The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

TL;DR

This paper proves that elliptic measures for certain divergence form operators with non-smooth, BMO anti-symmetric coefficients are absolutely continuous with respect to Lebesgue measure, ensuring unique solvability of the Dirichlet problem in the upper half-space.

Contribution

It establishes the first result on absolute continuity of elliptic measure for operators with non-smooth coefficients having a BMO anti-symmetric part, extending previous regularity results.

Findings

Elliptic measure belongs to the A_infty class relative to Lebesgue measure.

Dirichlet problem is uniquely solvable for boundary data in L^p.

Results hold for operators with unbounded, non-smooth coefficients.

Abstract

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation ${\rm div}(A\nabla u)=0$ in the upper half-space $(x,t)\in\mathbb{R}^{n+1}_+$ is uniquely solvable when $n\ge2$ and the boundary data is in $L^p(\mathbb{R}^n,dx)$ for some $p\in (1,\infty)$. This result is equivalent to saying that the elliptic measure associated to $L$ belongs to the $A_\infty$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity.