Level set method for solving Poisson's equation with discontinuous nonlinearities

Abstract

We study semi-linear elliptic free boundary problems in which the forcing term is discontinuous; i.e., a Poisson's equation where the forcing term is the Heaviside function applied to the unknown minus a constant. This approach uses level sets to construct a monotonic sequence of iterates which converge to a class of solutions to the free boundary problem. The monotonicity of the construction based on nested sets provides insight into the connectivity of the free boundary sets associated with the solutions.