Perturbation formula of eigenvalues in singularly perturbed domain
The author deals with partial degeneration of a domain (the Dumbbell shaped domain) and an asymptotic behavior of a certain class of eigenvalues of the Laplacian with Neumann boundary conditions. For small \(\zeta>0\), the domain \(\Omega(\zeta)\) is expressed as follows \(\Omega(\zeta)=D_ 1\cup D_ 2\cup Q(\zeta)\subset\mathbb{R}^ n\) where \(Q(\zeta)\) is a thin cylindrical region which approaches a line segment \(L\) as \(\zeta\to 0\). The set of the eigenvalues \(\{\mu(\zeta)\}^ \infty_{m=1}\) of Laplacian (with Neumann boundary conditions) is expressed as follows, \[ \{\mu_ k(\zeta)\}^ \infty_{k=1}=\{\omega_ k(\zeta)\}^ \infty_{k=1}\cup\{\lambda_ k (\zeta)\}^ \infty_{k=1},\;\lim_{\zeta\to 0}\omega_ k(\zeta)=\omega_ k,\;\lim_{\zeta\to 0}\lambda_ k(\zeta)=\lambda_ k,\;(k\geq 1), \] where \(\omega_ k\) is the \(k\)th eigenvalue of \(-\Delta\) in \(D_ 1\cup D_ 2\) (Neumann boundary conditions) and \(\lambda_ k\) is the \(k\)th eigenvalue of \(-d^ 2/dz^ 2\) in \(L\) with Dirichlet boundary conditions. The purpose of this paper is to give some elaborate characterization of \(\omega_ k(\zeta)\) when \(\zeta\to 0\). Precisely speaking the value of limit \(\lim_{\zeta\to 0}{\omega_ k(\zeta)-\omega_ k\over\zeta^{n- 1}}\) is given for any \(k\geq 1\). For the study of the asymptotic behavior of the eigenfunctions of the above eigenvalue problem [see the author, J. Differ. Equations 77, No. 2, 322-350 (1989; Zbl 0703.35138)].
scientific article; zbMATH DE number 469066