In this paper we are concerned with the well posedness in $H^{1,p}_0 (\Omega) $ of the Dirichlet problem for the divergence form elliptic equation $$ Lu \equiv - \left(a_{ij} u_{x_i} \right)_{x_j} = \dive {\bf f} \eqno{(\num)} $$ in a bounded open set $\Omega \subset \RR^n .$ In particular we are interested in estimates like $$ \| \nabla u \|_p \leq c \| {\bf f} \|_p \qquad \forall p \in ]1, \infty [ \eqno{(\num)} $$ where $c$ is a constant independent from $u$ and ${\bf f}$ (see theorem 2.1). \capo In the case of discontinuous coefficients $ a_{ij},$ Meyers (see [M]) provided an example in which $ {\bf f} \in \left[ L^p (\Omega) \right]^n $ for $ 1<\infty $ but $ |\nabla u| $ does not and then an estimate like (1.2) in general cannot hold true. Calderon \& Zygmund via singular integrals technique (see e.g. [N]) proved that (1.2) holds true for every $ p\in ]1, \infty [ $ if $a_{ij}(x) = \delta_{ij}$ and later their results were extended by Morrey, Simader and Campanato (see respectively [MO], [Si] and [C]) to the case of continuous coefficients $ a_{ij} .$ \capo The purpose of this paper is to prove (1.2) when $a_{ij}$ are in the class $VMO$ (see section 2 for definitions). We explicitely point out that $C^0$ is strictly contained in $VMO.$ Our technique is based on some representation formulas in term of singular integral operators and commutators of the kind already considered in [CFL1] and [CFL2]. As a corollary, we obtain h\"older continuity of the solution $u$ for $ p>n.$ To conclude this introduction we mention the very interesting paper [A] where, with a different technique, similar results are obtained for some linear second order elliptic systems with discontinuous coefficients. In [A] the coefficients of the leading part are supposed to be in a class of possibly discontinuous functions which is a proper subset of $VMO.$ Also we point that an essential estimate in [A] (the $BMO$ estimate) can be shown to be false even for continuous coefficients (see [A]). Hence the methods in [A] cannot be adapted to cover the general $VMO$ case. In a forthcoming paper (see [CDF]) we will extend the present results to elliptic systems in divergence form. \capo We thank F.Chiarenza and M.Frasca for some helpful conversations on the topic of this work.
Lp Estimates for Divergence Form Elliptic Equations with Discontinuous Coefficients
DI FAZIO, Giuseppe
1996-01-01
Abstract
In this paper we are concerned with the well posedness in $H^{1,p}_0 (\Omega) $ of the Dirichlet problem for the divergence form elliptic equation $$ Lu \equiv - \left(a_{ij} u_{x_i} \right)_{x_j} = \dive {\bf f} \eqno{(\num)} $$ in a bounded open set $\Omega \subset \RR^n .$ In particular we are interested in estimates like $$ \| \nabla u \|_p \leq c \| {\bf f} \|_p \qquad \forall p \in ]1, \infty [ \eqno{(\num)} $$ where $c$ is a constant independent from $u$ and ${\bf f}$ (see theorem 2.1). \capo In the case of discontinuous coefficients $ a_{ij},$ Meyers (see [M]) provided an example in which $ {\bf f} \in \left[ L^p (\Omega) \right]^n $ for $ 1<\infty $ but $ |\nabla u| $ does not and then an estimate like (1.2) in general cannot hold true. Calderon \& Zygmund via singular integrals technique (see e.g. [N]) proved that (1.2) holds true for every $ p\in ]1, \infty [ $ if $a_{ij}(x) = \delta_{ij}$ and later their results were extended by Morrey, Simader and Campanato (see respectively [MO], [Si] and [C]) to the case of continuous coefficients $ a_{ij} .$ \capo The purpose of this paper is to prove (1.2) when $a_{ij}$ are in the class $VMO$ (see section 2 for definitions). We explicitely point out that $C^0$ is strictly contained in $VMO.$ Our technique is based on some representation formulas in term of singular integral operators and commutators of the kind already considered in [CFL1] and [CFL2]. As a corollary, we obtain h\"older continuity of the solution $u$ for $ p>n.$ To conclude this introduction we mention the very interesting paper [A] where, with a different technique, similar results are obtained for some linear second order elliptic systems with discontinuous coefficients. In [A] the coefficients of the leading part are supposed to be in a class of possibly discontinuous functions which is a proper subset of $VMO.$ Also we point that an essential estimate in [A] (the $BMO$ estimate) can be shown to be false even for continuous coefficients (see [A]). Hence the methods in [A] cannot be adapted to cover the general $VMO$ case. In a forthcoming paper (see [CDF]) we will extend the present results to elliptic systems in divergence form. \capo We thank F.Chiarenza and M.Frasca for some helpful conversations on the topic of this work.File | Dimensione | Formato | |
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