Almost Everywhere Regularity for the Free Boundary of the Normalized p-harmonic Obstacle problem $p>2$

TL;DR

This paper proves that for the normalized p-harmonic obstacle problem with p>2, the free boundary is locally a $C^{1,eta}$-smooth surface at almost every free boundary point, enhancing understanding of regularity in nonlinear obstacle problems.

Contribution

It establishes almost everywhere regularity of the free boundary for the normalized p-harmonic obstacle problem with p>2, showing $C^{1,eta}$ smoothness at nearly all free boundary points.

Findings

Almost every free boundary point has a neighborhood where the boundary is $C^{1,eta}$.

The regularity result applies to points with respect to the $(n-1)$-Hausdorff measure.

The free boundary exhibits smoothness at nearly all points in the measure sense.

Abstract

Let $u$ be a solution to the normalized p-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2<p<\infty$, $u\ge 0$ and $$ \d\left( |\nabla u|^{p-2}\nabla u\right)=\chi_{\{u>0\}}\textrm{ in }B_1(0) $$ where $u(x)\ge 0$ and $\chi_A$ is the characteristic function of the set $A$. Our main result is that for almost every free boundary point, with respect to the $(n-1)-$Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,\beta}-$graph. That is, for $\H^{n-1}-$a.e. point $x^0\in \partial \{u>0\}\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\cap \partial \{u>0\}\in C^{1,\beta}$.