Content-Type: multipart/mixed; boundary="-------------0004041757674" This is a multi-part message in MIME format. ---------------0004041757674 Content-Type: text/plain; name="00-154.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-154.comments" To appear in Nonlinear Studies ---------------0004041757674 Content-Type: text/plain; name="00-154.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-154.keywords" fixed point methods - quasilinear equations - Dirichlet conditions ---------------0004041757674 Content-Type: application/x-tex; name="quasi.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="quasi.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\lra{\longrightarrow} \def\noi{\noindent} \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\cor{\allowmathbreak} \title Solutions to general quasilinear elliptic second order problems \endtitle \title P. Amster, M.M. Cassinelli and M. C. Mariani \endtitle \vglue 2truecm \newdimen\normalbaselineskip \normalbaselineskip=10pt \normalbaselines \pagewidth{12.5cm} \pageheight{17cm} \lema{Abstract:} We study the second order quasilinear equation (1) $Qu=f(x,u, D_1u,...,D_nu)$ with Dirichlet conditions. We obtain existence and uniqueness results in the Sobolev Space $W^{2,p}$. We also prove that, in certain cases, the set of solutions of (1) is a connected subset of $W^{2,p}$. \newdimen\normalbaselineskip \normalbaselineskip=15pt \normalbaselines \pagewidth{13cm} \pageheight{19cm} \bigskip \tit{Statement of the problem and known results} We consider the Dirichlet problem in a bounded $C^{1,1}$ domain $\Omega \subset R^n$ for $u:\overline \Omega\lra R$ satisfying $$ \text{(2)} \cases Qu = f(x,u,D_1u,...,D_nu) \text{ in } \Omega &\\ u=g \quad \text{ in }\partial \Omega \endcases $$ where $Q$ is the quasilinear elliptic operator given by $Qu=\sum a_{ij}D_{ij}u+\sum b_{j}D_{j}u+cu$, with $a_{ij}(x,u,D_1u,...,D_nu)$ continuous and $C^1$ in $u,D_1u,...,D_nu$, $b_{j}(x,u)$ continuous and $C^1$ in $u$, and $c \in L^\infty(\Omega)$. We'll assume that $Q$ is strictly elliptic, i.e. if $\lambda_i$ and $\lambda_s$ are respectively the minimum and the maximum eigenvalues of the matrix $a_{ij}(x,u,D_1u,...,D_nu) $, then $\lambda_i\ge \lambda >0$. We'll assume also that $f:\overline \Omega \times R \times R^n\lra R$ is continuous and $g \in W^{2,p}(\Omega)$ for some $n c(\overline u)$, $w \in W^{2,p} \cap W_0^{1,p}$. For $\Vert \overline v - \overline u\Vert_{1,\infty}\le R$, we have: $$\Vert L_{\overline u}w \Vert_p \ge \frac 1{c(\overline u)} \Vert w \Vert_{2,p}- \Vert (L_{\overline u}- L_{\overline v}) w \Vert_{p}$$ As $$\Vert (L_{\overline u}- L_{\overline v})w \Vert_{p} \le \Vert\sum (a_{ij}(x,\overline u,D_1\overline u,...,D_n\overline u)- a_{ij}(x,\overline v,D_1\overline v,...,D_n\overline v)) D_{ij}w+$$ $$\sum (b_{j}(x, \overline u)-b_{j}(x, \overline v))D_{j}w\Vert_p\le kR\Vert w \Vert_{2,p} $$ for some constant $k$, we obtain: $$\Vert L_{\overline u}w \Vert_p \ge (\frac 1{c(\overline u)}-kR) \Vert w \Vert_{2,p}$$ and choosing $R$ such that $kR < \frac 1{c(\overline u)}$ it follows that $\frac 1{c(\overline v)}\ge \frac {1-kRc(\overline u)} {c(\overline u)}$. Letting $R\lra 0$ we conclude that $c(\overline v) \le \frac {c(\overline u)}{1-kRc(\overline u)}0$ such that $kR<\frac 1{c(\overline u)}$ where $k$ is the constant of lemma 1. Fixing $\overline c$ such that $\overline c\ge c(\overline v)$ for any $\overline v \in B_R(\overline u)$, we obtain: $$\Vert T\overline u - T\overline v\Vert_{1,\infty}\le c_1\Vert T\overline u - T\overline v\Vert_{2,p}\le \overline cc_1 \Vert L_{\overline v}(T\overline u - T\overline v)\Vert_{p}\le$$ $$\overline cc_1 (\Vert L_{\overline u}T\overline u - L_{\overline v}T\overline u\Vert_{p} + \Vert (L_{\overline u}- L_{\overline v})T\overline u\Vert_{p}) \le$$ $$\overline cc_1(\Vert f(x,\overline u,D_1\overline u,...,D_n\overline u)- f(x,\overline v,D_1\overline v,...,D_n\overline v)\Vert_p +k\Vert T\overline u\Vert_{2,p} \Vert \overline u-\overline v\Vert_{1,\infty}).$$ Then continuity follows. Furthermore, as $T(B_R(\overline u))\subset W^{2,p}$ is bounded, by the compactness of the imbedding $W^{2,p}\hookrightarrow C^1$, we conclude that the closure of $T(B_R(\overline u))$ is compact. \medskip The next result establishes the existence of a solution in a neighborhood of any $u$ satisfying $Qu=f(x,u, D_1u,...,D_nu)$, if the boundary data $g$ is close enough to $u$. More generally, we can also allow small perturbations of $f$: \lema{Theorem 3} Let $g_0 \in W^{2,p}$, $u_0 \in W^{2,\infty}$ satisfy $Qu_0 = f_0(x,u_0,D_1u_0,...,D_nu_0)$ in $\Omega$, $u_0=g_0$ on $\partial\Omega$. We define $$\psi(x):=\sum_{i,j} \frac {\partial a_{ij}}{\partial u}(x, u_0,D_1 u_0,...,D_n u_0) D_{ij}u_0 +\sum_j \frac {\partial b_j}{\partial u}(x,u_0)D_{j}u_0$$ \noi and assume that one of the following conditions holds: i) $f_0$ is Lipschitz in $u,D_1u,...,D_nu$ with constant small enough and $\psi + c\le 0$. ii) $f_0$ is $C^1$ in $u,D_1u,...,D_nu$ and $\psi +c\le \frac {\partial f_0}{\partial u}(\cdot ,u_0,D_1 u_0,...,D_n u_0)$. \noi Then problem (2) is solvable in $W^{2,p}$ for any $(f,g)$ close enough to $(f_0,g_0)$ in $C(\overline \Omega \times R\times R^n) \times W^{2,p}(\Omega)$. \demost{Proof} Let $(f,g)$ be in a neighborhood of $(f_0,g_0)$. We look for a solution of (2) for $(f,g)$, which is equivalent to find $u \in W^{2,p}(\Omega)$ such that $u=g$ in $\partial \Omega$ and $$L_uu - L_{u_0}u_0 = f(x, u,D_1 u,...,D_n u) - f(x, u_0,D_1 u_0,...,D_n u_0) \quad \text {in } \Omega$$ For $z=u-u_0$, we have that $$ L_uu - L_{u_0}u_0= \sum a_{ij}(x, u,D_1 u,...,D_n u)D_{ij}z+\sum b_{j}(x, u)D_{j}z+cz+$$ $$\sum (a_{ij}(x, u,D_1 u,...,D_n u)- a_{ij}(x, u_0,D_1 u_0,...,D_n u_0)) D_{ij}u_0+\sum (b_{j}(x, u) - b_{j}(x, u_0))D_{j}u_0$$ We may write $$ a_{ij}(x, u,D_1 u,...,D_n u)- a_{ij}(x, u_0,D_1 u_0,...,D_n u_0)= \frac {\partial a_{ij}}{\partial u}(x, u_0,D_1 u_0,...,D_n u_0)z$$ $$+\sum_k\frac {\partial a_{ij}}{\partial D_ku }(x, u_0,D_1 u_0,...,D_n u_0) D_kz- \rho_{ij}(x, z,D_1 z,...,D_n z)$$ $$ b_{j}(x, u) - b_{j}(x, u_0)=\frac {\partial b_j}{\partial u}(x,u_0) -\rho_j(x,z)$$ and define the quasilinear operator $$Mz= \sum a_{ij}(x,z+u_0,D_1(z+u_0),...,D_n (z+u_0))D_{ij}z +$$ $$\sum_k \left( b_{k}(x, z+u_0) +\sum_{i,j}\frac {\partial a_{ij}}{\partial D_ku }(x, u_0,D_1 u_0,...,D_n u_0) D_{ij}u_0\right) D_kz+$$ $$\left(\sum_{i,j}\frac {\partial a_{ij}}{\partial u}(x, u_0,D_1 u_0,...,D_n u_0) D_{ij}u_0 +\sum_j \frac {\partial b_j}{\partial u}(x,u_0)D_{j}u_0+c\right) z $$ Thus, the problem becomes $$ \cases Mz = \sum\rho_{ij}D_{ij}u_0+\sum\rho_{j}D_{j}u_0+F \text{ in } \Omega &\\ z=g-g_0 \quad \text{ in }\partial \Omega \endcases $$ where $F(x,z,D_1z,...,D_nz)=f(x,u,D_1u,...,D_nu)-f_0(x,u_0,D_1u_0,...,D_nu_0)$. As the operator $M$ satisfies the required conditions, lemma 1 holds, and we may define a continuous and locally compact operator $T:\overline z \lra z$ as in lemma 2. Then, for $R$ small we fix $\overline c$ such that $\overline c\ge c(\overline z)$ for any $\overline z \in B_R(0) \subset C^1$. If i) holds, we obtain: $$\Vert T(\overline z) \Vert_{1,\infty} \le \Vert g-g_0 \Vert_{1,\infty} +\Vert T(\overline z)-(g-g_0) \Vert_{1,\infty}\le $$ $$\Vert g-g_0 \Vert_{1,\infty} +\overline cc_1 \Vert L_{\overline z}(T(\overline z)-(g-g_0)) \Vert_{p}\le \Vert g-g_0 \Vert_{1,\infty} + $$ $$\overline cc_1( \Vert \sum\rho_{ij}D_{ij}(u_0)+\sum\rho_{j}D_{j}(u_0)+ F(x,\overline z,D_1\overline z,...,D_n\overline z) \Vert_p+ \Vert L_{\overline z}(g-g_0)) \Vert_{p}).$$ Moreover, $$ \Vert F(x,\overline z,D_1\overline z,...,D_n\overline z) \Vert_p\le$$ $$\Vert f(x,\overline z+u_0,D_1(\overline z+u_0),...,D_n(\overline z+u_0)) -f_0(x,\overline z+u_0,D_1(\overline z+u_0),...,D_n(\overline z+u_0)) \Vert_p+$$ $$\Vert f_0(x,\overline z+u_0,D_1(\overline z+u_0),...,D_n(\overline z+u_0)) -f_0(x,u_0,D_1u_0,...,D_nu_0) \Vert_p$$ and then taking $(f,g)$ close enough to $(f_0,g_0)$ it's easy to conclude that $$\Vert T(\overline z) \Vert_{1,\infty}\le R$$ for some $R$ small enough. Hence, $T(B_R(0)) \subset B_R(0)$ and the result follows from Schauder fixed point theorem. Under assumption ii), the proof is similar, writing $$ f_0(x, u,D_1 u,...,D_n u)- f_0(x, u_0,D_1 u_0,...,D_n u_0)= \frac {\partial f_0}{\partial u}(x, u_0,D_1 u_0,...,D_n u_0)z$$ $$+\sum_k\frac {\partial f_0}{\partial D_ku }(x, u_0,D_1 u_0,...,D_n u_0) D_kz+ \rho(x, u,D_1 u,...,D_n u)$$ and defining the operator $$M'z=Mz- \frac {\partial f_0}{\partial u}(\cdot , u_0,D_1 u_0,...,D_n u_0)z -\sum_k\frac {\partial f_0}{\partial D_ku }(\cdot , u_0,D_1 u_0,...,D_n u_0) D_kz.$$ \lema {Remarks:} a) In i) and ii) the condition on $\psi$ may be replaced by the assumption that $\Vert \psi +c \Vert_p$ (respectively $\Vert \psi +c -\frac {\partial f_0}{\partial u}(\cdot , u_0,D_1 u_0,...,D_n u_0) \Vert_p$) is small enough. b) Theorem 3 can be combined with the non-existence conditions in [GT] (see e.g. Theorem 14.12, and corollary 14.13 for the mean curvature equation). For instance, if $H>0$ satisfies the existence conditions of [GT, theorem 16.10] with $H'(y) = \frac n{n-1} H(y)$ at some point $y \in \partial \Omega$ where $H'$ is the mean curvature of $\partial \Omega$, there is a sequence $H_n\lra H$ satisfying the conditions of [GT, corollary 14.13]. Thus, there exists a sequence $g_n$ such that $\Vert g_n\Vert_\infty \lra 0$ and the Dirichlet problem is not solvable. From theorem 3 we conclude that no subsequence of $g_n$ converges in $W^{2,p}$. \rm \medskip In the same way we can prove the local uniqueness of the solutions of (2). More precisely: \lema{Theorem 4} Let $u_0 \in W^{2,\infty}$ be a solution of (2), $\psi$ as in the previous theorem and assume that one of the following conditions holds: i) $f$ is Lipschitz in $u,D_1u,...,D_nu$ with constant small enough and $\psi+c\le 0$. ii) $f$ is $C^1$ in $u,D_1u,...,D_nu$ and $\psi+c\le \frac {\partial f}{\partial u}(\cdot , u_0,D_1 u_0,...,D_n u_0). $ \noi Then $u_0$ is isolated in $C^1$. \demost{Proof} As in the previous theorem, for $f=f_0$ and $g=g_0$, we obtain for small $R$ and $\overline z \ne 0$ that $$\Vert T(\overline z) \Vert_{1,\infty} < \Vert \overline z\Vert_{1,\infty}$$ and the result holds. Furthermore, in certain cases we can prove the global uniqueness: \lema{Theorem 5} Let $u_0 \in W^{2,\infty}$ be a solution of (2) and assume that $a_{ij}$, $b_j$ do not depend of $u$ and $f$ is $C^1$ in $u,D_1u,...,D_nu$. Then, if $c\le \frac {\partial f}{\partial u}(y)$ for any $y \in R^{n+1}$, $u_0$ is unique in $C^1$. \demost{Proof} As in theorem 3, if $u$ is another solution of (2), we have that $L_uu - L_{u_0}u_0 = f(x, u,D_1 u,...,D_n u) - f(x, u_0,D_1 u_0,...,D_n u_0)$ and $u=u_0$ on $\partial \Omega$. We can choose $\varphi_{ij}^k $, $\varphi$, $ \varphi^k \in L^\infty$ such that $$ a_{ij}(x,D_1 u,...,D_n u)- a_{ij}(x,D_1 u_0,...,D_n u_0)= \sum_k\frac {\partial a_{ij}}{\partial D_ku }(x,\varphi_{ij}^1 ,...,\varphi_{ij}^n)D_kz$$ and $$ f(x,u,D_1 u,...,D_n u)-f(x,u_0,D_1 u_0,...,D_n u_0)= \frac {\partial f}{\partial u}(x,\varphi, \varphi^1,...,\varphi^n)z$$ $$+\sum_k\frac {\partial f}{\partial D_ku }(x, \varphi^1 ,...,\varphi^n)D_kz$$ where $z=u-u_0$. Thus, we obtain: $$Lz=0 \text { in }\Omega \qquad z=0 \text { in }\partial\Omega $$ for some linear elliptic operator satisfying the conditions of lemma 9.17 in [GT] and the result follows. \medskip \tit {Regularity of the solutions} \lema {Theorem 6} Let $u \in W^{2,p}(\Omega)$ be a solution of (2), and assume for some $k \ge 0$, $0<\alpha<1$ that $f \in C^{k,\alpha}$, $\partial \Omega \in C^{k+2,\beta}$, $c \in C^{k,\beta}$ and $g \in C^{k+2,\beta}$ where $\beta = \alpha (1- \dfrac np)$. Then $u \in C^{k+2,\beta}(\overline \Omega)$. \demost{Proof} Case $k=0$: by Sobolev immersion $u \in C^{1,1-n/p}(\overline \Omega)$. Then $Lu \in C^\beta(\overline \Omega)$ where the elliptic operator $L$ has $C^\beta$ coefficients. By theorem 6.14 in [GT], the equation $Lw = f $ en $\Omega$, $w = g$ en $\partial \Omega$ is uniquely solvable in $C^{2,\beta}(\overline \Omega)$, and the result follows from the uniqueness in theorem 9.15 in [GT]. The general case is now immediate, from theorem 6.19 in [8]. \tit {Properties of the set $\{ Qu = f \}$ } Let us assume that $\partial \Omega \in C^{2,\beta}$, $f \in C^\alpha$, and that $f$ and $Q$ satisfy the hypothesis of theorem 5. We consider the sets: $$M_f = \{ u \in W^{2,p}(\Omega) / Qu=f(x,u,D_1u,...,D_nu) \text{ in } \Omega \} $$ $$S_f = \{ g \in W^{2,p}(\Omega) \text { harmonic / (2) has a solution in } W^{2,p}(\Omega) \}$$ $M_f$ is closed $W^{2,p}(\Omega,R)$, since $M_f=F^{-1}(0)$ for some $F:W^{2,p}(\Omega,R)\lra L^p$ continuous. Moreover, by theorem 3 and 6 $S_f$ is open in the closed subspace $\Delta^{-1}(0) \subset W^{2,p}(\Omega)$. By theorems 5 and 6 we may define a one to one map $\lambda:M_f \cap C^{2,\beta}(\overline \Omega) \lra S_f \cap C^{2,\beta}(\overline \Omega)$, where $\lambda (u)$ is the only harmonic $g$ such that $u=g$ in $\partial \Omega$. It's immediate to see that $\lambda$ is continuous, since $$\Vert \lambda (u) - \lambda (v) \Vert_{2,p} \le \Vert \lambda (u) - u - (\lambda (v) -v) \Vert_{2,p}+ \Vert u - v \Vert_{2,p} \le c \Vert \Delta u - \Delta v \Vert_{2,p} +\Vert u - v \Vert_{2,p}. $$ \noi Actually, $\lambda$ may be thought as the restriction of a linear operator $\overline \lambda:W^{2,p}(\Omega)\lra \Delta^{-1}(0)$ which is onto and continuous (and hence, open). In order to see that $\lambda$ is an homeomorfism, it suffices to consider a sequence $g_n \lra g$, and note that if $\lambda (u)=g$, we may find, following the proof of theorem 3, for every $n$ big enough a solution $\overline u_n \in B_{R_n}(u)$ of the problem with boundary data $g_n$. By uniqueness, $\overline u_n = u_n$. Moreover, with the notations of theorem 3 ii) for $g=g_n$ and $f=f_0$ we have that if $\overline z \in B_{R_n}(0)$ then $$\|T'(\overline z)\|_{1,\infty}\le k \|g_n -g_0\|_{2,p} + c_1c \|\rho\|_p$$ As $c_1c \|\rho\|_p$ is independent of $g_n$ it's clear that we may take $R_n \lra 0$ when $g_n-g\lra 0$. \medskip As a simple consequence we have the following \lema{Theorem 7} Let us assume that $\partial \Omega \in C^{2,\beta}$, and that $Q$, $f \in C^\alpha$ satisfy the hypothesis of theorem 5. Then, if the problem (1) admits a solution for any $g$, $(M_f \cap C^{2,\beta}(\overline \Omega),\| \cdot \|_{2,p})$ is homeomorphic to the dense subspace $C^{2,\beta}\cap \Delta^{-1}(0) \subset (\Delta^{-1}(0),\| \cdot \|_{2,p})$. In particular, it is connected by arcs. \medskip \lema {Remark:} Theorem 7 holds for the solutions of the mean curvature equation for nonparametric surfaces with constant $H$, if the curvature $H'$ of $\partial \Omega$ verifies the condition $H'\ge \frac{n\vert H \vert}{n-1}$ (see [GT], corollary 16.11). \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. [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. [G] Giaquinta, M.: On the Dirichlet problem for surfaces of prescribed mean curvature. Manuscripta Math. 12, 73/86 (1974). [GT] Gilbarg, D. Trudinger, N. S. : Elliptic partial differential equations of second order, Springer- Verlag (1983). [Se] Serrin, J.: Gradient estimates for solutions of nonlinear elliptic and parabolic equations. In: Contributions to Nonlinear Functional Analysis, pp 565-601. New York, Academic Press (1971). [Si] Simon, L.: Equations of mean curvature type in 2 independent variables. Pacific J.Math (1977). [W] Wang Guofang: The Dirichlet problem for the equation of prescribed mean curvature, Analyse Nonlin\'eaire 9 (1992), 643-655. \bigskip \bigskip {\bf P.Amster, M.M.Cassinelli 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 or Prof. P.Amster, 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 - pamster\@dm.uba.ar \end ---------------0004041757674--