On the (growing) gap between Dirichlet and Neumann eigenvalues

We provide an answer to a question raised by Levine and Weinberger in their $1986$ paper concerning the difference between Dirichlet and Neumann eigenvalues of the Laplacian on bounded domains in ⁿ. More precisely, we show that for a certain class of domains there exists a sequence $p(k)$ such that λₖ≥ μk+ p(k) for sufficiently large k. This sequence, which is given explicitly, grows with k1-1/n as k goes to infinity, which we conjecture to be optimal, and may be chosen independently of the domain. We also prove the existence of a sequence, now not given explicitly and only of order k1-3/n but valid for bounded Lipschitz domains in ⁿ (n≥4), for which a similar inequality holds for all k. From these results and the analysis of some particular examples we formulate a conjecture for general Euclidean domains.