The Dirichlet–Neumann eigenvalue gap conjecture for bounded Euclidean domains

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, and let $\lambda_k(\Omega)$ and $\mu_k(\Omega)$ denote its Dirichlet and Neumann eigenvalues, respectively. For positive integers $k$, define

$$p(k)=\left\lfloor C_n\frac{n\omega_{n-1}}{2\omega_n^{1-2/n}}k^{1-1/n}\right\rfloor,$$

where $C_n\in(0,1]$ depends only on the dimension and $\omega_j$ denotes the volume of the unit ball in $\mathbb{R}^{j}$. The Dirichlet–Neumann eigenvalue gap conjecture. There exist such constants $C_n$ for which

$$\lambda_k(\Omega)\geq\mu_{k+p(k)}(\Omega)$$

for every positive integer $k$. Furthermore, the power $1-1/n$ is optimal. The conjecture is motivated by inequalities proved for Euclidean domains and by the role of non-periodicity and isoperimetric estimates; analogous inequalities can fail or reverse in some non-Euclidean settings, so the general Euclidean-domain case remains open.