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Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.

I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) Laplacian is below the $2$nd Neumann eigenvalue, i.e.

$$\lambda_1^{\operatorname{Neumann}}(\Delta) \le \lambda_1^{\operatorname{Dirichlet}}(\Delta)$$ is clear but when do we also have

$$\lambda_1^{\operatorname{Dirichlet}}(\Delta) \le \lambda_2^{\operatorname{Neumann}}(\Delta)? $$

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    $\begingroup$ You have always for $n \ge 2$ that $\lambda_{n+1}^{Neumann} <\lambda_n^{Dirichlet}$. The easiest prrof is due to Filonov (st. Petersburgh Math. J. 2005). $\endgroup$
    – Giorgio Metafune
    Mar 19, 2020 at 18:10

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