Content-Type: multipart/mixed; boundary="-------------0004041735718" This is a multi-part message in MIME format. ---------------0004041735718 Content-Type: text/plain; name="00-152.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-152.comments" To appear in Nonlinear Analysis ---------------0004041735718 Content-Type: text/plain; name="00-152.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-152.keywords" topological methods - nonlinear problems - prescribed mean curvature equation ---------------0004041735718 Content-Type: application/x-tex; name="general.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="general.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\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 The prescribed mean curvature equation with Dirichlet conditions \endtitle \title P. Amster and M. C. Mariani \endtitle \vglue 2truecm \newdimen\normalbaselineskip \normalbaselineskip=10pt \normalbaselines \pagewidth{12.5cm} \pageheight{17cm} \lema{Abstract:} In this paper we study H-systems with a Dirichlet boundary data $g$. We find solutions under different conditions on $H$ and $g$, and prove that weak solutions in $W^{1,\infty}$ are classic. \newdimen\normalbaselineskip \normalbaselineskip=15pt \normalbaselines \pagewidth{13cm} \pageheight{19cm} \bigskip \tit{Introduction} We consider the Dirichlet problem in a bounded $C^{1,1}$ domain $\Omega \subset R^2$ for a vector function $X: \overline \Omega \longrightarrow { R^3}$ which satisfies the equation of prescribed mean curvature $$ \text{(1)} \cases \Delta X=2H(u,v,X,X_u,X_v)X_u\land X_v \qquad \text{ in }\quad \Omega &\\ X=g\qquad \text{ in }\quad \partial \Omega & \endcases $$ where $X_u=\dfrac{\partial X}{\partial u},$ $X_v=\dfrac{\partial X}{\partial v},$ $\land$ denotes the exterior product in ${ R^3}$ and $H:\overline \Omega \times (R^3)^3 \longrightarrow { R}$ 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 [1-2-3-4-5-6-7-10]. In order to study problem (1) we assume that $g \in W^{2,p}(\Omega,R^3)$ for $2 0$ such that $$\inf_{\Vert \Delta \overline Y \Vert_p \le R, \overline Y = g \text { in } \partial \Omega} \Vert \Delta \overline Y - 2 H(u,v, \overline Y, \overline Y_u, \overline Y_v) \overline Y_u\land \overline Y_v \Vert_p = 0$$ iii) There exists $R > 0$ such that $$\inf_{\Vert \overline Y \Vert_{1,\infty} \le R, \overline Y = g \text { en } \partial \Omega} \Vert \overline Y - T(\overline Y)\Vert_{1,\infty} = 0$$ \demost{Proof} $i) \Longrightarrow ii)$ is obvious. In order to see $ii) \Longrightarrow iii)$, it suffices to note that if $\overline Y \in W^{2,p}(\Omega,R^3)$ verifies $\Vert \Delta \overline Y \Vert_p \le R, \overline Y = g \text { in } \partial \Omega$ and $g$ is harmonic then $$\Vert \overline Y \Vert_{1,\infty} \le c_1 \Vert \overline Y \Vert_{2,p} \le c_1 (\Vert \overline Y - g \Vert_{2,p} + \Vert g \Vert_{2,p}) \le c_1 (c\Vert \Delta (\overline Y - g) \Vert_p + \Vert g \Vert_{2,p}) \le$$ $$\le c_1 (cR + \Vert g \Vert_{2,p}) = R'$$ Moreover, $$\Vert \overline Y - T(\overline Y)\Vert_{1,\infty} \le c_1 \Vert \overline Y - T(\overline Y)\Vert_{2,p} \le c_1c \Vert \Delta (\overline Y - T(\overline Y))\Vert_p =$$ $$=c_1c \Vert \Delta \overline Y - 2 H(u,v, \overline Y, \overline Y_u, \overline Y_v) \overline Y_u\land \overline Y_v \Vert_p$$ and the result follows. Finally, we prove $iii) \Longrightarrow i)$: let us consider a sequence $\overline Y_n$ bounded in $C^1(\overline \Omega,R^3)$ such that $\Vert \overline Y_n - T(\overline Y_n)\Vert_{1,\infty} \lra 0$. Being $T$ compact we may assume that $T(\overline Y_n)$ converges to $\overline Y$ in $C^1(\overline \Omega,{R^3})$. Then $\overline Y_n \lra \overline Y$, and $\overline Y$ is a fixed point of $T$. \lema {Theorem 3} Let $c_1, c$ be the constants of the previous theorems, $2 0$ it holds, for any $\overline X \in B_R(g)$: $$2c_1c\Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_u \Vert _p \le \dfrac R{R+\Vert \nabla g \Vert_{\infty}}$$ \noi or $$2c_1c\Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_v \Vert _p \le \dfrac R{R+\Vert \nabla g \Vert_{\infty}}$$ Then there is at least one solution of (1) in $ B_R(g)$. \demost{Proof} We have, for $\Vert \overline X - g \Vert_{1,\infty}\le R$: $$\Vert X - g \Vert_{1,\infty}\le 2c_1c \Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_u\land \overline X_v\Vert_p$$ Then $$\Vert X - g \Vert_{1,\infty}\le 2c_1c \Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_u \Vert_p \Vert \overline X_v \Vert_\infty$$ $$\le 2c_1c\Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_u \Vert _p(R+\Vert \nabla g \Vert_{\infty})$$ and also $$\Vert X - g \Vert_{1,\infty}\le 2c_1c \Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_v \Vert_p \Vert \overline X_u \Vert_\infty$$ $$\le 2c_1c\Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_v \Vert_p(R+\Vert \nabla g \Vert_{\infty})$$ Thus $T(B_R) \subset B_R$ and the result follows by Schauder Theorem. \lema {Corollary 4} Let $c_1, c$ be the constants of the previous theorems, $2 0$ such that $k < \dfrac 1{2c_1c}$, and $$\Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_u \land \overline X_v \Vert_p \le k \Vert \overline X \Vert_{1,\infty}$$ \noi for every $\overline X$ such that $\Vert \overline X \Vert_{1,\infty} \ge R_0$. Then (1) has at least one solution in $ W^{2,p}(\Omega,R^3)$. \demost{Proof} In the same way as before, for harmonic $g$, if $T(\overline X)=X$ then $$\Vert X - g \Vert_{1,\infty}\le 2c_1c \Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_u\land \overline X_v\Vert_p $$ Then for $\Vert \overline X \Vert_{1,\infty} \ge R_0$, $$\Vert X - g \Vert_{1,\infty} \le 2c_1ck \Vert \overline X \Vert_{1,\infty} \le 2c_1ck \left(\Vert \overline X -g \Vert_{1,\infty} + \Vert g \Vert_{1,\infty}\right)$$ If also $\Vert \overline X - g \Vert_{1,\infty} \le R$, being $2c_1ck<1$ it follows that $$\Vert X - g \Vert_{1,\infty} \le 2c_1ck \left(R + \Vert g \Vert_{1,\infty}\right) \le R$$ when $R$ is big enough. On the other hand, if $\Vert \overline X \Vert_{1,\infty} \le R_0$ then $$\Vert X - g \Vert_{1,\infty}\le 2c_1c \Vert H(u,v, \overline X, \overline X_u, \overline X_v) \overline X_u\land \overline X_v\Vert_p \le R'$$ for some $R'$, and the result follows taking $R > R'$ big enough. \lema {Remark} As a particular case, theorem 6 is valid when the support of $H = H(u,v,x,y,z)$ is bounded in $y$ and $z$. It holds also for $H =\frac {H_1(u,v,X)}{1+\vert\nabla X \vert^2}$. \lema {Theorem 7} Let $(1_\sigma )$ be the equation given by $$ (1_\sigma ) \cases \Delta X=2\sigma H(u,v,X,X_u,X_v)X_u\land X_v \qquad \text{ in } \quad \Omega &\\ X=\sigma g\qquad \text{ in }\quad \partial \Omega & \endcases $$ \noi and suppose there exists $M>0$ such that for any $X \in W^{2,p}(\Omega,R^3)$ solution of $(1_\sigma )$ for some $\sigma \in (0,1)$, $\Vert X \Vert_{1,\infty} \le M$. Then (1) has a solution in $C^1(\overline \Omega,{R^3})$. \demost{Proof} For fixed $M' > M$, we define the operator $T'$, given by $$T'X = \cases TX \qquad \quad \text{ if } \Vert TX \Vert_{1,\infty} \le M' &\\ \dfrac {M'TX}{\Vert TX \Vert_{1,\infty}} \qquad \text{ if }\Vert TX \Vert_{1,\infty} > M' \endcases $$ It is clear that $T'$ is compact, and as $T'(B_{M'}(0)) \subset B_{M'}(0)$, $T'$ has a fixed point $X$. If $X$ is not a fixed point of $T$, then $X = \sigma TX$, with $\sigma = \dfrac {M'}{\Vert TX \Vert_{1,\infty}} < 1$, and $X$ is a solution of $(1_{\sigma})$. Moreover, $\Vert X \Vert_{1,\infty} = M'$, a contradiction. \lema {Theorem 8} Let $X_n \in W^{1,\infty}(\Omega,{R^3})$ be a bounded sequence such that $X_n$ is a weak solution of (1) for some $H_n \in C(\overline \Omega \times (R^3)^3),R)$, $g_n \in W^{2,p}(\Omega,R^3)$ such that $H_n \lra H$ and $g_n \lra g$. Then (1) has a solution in $W^{2,p}(\Omega,R^3)$ for $H$ and $g$. \demost{Proof} Let us note first that $X_n \in W^{2,p}(\Omega,R^3)$, since if $Z_n$ is the only element in $W^{2,p}(\Omega,R^3)$ verifying $$ \cases \Delta Z_n=2 H_n(u,v,X_n,X_{n_u},X_{n_v})X_{n_u}\land X_{n_v} \qquad \text{ in } \quad \Omega &\\ Z_n= g_n\qquad \text{ in }\quad \partial \Omega & \endcases $$ it holds weakly that $\Delta (Z_n-X_n) =0$ and being $Z_n-X_n=0$ in $\partial \Omega$, we conclude that $Z_n = X_n$. On the other hand, by compactness we may assume that $TX_n \lra X \in C^1(\overline \Omega,R^3)$. Moreover, $\Delta (TX_n - X_n) = $ $$2 (H(u,v, X_n, X_{n_u}, X_{n_v}) X_{n_u}\land X_{n_v} - H_n(u,v, X_n, X_{n_u}, X_{n_v}) X_{n_u}\land X_{n_v}) \lra 0$$ uniformly in $\overline \Omega$, and then $$\Vert TX_n - X_n \Vert_{1,\infty} \le \Vert g - g_n \Vert_{1,\infty} + c_1c (\Vert \Delta (TX_n - X_n) \Vert _p +\Vert \Delta (g - g_n) \Vert_p)\lra 0$$ We conclude that $X_n \lra X$ en $C^1(\overline \Omega,{R^3})$ and $TX = X$. \lema {Remark} It is known (see e.g [9]) that if $X$ is a weak solution of (1), then $X \in C(\overline \Omega,R^3) \cap C^2(\Omega,{R^3})$. In the previous theorem we saw that if $X \in W^{1,\infty}( \Omega,{R^3})$ then $X \in W^{2,p}(\Omega,R^3)$. We'll see now that under some extra hypothesis, $X \in C^2(\overline \Omega,R^3)$. \lema {Theorem 9} Let $X \in W^{1,\infty}(\Omega,R^3)$ be a weak solution of (1), and assume, for $k \ge 0$, that $\partial \Omega \in C^{k+2,\alpha}$, $H \in C^{k,\alpha}(\overline \Omega \times (R^3)^3,R)$ $g \in C^{k+2,\alpha}(\overline \Omega,R^3)$ for some $\alpha$, with $0 < \alpha \le 1-\dfrac 2p$. Then $X \in C^{k+2,\alpha}(\overline \Omega,R^3)$. \demost{Proof} Case $k=0$: by Sobolev imbedding and previous theorem, $X \in W^{2,p}(\Omega,R^3) \hookrightarrow C^{1,\alpha}(\overline \Omega,{R^3})$. Then $\Delta X = f \in C^\alpha(\overline \Omega,{R^3})$. By theorem 6.14 in [8], the equation $\Delta Z = f $ en $\Omega$, $Z = g$ en $\partial \Omega$ is uniquely solvable in $C^{2,\alpha}(\overline \Omega,R^3)$, and the result follows from the uniqueness in theorem 9.15 in [8]. The general case is now immediate, from theorem 6.19 in [8]. \tit{References} [1] Brezis, H. Coron, J. M:Multiple solutions of $H$ systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), 149-187. [2] Wang Guofang: The Dirichlet problem for the equation of prescribed mean curvature, Analyse Nonlin\'eaire 9 (1992), 643-655. [3] Struwe, M: Plateau 's problem and the calculus of variations, Lecture Notes Princeton Univ. Press (1988). [4] Lami Dozo, E. Mariani, M. C: A Dirichlet problem for an $H$ system with variable $H$. Manuscripta Math. 81 (1993), 1-14. [5] Struwe, M.: Multiple solutions to the Dirichlet problem for the equation of prescribed mean curvature, Preprint. [6] M. C. Mariani and D. Rial: Solutions to the mean cuvature equation by fixed point methods. Bull.Belgian Math. Soc. Simon Stevin 4 (1997) 617-620. [7] S.Hildebrandt: On the Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 23 (1970) 97-114. [8] Gilbarg, D. Trudinger, N. S. : Elliptic partial differential equations of second order, Springer- Verlag (1983). [9] Bethuel, F., Ghidaglia, J., Improved regularity of solutions to elliptic equations involving jacobians and applications. J.Math.Pures Appl. (9) 72 no.5, 1993. [10] 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. \bigskip \bigskip {\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 \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 ---------------0004041735718--