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Perturbation formula of eigenvalues in singularly perturbed domain. (English) Zbl 0785.35069
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)].
Reviewer: S.Jimbo (Tsushima)

##### MSC:
 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35B25 Singular perturbations in context of PDEs
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