The principal eigenvalue of a mixed local and nonlocal operator with drift

We study the eigenvalue problem involving the mixed local-nonlocal operator L: = - + (-) ^s+q~ in a bounded domain N, where a Dirichlet condition is posed on N. The field q stands for a drift or advection in the medium. We prove the existence of a principal eigenvalue and a principal eigenfunction for s (0, 1/2]. Moreover, we prove C^2, regularity, up to the boundary, of the solution to the problem Lu=f when coupled with a Dirichlet condition and 0<s<1/2. To prove the regularity and the existence of a principal eigenvalue, we use a continuation argument, Krein-Rutman theorem as well as a Hopf Lemma and a maximum principle for the operator L, which we derive in this paper.