MATH SEMINAR:The Wiener-type Criterion at ∞ for the Elliptic...03-05-2021
Speaker: Uğur Abdulla
Title: The Wiener-type Criterion at ∞ for the Elliptic and Parabolic PDEs and its Measure-Theoretical, Topological and Probabilistic Consequences
Date/Time: 5 May 2021 / 18:00 - 19:00
Abstract: Norbert Wiener’s celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results of 20th-century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories.This lecture presents my solution of the Kolmogorov’s Problem (1928) expressed in the Wiener-type criterion for the regularity of ∞ for the second-order elliptic and parabolic PDEs in arbitrary open sets.The new concept of regularity or irregularity of ∞ for the elliptic (or parabolic) PDEs is defined according to whether or not the harmonic (or caloric) measure of ∞ is null or positive.Wiener-type criterion precisely characterizes the regularity of the point at ∞ for the second-order elliptic and parabolic PDEs and broadly extends the Wiener test’s role in classical analysis. The Wiener test at ∞ arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at ∞ arises as a thinness criterion at ∞ in fine topology. Precisely, the open set is a deleted neighborhood of ∞ in fine topology if and only if ∞ is an irregular boundary point. In a probabilistic context, the Wiener test at ∞ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criterion for the removability of singularities and/or for unique continuation at the finite boundary point; thinness criterion near Martin boundary point at minimal-fine topology; asymptotic laws for the conditional Brownian motion. The talk will discuss some outstanding open problems and perspectives of the development of the potential theory of nonlinear elliptic and parabolic PDEs.
Bio: Professor Ugur Abdulla received his Ph.D. and Doctor of Science degrees from the USSR Academy of Sciences and Habilitation degree from the Saxon Ministry of Fine Arts of Germany. He was awarded the Royal Society Research Fellowship in the United Kingdom, and Alexander von Humboldt Fellowship, and Max-Planck Research Fellowships in Germany. He is a Professor at the Florida Institute of Technology. Dr. Abdulla was a Visiting Professor at the Massachusetts Institute of Technology, University of Nottingham, UK, and Universities of Bonn, Leipzig, and Max-Planck Institute of Mathematics in the Natural Sciences in Germany. Professor Abdulla is an expert on Partial Differential Equations, Potential Theory, and Optimal Control of Systems with Distributed Parameters with 58 research papers. Professor Abdulla solved long-standing open Kolmogorov's problem (1928) and proved the Wiener-type criterion at infinity for the second-order elliptic and parabolic PDEs.