On the (growing) gap between Dirichlet and Neumann eigenvalues

Abstract

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 ${\mathbb{R}}^n$. More precisely, we show that for a certain class of domains there exists a sequence $p(k)$ such that $\hat{I}{»}_k\ge \hat{I}{¼}_{k+p(k)}$ for sufficiently large $k$. This sequence, which is given explicitly and is independent of the domain, grows with $k^{1-1/n}$ as $k$ goes to infinity, which we conjecture to be optimal. We also prove the existence of a sequence, now not given explicitly and only of order $k^{1-3/n}$ but valid for bounded Lipschitz domains in $mathbb{R}^n(n\ge 4)$, for which a similar inequality holds for all $k$. We then frame these general results with some specific planar Euclidean examples such as rectangles and disks, for which we provide bounds valid for all eigenvalue orders.

author added, some major changes to the text, and addition of some results, including the study of the case of the disk, 16 pages, one figure