Existence and Regularity in the Fully Nonlinear One-Phase Free Boundary Problem

Abstract. We consider viscosity solutions to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show the existence of viscosity solutions via Perron’s method and we prove [Formula: see text] regularity of flat free boundaries via a quadratic improvement of flatness. Finally, we obtain the higher regularity of the free boundary via a hodograph transform.