Asymptotic expansion of eigenvalues of the Neumann problem in a domain with a thin bridge (Q1204745)

This article deals with the spectral problem \[ \begin{aligned} \Delta u(\varepsilon, x)+ \lambda(\varepsilon) u(\varepsilon,x) &= 0\qquad (x\in\Omega_ \varepsilon),\\ \partial_ n u(\varepsilon, x) &=0\qquad (x\in\partial\Omega),\\ (\partial u/ \partial\nu^ \pm) (\varepsilon, x) &=0\qquad (x\in G_ \varepsilon^ \pm), \end{aligned} \] where \(\Omega\) is a domain in \(\mathbb{R}^ 3\) with smooth boundary \(\partial\Omega\), \(G\) is a smooth two-dimensional submanifold of \(G\) with boundary \(\partial G\), \(\Omega_ \varepsilon= \Omega\setminus \omega_ \varepsilon\), \(\omega_ \varepsilon\) is a thin set over \(G\) of `thickness' \(\varepsilon\) in \(\Omega\), \(G_ \varepsilon^ \pm\) are `shores' of \(G_ \varepsilon\). The main goal of this article is to study the asymptotic behavior for eigenvalues \(\lambda(\varepsilon)\) of this problem. To this end the limit problems are determined and then asymptotics of three types \[ \lambda(\varepsilon)\sim \lambda_ 0+ \varepsilon\lambda_ 1, \quad \lambda(\varepsilon)\sim \lambda_ 0+ (\varepsilon \ln\varepsilon) \lambda_ 1, \quad \lambda(\varepsilon)\sim \lambda_ 0+ \varepsilon^{1/2} \lambda_ 1, \] where \(\lambda_ 0\) is an eigenvalue of the limit problem are presented.