Regularity of Infinity for Elliptic Equations with Measurable Coefficients and Its Consequences

TL;DR

This paper studies the regularity of the point at infinity for elliptic equations with measurable coefficients, establishing conditions for unique bounded solutions and linking regularity to Wiener tests and set thinness at infinity.

Contribution

It introduces a new notion of regularity at infinity for elliptic equations with measurable coefficients and provides a Wiener test criterion for solution uniqueness.

Findings

A necessary and sufficient Wiener test condition for regularity at infinity.

Equivalence of regularity at infinity with set thinness in fine topology.

Characterization of solution existence based on boundary regularity at infinity.

Abstract

This paper introduces a notion of regularity (or irregularity) of the point at infinity for the unbounded open subset of $\rr^{N}$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the A-harmonic measure of the point at infinity is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of $\rr^{N}, N\ge 3$ is established in terms of the Wiener test for the regularity of the point at infinity. It coincides with the Wiener test for the regularity of the point at infinity in the case of Laplace equation. From the topological point of view, the Wiener test at infinity presents thinness criteria of sets near infinity in fine topology. Precisely, the open set is a deleted neigborhood of the point at infinity in fine topology if and only if infinity is irregular.