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 $C^{1,\theta }$-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 $C^{1,\alpha }$, while for the regularity result we also require that $p>sq$.