$L^p$ estimates for divergence form elliptic equations with discontinuous coefficients

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<p<\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.