Content-Type: multipart/mixed; boundary="-------------0004051738213" This is a multi-part message in MIME format. ---------------0004051738213 Content-Type: text/plain; name="00-166.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-166.comments" To appear in Revista de la Union Matematica Argentina ---------------0004051738213 Content-Type: text/plain; name="00-166.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-166.keywords" nonparametric surfaces - variational calculus ---------------0004051738213 Content-Type: application/x-tex; name="umapaper.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="umapaper.tex" \input vanilla.sty %\input gordo %\input refp %\nopagenumbers \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\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\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}\;} %\tit{ } \bigskip \bigskip \bigskip \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 $02$ 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} \end ---------------0004051738213--