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 ${\mathbb{R}}^n$. More precisely, we show that for a certain class of domains there exists a sequence $p(k)$ such that ${\lambda }_k\ge {\mu }_{k+p(k)}$ for sufficiently large $k$. This sequence, which is given explicitly and is independent of the domain, grows with $k^{1-1\mathrm{/}n}$ 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 $k^{1-3\mathrm{/}n}$ 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.