Boundary value problems for elliptic operators satisfying Carleson condition

In this paper we present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a variety of Carleson conditions. More precisely, with $L=\mbox{div}(A\nabla)$, we assume the matrix $A$ is elliptic and satisfies a natural Carleson condition either in the form that ($|\nabla A(X)|\lesssim \mbox{dist}(X,\partial\Omega)^{-1}$ and $|\nabla A|(X)^2\mbox{dist}(X,\partial\Omega)\,dX$) or $\mbox{dist}(X,\partial\Omega)^{-1}\left(\mbox{osc}_{B(X,\delta(X)/2)}A\right)^2\,dX$ is a Carleson measure. We present two types of results, the first is the so-called "small Carleson" case where, for a given $1<p<\infty$, we prove solvability of the three considered boundary value problems under assumption the Carleson norm of the coefficients and the Lipschitz constant of the considered domain is sufficiently small. The second type of results ("large Carleson") relaxes the constraints to any Lipschitz domain and to the assumption that the Carleson norm of the coefficients is merely bounded. In this case we have $L^p$ solvability for a range of $p$'s in a subinterval of $(1,\infty)$. At the end of the paper we give a brief overview of recent results on domains beyond Lipschitz such as uniform domains or chord-arc domains.