Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with the smooth boundary $\partial \Omega$. Consider the eigenvalues $\lambda_k$ of the Laplace operator on $\Omega$ under zero Dirichlet boundary conditions. Let us denote by $m(\lambda_k)$ the multiplicity of $\lambda_k$.

It was proved by Hoffmann-Ostenhof, Michor and Nadirashvili ¹ that $m(\lambda_k) \leq 2k-3$ for any $k \geq 3$. Here, I would like to provide the following Pleijel-type remark about asymptotic behavior of $m(\lambda_k)$ as $k \to \infty$.

Remark. The following inequality is satisfied:

\begin{equation}\label{eq:1} \limsup_{k \to \infty} \frac{m(\lambda_k)}{k} < \frac{8}{j_0^2} - 6 \cdot 10^{-9} = 1.383320546…, \end{equation}

where $j_0$ is the first zero of the Bessel function $J_0$.

Moreover, $m(\lambda_k) \leq 2k-5$ provided

\begin{equation}\label{2} k > \frac{32 \pi j_0^4 |\Omega|}{(j_0^2-4)^2} \left(\inf \left\{\epsilon:~ |{x \in \Omega: d(x,\partial \Omega) < \epsilon}| \geq \frac{(j_0^2-4)|\Omega|}{2j_0^2}\right\}\right)^{-2}. \end{equation}

The inequality \eqref{eq:1} follows from Theorem B of ¹ in combination with the result of Pleijel ² and Bourgain ³.

The estimate \eqref{2} follows from Theorem B of ¹ in combination with Theorem 1 (iii) of van den Berg & Gittins ⁴.

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