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. Recent developments precisely characterize the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extend the role of the Wiener test 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 criteria at ∞ in fine topology. 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. Recent related publications are listed below:
Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and its Consequences, Calculus of Variations and Partial Differential Equations Volume 57:157, December 2018.
Regularity of ∞ for Elliptic Equations with Measurable Coefficients and Its Consequences, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 32, 10(2012), 3379-3397.
Wiener's Criterion at ∞ for the Heat equation and its Measure-theoretical Counterpart, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 44-51.
Wiener’s Criterion at ∞ for the Heat Equation, Advances in Differential Equations, 13, 5-6(2008), 457-488.
Wiener’s Criterion for the Unique Solvability of the Dirichlet Problem in Arbitrary Open Sets with Non-Compact Boundaries, Nonlinear Analysis, 67, 2(2007), 563-578.