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

Abstract

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