Content-Type: multipart/mixed; boundary="-------------0004051245836" This is a multi-part message in MIME format. ---------------0004051245836 Content-Type: text/plain; name="00-165.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-165.comments" To appear in Abstract and Applied Analysis ---------------0004051245836 Content-Type: text/plain; name="00-165.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-165.keywords" nonparametric surfaces - variational methods ---------------0004051245836 Content-Type: application/x-tex; name="gossez.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gossez.tex" \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\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}\;} \hyphenation{Iberoame-ricana} \title Existence and regularity of weak solutions to the prescribed \endtitle \title mean curvature equation for a nonparametric surface \endtitle \centerline {P. Amster. M. Cassinelli M. C. Mariani and D.F. Rial} \centerline {Departamento de Matem\'{a}tica, FCEyN-UBA } \vglue 2truecm \newdimen\normalbaselineskip \normalbaselineskip=10pt \normalbaselines \pagewidth{12.5cm} \pageheight{17cm} \lema{Abstract:} We give conditions on the boundary data and the function $h$ in order to obtain multiple solutions for the problem (1) below, using variational methods. We also study the regularity of the solutions. \newdimen\normalbaselineskip \normalbaselineskip=15pt \normalbaselines \pagewidth{13cm} \pageheight{19cm} \bigskip \tit{Introduction} The prescribed mean curvature equation with Dirichlet condition for a nonparametric surface $X:\Omega \longrightarrow R^3$, $X(u,v)=(u,v,f(u,v))$ is the quasilinear partial differential equation $$ \text{(1)} \cases (1+f_v^2)f_{uu}+(1+f_u^2)f_{vv}-2f_uf_vf_{uv}= 2h(u,v,f)\left( 1+\left| \nabla f\right| ^2\right) ^{\frac 32} \text{ in } \Omega &\\ f=g \quad \text{ in }\partial \Omega \endcases $$ where $\Omega $ is a bounded domain in $R^2$, $h:\overline \Omega \times R\longrightarrow R$ is continuous and $g\in H^1(\Omega )$. We call $f\in H^1(\Omega)$ a weak solution of (1) if $f\in g+H_0^1(\Omega )$ and for every $\varphi \in C_0^1(\Omega )$ $$\int_\Omega \left( ( 1+| \nabla f| ^2) ^{-1/2} \nabla f\nabla \varphi + 2h(u,v,f) \varphi\right) dudv=0 $$ It is known that for the parametric Plateau's problem, weak solutions can be obtained as critical points of a functional (see [2], [5], [6], [7], [9], [10]). The nonparametric case has been studied for $H=H(x,y)$ (and generally $H=H(x_1,...,x_n)$ for hypersurfaces in $R^{n+1}$) by Gilbarg, Trudinger, Simon, Serrin, among other authors. It has been proved [12] 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 $\epsilon > 0$. They also proved a non-existence result ([12], 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 $\Vert g \Vert_\infty \le \epsilon$ and that Dirichlet's problem is not solvable. We remark that the solutions obtained in [12] are classical. In this work we find weak solutions of the problem by variational methods. We will prove that for prescribed $h$ there exists an associated functional to $h$, and under some conditions on $h$ and $g$ we will find that this functional has a global minimum in a convex subset of $H^1(\Omega )$, which provides a weak solution of (1). We denote $H^1(\Omega )$ the usual Sobolev space, [1]. \bigskip \tit{The associated variational problem} \medskip Given a function $f\in C^2(\Omega )$, the generated nonparametric surface associated to this function is the graph of $f$ in $R^3$, parametrized as $X(u,v)=(u,v,f(u,v)). $ The mean curvature of this surface is $$h(u,v,f)={1\over 2} {Ef_{vv}-2Ff_{uv}+Gf_{uu}\over (1+f_u^2+f_v^2)^{3\over 2}} $$ where $E,F$ and $G$ are the coefficients of the first fundamental form [3], [8]. For prescribed $h$, weak solutions of (1) can be obtained as critical points of a functional: \medskip \lema{Proposition 1} Let $J_h:H^1(\Omega )\longrightarrow R$ be the functional defined by $$J_h(f)=\int\limits_\Omega \left( ( 1+| \nabla f| ^2) ^{1/2}+H(u,v,f)\right)dudv \tag2$$ where $H(u,v,z)=\int_0^z2h(u,v,t)dt$. Then (1) is the Euler Lagrange equation of (2). \lema{Remark} If $f\in T = g + H_0^1(\Omega )$ is a critical point of $J_h$, then $f$ is a weak solution of (1). \demost{Proof} For $\varphi \in C_0^1(\Omega )$, integrating by parts we obtain: $$dJ_h(f)(\varphi )=2\int_\Omega \left({1\over 2}{Ef_{vv}-2Ff_{uv}+Gf_{uu}\over (1+f_u^2+f_v^2)^{3\over 2}}-h(u,v,f)\right)\varphi dudv $$ \medskip \tit{Behavior of the functional $J_h$} In this section we will study the behavior of the functional $J_h$ restricted to $T$. For simplicity we write $J_h(f)=A(f)+B(f) $, with $$A(f)=\int_\Omega ( 1+| \nabla f| ^2) ^{1/2} dudv , \qquad \qquad B(f)=\int_\Omega H(u,v,f)dudv. $$ We'll assume that $h$ is bounded. \lema{Lemma 2} The functional $A:T\longrightarrow R$ is continuous and convex. \demost{Proof} Continuity can be proved by a simple computation. Let $a,b\ge 0$ such that $a+b=1$. By Cauchy inequality, it follows that $$\sqrt{1+| \nabla (af+bf_0)| ^2}\le a\sqrt{1+| \nabla f|^2}+b\sqrt{1+|\nabla f_0|^2} $$ and convexity holds. \lema{Remark} As $A$ is continuous and convex, then it is weakly lower semicontinuous in $T$. \lema{Lemma 3} The functional $B$ is weakly lower semicontinuous in $T$. \demost{Proof} Since $h$ is bounded, we have that $$| H(u,v,z)| \le c| z| +d $$ >From the compact immersion $H_0^1( \Omega) \hookrightarrow L^1(\Omega)$ and the continuity of Nemytski operator associated to $H$ in $L^1( \Omega)$, we conclude that $B$ is weakly lower semicontinuous in $T$ (see [4] and [11]). \bigskip \tit{Weak solutions as critical points of $J_h$} \medskip Let us assume that $g \in W^{1,\infty}$, and consider for each $k>0$, the following subset of $T$: $$ \overline M_k=\{ f\in T:\quad\quad \| \nabla (f-g)\| _\infty \le k \}. $$ $\overline M_k$ is nonempty, closed, convex, bounded, then it is weakly compact. \lema {Remark:} as $g \in W^{1,\infty}$, taking $p>2$ we obtain, for any $f \in \overline M_k$: $$\Vert f-g \Vert_p \le c \Vert \nabla (f-g) \Vert_p $$ Then, by Sobolev imbedding, $\Vert f - g \Vert_\infty \le c_1\Vert f - g \Vert_{1,p} \le \overline c k$ for some constant $\overline c$. We deduce that $f \in W^{1,\infty}$ and $f(\Omega) \subset K$ for some fixed compact $K \subset R$. Thus, the assumption $\Vert h \Vert_\infty < \infty$ will not be needed. \rm Let $\rho $ be the slope of $J_h$ in $\overline M_k$ defined by: $$ \rho (f_0,\overline M_k)=\sup \{dJ_h(f_0)(f_0-f);f\in \overline M_k\} $$ (see [6] and [9]), then the following result holds: \lema{Lemma 4} If $f_0\in \overline M_k$ verifies $$J_h( f_0) =\inf \{ J_h( f) :f\in \overline M_k\} $$ then $\rho (f_0,\overline M_k)=0$. \demost {Proof} $$dJ_h(f_0)(f-f_0)=\lim_{\varepsilon \longrightarrow 0} {J_h(f_0+\varepsilon (f-f_0))-J_h(f_0)\over\varepsilon} = \lim_{\varepsilon \longrightarrow 0}{J_h((1-\varepsilon)f_0+\varepsilon f) -J_h(f_0)\over\varepsilon }.$$ When $0<\varepsilon <1$ we have that $(1-\varepsilon )f_0+\varepsilon f\in \overline M_k$, and then $dJ_h(f_0)(f_0-f)\le 0$ for all $f\in \overline M_k $. As $dJ_h(f_0)(f_0-f_0)=0$, we conclude that $\rho ( f_0,\overline M_k) =0$. \lema {Remark:} Being $J_h$ weakly semicontinuous and $\overline M_k$ a weakly compact subset of $T$, $J_h$ achieves a minimum $f_0$ in $\overline M_k$. By lemma 4, $\rho(f_0,\overline M_k)=0$. \rm As in [6], if $f_0$ has zero slope, we call it a $\rho$-critical point. The following result gives sufficient conditions to assure that if $f_0$ is a $\rho$-critical point, then it is a critical point of $J_h$. \lema{Theorem 5} Let $f_0\in \overline M_k$ such that $\rho(f_0,\overline M_k)=0$, and assume that one of the following conditions holds: i) $dJ_h( f_0)(f_0 -g) \ge 0$ ii) $\Vert \nabla (f_0 - g) \Vert_\infty < k$ \noi Then $dJ_h( f_0)=0$. \demost{Proof} As $\rho (f_0,\overline M_k)=0$, we have that $dJ_h(f_0)(f_0-f)\le 0$ and then $dJ_h(f_0)(f_0-g)\le dJ_h(f_0)(f-g)$ for any $f\in \overline M_k$. We'll prove that $dJ_h(f_0)(\varphi )=0$ for any $\varphi \in C_0^1$. Let $\widetilde{\varphi }= {k\varphi \over 2\| \nabla \varphi \| _\infty }$, then $\pm\widetilde{\varphi }+g\in \overline M_k$, and then $dJ_h(f_0)(f_0-g)\le \pm dJ_h(f_0)(\widetilde{\varphi })$. Let us suppose that $dJ_h(f_0)(\widetilde{\varphi })\ne 0$, then $dJ_h(f_0)(f_0-g)<0$. If i) holds, we immediately get a contradiction. On the other hand, if ii) holds, there exists $r >1$ such that $g + r(f_0-g) \in \overline M_k$. Then $ dJ_h(f_0)(f_0-g) \le r dJ_h(f_0)(f_0-g)$, a contradiction. \lema {Examples} Let us assume that $\int_\Omega \dfrac {\nabla(f-g)\nabla g}{\sqrt{1+\vert \nabla f \vert^2}}dudv \ge 0$ for any $f \in \overline M_k$. Then condition i) of theorem 5 is fulfilled for example if: a) $| h(u,v,z)| \le c( z- g(u,v))_{+}$ for every $(u,v)\in \Omega$, $z\in {R}^3$, for some constant $c$ small enough. b) $\int_\Omega h(u,v,f)(f-g)dudv \ge 0$ for every $f \in \overline M_k$. As a particular case, we may take $h(u,v,z) = c(z-g(u,v))$ for any $c \ge 0$. c) $h(u,v,z) = -c(z-g(u,v))$ for some $c > 0$ small enough. Indeed, in all the examples the inequality $dJ_h(f)(f-g) \ge 0$ holds for any $f \in \overline M_k$, since $$dJ_h(f)(f-g)=\int_\Omega \left({\nabla f\nabla ( f-g) \over\sqrt{1+| \nabla f|^2}} +2h(u,v,f) (f-g)\right)dudv= $$ $$\int_\Omega \left({| \nabla (f-g) | ^2\over \sqrt{1+| \nabla f| ^2}}+2h(f-g)\right) dudv+ \int_\Omega {\nabla(f-g)\nabla g\over\sqrt{1+|\nabla f|^2}}dudv \ge $$ $$\int_\Omega \left({| \nabla (f-g) | ^2\over \sqrt{1+| \nabla f| ^2}}+2h(f-g)\right)dudv.$$ Then the result follows immediately in example b). In examples a) and c), being $\Vert \nabla( f-g) \Vert_\infty \le k$ we can choose $\widetilde{k}$ such that $\sqrt{1+\Vert \nabla f \Vert_\infty ^2}\le \tilde k$. Then $$\int_\Omega \left( {|\nabla ( f-g) | ^2\over \sqrt{1+|\nabla f|^2}}+2h(u,v,f)(f-g)\right)dudv \ge \int_\Omega \left(\frac {|\nabla (f-g)| ^2}{\tilde k} -2c(f-g) ^2\right)dudv$$ $$\ge {1\over\tilde{k}} \|\nabla(f-g)\|_2^2-2cc_1^2\|\nabla (f-g)\|_2^2=({1\over\tilde{k}}-2cc_1^2) \|\nabla(f-g)\|_2^2$$ where $c_1$ is the Poincar\'{e}'s constant associated to $\Omega.$ Thus, the result holds for $c \le \frac 1{2\tilde{k} c_1^2}$. \lema {Remark} As it happens in the preceding examples, it can be proved that if $dJ_h(f)(f-g) \ge 0$ for any $f \in \overline M_k$, then $g$ is a weak solution of (1). Indeed, if $dJ_h(g) \ne 0$, from theorem 5 it follows that $\rho (g,\overline M_k)>0$. As $J_h$ achieves a minimum in every $\overline M_k$, we may take $k \ge k_n \longrightarrow 0$, and $f_n$ such that $\rho (f_n,\overline M_{k_n}) =0$. As $\overline M_{k_n}\subset \overline M_k$, condition i) in theorem 5 holds and then $dJ_h(f_n)=0$. It is immediate that $f_n \longrightarrow g $ in $W^{1,\infty}$ and then it follows easily that $dJ_h(g)=0$. Furthermore, for constant $g$ we can see that if $dJ_h(f)(f-g) \ge 0$ for any $f \in \overline M_k$, then $g$ is a global minimum of $J_h$ in $\overline M_k$: let us define $\varphi(t)=J_h(tf+(1-t)g)$, then $\varphi'(t)=dJ_h(tf+(1-t)g)(f-g)$. As $0 \le dJ_h(tf+(1-t)g)(tf+(1-t)g-g) = tdJ_h(tf+(1-t)g)(f-g)$ it follows that $J_h(f)-J_h(g)=\varphi(1)-\varphi(0)=\varphi'(c) \ge 0$. \medskip \tit{Multiple Solutions} In this section we study the multiplicity of weak solutions of (1). Let us consider: $$\overline N_k=\{f\in \overline M_k\cap H^2: \|\frac{\partial^2 f} {\partial x_{i}\partial x_{j}}\|_2\le k\} $$ $\overline N_k$ is a nonempty, closed, bounded, and convex subset of $T$, therefore $\overline N_k$ is weakly compact. \rm Then we obtain the following theorem, which is a variant of the Mountain Pass Lemma: \lema{Theorem 6} Let $f_0\in \overline N_k$ be a local minimum of $J_h $ and assume that $J_h(f_1)0$ we take $g_n\in \overline N_k$ such that $\rho (f_n, \overline N_k)-\frac \epsilon 20$. Then, $g_0$ is a minimum of $J_h$ in $\overline M_{k_1}$ for $k_1$ small enough, and a local minimum in $M_k$ for any $k \ge k_1$. Moreover, taking $\Omega = B_R$, $f(u,v) = g_0 + R^2 - (u^2+v^2)$, it follows that $$J_h(f) - J_h(g_0) = 2\pi (o(R^3) - \frac c6 R^6)$$ and taking $k=2\sqrt \pi R$ it holds that $f \in \overline N_k$. Hence, if $R$ is big enough it follows that $g_0$ is not a global minimum in $\overline N_k$. Furthermore, we see that the proof of lemma 4 may be repeated in $\overline N_k$, and then the minimum of $J_h$ in $\overline N_{k}$ is a $\rho$-critical point. From Theorem 6 there is a third $\rho$-critical point which is not a local minimum of $J_h$. \tit{Regularity} As we proved, problem (1) admits (for an apropiated $k>0$) a weak solution in a subset $\overline{M}(k)=\left\{ f\in T\quad /\quad \left\| \nabla (f-g)\right\| _\infty \le k\right\}.$ Let us consider $p>2$, and $f_0 \in W^{2,p}(\Omega) \hookrightarrow C^1(\overline \Omega)$ a weak solution of (1). Then $ L_{f_0}f_0 = 2 h(u,v,f_0)(1+ \nabla f_0^2)^{3/2} \text{ in } \Omega$ where for any $f \in C^1(\overline \Omega)$ $L_{f}:W^{2,p}\longrightarrow L^p$ is the strictly elliptic operator given by $$L_{f}\phi= (1+f_v^2)\phi_{uu}+(1+f_u^2)\phi_{vv}-2f_uf_v\phi_{uv}$$ In order to prove the regularity of $f_0$, we study the equation (2) $ L_{f_0}\phi = 2 h(u,v,f_0)(1+ \nabla f_0^2)^{3/2} \text{ in } \Omega$, $\phi = g \text { in } \partial \Omega$. \lema {Proposition 10} Let us assume that $\partial \Omega \in C^{2,\alpha}$, $g \in C^{2,\alpha}$ and $h \in C^\alpha$ for some $0 < \alpha \le 1- \dfrac 2p$. Then, if $\phi \in W^{2,p} $ is a strong solution of (2), $\phi \in C^{2,\alpha}(\overline{\Omega })$. \demost{Proof} By Sobolev imbedding $\phi \in C^{1,\alpha}(\overline \Omega)$. Then $L_{f_0}\phi \in C^\alpha(\overline \Omega)$ and the coefficients of the operator $L_{f_0}$ belong to $C^\alpha$. By theorem 6.14 in [8], the equation $Lw = L_{f_0}\phi$ in $\Omega$, $w = g$ in $\partial \Omega$ is uniquely solvable in $C^{2,\alpha}(\overline \Omega)$, and the result follows from the uniqueness in theorem 9.15 in [8]. \lema {Remark} As a simple consequence, we obtain that $f_0 \in C^{2,\alpha} (\overline\Omega)$, by the uniqueness in $W^{2,p}$ given by theorem 9.15 in [8]. \lema {Corollary 11} Let us assume that $\partial \Omega \in C^{k+2,\alpha}$, $g \in C^{k+2,\alpha}$ and $h \in C^{k,\alpha}$ for some $0 < \alpha \le 1- \dfrac 2p$. Then $f_0 \in C^{k+2,\alpha} (\overline \Omega)$. \demost {Proof} It is immediate from Proposition 1 and theorem 6.19 in [8]. \medskip \centerline{ACKNOWLEDGEMENT} The authors thank specially Prof. J.P.Gossez for the careful reading of the manuscript and his suggestions and remarks. \bigskip \tit{References} [1] Adams R. A., Sobolev Spaces. Academic Press, 1975. [2] 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. [3] Do Carmo M., Differential Geometric of Curves and Surfaces. Prentice-Hall, 1976. [4] Goldstein Costa D., T\'opicos em Analise n\~ao Linear e Aplicac\~oes as Ecuac\~{o}es Diferenciais. I.M.PA., 1986. [5] Hildebrandt S., On th Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 23 (1970), 97- 114. [6] Lami Dozo E. and Mariani M.C., A Dirichlet problem for an H-system with variable $H$. Manuscripta Mathematica 81 (1993), 1 - 14. [7] 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. [8] Osserman R., A survey of minimal surfaces, Van Nostrand Reinhold Company, 1969. [9] Struwe M., Plateau's Problem and the Calculus of Variations. Math. Notes 35, Princeton University Press. Princeton 1989. [10] Struwe M., Non uniqueness in the Plateau problem for surfaces of constant mean curvature, Arch. Rat. Mech. Anal. 93 (1986), 135-157. [11] Vainberg M. M., Variational Methods in the Theory of Non Linear Operators, Holden-Day, San Francisco, 1964. [12] Gilbarg, D. Trudinger, N. S. : Elliptic partial differential equations of second order, Springer- Verlag (1983). \tit{P. Amster${}^*$, M. M. Cassinelli${}^*$, M. C. Mariani ${}^*$ and D. F. Rial} Dpto. de Matem\'{a}tica, Fac. de Cs. Exactas y Naturales, UBA. PAB I, Cdad. Universitaria, 1428. Capital, Argentina. ${}^*$ CONICET \tit{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. \tit{email:} mcmarian\@dm.uba.ar \end ---------------0004051245836--