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 that $\hbox{div} (|\nabla u|^{p-2}\nabla u) \leq 0, \qquad u\geq ϕ, \qquad \hbox{in } B_1,$ with $ϕ\in C^{1,1}(B_1)$ and given boundary datum on $\partial B_1$. 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 $L^\infty$. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is $C^{1,\frac{1}{p-1}}$ in the spatial variables.

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