A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing integral(2) G (vertical bar del u vertical bar) + lambda chi({u > 0}) dx in the class of functions W-1.G(Omega) with u - phi(0) is an element of W-0(1.G) (Omega), for a given phi(0) >= 0 and bounded. W-1.G (Omega) is the class of weakly differentiable functions with integral(Omega)G (vertical bar del u vertical bar) dx < infinity. The conditions on the function G allow for a different behavior at 0 and at infinity. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Omega boolean AND partial derivative{u > 0}, 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. (C) 2008 Elsevier Inc. All rights reserved.