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 \(p > s q\).