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The Friedland-Hayman inequality is a sharp inequality concerning the growthrates of homogeneous, harmonic functions with Dirichlet boundary conditions oncomplementary cones dividing Euclidean space into two parts. In this paper, weprove an analogous inequality in which one divides a convex cone into twoparts, placing Neumann conditions on the boundary of the convex cone, andDirichlet conditions on the interface. This analogous inequality was alreadyproved by us jointly with Sarah Raynor. Here we present a new proof thatpermits us to characterize the case of equality. In keeping with the two-phasefree boundary theory introduced by Alt, Caffarelli, and Friedman, such animprovement can be expected to yield further regularity in free boundaryproblems.

The Friedland-Hayman inequality and Caffarelli's contraction theorem