\input vanilla.sty %\input gordo %\input refp \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\Rp{\pmb{R}} \def\lra{\longrightarrow} \def\sen{\;\text{sen}\;} \def\noi{\noindent} \def\br{(B,\Rp ^3)} \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\ds{\;\text{d}s\;} \def\cor{\allowmathbreak} \def\mk{\overline{M(k)}} \def\id{\;\text{id}\;} \def\tr{\;\text{Tr}\;} \def\inb{\int\sb B } \def\dde{\sober{$d$}/{$d\va$}} \def\D{\pmb{D}} \def\flecha{\Longleftrightarrow} \def\C{\pmb{C}} \def\div{\;\text{div}\;} \title Existence and uniqueness of H-systems's solutions with Dirichlet conditions \endtitle \title P. Amster, M. C. Mariani and D. F. Rial \endtitle \vglue 2truecm \newdimen\normalbaselineskip \normalbaselineskip=10pt \normalbaselines \pagewidth{12.5cm} \pageheight{17cm} \lema{Abstract:}We give conditions on the boundary data in order to obtain a solution for the problem (1) below and we prove that in certain cases the solutions are isolated. The condition $\Vert H\Vert_{\infty}\Vert g\Vert_{\infty}\le 1$ is not necessary. \newdimen\normalbaselineskip \normalbaselineskip=15pt \normalbaselines \pagewidth{13cm} \pageheight{19cm} \bigskip \tit{Introduction} We consider the Dirichlet problem in the unit disc $B=\{ ( u,v) \in { R^2};u^2+v^2< 1\} $ for a vector function $X: \overline{B}\longrightarrow { R^3}$ which satisfies the equation of prescribed mean curvature $$ \text{(1)} \cases \Delta X=2H(u,v)X_u\land X_v \qquad \text{ in }\quad B &\\ X=g\qquad \text{ in }\quad \partial B & \endcases $$ where $X_u=\dfrac{\partial X}{\partial u},$ $X_v=\dfrac{\partial X}{\partial v},$ $\land$ denotes the exterior product in ${ R^3}$ and $H:{\overline B}% \longrightarrow { R}$ is a given continuous function. The problem above arises in the Plateau and Dirichlet problems for the prescribed mean curvature equation that has been studied in [1-3-4-5-6-7-8]. The main results are the following theorems: \lema {Theorem 1} Let be $2
2$ then $X$ is isolated in $W^{1,p}$. Moreover, if $\overline\Omega$ is a bounded subset of $W^{1,p}$ the number of solutions of (1) in $\overline\Omega$ is finite. \lema{Remark} For $H$ constant Hildebrandt found a solution of (1) in $W^{1,2}$, for the case $\vert H\vert\Vert g\Vert_{\infty}\le 1$ [3]. Theorem 1 gives a solution for $p>2$ when $g$ is close to a constant $a$. Since the equation in $B$ depends only on the derivatives of $X$, we may suppose $a = 0$ \tit{1. Solution by fixed point methods} The systems (2) and (3) are equivalent to (1) with $X=X_0+Y$ $$\text{(2) } \cases \Delta X_0=0 \text{ in } \quad B &\\ X_0=g\qquad \text{ in } \quad \partial B & \endcases $$ $$\text{(3) } \cases \Delta Y=F(X_0,Y) \qquad \text{ in } \quad B &\\ Y=0\qquad \text{ in } \quad \partial B & \endcases $$ where $F$ is given by $$F( X_0,Y)=2H(u,v) [(X_{0u}\land Y_v+Y_u\land X_{0v})+(Y_{u}\land Y_v+ X_{0u}\land X_{0v})] $$ For $g \in W^{2,p}\left( B,{ R^3}\right)$, (2) admits a unique solution in $W^{2,p}\left( B,{ R^3}\right)$. We can rewrite (3) as: $$\cases L(X_0) Y=F_1(X_0,Y) \qquad \text{ in } \quad B &\\ Y=0\qquad \text{ in } \quad \partial B & \endcases $$ where $L(X_0)$ is the linear operator $$L(X_0) Y=\Delta Y-2H(u,v) (X_{0u}\land Y_v+Y_u\land X_{0v})$$ and $F_1$ is defined by $$F_1(X_0,Y)=2H(u,v)(Y_{u}\land Y_v+X_{0u}\land X_{0v}).$$ \lema{Remark} $L(X_0)$ is strictly elliptic and for $p > 2$ its coefficients are bounded since $X_0\in W^{1,\infty}\left( B,{ R^3}\right) $ and $H\in C(\overline B)$. We will use the following technical lemmas: \lema{Lemma 4} Let be $X_0\in W^{2,p}(B,{ R^3})$, then i) If $2
1$, $X_0$, $Y_1$, $Y_2 \in W^{1,p}\cap W^{1,\infty}$ $$\left\| F_1( X_0,Y_1)\right\| _{p}\le 2\Vert H\Vert _{\infty}(\Vert X_0\Vert_{1,\infty}.\Vert X_0\Vert_{1,p}+\left\| Y_1\right\| _{1,\infty}.\Vert Y_1\Vert_{1,p}) \tag6 $$ $$\left\| F_1\left( X_0,Y_1\right) -F_1\left( X_0,Y_2\right) \right\| _{p}\leq 4\Vert H\Vert _{\infty} R \left( \left\| Y_1-Y_2\right\| _{1,p}\right)\tag7$$ for $Y_1, Y_2\in B_R(0) \subset W^{1,\infty}$. \demost {Proof} $$\left\| F_1( X_0,Y_1)\right\| _{p/2}\le 2\Vert H\Vert _{\infty}(\Vert Y_{1u}\land Y_{1v} \Vert_{p/2}+ \Vert X_{0u}\land X_{0v}\Vert_{p/2})$$ and $$\Vert Y_{1u}\land Y_{1v}\Vert_{p/2}^{p/2}\le \int \vert Y_{1u}\vert^{p/2} \vert Y_{1v}\vert^{p/2}\le \Vert Y_{1u}\Vert_p^{p/2}\Vert Y_{1v}\Vert_p^{p/2}$$ The same inequality holds for $X_0$, and in order to prove (5), we also have that $$\Vert Y_{1u}\land Y_{1v}-Y_{2u}\land Y_{2v}\Vert_{p/2}\le \Vert Y_{1u}\land (Y_{1v}-Y_{2v})\Vert_{p/2}+\Vert Y_{2v}\land (Y_{1u}-Y_{2u})\Vert_{p/2} $$ and the proof follows. In the same way we obtain (6) and (7), using that $\Vert a\land b \Vert_p \le \Vert a\Vert_\infty. \Vert b\Vert_p $ $\clubsuit $ \medskip \lema{Lemma 5} If $\Vert X_{0}\Vert_{1,p}\le\delta$, $2
2$, then the following problem $$\text{(8) } \cases L(X_0) Y=F_1(X_0,\overline Y) \qquad \text{ in }\quad B &\\ Y=0\qquad \text{ in } \quad \partial B & \endcases $$ define a continuous map $T:\overline Y\longrightarrow Y$ in $W_0^{1,p}$. Furthermore, if $\Vert X_0\Vert_{1,p}$ is small enough, there exists a number $R$ such that $T$ is a contraction in $B_R(0) \subset W^{1,p} $. \demost {Proof} >From Theorem 9.15 in [2], problem (8) admits a unique solution $Y \in W^{2,p/2}$, so the map $T$ is well defined. Also we have from (5) that if $\overline Y,\overline Z\in B_R \subset W^{1,p}$ then $$\left\| Y-Z\right\| _{2,p/2}\leq C( \Vert L(X_0)(Y-Z)\Vert_{p/2}=C \Vert F_1(X_0,\overline Y)-F_1(X_0,\overline Z)\Vert_{p/2}\le 4C\Vert H\Vert _{\infty}R\Vert \overline Y-\overline Z\Vert_{1,p}$$ and using the Sobolev inmersion of $W^{2,p/2}$ in $W^{1,p}$ the continuity of $T$ follows. Also we have from (4) $$\left\| Y\right\| _{1,p}\leq\overline c\left\| Y\right\| _{2,p/2} \leq C\overline c\left\| L(X_0)Y\right\| _{p/2} \leq 2C\overline c \Vert H\Vert _{\infty}(\Vert X_0\Vert_{1,p}^2+\left\|\overline Y\right\| _{1,p}^2) $$ Choosing $\Vert X_0\Vert_{1,p}$ and $R$ small enough, we obtain $$4C\Vert H\Vert _{\infty}R<1$$ $$\left\| Y\right\| _{1,p}\leq R$$ and $T$ is a contraction in $B_R$. $\clubsuit $ \lema{Remark} Note that if $Y$ is a fixed point of $T$, then $Y \in W^{2,p}$, and we obtain Theorem 1 as an immediate consequence of Proposition 6. \medskip \medskip \tit{2. Local unicity of the solutions} Now we will prove theorem 2. Let $X_0$ be a solution of (1). If $Y$ is another solution of (1) then $Z=Y-X_0$ satisfies $$ \cases L(X_0) Z=2H(u,v)Z_u\land Z_v \text{ in } \quad B &\\ Z=0\qquad \text{ in } \quad \partial B & \endcases $$ We consider the associated problem $$ \cases L(X_0) Z=2H(u,v)\overline Z_u\land\overline Z_v \text{ in } \quad B &\\ Z=0\qquad \text{ in } \quad \partial B & \endcases $$ For ii), in the same way as before, we obtain $$\left\| Z\right\| _{1,p}\leq 2 C(X_0)\overline c \Vert H\Vert _{\infty}\Vert \overline Z\Vert^2_{1,p}$$ and if $\overline Z_1,\overline Z_2\in B_R(0) \subset W^{1,p}$ then $$\left\| Z_1-Z_2\right\| _{1,p}\leq 4 C(X_0)\overline c \Vert H\Vert _{\infty}R\Vert \overline Z_1-\overline Z_2\Vert_{1,p}$$ If we choose $R$ verifying $$4 C(X_0)\overline c\Vert H\Vert _{\infty}R<1$$ the map $\overline T:\overline Z\longrightarrow Z$ is a contraction in $B_R$, so $X_0$ is isolated. \lema {Remark} >From Sobolev inmersions $\overline T$ is compact in $W^{1,p}$ and being any solution of (1) a fixed point of $\overline T$, we conclude that the number of solutions in $B_R$ is finite for every $R$. In the same way we prove i) using (6) and (7). $\clubsuit $ \lema{Remark} The number $R$ in the theorems above can be estimated in terms of the Sobolev inmersions constants, $\Vert H\Vert _{\infty}$ and $\Vert g\Vert _{2,p}$. \tit{References} [1] H. Brezis, J. Coron, Multiple solutions of $H$ systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), 149-187. [2] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer- Verlag (1983). [3] S. Hildebrandt, On the Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 23 (1970) 97-114. [4] E. Lami Dozo, M.C. Mariani, A Dirichlet problem for an $H$ system with variable $H$. Manuscripta Math. 81 (1993), 1-14. [5] M. C. Mariani, D. Rial, Solutions to the mean cuvature equation by fixed point methods. Bulletin of The Belgian Mathematical Society - Simon Stevin (1997). [6] M. Struwe, Plateau 's problem and the calculus of variations, Lecture Notes Princeton Univ. Press (1988). [7] M. Struwe, Multiple solutions to the Dirichlet problem for the equation of prescribed mean curvature, Research papers published in honor of J. Moser's 60 Birthday, Rabinowitz, P and Ehnder, E. (Eds.), Boston, Academic Press, 639-666 (1990). [8] Wang Guofang: The Dirichlet problem for the equation of prescribed mean curvature, Analyse Nonlin\'eaire 9 (1992), 643-655. \bigskip \bigskip {\bf P.Amster and D. F. Rial} Dpto. de Matem\'atica Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Capital, Argentina {\bf M. C. Mariani} Dpto. de Matem\'atica Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Capital, Argentina CONICET \bigskip {\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} \end