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Explore advanced mathematical analysis in this 54-minute lecture examining recent developments in the nonlocal one-phase free boundary problem. Discover optimal regularity and non-degeneracy properties of solutions, learn about improvement of flatness results, and understand higher regularity behavior of free boundaries near regular points. Examine how these findings lead to proofs of free boundary smoothness in open, dense sets and complete smoothness in two-dimensional cases. Study the problem framework for general integro-differential operators of order 2s and master purely nonlocal techniques that connect deeply to nonlocal equations with local boundary conditions. Gain insights from collaborative research findings that advance the theoretical understanding of free boundary problems in nonlocal settings.
