$C^{1,\alpha}$ estimates for the fully nonlinear Signorini problem

TL;DR

This paper proves local $C^{1,eta}$ regularity estimates for solutions to the fully nonlinear Signorini problem, extending previous results to non-symmetric cases and general Lipschitz domains.

Contribution

It establishes new local regularity estimates for fully nonlinear Signorini problems without symmetry assumptions and in Lipschitz domains.

Findings

Solutions are $C^{1,eta}$ regular on each side of the obstacle.

Estimates do not depend on boundary data, only on local properties.

Results extend previous symmetry-dependent regularity results.

Abstract

We study the regularity of solutions to the fully nonlinear thin obstacle problem. We establish local $C^{1,\alpha}$ estimates on each side of the smooth obstacle, for some small $\alpha > 0$. Our results extend those of Milakis-Silvestre in two ways: first, we do not assume solutions nor operators to be symmetric, and second, our estimates are local, in the sense that do not rely on the boundary data. As a consequence, we prove $C^{1,\alpha}$ regularity even when the problem is posed in general Lipschitz domains.