The Boundary value problems for second order elliptic operators satisfying a Carleson condition

Abstract

Let $\Omega$ be a Lipschitz domain in ${\mathbb{R}}^n$ $n\ge 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in $H^{1,p}(\partial \Omega )$ and of the Neumann problem with $L^p(\partial \Omega )$ data for the operator $L$ on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator $L$ to be rough obeying a certain Carleson condition with small norm. These results complete the results of [5] where $L^p(\partial \Omega )$ Dirichlet problem was considered under the same assumptions and [6] where the regularity and Neumann problems were considered on two dimensional domains.

References in corpus (1)

• Partial regularity and smooth topology-preserving approximations of rough domains

Cited by in corpus (4)

• Layer potentials for general linear elliptic systems
• The Dirichlet boundary problem for second order parabolic operators satisfying Carleson condition
• BMO solvability and the $A_{\infty }$ condition for second order parabolic operators
• Regularity theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem