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 $0=t_0c(u_0,\alpha_0,\beta_0)$. 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--