Boundary value problems for elliptic operators satisfying Carleson condition

TL;DR

This paper investigates the solvability of boundary value problems for elliptic operators with coefficients satisfying Carleson conditions on Lipschitz domains, providing results for both small and large Carleson norm scenarios.

Contribution

It introduces new solvability results for elliptic boundary value problems under Carleson measure conditions, extending to large Carleson norms and broader domain classes.

Findings

Solvability of Dirichlet, Regularity, and Neumann problems under small Carleson norm.

Extension of solvability results to large Carleson norm cases.

Overview of recent advances for non-Lipschitz domains.

Abstract

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.