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

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