The Friedland-Hayman inequality and Caffarelli's contraction theorem

TL;DR

This paper extends the Friedland-Hayman inequality to convex cones with mixed boundary conditions, providing a new proof that characterizes equality cases and potentially enhances regularity results in free boundary problems.

Contribution

We present a new proof of an inequality analogous to Friedland-Hayman for convex cones with mixed boundary conditions, including a characterization of equality cases.

Findings

Established an inequality for convex cones with Neumann and Dirichlet conditions.

Characterized the cases of equality in the new inequality.

Potential implications for regularity in free boundary problems.

Abstract

The Friedland-Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone, and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.