Optimal regularity for the fully nonlinear thin obstacle problem

TL;DR

This paper establishes the optimal regularity of solutions to the fully nonlinear thin obstacle problem, proving existence of an optimal exponent and regularity results, including special cases with rotational invariance.

Contribution

It introduces the optimal regularity exponent for solutions and proves uniqueness of blow-ups and solution expansions at regular points.

Findings

Existence of an optimal exponent lpha_F for regularity.

Solutions are ^{1,lpha_F} near the obstacle.

If the operator is rotationally invariant, solutions are ^{1,1/2}.

Abstract

In this work we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent $\alpha_F$ such that $u$ is $C^{1,\alpha_F}$ on either side of the obstacle. In order to do that, we prove the uniqueness of blow-ups at regular points, as well as an expansion for the solution there. Finally, we also prove that if the operator is rotationally invariant, then $\alpha_F\ge \frac12$ and the solution is always $C^{1,1/2}$.