A minimum problem with free boundary in Orlicz spaces

Abstract

We consider the minimization problem: int_Omega ?G(|nabla u), dx +|{u>0}|) in u-u_0 in W_0^{1,G}(Omega) for a given u_0 in W_0^{1,G}(Omega). The conditions on the function G allow for different behavior at 0 and infinity. We prove that every solution is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C^{1,alpha} regularity of their free boundaries near flat free boundary points.