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

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 $\zeta$.