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.


  1. Hoffmann-Ostenhof, T., Michor, P. W., & Nadirashvili, N. (1999). Bounds on the multiplicity of eigenvalues for fixed membranes. Geometric & Functional Analysis GAFA, 9(6), 1169-1188. arXiv:9801090  2 3

  2. Pleijel, Å. (1956). Remarks on Courant’s nodal line theorem. Communications on pure and applied mathematics, 9(3), 543-550. DOI:10.1002/cpa.3160090324 

  3. Bourgain, J. (2015). On Pleijel’s nodal domain theorem. International Mathematics Research Notices, 2015(6), 1601–1612. DOI:10.1093/imrn/rnt241 arXiv:1308.4422 

  4. Berg, M. V. D., & Gittins, K. (2016). On the number of Courant-sharp Dirichlet eigenvalues. Journal of Spectral Theory, 6(4), 735-x745. arXiv:1602.08376