Estimates for the ratio of the first two eigenvalues of the Dirichlet-Laplace operator with a drift

Abstract Let Ω ⊂ R be an open and bounded set. Consider the eigenvalue problem of the Laplace operator with a drift term −∆u−x ·∇u = λu in Ω subject to the homogeneous Dirichlet boundary condition (u = 0 on ∂Ω). Denote by λ1(Ω) and λ2(Ω) the first two eigenvalues of the problem. We show that λ2(Ω)λ1(Ω) ≤ 1 + 4N−1. In particular, we complement a similar result obtained by Thompson [Stud. Appl. Math. 48 (1969) 281–283] for the classical eigenvalue problem of the Laplace operator.