Caltech/UCLA Joint Analysis Seminar

The Friedland-Hayman inequality provides a lower bound on the first Dirichlet eigenvalues of complementary subsets of the sphere. In this talk, we will describe a variant of this inequality to geodesically convex subsets of the sphere with mixed Dirichlet-Neumann boundary conditions. Using this inequality, we prove an almost-monotonicity formula and Lipschitz continuity up to the boundary for the minimizer of a two-phase free boundary problem. This is joint work with David Jerison and Sarah Raynor.