| Abstract: |
| Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and an associated eigenfunction, respectively, of the Dirichlet Laplacian on a bounded domain $\Omega \subset \mathbb{R}^2$. Denote by $\mu(\varphi_k)$ the number of nodal domains of $\varphi_k$. Courant`s nodal domain theorem asserts that $\mu(\varphi_k) \leq k$ for any $k$. Pleijel obtained the following refinement of this fact:
\begin{equation*}
Pl(\Omega) := \limsup_{k \to \infty} \frac{\mu(\varphi_k)}{k} \leq \frac{4}{j_{0,1}^2} = 0.69166\ldots
\end{equation*}
Here, $Pl(\Omega)$ is called Pleijel constant of $\Omega$, and $j_{0,1}$ stands for the first zero of the Bessel function $J_0$.
In the present talk, we discuss explicit expressions and values of the Pleijel constant $Pl(\Omega)$ for several domains $\Omega$ with separable geometries, such as a disk, annuli, and their sectors. Consideration of the case of annuli required the development of the theory of zeros of cross-products of Bessel functions, and revealed natural but open problems on multiplicity of corresponding eigenvalues. The talk is based on the preprints arXiv:1802.04357 and arXiv:1803.09972. |
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