The Wiener Test for the Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and Its Consequences

TL;DR

This paper develops a Wiener test to determine when logarithmic singularities are removable for elliptic PDEs with measurable coefficients, linking boundary regularity, potential theory, and probabilistic behavior of log-Brownian motion.

Contribution

It introduces the concept of log-regularity and establishes a Wiener test for the removability of logarithmic singularities in elliptic PDEs with measurable coefficients.

Findings

Provides necessary and sufficient conditions for logarithmic singularity removability.

Connects boundary regularity with minimal thinness in minimal fine topology.

Describes asymptotic behavior of log-Brownian motion near boundary points.

Abstract

This paper introduces the notion of $log$-regularity (or $log$-irregularity) of the boundary point $\zeta$ (possibly $\zeta=\infty$) of the arbitrary open subset $\Omega$ of the Greenian deleted neigborhood of $\zeta$ in $R^2$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the $log$-harmonic measure of $\zeta$ is null (or positive). A necessary and sufficient condition for the removability of the logarithmic singularity, that is to say for the existence of a unique solution to the Dirichlet problem in $\Omega$ in a class $O(\log |\cdot - \zeta|)$ is established in terms of the Wiener test for the $log$-regularity of $\zeta$. From a topological point of view, the Wiener test at $\zeta$ presents the minimal thinness criteria of sets near $\zeta$ in minimal fine topology. Precisely, the open set $\Omega$ is a deleted neigborhood of $\zeta$ in minimal fine topology if and only if $\zeta$ is $log$-irregular. From the probabilistic point of view, the Wiener test presents asymptotic law for the $log$-Brownian motion near $\zeta$ conditioned on the logarithmic kernel with pole at $\zeta$.