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

TL;DR

This work analyzes the Dirichlet problem for the elliptic operator $L^*u=\operatorname{div}^2(\mathbf A u)-\operatorname{div}(\boldsymbol{b}u)+cu$ in double divergence form, under the assumption that the leading coefficients have Dini mean oscillation. It establishes that regular boundary points for $L^*$ coincide with those for the Laplacian via the Wiener criterion, in both $d\ge 3$ and $d=2$, by developing a Perron framework, Green's function estimates, and a robust potential-capacity theory for $L^*$. The results yield solvability of the Dirichlet problem in $C(\overline{\Omega})$ for bounded domains when the domain is Laplace-regular, and provide explicit Wiener-type criteria for boundary regularity in all dimensions. A key outcome is the quantitative comparison between $L^*$-capacity and Laplacian capacity, which underpins the equivalence of regularity and the Wiener criterion. Overall, the paper extends classical Wiener theory to elliptic equations in double divergence form, with relevance to stationary Fokker-Planck-Kolmogorov equations and related diffusion models.

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.

Theorems & Definitions (77)

• Theorem 1: Theorems \ref{['thm0800sat']} and \ref{['thm1510mon']}
• Theorem 2: Theorem \ref{['thm0802sat']} and \ref{['thm1515mon']}
• Definition 3.1
• Theorem 3.3
• proof
• Theorem 3.7: Harnack inequality
• Lemma 3.8
• proof
• Lemma 3.9
• proof