A free boundary problem for the p(x)p(x)-Laplacian

We consider the optimization problem of minimizing ∫Ω1p(x)|∇u|p(x)+λ(x)χ{u>0}dx in the class of functions W1,p(⋅)(Ω)W1,p(⋅)(Ω) with u−φ0∈W01,p(⋅)(Ω), for a given φ0≥0φ0≥0 and bounded. W1,p(⋅)(Ω)W1,p(⋅)(Ω) is the class of weakly differentiable functions with ∫Ω|∇u|p(x)dx<∞. We prove that every solution uu is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}Ω∩∂{u>0}, is a regular surface.