Asymptotics of the eigenvalues of the Dirichlet-Laplace problem in a domain with thin tube excluded (Q2821870)

Let \(\Omega\subset\mathbb R^3\) be a Lipschitz domain and \(\gamma:\mathbb R\to\Omega\) be a \(\mathcal C^1\)-function that is \(L\)-periodic for some \(L>0\), such that \(0<\alpha_-\leq |\frac{d\gamma}{dz}(z)|\leq \alpha_+,\forall z\in\mathbb R,\) for fixed constants \(\alpha\pm\). Then \(\Gamma=\gamma(\mathbb R)\) is a Lipschitz curve. The author considers a family of Lipschitz domains \((\Xi_{\delta})_{\delta>0}\), such that NEWLINE\[NEWLINE\underset{x\in\bar{\Xi}_{\delta}}{\sup}d(x,\Gamma)\leq C \deltaNEWLINE\]NEWLINE where \(C\) is a positive constant independent of \(\delta\). He sets \(\Omega_{\delta}=\Omega\setminus \bar{\Xi}_{\delta}\). He considers the operators \(A_{\delta} :H^1_{o}(\Omega_{\delta})\to H^{-1}(\Omega_{\delta}), \delta>0\) [resp. \(A_{o}:H^1_{o}(\Omega) \to H^{-1}(\Omega)\)] defined by NEWLINE\[NEWLINE \langle A_{\delta} (u),v\rangle =(\nabla u,\nabla v)_{\Omega},\quad \forall u,v\in H^1_{o}(\Omega_{\delta} )NEWLINE\]NEWLINE [resp. NEWLINE\[NEWLINE\langle A_{o} (u),v\rangle =(\nabla u,\nabla v)_{\Omega},\quad \forall u,v\in H^1_{o}(\Omega )].NEWLINE\]NEWLINE He introduces suitable weighted Sobolev spaces \(V^1_{\beta,\delta}(\omega),\) whose norms depend on \(\delta\), where \(\beta\in\mathbb R\), and \(\omega\) is a Lipschitz domain. He shows the existence of \(\beta^\ast>0\) such that, for each \(\beta\) satisfying \(|\beta|<\beta^\ast\) there exist constants \(c_{\beta},\delta_{o}>0,\) independent of \(\delta,\) such that NEWLINE\[NEWLINE\underset{f\in L^2(\Omega\setminus \{0\}}{)}{\sup} \frac{||A_{o}^{-1}(f)-A_{\delta}^{-1}(f)||_{V_{\beta,\delta}^1(\Omega)}}{||f||_{L^2(\Omega)}} \leq \frac{c_{\beta}}{|\log \delta|^{\beta}}\quad \forall \delta\in (0,\delta_{o})\eqno{(1)}NEWLINE\]NEWLINE (Proposition 5.1). He obtains a sharper result when, in (1), the \(V_{\beta,\delta}^1\)-norm is replaced by the \( L^2\)-norm. He deduces that the spectra of \(A_{\delta}^{-1}\) and \(A_{o}^{-1}\) are closed to each other, as \(\delta\to 0\) (Proposition 5.2).