Content-Type: multipart/mixed; boundary="-------------0004041811453"

This is a multi-part message in MIME format.

---------------0004041811453
Content-Type: text/plain; name="00-156.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="00-156.comments"

To appear in the Belgian Bull. Simon Stevin
---------------0004041811453
Content-Type: text/plain; name="00-156.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="00-156.keywords"

quasilinear equations - iterative methods
---------------0004041811453
Content-Type: application/x-tex; name="mmartha.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="mmartha.tex"

\input vanilla.sty
\pagewidth{13cm}
\pageheight{19cm}
\normalbaselineskip=15pt
\normalbaselines
\parindent=0pt
\parskip=3pt
\mathsurround=1.7pt
\overfullrule=0pt
\scaledocument{\magstep1}
\def\tit#1{\bigskip \bf\noindent #1 \medskip\rm}
\def\lema#1{\medskip\smc\noindent #1\quad\sl}
\def\demost#1{\smallskip\noindent\underbar{\it #1}\quad\rm}
\def\note#1{\medskip \smc\noindent #1\quad\rm}
\def\lra{\longrightarrow}
\def\noi{\noindent}
\def\li#1#2{\smash{\mathop{#1}\limits\sb{#2}}}
\def\lie#1#2#3{\smash{\mathop{#1}\limits\sb{{\scriptstyle
#2}\atop{\scriptstyle#3}}}}
\def\cor{\allowmathbreak}
\def\inb{\int\sb B }
\def\dde{\sober{$d$}/{$d\va$}}
\def\flecha{\Longleftrightarrow}
\def\R{\text{{\rm I}\!{\rm R}}}
\def\N{\text{{\rm I}\!{\rm N}}}
\def \ee{\varepsilon}

\tit{ }
\bigskip

\bigskip

\bigskip

\bigskip

\centerline{\bf SOLUTIONS TO QUASILINEAR EQUATIONS}

\bigskip

\centerline{\bf BY AN ITERATIVE METHOD} 

\bigskip

\centerline{ \bf P. Amster, M. M. Cassinelli and M. C. Mariani}

\centerline{FCEyN - Universidad de Buenos Aires}

\bigskip
 
{\bf ABSTRACT}

We apply an iterative method in order to
construct a solution to the mean curvature equation 
for nonparametric surfaces.

\bigskip

{\bf INTRODUCTION}

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

$$
\text{(1)}
\cases
(1+u_y^2)u_{xx}+(1+u_x^2)u_{yy}-2u_xu_yu_{xy}= 
2h(u)\left( 1+\left|
\nabla u\right| ^2\right) ^{\frac 32} \text{ in } \Omega &\\
u=g \quad \text{ in }\partial \Omega &
\endcases
$$
where $\Omega $ is a bounded domain in $\R^2$, and
$h:\R \longrightarrow \R$ 
is a given continuous function. 
This problem and the general parametric case have been studied
by several authors, see e.g. [2-5,6,7,9-13].

\bigskip

{\bf SOLUTIONS BY AN ITERATIVE METHOD}

We'll apply an iterative method inspired 
in the Newton Imbedding procedure [8].

For this purpose, let us 
define for each $v \in C^{1}(\overline\Omega)$ the bounded
linear operator
$Q_v:W^{2,p}(\Omega)\to L^{p}(\Omega)$ given by
$$Q_v u= {1\over 2(1+\nabla v^2)^{3\over 2}}((1+v_y^2)u_{xx}+
(1+v_x^2)u_{yy}-2v_xv_yu_{xy})$$

{\bf Remark}: $u\in W^{2,p}(\Omega)$ 
is a solution of (1) if and only if 

$$
\cases
Q_{u}u =
h(u)\text{ in } \Omega
 &\\
u=g \quad \text{ in }\partial \Omega &
\endcases
$$

We'll assume that $h\in C^2(\R)$, $h'\ge 0$, 
$g\in C^{2,\gamma}(\overline \Omega)$
for $0<\gamma <1$,
and $\partial\Omega\in C^{2,\gamma}$.
The aim of the method is to start with $ u_0$ 
solution of
$$
 (2_t)
\cases
Q_{u_0}u_0 =
th(u_0) \text{ in } \Omega
 &\\
u_0=g \quad \text{ in }\partial \Omega &
\endcases
$$
and then find a step
 $\varepsilon $ such that a solution of the problem
$$
(2_{ t +\varepsilon })
\cases
Q_{u}u =(t +\varepsilon)h(u)\text{ in } \Omega
 &\\
u=g \quad \text{ in }\partial \Omega &
\endcases
$$
may be obtained as a limit of a sequence 
$\{ u_n\}_{n\in \N} \subset W^{2,p}(\Omega)$ for some 
$p$ such that $\gamma <1-\frac 2p$.

{\bf Remark:}
If the curvature of $\partial\Omega$ is 
positive then (1) is solvable for 
$h=0$ [5]. 
Thus, by this method it's possible to find 
a sequence $0=t_0<t_1<t_2<... .$ such that ($2_{t_j}$) admits a solution 
for every $t_j$.

In order to define the sequence $\{ u_n\}_{n\in \N} $ we'll use the following 
results:

\lema {Lemma 1}

Let $u,u_0\in C^{1}(\overline\Omega)$. 
Then 
$$\Vert(Q_u-Q_{u_0})v \Vert_p \le \sqrt 3 \Vert u-u_0 \Vert_{1,\infty}
\Vert v \Vert_{2,p}$$
for any $v\in W^{2,p}(\Omega)$ (i.e. $Q:C^{1}(\overline\Omega)\to 
\Cal L (W^{2,p}(\Omega),L^{p}(\Omega))$ is Lipschitz continuous with 
constant $k\le\sqrt 3$).

\demost {Proof}

Let $F_i(a_1,a_2)=\frac {1+a_i^2}{2(1+a_1^2+a_2^2)^{3/2}}$, 
$G(a_1,a_2)=\frac {a_1a_2}{(1+a_1^2+a_2^2)^{3/2}}$. By simple 
computation
we obtain: 
$$|\frac {\partial F_i}{\partial a_j}| \le 
\cases 
\frac 2{3\sqrt 3} \qquad \text {if } i=j &\\
\frac 1{\sqrt 3} \qquad \text { if } i\ne j &
\endcases
$$
and 
$$|\frac {\partial G}{\partial a_j}| \le \frac 4{3\sqrt 3}$$
Thus, 
$$\Vert(Q_u-Q_{u_0})v \Vert_p =
\Vert (F_2(\nabla u)- F_2(\nabla u_0))v_{xx}+
(F_1(\nabla u)- F_1(\nabla u_0))v_{yy}-
 (G(\nabla u)- G(\nabla u_0))v_{xy}\Vert_p$$
$$=\Vert \nabla F_2(\xi_1)\nabla (u-u_0)v_{xx}+
\nabla F_1(\xi_2)\nabla (u-u_0)v_{yy}-
\nabla G(\xi_3)\nabla (u-u_0)v_{xy}\Vert_p$$ 
and the result follows.

\medskip

We recall now the following apriori bound (see e.g. [5], lemma
9.17): let  
$u\in C^{1}(\overline\Omega)$ and $L:W^{2,p}(\Omega)\to 
L^{p}(\Omega)$ the operator given by 
$Lv=Q_uv+\alpha\nabla v+\beta v$, 
where
$\alpha \in L^\infty(\Omega,\R^2)$ and 
$\beta \in L^\infty(\Omega)$ is nonnegative. 
Then $L|_{W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)}$ is 
bounded by below, i.e. there exists a constant $c=c(u)$
such that
$$\text {(3)}\qquad\qquad \Vert v \Vert_{2,p} \le c \Vert Lv \Vert_p$$
for any $v \in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$. 
We'll see that $c$ may be choosen uniformly in a neighborhood of any
$(u,\alpha,\beta)$. In other words, if 
$E=C^1(\overline \Omega)\times L^\infty(\Omega,\R^2) 
\times L^\infty(\Omega)$ 
with the 
norm $\Vert (u,\alpha,\beta)\Vert = max \{ \Vert u \Vert_{1,\infty},
\Vert \alpha \Vert_{p},\Vert \beta \Vert_{p}\}$, then:

\lema {Lemma 2}

Let $c(u,\alpha,\beta)$ be the minimum such that (3) holds. Then
$c:E\to \R$ 
is upper semicontinuous.

\demost {Proof}

Let $(u_0,\alpha_0,\beta_0), (u,\alpha,\beta) \in E$ 
and $t>c(u_0,\alpha_0,\beta_0)$. 
Then, 
for $v\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$,
$$\Vert L_{u,\alpha,\beta)}v \Vert_p\ge 
\Vert L_{(u_0,\alpha_0,\beta_0)}v \Vert_p - 
\Vert (Q_u-Q_{u_0})v \Vert_p-\Vert (\alpha-\alpha_0)\nabla v \Vert_p-
\Vert (\beta-\beta_0)v \Vert_p
\ge$$
$$\frac 1{c(u_0)}\Vert v \Vert_{2,p} -
\sqrt 3 \Vert u-u_0 \Vert_{1,\infty}\Vert v \Vert_{2,p}-
c_1 \Vert \alpha-\alpha_0 \Vert_p\Vert v \Vert_{2,p}-
c_0 \Vert \beta-\beta_0 \Vert_p\Vert v \Vert_{2,p}
 $$
where $c_1$ and $c_0$ are the constants of the imbeddings of
$W^{2,p}(\Omega)$ in $C^1(\overline \Omega)$ and 
$C(\overline \Omega)$ respectively (see e.g. [1] or [5]).

Hence, for $\Vert u-u_0 \Vert_{1,\infty}+
c_1 \Vert \alpha-\alpha_0 \Vert_p+
c_0 \Vert \beta-\beta_0 \Vert_p
\le
\frac 1{\sqrt 3 c(u_0)}$ small enough, 
$$ \frac 1t < \frac 1{c(u_0)}-
\sqrt 3 \Vert u-u_0 \Vert_{1,\infty}-
c_1 \Vert \alpha-\alpha_0 \Vert_p-
c_0 \Vert \beta-\beta_0 \Vert_p
=\frac 1c \le \frac 1{c(u)}$$ 
and the result holds.

\medskip

Let $u_0 \in W^{2,p}(\Omega)$ be a solution of ($2_{t_0}$)
for some $t_0$. 
We define recursively the sequence $\{ u_n\}_{n\in\N}$, 
where $u_{n+1}$ is the solution of the
quasilinear problem
$$
\text{(4)}
\cases
Q_{u_{n+1}}u_{n+1} = 
(t_0 +\varepsilon)(h'(u_n)(u_{n+1}-u_n)+h(u_n))\quad\text{ in } \Omega
 &\\
u_{n+1}=g \quad \text{ in }\partial \Omega &
\endcases
$$
In order to prove
that the sequence is well 
defined for $\ee$ small enough, 
we'll state the following regularity result, which 
shows that $u_n \in 
C^{2,\gamma}(\overline \Omega)$ for every $n$:

\lema {Lemma 3}

Let $u\in W^{2,p}(\Omega)$ be a solution of 
$$\cases 
Q_uu=F(x,y,u) \quad\text{ in } \Omega &\\
u=g \quad \text{ in }\partial \Omega&
\endcases$$
where $F\in C^\gamma(\overline \Omega\times \R)$. 
Then $u\in C^{2,\gamma}(\overline \Omega)$.
\demost {Proof}

As 
$W^{2,p}(\Omega)\hookrightarrow C^{1,\gamma}(\overline \Omega)$, the problem
$$\cases 
Q_uz=F(x,y,u) \quad\text{ in } \Omega &\\
z=g \quad \text{ in }\partial \Omega&
\endcases$$
admits a unique solution $z\in C^{2,\gamma}(\overline \Omega)$, and by the uniqueness
in $W^{2,p}(\Omega)$ we conclude that $z=u$.

\lema {Theorem 4} 
 
There exists $\ee >0$ such that $\{ u_n\}_{n\in \N}$ 
is well defined, and converges in $W^{2,p}(\Omega)$ 
to a solution of $(2_{t_0+\ee})$.

\demost{Proof}

Let us first note that for fixed $v\in B_R(u_0)\subset W^{2,p}(\Omega)$ 
and
$u \in W^{2,p}(\Omega)$, we have:
$$Q_uu-Q_vv=
Q_u(u-v)+\left( DF_2(\nabla v)v_{xx}+ DF_1(\nabla v )v_{yy}
- DG(\nabla v )v_{xy}\right) \nabla (u-v)+r(\nabla u)$$
where the remainder $r$ satisfies:
$$\vert r(\nabla u) \vert \le \overline c \vert \nabla (u - v) \vert^2$$
for some constant $\overline c$ independent of $u$ and $v$. 
Moreover, if $\xi\in L^\infty(\Omega,\R^2)$ is a mean value 
between $\nabla u$ and $\nabla v$, 
and $ L_{v,\xi,u}$
the linear operator
given by
$$L_{v,\xi,u}w=
Q_{u}w+
\left( DF_2(\xi)v_{xx}+ DF_1(\xi) v_{yy}
- DG(\xi) v_{xy}\right) \nabla w-(t_0+\ee)h'(v)w$$
then by lemma 2 there exist constants $c,R$ such that if $v\in C^2(\overline \Omega)$, 
$\Vert v-u_0 \Vert_{2,p} \le R$ and
$\Vert u-u_0 \Vert_{1,\infty} \le c_1
R$,
then
$$\Vert w \Vert_{2,p}\le c \Vert L_{v,\xi,u}w\Vert_p$$
for every $w\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$.

Choosing $R$ and $\ee$ small enough,
we'll see that (4) is uniquely solvable.
Indeed, uniqueness follows from the assumption $h'\ge 0$
(using for example [5], theorem 10.2), and existence may be proved 
by fixed point methods in the following way: 
for $u_1$, writing $z=u_1-u_0$ and $L_z=L_{u_0,\nabla u_0,z+u_0}$, 
problem (4) is equivalent to
$$
\cases
L_{z}z=\ee h(u_0)+r(\nabla (z))
 \text{ in } \Omega
 &\\
z=0 \quad \text{ in }\partial \Omega &
\endcases
$$
Let $T:C^1(\overline \Omega)\to C^1(\overline \Omega)$
be the continuous 
operator defined by $Tz=w$, where $w\in W^{2,p}(\Omega)$ 
is the unique solution of the linear problem
$$
\cases
L_zw=\ee h(u_0)+r(\nabla (z))
 \text{ in } \Omega
 &\\
w=0 \quad \text{ in }\partial \Omega &
\endcases
$$
Then for $\Vert  z \Vert_{1,\infty}\le \overline R\le c_0R$ 
and a compact set $K$ containing a neighborhood of 
$u_0(\overline\Omega)$ 
we have:
$$\Vert  Tz \Vert _{2,p}\le \Vert L_z(Tz)\Vert_p = 
c \Vert \ee h(u_0)+r(\nabla z)\Vert_p 
\le c(\ee \Vert h \Vert_{\infty,K}+\overline c \overline R^2)$$ 
and 
by the compactness of the imbedding 
$W^{2,p}(\Omega) \hookrightarrow C^1(\overline \Omega)$ 
we conclude that the closure of 
$T(\{\Vert z \Vert_{1,\infty} \le \overline R\})$ 
is compact. Furthermore, 
$$\Vert  Tz \Vert _{1,\infty}\le
c_0c(\ee \Vert h 
\Vert_{\infty,K}+\overline c \overline R^2)\le \overline R$$
if $\ee$ and $\overline R$ 
are small enough. By Schauder theorem, we conclude 
that $T$ has a fixed point $z$, 
and then $u_1=z+u_0$ is a solution of (4).

Let us 
assume that the sequence is well defined up to $u_{n+1}$. 
Then, for $n>0$ 
$$Q_{u_{n+1}} u_{n+1}-Q_{u_{n}} u_n-(t_0 +\varepsilon)
h'(u_n)(u_{n+1}-u_n)=$$
$$(t_0 +\varepsilon)[h(u_n)-h_(u_{n-1})-h'(u_{n-1})(u_n-u_{n-1})]=
(t_0 +\varepsilon) {h''(s)\over 2}(u_n-u_{n-1})^2$$
for some mean value $s \in L^\infty(\Omega)$.

Moreover, if
$u_j \in B_R(u_0)\subset W^{2,p}(\Omega)$ 
for $j=1,...,n+1$
then
$$\|u_{n+1}-u_n\|_{2,p}\le c\|Q_{u_{n+1}}
u_{n+1}-Q_{u_{n}}u_n-(t_0 +\varepsilon)h'(u_n)
(u_{n+1}-u_n)\|_p,$$
and we conclude that
$$\|u_{n+1}-u_n\|_{2,p} \le c{(t_0 +\varepsilon )\over 2}
\|h''\|_{\infty,K}
\|u_n-u_{n-1}\|_p\|u_n-u_{n-1}\|_\infty \le $$
$$ \le {cc_0\over2}(t_0 +\varepsilon )
\|h''\|_{\infty,K}\|u_n-u_{n-1}\|_{2,p}^2$$
for $n>0$.

Thus, by induction
$$\|u_{n+1}-u_n\|_{2,p} \le({cc_0\over2}(t_0 +\varepsilon )
\|h''\|_{\infty,K}
\|u_1-u_0\|_{2,p})^{2^n-1}\|u_1-u_0\|_{2,p}$$
and as
$$\|u_1-u_0\|_{2,p}\le c\varepsilon \|h(u_0)\|_p,$$
if
$\varepsilon$ satisfies 
$$c(\varepsilon)=
{c^2c_0\over 2}(t_0 +\varepsilon )
\|h''\|_{\infty,K}
\varepsilon \|h(u_0)\|_p <1$$
then
$$\Vert u_{n+1} - u_0 \Vert_{2,p} \le 
\sum_{0\le j 
\le n}\Vert u_{j+1} - u_j \Vert_{2,p}
\le \frac {c\ee\Vert h(u_0) \Vert_{p}}{1- c(\varepsilon)}.$$

Choosing $\ee$ small, 
$\Vert u_1-u_0 \Vert_{2,p}\le R$, 
and then we may assume as inductive 
hypothesis that the sequence is well defined up to $u_{n}$ and that 
$u_{n}\in B_R(u_0)$. 
As before, if $z=u_{n+1}-u_n$, problem (4) is equivalent to
$$
\cases
L_zz=(t_0 +\varepsilon) {h''(s)\over 2}(u_n-u_{n-1})^2
+ r(\nabla (z))
 \text{ in } \Omega
 &\\
z=0 \quad \text{ in }\partial \Omega &
\endcases
$$
where $L_z:= L_{u_n,\nabla u_n,z+u_n}$ 
and defining an operator 
$T:C^1(\overline \Omega)\to C^1(\overline \Omega)$ 
we obtain for $\Vert z \Vert_{1,\infty}\le \overline R \le c_0R$:
$$\Vert Tz \Vert_{1,\infty} 
\le c_0c\left( c(\ee)^{2^n-1}\ee \Vert h(u_0)\Vert_p+
\overline c \overline R^2\right)$$
Then, it suffices to consider for example 
$\ee\le (c_0R)^2$ such that $c(\ee) < 1$ and
$$c_0c(\Vert h(u_0)\Vert_p+\overline c)\sqrt \ee \le 1,$$
since in that case taking $\overline R=\sqrt\ee$ we obtain 
$\Vert Tz \Vert_{1,\infty}\le \overline R$, and the existence of
$u_{n+1}$ can be deduced from Schauder theorem. 

Furthermore, as 
$\|u_{n+1}-u_n\|_{2,p} \le
c(\ee)^{2^n-1}\|u_1-u_0\|_{2,p}$, $\{u_n\}_{n\in \N}$ is a Cauchy 
sequence in $W^{2,p}(\Omega)$, and the proof is complete.

{\bf Remark:}

A sequence $\{u_n\}_{n\in \N}$ 
may be also defined recursively by 
the {\sl linear} problems
$$
\cases
Q_{u_{n}}u_{n+1} = 
(t_0 +\varepsilon)(h'(u_n)(u_{n+1}-u_n)+h(u_n))
\qquad \text { in } \Omega &\\
u_{n+1}=g \qquad \text { in } \partial \Omega &
\endcases 
$$
In this case, convergence can be
guaranteed for $\varepsilon$ small enough if
$\Vert u_0 \Vert_{2,p}$ is small.

\bigskip

{\bf REFERENCES}

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

[2] 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.

[3]  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.

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

[5] Gilbarg D., Trudinger, N. S. : Elliptic partial differential
equations of second order, Springer- Verlag (1983).
 
[6]  Goldstein Costa D., T\'opicos em Analise n\~ao
Linear e Aplicac\~oes as Ecuac\~{o}es Diferenciais. I.M.P.A., 1986. 

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

[8] Hsiao G., Lectures on variational methods for boundary integral
equations: theory and applications. Publicaciones de la primera escuela
de verano del Fondap en Matem\'aticas Aplicadas, Univ. de Concepci\'on,
Chile, 1998.

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

[10]  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.

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

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

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

[14] Vainberg M. M., 
Variational Methods in the Theory of Nonlinear Operators, Holden-Day, San Francisco, 1964.


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


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

${}^*$ CONICET


{\bf 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.

\medskip
{\bf email:} mcmarian\@dm.uba.ar

\end
---------------0004041811453--