A Minimum Problem with Free Boundary in Orlicz Spaces

We consider the optimization problem of minimizing R Ω G(|∇u|) + λχ{u>0} dx in the class of functions W (Ω) with u − φ0 ∈ W 1,G 0 (Ω), for a given φ0 ≥ 0 and bounded. W (Ω) is the class of weakly differentiable functions with R Ω G(|∇u|) dx < ∞. The conditions on the function G allow for a different behavior at 0 and at ∞. 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, Ω∩ ∂{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 regularity of their free boundaries near “flat” free boundary points.