[Plenary Lecture] Kolmogorov Problem and Wiener-type Criteria in Potential Theory

2nd OIST-Oxford-SLMath Summer Graduate School on Analysis and Partial Differential Equations

Time: July 1 (Wed) 1:00 pm - 2:00 pm

Speaker: Prof. Ugur Abdulla

Title: Kolmogorov Problem and Wiener-type Criteria in Potential Theory

Abstract: The central problem in the Analysis of PDEs is understanding the nature of singularities that arise in natural phenomena. This talk will present a full characterization of the fundamental boundary singularity, and equivalently, the unique solvability of the singular Dirichlet problem for the elliptic and parabolic PDEs. The results are threefold. We prove a new Wiener-type criterion for the geometric characterization of the removability of the fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of the complementary set near the singularity point. In the special case when the surface of revolution forms the boundary of the open set near the singularity point, we establish a Kolmogorov-Petrovsky-type test to characterize the removability of the singularity and uniqueness. Finally, in the special case when a continuous graph locally represents the boundary of the open set, the minimal thinness criterion for the removability of the singularity is expressed in terms of the minimal regularity of the boundary manifold at the singularity point. From the probabilistic point of view, the criteria present an asymptotic law for conditional Brownian motion. In the topological context, the criteria present a full characterization of the neighborhood base of the boundary singularity point in the minimal-fine topology. In the more general framework, the talk will outline a program for the full characterization of the singularities formed by the elliptic and parabolic PDEs.