| Abstract: |
| In this talk, I will survey recent and ongoing work on two-phase and multi-phase free boundary regularity problems for harmonic measure (Laplace`s equation) and caloric measure (the heat equation). Although these problems are non-variational, the techniques developed to analyze them are inspired by calculus of variations and geometric measure theory. For harmonic measure in two-phases, we now have a thorough understanding of blow-ups, dimension, and global regularity of the regular set (where the boundary looks asymptotically flat) and the singular set (where the boundary looks asymptotically like the zero set of a homogeneous harmonic polynomial). The next challenge is to analyze the free boundary for multi-phase problems for harmonic measure and for two-phase problems for elliptic and caloric measures. These are joint works with M. Engelstein and T. Toro, with M. Akman, and with A. Genschaw. |
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