A free boundary problem in Orlicz spaces related to mean curvature

Abstract

In this paper we address a one phase minimization problem for a functional that includes the perimeter of the positivity set. It also includes three terms, the first one is ∫fu and the second ∫u>0h where f and h are bounded functions. The third term is ∫G(|∇u|) where G is a smooth convex function. This term generalizes the integral of the |∇u|p. As a consequence of our results we find that, when f≤0, there exists a nonnegative minimizer. Moreover, every nonnegative minimizer is Lipschitz continuous, it is a solution to ΔGu=f in {u>0} and satisfies that H=Φ(|∇u|)−h on the reduced free boundary, ∂red{u>0} which, as a consequence, is proved to be as smooth as the data allow. Here Φ(t)=tg(t)−G(t) (g=G′) and H is the mean curvature of the free boundary.