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

We study the eigenvalue problem involving the mixed local-nonlocal operator L:= -Δ +(-Δ)ˢ+q·∇~ in a bounded domain Ω⊂N, where a Dirichlet condition is posed on NΩ. 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 C2,α 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.