Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form

TL;DR

This paper investigates the Dirichlet problem for a class of elliptic operators in double divergence form, establishing conditions under which boundary regularity matches that of the Laplace operator using Wiener criterion.

Contribution

It proves the equivalence of boundary regularity for elliptic operators with Dini mean oscillation coefficients and the Laplace operator, extending classical Wiener criterion results.

Findings

Boundary points regularity characterized by Wiener criterion

Equivalence of regularity conditions for $L^*$ and Laplace operator

Results applicable to stationary Fokker-Planck-Kolmogorov equations

Abstract

We study the Dirichlet problem for a second-order elliptic operator $L^*$ in double divergence form, also known as the stationary Fokker-Planck-Kolmogorov equation. Assuming that the leading coefficients have Dini mean oscillation, we establish the equivalence between regular boundary points for the operator $L^*$ and those for the Laplace operator, as characterized by the classical Wiener criterion.