Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

TL;DR

This paper develops a perturbation theory for second order elliptic operators with unbounded antisymmetric parts in BMO, establishing solvability of the $L^p$ Dirichlet problem under Carleson conditions on bounded chord arc and Lipschitz domains.

Contribution

It extends perturbation results to operators with unbounded antisymmetric parts in BMO, generalizing previous symmetric matrix cases to broader classes of elliptic operators.

Findings

Solvability of $L^q$ Dirichlet problem under Carleson condition

Small Carleson norm implies $q=p$ solvability

Application to operators on Lipschitz domains

Abstract

In the present paper we study perturbation theory for the $L^p$ Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $L_0 = \mbox{div}(A_0\nabla)$ and $L_1 = \mbox{div}(A_1\nabla)$ such that the $L^p$ Dirichlet problem for $L_0$ is solvable for some $p>1$; we show that if $A_0 - A_1$ satisfies certain Carleson condition, then the $ L^q$ Dirichlet problem for $L_1$ is solvable for some $q \geq p$. Moreover if the Carleson norm is small then we may take $q=p$. We use the approach first introduced in Fefferman-Kenig-Pipher '91 on the unit ball, and build on Milakis-Pipher-Toro '11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $L^p$ Dirichlet problem on a bounded Lipschitz domain for an operator $L = \mbox{div}(A\nabla)$, where $A$ satisfies a Carleson condition similar to the one assumed in Kenig-Pipher '01 and Dindo\v{s}-Petermichl-Pipher '07 but with unbounded antisymmetric part.