Boundary behavior of solutions of elliptic operators in divergence form with a BMO anti-symmetric part

TL;DR

This paper studies the boundary behavior of solutions to divergence-form elliptic operators with BMO anti-symmetric parts in NTA domains, establishing regularity, measure existence, and Dirichlet problem well-posedness.

Contribution

It extends boundary regularity and measure theory results to elliptic operators with BMO anti-symmetric parts in general NTA domains, including Lipschitz domains.

Findings

Solutions are Hölder continuous at the boundary.

Existence of elliptic measures for these operators.

Well-posedness of Dirichlet problems in NTA domains.

Abstract

In this paper, we investigate the boundary behavior of solutions of divergence-form operators with an elliptic symmetric part and a $BMO$ anti-symmetric part. Our results will hold in non-tangentially accessible (NTA) domains; these general domains were introduced by Jerison and Kenig and include the class of Lipschitz domains. We establish the H\"older continuity of the solutions at the boundary, existence of elliptic measures $\omega_L$ associated to such operators, and the well-posedness of the continuous Dirichlet problem as well as the $L^p(d\omega)$ Dirichlet problem in NTA domains. The equivalence in the $L^p$ norm of the square function and the non-tangential maximal function under certain conditions remains valid. When specialized to Lipschitz domains, it is then possible to extend, to these operators, various criteria for determining mutual absolute continuity of elliptic measure with surface measure.