Fine boundary regularity for fully nonlinear mixed local-nonlocal problems

Abstract

We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded C2 domain ohm subset of Rd, let u is an element of C(Rd) 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 H & ouml;lder regularity of Du up to the boundary.