Neumann Fixed Boundary Regularity for an Elliptic Free Boundary Problem
by
Sarah Raynor
Wake Forest University

We examine the regularity properties of solutions to an elliptic free boundary problem near a Neumann fixed boundary. Consider a nonnegative function u, defined variationally, which is harmonic where it is positive and satisfies a gradient jump condition weakly along the free boundary ∂{ u > 0 }. Our main result is that u is Lipschitz continuous. Additionally, we prove various basic properties of such a minimizer near a portion of the fixed boundary on which ∂u/∂n = 0 weakly. Our results include up-to-the boundary gradient estimates on harmonic functions with Neumann boundary conditions on convex domains, which have independent interest.

Date received: September 25, 2007