The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary

TL;DR

This paper establishes optimal regularity results and free boundary regularity for the variable coefficient thin obstacle problem in low regularity settings, extending previous work to more general metrics and obstacles.

Contribution

It introduces new techniques for analyzing the regular free boundary and solution regularity in variable coefficient thin obstacle problems with low regularity data.

Findings

Proves $C^{1,eta}$ regularity of the free boundary.

Derives the asymptotic expansion of solutions near free boundary points.

Establishes optimal regularity of solutions for $W^{1,p}$ metrics and $W^{2,p}$ obstacles.

Abstract

This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,\alpha}$ regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1-\frac{n+1}{p}, \frac{1}{2}\}}$ regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in \cite{KRS14} and the present article, we then also discuss the case of non-zero and interior thin obstacles. We obtain the optimal regularity of the solutions and the regularity of the regular free boundary for $W^{1,p}$ metrics and $W^{2,p}$ obstacles with $p\in (2(n+1),\infty]$.