Content-Type: multipart/mixed; boundary="-------------0004041804101" This is a multi-part message in MIME format. ---------------0004041804101 Content-Type: text/plain; name="00-155.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-155.comments" To appear in the Belgian Bull. Simon Stevin ---------------0004041804101 Content-Type: text/plain; name="00-155.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-155.keywords" prescribed mean curvature equation - Green function ---------------0004041804101 Content-Type: application/x-tex; name="green.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="green.tex" \input vanilla.sty \pagewidth{13cm} \pageheight{19cm} \normalbaselineskip=15pt \normalbaselines \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\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\noi{\noindent} \def\lra{\longrightarrow} \def\R{\text{{\rm I}\!{\rm R}}} \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 } \vglue 1.5truecm \newdimen\normalbaselineskip \normalbaselineskip=10pt \normalbaselines \pagewidth{12.5cm} \pageheight{17cm} \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. \newdimen\normalbaselineskip \normalbaselineskip=15pt \normalbaselines \pagewidth{13cm} \pageheight{19cm} \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 $A,B,C \in L^{\infty}(\Omega,\R^{3\times 3})$, $20$, 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 \end ---------------0004041804101--