A FREE BOUNDARY PROBLEM FOR THE p(x)-LAPLACIAN

We consider the optimization problem of minimizing; R; Ω; |∇u|p(x) + λχ{u>0} dx in the class of functions W 1,p(·)(Ω) with u − ϕ0 ∈ W 1,p(·)0 (Ω), for a given ϕ0 ≥ 0 and bounded. W 1,p(·)(Ω) is the class of weakly differentiable functions with R Ω |∇u|p(x) dx < ∞. 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