Kolmogorov Problem and Wiener-type Criterion for the Removability of the Fundamental Singularity for the Parabolic PDEs

2:00-2:50pm
3:00-3:50pm

Title: Kolmogorov Problem and Wiener-type Criterion for the Removability of the Fundamental Singularity for the Parabolic PDEs 

Speaker: Ugur G. Abdulla, Okinawa Institute of Science and Technology, Japan

Abstract: The major problem in the Analysis of PDEs is the characterization of singularities reflecting the natural phenomena. I will present my solution of the Kolmogorov's Problem (1928) expressed in terms of the new Wiener-type criterion for the removability of the fundamental singularity for the heat equation. The new concept of regularity or irregularity of singularity point for the parabolic (or elliptic) PDEs is defined according to whether or not the caloric (or harmonic) measure of the singularity point is null or positive. The new Wiener-type criterion precisely characterizes the uniqueness of boundary value problems with singular data, reveal the nature of the harmonic or caloric measure of the singularity point, asymptotic laws for the conditional Brownian motion, and criteria for thinness in minimal-fine topology.

The main goal of these two lectures is to sketch the proof of the new Wiener-type criterion for the removability of the fundamental singularity and, equivalently, the unique solvability of the singular Dirichlet problem for the heat equation. This work fully characterizes the removability of non-isolated boundary singularities through the fine-topological thinness of the exterior set near the singularity point. A significant tool in this characterization is the new concept of h-capacity of Borel sets, which measures thinness and establishes singularity behavior via the divergence of weighted sums of h-capacities within nested shells.