A free boundary problem in Orlicz spaces related to mean curvature. (November 2021) - Wolanski, N.
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 ∫ f u and the second ∫ u > 0 h 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 Δ G u = f in { u > 0 } and satisfies that H = Φ ( | ∇ u | ) − h on the reduced free boundary, ∂ r e d { u > 0 } which, as a consequence, is proved to be as smooth as the data allow. Here Φ ( t ) = t g ( t ) − G ( t ) ( g = G ′ ) and H is the mean curvature of the free boundary.