Fine boundary regularity for fully nonlinear mixed local-nonlocal problems

We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded $C^2$ domain $\Omega \subset R^d$, let $u\in C\left(R^d\right)$ be a viscosity solution of such Dirichlet problem. We obtain global Lipschitz regularity and fine boundary regularity for u by constructing appropriate sub and supersolutions coupled with a Harnack type inequality. We apply these results to obtain Hölder regularity of Du up to the boundary.