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 $\mathrm{\Omega }\subset {\mathbb{R}}^d,$ let $u\in C({\mathbb{R}}^d)$ 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.