  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.
