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

We study the eigenvalue problem involving the mixed local-nonlocal operator $L:=-\Delta +{(-\Delta )}^s+q\cdot \nabla +a\left(x\right)\mathrm{Id}$ in a bounded domain $\Omega \subset R^N$, where a Dirichlet condition is posed on $R^N\setminus \Omega$. The vector 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 the $L^p$ theory for L obtained via 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.