Optimal regularity for the obstacle problem for the p-Laplacian

In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such thatdiv(|∇u|p−2∇u)≤0,u≥ϕ,in B1, with ϕ∈C1,1(B1) and given boundary datum on ∂B1. We prove that the solution is uniformly C1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L∞. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C1,1p−1 in the spatial variables.