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

## TL;DR

This paper investigates the regularity of the free boundary in the $p$-Laplacian obstacle problem, especially at points where the gradient of the obstacle vanishes, revealing non-smoothness and conditions for rectifiability.

## Contribution

It provides the first analysis of free boundary regularity at points with zero gradient of the obstacle and constructs explicit solutions illustrating non-$C^1$ boundaries.

## Key findings

- Free boundary not $C^1$ at points where $
abla =0$ for $p
eq 2$.
- Explicit global 2-homogeneous solutions with corners in the free boundary.
- Under a concavity condition, free boundary is countably $(n-1)$-rectifiable.

## 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 $\Omega\subset\mathbb R^n$. 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.