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Session S33 - Spectral Geometry

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A Friedland-Hayman inequality for convex cones

Thomas Beck

Fordham University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Friedland-Hayman inequality concerns the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this talk, we will describe a variant of this inequality where one divides a convex cone into two parts, with Neumann conditions on the boundary of the cone, and Dirichlet conditions on the shared interface. This inequality plays a crucial role in the boundary regularity of a two-phase free boundary problem in a convex domain. The proof, which uses a variant of Caffarelli's contraction theorem for the Brenier optimal transport mapping, allows us to characterize the case of equality.

Joint work with David Jerison (MIT, USA) and Sarah Raynor (Wake Forest University, USA).

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