Arrieta, José M.; Hale, Jack K.; Han, Qing Eigenvalue problems for nonsmoothly perturbed domains. (English) Zbl 0736.35073 J. Differ. Equations 91, No. 1, 24-52 (1991). Let \(\Omega_ 0\), \(D\subset\mathbb{R}^ N\) be bounded, connected and smooth domains and \(R\) a connected set such that there exist \(\alpha,\beta\) satisfying \[ \{(x,y)\in\mathbb{R}\times\mathbb{R}^{N-1}: | x|<\alpha, | y|<\beta\}\cap\Omega_ 0=\{(x,y)\in\mathbb{R}\times\mathbb{R}^{N-1}:- \alpha<x<0,\;| y|<\beta\} \] and \[ \{(x,y)\in\mathbb{R}\times\mathbb{R}^{N-1}: 0<x<2\alpha, | y|<\beta\}\cap D=\{(x,y)\in\mathbb{R}\times\mathbb{R}^{N- 1}:\alpha<x<2\alpha,| y|<\beta\}, \]\[ \Omega_ 0\cap D=\emptyset,\quad R\subset\{(x,y)\in\mathbb{R}\times\mathbb{R}^{N-1}: 0\leq x\leq\alpha, | y|\leq\beta\}, \] \(\Omega_ 0\cup D\cup R\) is a bounded, connected and smooth domain. Let \(\varepsilon,\eta>0\), \(R_ \varepsilon=\{(\varepsilon x,\varepsilon^ \eta y):\allowbreak (x,y)\in R\}\) and \(D_ \varepsilon=\{\varepsilon x,\varepsilon y):(x,y)\in D\}\), \(\Omega_ \varepsilon=\Omega_ 0\cup D_ \varepsilon\cup R_ \varepsilon\), \(S_ \gamma=\{(x,y)\in\mathbb{R}\times\mathbb{R}^{N-1}:\allowbreak x^ 2+| y|^ 2\leq\gamma^ 2\}\cap\overline{\Omega_ 0}\). For \(0\leq\varepsilon<\varepsilon_ 0\) let \((\lambda^ \varepsilon_ m)_{m\in\mathbb{N}}\) be the sequence of eigenvalues arranged in increasing order and with multiplicity and for \(0<\varepsilon<\varepsilon_ 0\) let \((\omega^ \varepsilon_ m)_{m\in\mathbb{N}}\) be a corresponding sequence of orthonormal eigenfunctions for the problem \[ -\Delta u=\lambda u\hbox { in } \Omega_ \varepsilon,\quad \partial_ nu=0 \hbox { on } \partial\Omega_ \varepsilon. \] If \(\eta>(N+1)/(N-1)\) then \(\lim_{\varepsilon\to0}\lambda^ \varepsilon_ 1=0,\;\lim_{\varepsilon\to0}\lambda^ \varepsilon_ m=\lambda^ 0_{m-1} \hbox { for } m\geq2;\) \[ \lim_{\varepsilon\to0}\omega^ \varepsilon_ 1=0 \hbox { in } H^ 1(\Omega_ 0),\;\lim_{\varepsilon\to0} |\omega^ \varepsilon_ 1|_{L^ 2(R_ \varepsilon)}=0,\;\lim_{\varepsilon\to0} |\omega^ \varepsilon_ 1|_{L^ 2(D_ \varepsilon)}=1,\;\lim_{\varepsilon\to0}\left((\int_{D_ \varepsilon}\omega^ \varepsilon_ 1dx)^ 2/| D_ \varepsilon|\right)=1; \] for any sequence of positive numbers \((\varepsilon_ k)_{k\in\mathbb{N}}\), with \(\varepsilon_ k\to0\), there exist a subsequence \((\delta_ k)_{k\in\mathbb{N}}\) and a complete system of orthogonal eigenfunctions \(\omega^ 0_ m)_{m\in\mathbb{N}}\) for the problem \[ -\Delta u=\lambda u\hbox{ in } \Omega_ 0, \quad\partial_ nu=0 \hbox { on } \partial\Omega_ 0, \] such that \(\omega_ m^{\delta_ k}\to\omega^ 0_{m-1}\) in \(H^ 1(\Omega_ 0)\), \(|\omega_ m^{\delta_ k}|_{H^ 1(D_{\delta_ k}\cup R_{\delta_ k})}\to0\) for \(m\geq2\); if \(\Omega_ 0\) is a \(C^ \infty\) domain, for any \(\ell\geq1\) and \(\gamma\in]0,\gamma_ 0[\) we have \(\lim_{\varepsilon\to0}\omega^ \varepsilon_ 1=0\) in \(H^ \ell(\Omega_ 0\backslash S_ \gamma)\), \(\omega_ m^{\delta_ k}\to\omega^ 0_{m-1}\) in \(H^ \ell(\Omega_ 0\backslash S_ \gamma)\) for \(m\geq2\). The same questions for the mixed boundary value problem and for the Neumann problem in domains with thin channels are also investigated. Reviewer: G.Bottaro (Genova) Cited in 22 Documents MSC: 35P05 General topics in linear spectral theory for PDEs 35R25 Ill-posed problems for PDEs Keywords:complete system of orthogonal eigenfunctions; mixed boundary value problem; Neumann problem; domains with thin channels PDFBibTeX XMLCite \textit{J. M. Arrieta} et al., J. Differ. Equations 91, No. 1, 24--52 (1991; Zbl 0736.35073) Full Text: DOI References: [1] Adams, R. A., Sobolev Spaces (1977), Academic Press: Academic Press New York · Zbl 0385.46024 [2] Babuska, I.; Vyborny, R., Continuous dependence of eigenvalues on the domains, Czechoslovak Math. J., 15, 169-178 (1965) · Zbl 0137.32302 [3] Beale, J. T., Scattering frequencies of resonators, Comm. Pure Appl. 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