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Potential Theory

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:

Potential Theory: What We Do
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