Free boundary partial regularity in the thin obstacle problem

TL;DR

This paper establishes partial regularity results for the free boundary in the thin obstacle problem, showing that solutions are well-approximated by their blow-ups except on a small set, leading to $C^{1,1}$ regularity up to codimension 3.

Contribution

It proves that at most free boundary points, solutions exhibit higher order expansions, implying $C^{1,1}$ regularity except on a small codimension 3 set.

Findings

Solutions differ from blow-ups by higher order terms at most free boundary points.

The free boundary is $C^{1,1}$-regular outside a codimension 3 set.

Partial regularity holds for the free boundary in the thin obstacle problem.

Abstract

For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at all free boundary points, with the exception of a $(n-3)$-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion entails a $C^{1,1}$-type free boundary regularity result, up to a codimension 3 set.