Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones

We investigate a fully nonlinear two-phase free boundary problem with a Neumann boundary condition on the boundary of a general convex set K ⊂ Rⁿ with corners. We show that the interior regularity theory developed by Caffarelli for the classical two-phase problem in his pioneer works {C1,C2}, can be extended up to the boundary for the Neumann boundary condition under very mild regularity assumptions on the convex domain K. To start, we establish a general existence theorem for the Dirichlet two-phase problem driven by two different fully nonlinear operators, which is a result of independent interest.