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

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 [9] 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.