Optimal Regularity for the Thin Obstacle Problem with $C^{0,\alpha}$ Coefficients

TL;DR

This paper establishes optimal regularity results for solutions to the thin obstacle problem with low regularity coefficients and obstacles, and analyzes the structure of the free boundary under these conditions.

Contribution

It proves the optimal $C^{1,eta}$ regularity of solutions with $C^{0,eta}$ coefficients and $C^{1,eta}$ obstacles, extending previous regularity results to lower regularity settings.

Findings

Solutions are $C^{1,eta}$ regular with $eta= ext{min}\{ ext{ extalpha},1/2 ext}$ under low regularity assumptions.

The free boundary is shown to be a $C^{1, ext{ extgamma}}$ manifold for some $ ext{ extgamma} ext extgreater 0$.

The methods combine linearization and epiperimetric inequalities to handle low regularity coefficients.

Abstract

In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson \cite{An16} and the epiperimetric inequality from \cite{FS16}, \cite{GPSVG15}, we prove the optimal $C^{1,\min\{\alpha,1/2\}}$ regularity of solutions in the presence of $C^{0,\alpha}$ coefficients $a^{ij}$ and $C^{1,\alpha}$ obstacles $\phi$. Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $C^{1,\gamma}$ manifold for some $\gamma \in (0,1)$.