A minimum problem with free boundary for the p(x) - Laplace operator

In this paper we consider the problem of minimizing the functional J(u) = ∫_Ω(1/p(x)|∇u|^p(x) + Q(x)Χ[u≥0])dx. We prove Lipschitz continuity for each minimizer u and establish the nondegeneracy at the free boundary (∂[u ≥ 0]) ∩ Ω and the locally uniform positive density of the sets [u ≥ 0] and [u = 0]. In particular we obtain that the Lebesgue measure of the free boundary is zero.