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

created by figalli on 22 Mar 2017

[BibTeX]

Accepted Paper

Inserted: 22 mar 2017
Last Updated: 22 mar 2017

Journal: J. Differential Equations
Year: 2017

Abstract:

We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\min\bigl\{-\Delta_p u,\,u-\varphi\bigr\}=0$ in $\Omega\subset\R^n$. Here, $\Delta_p u=\textrm{div}\bigl( \nabla u ^{p-2}\nabla u\bigr)$, and $p\in(1,2)\cup(2,\infty)$.

Near those free boundary points where $\nabla \varphi\neq0$, the operator $\Delta_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\nabla \varphi=0$ 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 $\nabla \varphi=0$. On the one hand, for every $p\neq2$ 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 $\nabla \varphi=0$. On the other hand, under the concavity'' assumption $\nabla \varphi ^{2-p}\Delta_p \varphi<0$, we show the free boundary is countably $(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.