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

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.