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In this 35-minute talk from the Workshop on "Degenerate and Singular PDEs" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), explore the Zaremba problem as a mixed boundary value problem where Dirichlet conditions are imposed on sets with positive capacity according to Mazya's condition, rather than necessarily on sets of positive measure. Learn how this formulation allows Dirichlet conditions to be defined on fractal-like sets, including Cantor-type sets. Discover established results for the local Zaremba problem for both the Laplacian and p-Laplacian with degenerate weights. Examine the proposed well-posed formulation of the nonlocal Zaremba problem and the foundational existence and regularity results derived from it. The presentation covers joint projects with Ho-Sik Lee (Bielefeld) and ongoing work with Guy Foghem (Dresden).
