Eigenvalues of the Neumann problem in a domain with a thin channel
The author constructs asymptotics of eigenvalues of the Neumann problem for the Laplace operator \[ \begin{aligned} -\Delta\Psi^{\varepsilon} &= \lambda^{\varepsilon}\Psi^{\epsilon}\quad \text{ at} \quad x \in \Omega_{\varepsilon} \\ \frac{\partial\Psi^{\varepsilon}}{\partial\nu}&= 0 \quad \text{at}\quad x \in \partial\Omega_{\varepsilon}\end{aligned} \] in a three-dimensional domain of dumb bell type with a thin channel \[ \Omega_{\epsilon} = \Omega^{in} \cup \Omega^{ex} \cup \kappa_{\epsilon}, \] where \( 0 < \varepsilon \ll 1\), and \(\nu\) is an outward normal.
scientific article; zbMATH DE number 5115903