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, $\min \{-{\mathrm{\Delta }}_{p}u,u-\phi \}=0$ in $\mathrm{\Omega }\subset {\mathbb{R}}^n$. Here, $\Delta_p u=$\text{div}$\bigl(|\nabla u|^{p-2}\nabla u\bigr)$,and$p\in(1,2)\cup(2,\infty)$. Near those free boundary points where $\mathrm{\nabla }\phi \mathrm{\ne }0$, the operator ${\mathrm{\Delta }}_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\mathrm{\nabla }\phi =0$ then ${\mathrm{\Delta }}_p$ 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 $\mathrm{\nabla }\phi =0$. On the one hand, for every $p\mathrm{\ne }2$ 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 $\mathrm{\nabla }\phi =0$. On the other hand, under the “concavity” assumption $|\nabla \varphi|^{2-p}\Delta_p \varphi<0$,weshowthefreeboundaryiscountably$(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.