vert_{L^\infty(\overline \Omega \times [\epsilon ,\epsilon ])}
$ 
are small enough.                                             
Then there exists $R\leq \epsilon$ such that if
$$ \left\| \overline v+\varphi \right\| _{1,\infty }\leq R $$
then

$$
\left\| T(\overline v)+\varphi \right\| _{1,\infty } \leq R 
$$

Furthermore, the operator $T$ is continuous in the closed ball 
$\overline{B_R(-\varphi )},$ and its range is a precompact set.

\smallskip 

{\bf Proof}
\rm

Assume that $\left\| \overline v+\varphi \right\| _{1,\infty }\le R < 
\frac 1{2\sqrt{C_1}}$. Then
$$\left\| v+\varphi \right\| _{1,\infty }\le 
k \left\| v \right\|
_{2,p} + \Vert \varphi \Vert_{1,\infty} \le Ck\left\| { L}_{\overline v} (v) \right\|_p 
+ \Vert \varphi \Vert_{1,\infty} 
= \dfrac{C_1k}{1-4C_1R^2}\left\| { F}(\overline v)\right\| _p
+ \Vert \varphi \Vert_{1,\infty} 
$$
and 
$$\left\| { F}(\overline v)\right\| _p\leq 2(1+\left\| \overline v+\varphi \right\|
_{1,\infty }^2)^{3/2}\left\| H(x,y,\overline v+\varphi )\right\| _p+2(1+ 2 \left\| \overline
v+\varphi \right\| _{1,\infty }^2)\left\| \varphi \right\| _{2,p}\leq $$

$$ \le 2(1+R^2)^{3/2}
\vert\Omega \vert^{1/p}
\left\| H\right\|_\infty
+2(1+2R^2)\left\| \varphi \right\| _{2,p}$$
 
We look for a number $R$ such that
$$\frac{2C_1k}{1-4C_1R^2}((1+R^2)^{3/2}
\vert\Omega \vert^{1/p}
\left\| H\right\|_\infty
+(1+2R^2)\left\| \varphi 
\right\| _{2,p} ) + \Vert \varphi \Vert_{1,\infty} \leq R 
$$
or, equivalently, such that $f(R) \le 0$, where   

$$f(R) = \frac{2C_1k}{1-4C_1R^2} 
((1+R^2)^{3/2}\vert\Omega \vert^{1/p}
\left\| H\right\|_\infty
+(1+2R^2)\left\| \varphi 
\right\| _{2,p} ) + \Vert \varphi \Vert_{1,\infty} - R $$

It is clear that $f \le \dfrac P{1-4C_1R^2}$ in the interval
$ (0,\dfrac 1{2\sqrt C_1})$, where 

$$
P(R)= 2C_1k((1+ \dfrac1{4C_1})^{3/2} 
\vert\Omega \vert^{1/p}
\left\| H\right\|_\infty
+(1+ \dfrac 1{2C_1})\left\| \varphi 
\right\| _{2,p}) + \Vert \varphi \Vert_{1,\infty} -R + 4C_1 R^3$$ 

$P$ achieves a minimum in $R_0 = \dfrac 1{\sqrt {12C_1}}$ and $P(R_0) \le 0$ if
$\Vert \varphi\Vert_{2,p}$ 
and 
$\Vert H\Vert_{\infty}$ 
are small enough. Then 
$f(R_0) \le 0$.

In order to complete the proof we must
 see that $T$ is continuous 
and compact. 
Indeed, for $\overline u$, $\overline v \in \overline {B_R(-\varphi)}$:
$$
\left\| u-v\right\| _{2,p }\leq
C\left\| { L}_{\overline u}\left( u-v\right) \right\| _p\leq 
C\left( \left\| F(\overline u)-F(\overline v)\right\| _p+
\left\| { L}_{\overline v}(v)-L_{\overline u}(v)\right\| _p\right).  
$$
But
$$\left\| F(\overline u)-F(\overline v)\right\|_p \le 
 \Vert 2H(x,y,\overline u+\varphi) - 2H(x,y,\overline v+\varphi))
\left( 1+\left| \nabla (\overline u+\varphi )
\right| ^2\right) ^{\frac 32} \Vert_p$$
$$ +\Vert 2H(x,y,\overline v+\varphi) (\left( 1+\left| \nabla (\overline u+\varphi)
\right| ^2\right) ^{\frac 32} - \left( 1+\left| \nabla (\overline v+\varphi)
\right| ^2\right) ^{\frac 32}) \Vert_p $$
$$+\Vert((\overline u_y+\varphi _y)^2-(\overline v_y+\varphi _y)^2)\varphi _{xx}\Vert_p
  +\Vert ((\overline u_x+\varphi_x)^2-(\overline v_x+\varphi_x)^2)\varphi _{yy}\Vert_p
+2 \Vert((\overline u_x+\varphi _x)(\overline u_y+\varphi _y)-$$
$$(\overline v_x+\varphi _x)(\overline v_y+\varphi _y))\varphi _{xy}\Vert_p
 \le 2\Vert H(x,y,\overline u+\varphi) - H(x,y,\overline v+\varphi)\Vert_p
(1+R^2)^{3/2}$$
$$ +6R (1+ R^2)^{\frac 12}
(R+\Vert \varphi \Vert_{1,\infty}) \Vert \overline u-\overline v \Vert_{1,\infty}
\vert \Omega \vert^{1/p}\Vert H\Vert_\infty + 8R \Vert \overline u- \overline v \Vert_{1,\infty} \Vert \varphi \Vert_{2,p}$$
and 
$$\left\| L_{\overline v}(v)-L_{\overline u}(v)\right\| _p \le 
8R \Vert  \overline u- \overline v \Vert_{1,\infty} \Vert v \Vert_{2,p}$$

Being $H$ uniformely continuous and
$\Vert u-v \Vert_{1,\infty} \le k \Vert u-v \Vert_{2,p}$, the continuity follows.

Moreover, fixing any $\overline v \in C^1(\overline \Omega)$ we see that 
 $\overline{T(B_R(-\varphi )) }$ is bounded in 
$W^{2,p}$, and the result follows from the compactness of
the imbedding
$W^{2,p}(\Omega)\hookrightarrow C^1(\overline \Omega )$.

\lema{Remark}

In the situation of 
proposition 3, 
if we write $P(R)=4C_1R^3-R+a$, the smallness of $H$ and $\varphi$
can be stated in the more precise condition $a \le 
\frac 2{3\sqrt {12C_1}}$.
\medskip
\rm
{\bf Proof of theorem 1:} From proposition 3 we know that the
operator $T$ satisfies the assumptions 
of Schauder 
fixed point theorem (see [GT] corollary 11.2). 
Thus, we obtain a fixed point $w\in W^{2,p}(\Omega
)\cap W_0^{1,p}(\Omega )$ for the operator $T$,  
which corresponds to a 
solution of equation (2).


\tit {References}

[A] Adams, R: Sobolev Spaces, Academic Press, (1975).

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

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

[DST] D\'\i az J., Saa J., Thiel U: Sobre la ecuaci\'on de
curvatura media prescripta y otras ecuaciones cuasilineales
el\'\i pticas con soluciones anul\'andose localmente,
Revista de la Uni\'on Matem\'atica Argentina vol.35, 1989, 175-206.

[GT] Gilbarg D. and Trudinger N: Elliptic Partial
Differential Equations of Second Order, Springer-Verlag. Second Edition

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

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

[MR] Mariani M.C, Rial D: Solution to the mean curvature
equation by fixed point methods. To appear in Bulletin of The Belgian
Mathematical Society - Simon Stevin. 

[S1] Struwe M:  Plateau's problem and the calculus of
variations, Lecture Notes, Princeton Univ. Press (1988).

[S2] Struwe M: Multiple solutions to the Dirichlet problem
for the equation of prescribed mean curvature, Preprint.

[WG] Wang Guofang, The Dirichlet problem for the equation of
prescribed mean curvature, Preprint.


\bigskip

{\bf J.P.Borgna and D. F. Rial}

Dpto. de Matem\'atica
Fac. de Cs. Exactas y Naturales, UBA
Pab. I, Ciudad Universitaria (1428) Capital, Argentina

{\bf P.Amster and M. C. Mariani}

Dpto. de Matem\'atica
Fac. de Cs. Exactas y Naturales, UBA
Pab. I, Ciudad Universitaria (1428) Capital, Argentina

CONICET

{\bf Address for correspondence:} Prof.  M. C. Mariani,
Dpto. de Matem\'atica
Fac. de Cs. Exactas y Naturales, UBA
Pab. I, Ciudad Universitaria (1428) Capital, Argentina

{\bf E-mail: mcmarian\@dm.uba.ar}

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