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# Thomas Beck, Fordham University – Analysis & PDE Seminar

## August 24, 2022 @ 3:00 pm - 4:00 pm

**Mode:** In-person

**Title:** A Friedland-Hayman inequality and two-phase free boundary problems

**Abstract:** The Friedland-Hayman inequality provides a lower bound on the first Dirichlet eigenvalues of complementary subsets of the sphere. Inthis talk, we will discuss a variant of this inequality for convex subsets of the sphere, with mixed Dirichlet-Neumann boundary conditions. The proof of this inequality, together with the case of equality, uses a version of Caffarelliâ€™s contraction theorem for the Brenier optimal transport mapping. We will then show how this inequality appears in the boundary regularity theory of a two-phase free boundary problem in a convex domain. This is joint work with David Jerison and Sarah Raynor.