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|0\), \(R_ \varepsilon=\{(\varepsilon x,\varepsilon^ \eta y): (x,y)\in R\}\) and \(D_ \varepsilon=\{\varepsilon x,\varepsilon y):(x,y)\in D\}\), \(\Omega_ \varepsilon=\Omega_ 0\cup D_ \varepsilon\cup R_ \varepsilon\), \(S_ \gamma=\{(x,y)\in\mathbb{R}\times\mathbb{R}^{N-1}: x^ 2+| y|^ 2\leq\gamma^ 2\}\cap\overline{\Omega_ 0}\). For \(0\leq\varepsilon(N+1)/(N-1)\) 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.