Content-Type: multipart/mixed; boundary="-------------0004041843187" This is a multi-part message in MIME format. ---------------0004041843187 Content-Type: text/plain; name="00-157.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-157.comments" To appear in Proyecciones ---------------0004041843187 Content-Type: text/plain; name="00-157.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-157.keywords" variational methods - prescribed mean curvature equation ---------------0004041843187 Content-Type: application/x-tex; name="hildebr.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hildebr.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\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\lra{\longrightarrow} \def\C{\pmb{C}} \def\div{\text {\rm div}} \def\su{ \underline{u}} \def\R {I\!\!R } \centerline{\bf Solutions to the prescribed mean curvature equation} \bigskip \centerline{\bf with Dirichlet conditions by variational methods } \bigskip \medskip \centerline {P. Amster, P. De N\'apoli and M. C. Mariani} \centerline {Universidad de Buenos Aires} \vglue 2truecm \newdimen\normalbaselineskip \normalbaselineskip=10pt \normalbaselines \pagewidth{12.5cm} \pageheight{17cm} \lema{Abstract:} We apply variational methods in order to prove that the nonlinear system (1) admits at least one regular solution. \newdimen\normalbaselineskip \normalbaselineskip=15pt \normalbaselines \pagewidth{13cm} \pageheight{19cm} \bigskip \tit{Introduction} We consider the problem of finding $u \in H^1(\Omega,\R^3)$ such that $$\text {(1)} \cases \triangle u = 2 H(u) u_x \wedge u_y \text{ in } \Omega &\\ u = \gamma \text{ in } \partial \Omega \endcases $$ where $H$ is a given continuously differentiable function and $\Omega$ is an open subset of $\R^2$ with $C^1$ boundary. The system of differential equations above is the Dirichlet problem for the prescribed mean curvature equation in isothermal coordinates, also called H-system, which arises in the generalized Plateau's problem (see [O]). Hildebrant, Wente (see [W1,W2]) and others have studied the problem for constant $H$. For this case, Brezis-Coron and Struwe have shown by variational methods (see [BC], [S]) that if the boundary data is small and non constant, then there are at least two weak solutions. The problem has been also studied by fixed point methods (see [AMR]). The goal of this work is to show that also for non constant $H$, the problem may be solved by variational methods, under rather general conditions on $H$ and the boundary data. Our main result is the following: \lema{Theorem 1} Let $\gamma \in H^{1/2}(\partial \Omega,\R^3) \cap L^\infty(\partial \Omega,\R^3)$, $R= \| \gamma -c\|_{L^\infty(\partial\Omega,\R^3)}$ for some constant $c$, and $B \subset \R^3$ the closed ball of radius R centered at $c$. Then, if $H \in C^1(B)$ satisfies $$ \| H \|_{L^\infty(B)} < \frac{1}{R}$$ then problem (1) admits a weak solution $u \in H^1(\Omega,\R^3) \cap L^\infty(\Omega,\R^3)$, with \linebreak $\| u-c \|_{L^\infty(B)} \le R$. \rm {\bf Remark:} By a result of Bethuel ([B]) weak solutions of the mean curvature equation are $C^{2,\alpha}(\Omega)$ for $H$ bounded and Lipschitz continuous. Moreover, by a result of Chanillo and Li ([CL]) the solutions are continuous up to the boundary. \tit{ Variational formulation of the problem } For simplicity, we may assume that $c=0$ and extend $H$ to a function of class $C^1(\R^3)$ such that $\| H \|_{L^\infty(\R^3)} < \frac{1}{R}$. We define the vector field $q:\R^3 \to \R^3$: $$ q(x,y,z) = \frac{4}{3} (\int_0^x H(t,y,z) dt, \int_0^y H(x,t,z) dt,\int_0^z H(x,y,t) dt) $$ It is immediate that $\div \ q = 4 H$ and $ |q(x,y,z)| \leq \frac{4}{3} \|H\|_{L^\infty} |(x,y,z)|$. We'll work in the Banach space $ H^1(\Omega,\R^3) \cap L^\infty(\Omega,\R^3) $, with the norm: $$ \| u \| = \| u \|_{H^1} + \| u \|_{L^\infty} $$ and consider energy functional $$ E(u) = D(u) + Q(u) $$ where $D$ is the Dirichlet's integral and $Q$ is the functional $$ Q(u) = \int_{\Omega} q(u) \cdot (u_x \wedge u_y)$$ We'll use the estimatives $$ |Q(u)| \leq \int_{\Omega} |q(u)| \cdot |u_x \wedge u_y)| \leq \frac{2}{3} \|H\|_{L^\infty} \|u\|_{L^\infty}\int_{\Omega} |\nabla u|^2 \tag2$$ $$ E(u) \geq \left( 1 - \frac{2}{3} \|H\|_{L^\infty} \|u\|_{L^\infty}\right) \int_{\Omega} |\nabla u|^2 \tag3 $$ \tit {Technical lemmas} In this section we present some technical facts. \lema{Lemma 2} $Q$ is continuous in $H^1 \cap L^\infty$. \lema{Lemma 3} We consider the functional $$ I(u,h) = \int_{\Omega} H(u) \cdot (u_x \wedge u_y ) \cdot h $$ where $u \in H^1 \cap L^\infty$ and $h \in L^\infty$. Then $I$ is continuous. \demost{Proof} Lemma 2 and Lemma 3 follow from simple computations. \lema {Lemma 4} The functional $E$ is Gateaux differentiable in $H^1 \cap L^\infty$ for variations in $H^1_0 \cap L^\infty$, and $$ dE(u)(h) = 2 \int_{\Omega} \nabla u \cdot \nabla h + \int_{\Omega} 4H(u) \cdot (u_x \wedge u_y) \cdot h $$ for every $u \in H^1 \cap L^\infty, h \in H^1_0 \cap L^\infty$. \demost{Proof} Let us first take $ u,h \in C^1 $ and define a function $U:\Omega \times \R \to \R^3$ by: $$ U(x,y,t) = u(x,y) + t \cdot h(x,y) $$ We'll write $u^t(x,y) = U(x,y,t)$. We recall that the field $q$ in $\R^3$ is associated to the 2-form $\omega$ given by: $$ \omega(X,Y) = (X \wedge Y) \cdot q(u) $$ with $$ d\omega = \div (q) du^1 \wedge du^2 \wedge du^3 $$ Applying Stokes theorem to the cylinder $C = \Omega \times [0,\varepsilon ]$, we obtain: $$ \int_{C} d(U^{*}\omega) = \int_{\partial C} U^{*}\omega $$ Moreover, $$ d(U^{*}\omega) = U^{*}(d\omega) = U^{*}(\div q(u) du^1 \wedge du^2 \wedge du^3) $$ $$ = 4H(U) \cdot \frac{\partial(U^1,U^2,U^3)}{\partial(x,y,t)} dx \wedge dy \wedge dt = 4H(u^t) \cdot (u^t_x \wedge u^t_y) \cdot h $$ We have: $$ U^{*}\omega = \omega (\frac{\partial U}{\partial x},\frac {\partial U}{\partial y} ) dx \wedge dy = q(u^t) \cdot (u^t_x \wedge u^t_y) $$ on $\Omega \times \{\varepsilon\}$ and $\Omega \times \{ 0\}$ and $$ U^{*}\omega = \omega(\frac{\partial U}{\partial \sigma},\frac{\partial U}{\partial t}) d\sigma \wedge dt = q(u^t) \cdot ( \frac{ \partial U}{\partial \sigma} \wedge h) d\sigma \wedge dt $$ on the lateral surface $\partial \Omega \times [0,\varepsilon ]$, where $\sigma$ is the unit tangent vector on $\partial \sigma$. Thus, $$ \int_0^\varepsilon \int_{\Omega} 4H(u^t) \cdot (u^t_x \wedge u^t_y) \cdot h dx \wedge dy \wedge dt = $$ $$ \int_{\Omega \times \{\varepsilon\}} q(u^\varepsilon) \cdot (u^\varepsilon_x \wedge u^\varepsilon_y) dx \wedge dy \wedge dt - \int_{\Omega \times \{ 0\} } q(u) \wedge (u_x \wedge u_y) \cdot h $$ $$ + \int_{\partial \Omega \times [0,\varepsilon]} q(u) \cdot ( \frac{ \partial U}{\partial \sigma} \wedge h) d\sigma \wedge dt $$ Assuming that $h=0$ on the boundary of $\Omega$, we get: $$ \int_0^\varepsilon \int_{\Omega} 4H(u^t) \cdot (u^t_x \wedge u^t_y) \cdot h = Q(u^\varepsilon) - Q(u).$$ By density, this formula holds for $u \in H^1 \cap L^\infty$ and $ h \in H^1_0 \cap L^\infty $. By dominated convergence we conclude that $$ dQ(u)(h) = \int_{\Omega} 4H(u) \cdot (u_x \wedge u_y) \cdot h $$ which completes the proof. {\bf Remark:} By lemmas 2 and 3 $dE$ is continuous, i.e. $E \in C^1(H^1_0 \cap L^\infty)$. \newpage \lema {Corollary 5} If $u$ is a critical point of functional $E$ for variations in $H^1_0 \cap L^\infty$ then $u$ is a weak solution of (1). \rm \tit{Proof of Theorem 1} We fix $R^{\prime} > R$ such that $ \| H \|_{L^\infty} R^{\prime} < 1 $ and consider the nonempty set: $$ K = \{ u \in H^1(\Omega,\R^3): u = \gamma \text{ in } \partial \Omega , \| u \| \leq R^{\prime} \} $$ Then we obtain \lema{Lemma 6} $E$ achieves a minimum in $K$. \demost{Proof} By (3), the functional $E$ is bounded from below in $K$. Let $ E_0 = \inf_{v \in K} E(v) $ and consider a minimizing sequence $(u^n)$ By (3) and Poincar\'e's inequality $u^n$ is bounded in $H^1$, and then we may suppose that $$ u^n \to \su \text{ in } H^1 \text{weakly} $$ and $$ u^n \to \su \text{ a.e. } $$ By the continuity of $q$ we conclude that: $$ q(u^n) \to q(\su) \text{ a.e. } $$ and since $|q(u^n)| \leq \frac{4}{3} \|H \|_{L^\infty} $, we may also assume that: $$ q(u^n) \to q(\su) \text{ in } L^\infty \text{ weak* }$$ Set $ \theta^n = u^n - u \in H^1_0$. Then $ \| \theta^n \|_{L^\infty} \leq 2R^{\prime} $, and $$ E(u^n) = \int_{\Omega} |\nabla \su|^2 + 2 \int_{\Omega} \nabla u \cdot \nabla \theta^n$$ $$ + \int_{\Omega} |\nabla \theta^n|^2 + \int_{\Omega} q(u^n) \cdot (\su_x + \theta^n_x) \wedge (\su_y + \theta^n_y) $$ Thus, $$\int_{\Omega} \nabla u \cdot \nabla \theta^n \to 0 $$ Moreover, being $$\int_{\Omega} q(u^n) \cdot (\su_x \wedge \theta^n_y) = - \int_{\Omega} \theta^n_y \cdot (q(u^n) \wedge \su_x) $$ then $$\int_{\Omega} q(u^n) \cdot (\su_x \wedge \theta^n_y) \to 0,$$ since $ \theta^n_y \to 0 $ weakly in $L^2$, and $$ |q(u^n) \wedge \su_x|^2 \leq \left( \frac{4}{3} |u^n|\right )^2 \cdot |\su_x|^2 \leq \left( \frac{4}{3} R^{\prime} \right) |\su_x|^2.$$ In the same way, we conclude that $$ \int_{\Omega} q(u^n) \cdot (\theta^n_x \wedge \su_y) \to 0 $$ We also have the estimative: $$ |\int_{\Omega} q(u^n) \cdot \theta^n_x \wedge \theta^n_y| \leq \int_{\Omega} 4 \|H\|_{L^\infty} \|u^n)\|_{L^\infty} |\theta^n_x \wedge \theta^n_y| $$ $$ \leq \frac{2}{3} \|H\|_{L^\infty} R^{\prime} \int_{\Omega} |\nabla \theta^n|^2 \leq \frac{2}{3} \int_{\Omega} |\nabla \theta^n|^2 $$ Finally as $ q(u^n) \to q(\su) $ weak* in $L^\infty$: $$ \int_{\Omega} q(u^n) \cdot (\su_x \wedge \su_y) = \int_{\Omega} q(\su) \cdot (\su_x \wedge \su_y) + o(1) $$ Then, $$ E(u^n) \geq E(u) + o(1) + \frac{1}{3} \int_{\Omega} |\nabla \theta^n|^2 $$ and we can conclude that $E(u) = E_0 $ and $ \int_{\Omega} |\nabla \theta^n|^2 \to 0$. \lema{Lemma 7} Let $u \in K $ satisfy $E(u) = \inf_{v \in K} E(v)$. Then $u$ is a weak solution of (1). \demost{Proof} Let us fix a nonnegative test function $ \eta \in {\cal D}(\Omega)$. Then $ u(1-\varepsilon \eta) \in K $ for $\varepsilon \geq 0 $ small enough. As the function $ \varphi(\varepsilon) = E(u(1-\varepsilon \cdot \eta))$ has a minimum in $ \varepsilon = 0 $, then $ \varphi^{\prime}(0) = dE(u)(-u \cdot \eta) \geq 0 $, i.e. $$ 2 \int_{\Omega} \nabla u \cdot \nabla (-\eta \cdot u) + \int_{\Omega} 4H(u) (u_x \wedge u_y) \cdot (-\eta u) \geq 0 \; \forall \; \eta \in {\cal D}(\Omega) $$ By Green's formula, $$ \int_{\Omega} \triangle(|u|^2) \cdot \eta = - \int_{\Omega} \nabla(|u|^2) \nabla(\eta) = -\int_{\Omega} 2 ( \nabla u \cdot u ) \cdot \nabla \eta $$ It follows that $$ -\frac{1}{2} \triangle |u|^2 + |\nabla u |^2 + 2H(u) u \cdot u_x \wedge u_y \leq 0 \text { in } {\cal D}^{\prime}(\Omega) $$ Then, as $$ |2H(u) u \cdot ( u_x \wedge u_y)| \leq \| H \|_{L^\infty} R^{\prime} |\nabla u|^2 $$ it follows that $$ -\triangle |u|^2 \leq 0 \text { in } { \cal D}^{\prime}(\Omega) $$ and then by Stampacchia's maximum principle (see [GT]) we get: $$ sup_{\Omega} |u| = sup_{\partial \Omega } |u| = R $$ Since $ R < R^{\prime} $, $u$ is interior in $K$. It follows that $dE(u)(v) = 0$ for any $v \in H^1_0 \cap L^\infty $, and by corollary 5 $u$ is a weak solution of (1). \bigskip \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. [B] Bethuel, F: Un r\'esultat de regularit\'e pour les solutions de l'equation des surfaces \`a courboure moyenne prescrite. R.Acad. Sci. Paris S\'er. I Math 314 (1992) $n^{o}$ 13, 1003-100. [B-C] Brezis, H., Coron, J., Multiple Solutions of H-Systems and Rellich's conjecture. Comm.Pure Appl. Math 37, 1984, 149-187. [CL] Chanillo S., Li Y.: Continuity of Solutions of Uniformly Elliptic Equations in $\R^2$. Manuscripta math. 77,415-433 (1992) [GT] Gilbarg, D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag 1977 [O] Osserman, R: A Survey of Minimal Surfaces. Van Nostrand Reinhold Company, 1969 [S] Struwe, M.: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, p. 180. [W1] Wente, H.: An Existence Theorem for Surfaces of Constant Mean Curvature. Journal of Mathematical Analysis and Applications 26, p. 318-344 (1969) [W2] Wente, H.: The differential equation $\Delta X = 2H (X_u \wedge X_v)$ with vanishing boundary values. Proceedings of the American Mathematical Society 50 (1975), 131-7. \bigskip {\bf P.Amster$^*$, P. De N\'apoli and M. C. Mariani$^*$} \noi Dpto. de Matem\'atica, Fac. de Cs. Exactas y Naturales, Universidad de Buenos Aires. \noi $^*$ Consejo Nacional de Investigaciones Cient\'\i ficas y T\'ecnicas (CONICET) \bigskip {\bf Address for correspondence:} \noi Prof. P. Amster and M. C. Mariani, \noi Dpto. de Matem\'atica - Fac. de Cs. Exactas y Naturales, UBA \noi Pab. I, Ciudad Universitaria (1428) \noi Buenos Aires, Argentina {\bf E-mail:} pamster\@dm.uba.ar - mcmarian\@dm.uba.ar \end ---------------0004041843187--