Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

We address some regularity issues for mixed local-nonlocal quasilinear operators modeled upon the sum of a p-Laplacian and of a fractional (s,q)-Laplacian. Under suitable assumptions on the right-hand sides and the outer data, we show that weak solutions of the Dirichlet problem are C1,θ-regular up to the boundary. In addition, we establish a Hopf type lemma for positive supersolutions. Both results hold assuming the boundary of the reference domain to be merely of class C1,α, while for the regularity result we also require that p>sq.