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 $\lambda_{k}\geq \mu_{k+ p(k)}$ for sufficiently large $k$. This sequence, which is given explicitly and is independent of the domain, grows with $k^{1-1/n}$ 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 $mathbb{R}^{n} (n\geq4)$, 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.