The Wiener Criterion at $\infty$ for Degenerate Elliptic Equations

TL;DR

The paper addresses the unique solvability of the Dirichlet problem for a degenerate elliptic operator with a power-like weight $|x|^\gamma$ in arbitrary open sets, establishing a Wiener criterion at infinity that characterises regularity of the boundary point $\infty$ via the $\mathcal{A}$-harmonic measure. It develops a measure-theoretic and topological framework using $\mathcal{A}$-capacity, $\mathcal{A}$-fine topology, and Perron–Wiener–Brelot (PWB) theory to relate solvability to the exterior geometry and to Wiener-type integrals. The main result proves the equivalence of $\infty$-regularity with unique bounded solutions, $\infty$-irregularity with infinitely many solutions, $\mathcal{A}$-thickness/thinness of $\Omega^c$ at $\infty$, and divergence/convergence of the Wiener integral and Wiener sum built from $\operatorname{cap}(\cdot,\cdot)$. This extends the classical Wiener criterion to weighted degenerate elliptic equations and provides geometric, capacity-based criteria for solvability at infinity. The work thereby generalises previous unweighted results and connects measure-theoretic and fine-topological perspectives in the weighted, degenerate setting.

Abstract

This paper establishes a Wiener criterion at $\infty$ to characterise the unique solvability of the Dirichlet problem for degenerate elliptic equations with power-like weights in arbitrary open sets. In the measure-theoretical context, the criterion determines whether the $\A$-harmonic measure of $\infty$ is null or positive. From the topological point of view, it presents a test for the thinness of the exterior set at $\infty$ in the $\A$-fine topology.

Theorems & Definitions (12)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3