A Discrete Bernoulli Free Boundary Problem

Abstract: We consider a new type of free boundary problem for the p-Laplace operator. This problem is strictly related to the well-known Bernoulli problem: in this new formulation, the classical boundary gradient condition is replaced by a condition on the distance between two different level surfaces of the solution. Under suitable scaling this new formulation converges to the classical Bernoulli problem. We establish existence and qualitative theory in convex and non-convex regions. This is a joint work with M.d.M Gonzalez and H. Shahgholian.