Abstract:  I will discuss the regularity of solutions to a class of semilinear free boundary problems in which admissible functions have a topological constraint, or spanning condition, on their 1-level set. This constraint forces the 1-level set, which is a free boundary, to behave like a surface with singularities, attached to a fixed boundary frame, in the spirit of the set-theoretic Plateau problem. Two such free boundary problems that have been well-studied are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn approximation of the set-theoretic Plateau problem. We establish optimal Lipschitz regularity for solutions, and analytic regularity for the free boundaries away from a codimension two singular set. We further characterize the singularity models for these problems as conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems. This is joint work with Mike Novack and Daniel Restrepo.
 

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