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.


Motivation and main result
For each positive integer N denote by R N the N -dimensional Euclidean space and by | · | N the Euclidean norm in R N . Let Ω ⊂ R N be an open and bounded set.

The eigenvalue problem of the Dirichlet-Laplace operator
The eigenvalue problem for the Dirichlet-Laplace operator on Ω reads as follows −∆u = λu in Ω, It is well-known that the spectrum of problem (1) consists of an increasing and unbounded sequence of positive real numbers (see, e.g. [10,Theorem 8.2.1]). Denote by µ 1 (Ω) and µ 2 (Ω) the first two eigenvalues of problem (1). Let us also denote by Ω a ball from R N which has the same volume as Ω (i.e., |Ω | = |Ω|). In 1955-1956, Payne, Pólya and Weinberger [7] showed that when N = 2 and conjectured that the right-hand side could be replaced by This result was extended to all dimensions in 1969 by Thompson [8], who showed that and, again, it was conjectured that the right-hand side could be replaced by In the case N = 2 important advances on this problem were obtained by Brands [3], de Vries [5], Chiti [4], and the conjecture was finally settled positively by Asbaugh and Benguria [1]

The eigenvalue problem of the Dirichlet-Laplace operator with a drift
Consider the eigenvalue problem of the Dirichlet-Laplace operator with a drift term −∆u − x · ∇u = λu in Ω, where λ is a real parameter. We say that λ is an eigenvalue of problem (2) if there exists u λ ∈ X 0 \ {0} such that where dm N := e |x| 2 N /2 dx and X 0 := H 1 0 (Ω; dm N ). Function u λ from the above definition is called an eigenfunction corresponding to the eigenvalue λ.
Using [2, relation (2.10) on page 715] (see also [9,Théorème 8.7] with H = X 0 applied for the particular case induced by problem (2)), we deduce that the first eigenvalue of problem (2) has the following variational characterization and there exists a corresponding eigenfunction corresponding to λ 1 (Ω), and Moreover, the second eigenvalue of problem (2) has the following variational characterization The goal of this paper is to prove the following theorem: Theorem 1.1. The following estimate holds true Consequently, we show that the result by Thompson [8] established in the case of the Laplace operator continues to hold true in the case of the Laplace operator with the drift x · ∇·. Note that even if the two cases seem to be very similar we can point out differences between them. For example, it is easy to check that the ratio µ2(Ω) µ1(Ω) is invariant on rescaled domains. More precisely, if for some t > 0 we denote Ω t := tΩ = {tx : x ∈ Ω} then This equality holds since a simple change of variable shows that µ i (Ω t ) = t −2 µ i (Ω) for i ∈ {1, 2}. Such a relation fails to hold true in the case of λ i (Ω t ) and consequently in general In other words, we want to point out the fact that in the case of the Laplace operator with the drift x · ∇· the dependence of the ratio λ2(Ω) λ1(Ω) on the domain Ω is more involved than in the case of the ratio µ2(Ω) µ1(Ω) . Despite this fact, we can establish the same bound from above (1 + 4N −1 ) which depends only on the dimension of the Euclidean space and not on the domain Ω on which the eigenvalue problem is analysed.
The proof of Theorem 1.1 is now complete.