Solutions to a two-dimensional, Neumann free boundary problem

We explore regularity properties of solutions to a two-phase elliptic free boundary problem near a Neumann fixed boundary in two dimensions. Consider a function u, which is harmonic where it is not zero and satisfies a gradient jump condition weakly along the free boundary. Our main result is that u is Lipschitz continuous up to the Neumann fixed boundary. We also present a numerical exploration of the way in which the free and fixed boundaries interact.