$W^{2,2}$-solvability of the Dirichlet problem for a class of elliptic equations with discontinuous coefficients

Abstract

We study the Dirichlet problem for the second order elliptic equation

in a bounded regular domain $\Omega \subset {\mathbb{R}}^N,N>2$. We assume that $f\in L^2$ and that the coefficients $a_{ij}$ are measurable and bounded functions with the first derivatives in the Marcinkiewicz class weak $-L^N$ and having a sufficiently small distance to $L^{\infty }$. Under these assumptions we prove the solvability of the problem in $W^{2,2}\cap W_0^{1,2^{\ast }}$, where $2^{\ast }=\frac{2N}{N-2}$. An higher integrability result for the gradient of the solution is achieved when $f\in L^p,p>2$.