Lipschitz regularity for the parabolic $(s,p)-$obstacle problem

Abstract

We study the obstacle problem for the parabolic fractional $p-$Laplace equation \[\partial_t u+(-Δ_p)^su = 0\] in the degenerate range $2<p<2/(1-s)$. We prove that viscosity solutions are locally Lipschitz continuous in space and Hölder continuous in time. If, in addition, $p>1/(1-s)$, the time regularity improves to Lipschitz continuity.