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

We study the eigenvalue problem involving the mixed local-nonlocal operator $ L:= -\Delta +(-\Delta)^{s}+q\cdot\nabla$~ in a bounded domain $\Omega\subset\R^N,$ where a Dirichlet condition is posed on $\R^N\setminus\Omega.$ 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 $s\in (0,1/2]$. Moreover, we prove $C^{2,\alpha}$ 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.