On the regularity of the free boundary in the $p$-Laplacian obstacle problem

We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $minbigl{-Delta_p u,,u-varphibigr}=0$ in $Omegasubsetmathbb R^n$. Here, $Delta_p u=textrm{div}bigl(| abla u|^{p-2} abla ubigr)$, and $pin(1,2)cup(2,infty)$. Near those free boundary points where $ abla varphi eq0$, the operator $Delta_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $ abla varphi=0$ then $Delta_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where $ abla varphi=0$. On the one hand, for every $p eq2$ we construct explicit global $2$-homogeneous solutions to the $p$-Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not $C^1$ at points where $ abla varphi=0$. On the other hand, under the concavity assumption $| abla varphi|^{2-p}Delta_p varphi<0$, we show the free boundary is countably $(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.