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

Abstract

We consider the optimization problem of minimizing ∫Ω frac(1, p (x)) | ∇ u |p (x) + λ (x) χ{u > 0} d x in the class of functions W1, p ({dot operator}) (Ω) with u - φ0 ∈ W01, p ({dot operator}) (Ω), for a given φ0 ≥ 0 and bounded. W1, p ({dot operator}) (Ω) is the class of weakly differentiable functions with ∫Ω | ∇ u |p (x) d x < ∞. 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.