Optimal regularity for the obstacle problem for the $p$-Laplacian

Abstract

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 that) and given boundary datum on ∂B 1 . We prove that the solution is uniformly C 1,1 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 C 1, 1 p−1 in the spatial variables.