Neumann eigenvalue problems on exterior perturbations of the domain
The study of singular geometric perturbation of elliptic boundary problems in the sense that the domain \(\Omega_\varepsilon\) may be viewed as a \(C^0\)-perturbation of \(\Omega\) but not \(C^1\) has a long history. A significant example being given in the book by Courant and Hilbert and recent results may be seen in the papers by \textit{R. Hempel}, \textit{L. A. Seco} and \textit{B. Simon} [J. Funct. Anal. 102, No. 2, 448-483 (1991; Zbl 0741.35043)] and other authors. The author consolidates these results in a new and fairly general framework. In his sense introduce a pair \(\{(\Omega, \Omega_\varepsilon)\}\) where \(\Omega \subset \Omega_\varepsilon \subset \mathbb{R}^d\), bounded and smooth and \(m_d(\Omega_\varepsilon \backslash\Omega) @>\varepsilon \downarrow 0>> 0\). Define \(R_\varepsilon = \Omega_\varepsilon \backslash \overline \Omega\) and \(\Gamma_\varepsilon = \partial \Omega \cap \partial R_\varepsilon\). Introduce the operator \(-\Delta^{R_\varepsilon}_{D (\Gamma_\varepsilon) N}\) as being the Laplace operator with a Dirichlet boundary condition on \(\Gamma_\varepsilon\) and a Neumann condition on \(\partial R_\varepsilon \backslash \Gamma_\varepsilon\). Suppose that \(V_\varepsilon \in L^\infty (\Omega_\varepsilon)\) and \(|V_\varepsilon |_{L^\infty (\Omega_\varepsilon)} \leq C\), \(C\) independent of \(\varepsilon\). Then introduce the operators \[ A_N^{\Omega_\varepsilon} = - \Delta_N^{\Omega_\varepsilon} + V_\varepsilon, \quad A^\Omega_N = - \Delta^\Omega_N + V_\varepsilon, \quad A_{D (\Gamma_\varepsilon) N} = - \Delta_{D (\Gamma_\varepsilon) N} + V_\varepsilon, \] with a natural notation and eigenvalues \(\lambda^\varepsilon_n (\Omega_\varepsilon)\) \(\Gamma (\Omega, \varepsilon)\), \(\tau_n (R_\varepsilon)\), respectively. The following implicit geometric condition is introduced, namely Hypothesis H. If \(u_\varepsilon \in H^1(\Omega_\varepsilon)\) with \(|u_\varepsilon |_{H^1 (\Omega_\varepsilon)} \leq R\), \(R\) independent of \(\varepsilon\), then there exists a \(\overline u_\varepsilon \in H^1_{\Gamma_\varepsilon} (R_\varepsilon)\) satisfying \[ |u_\varepsilon - \overline u_\varepsilon |_{L^2 (R_\varepsilon)} @>\varepsilon \downarrow 0>> 0, \quad \text{and} \quad |\nabla \overline u_\varepsilon |_{L^2 (R_\varepsilon)} \leq |\nabla u_\varepsilon |_{L^2 (\Omega_\varepsilon)} + o(1). \] For a given \(\varepsilon_0 > 0\) and a subset \(I_0 \subset (0,\varepsilon_0)\) such that 0 is an accumulation point of \(I_0\) one says that \(x_0 \in\mathbb{R}^+\) divides the spectrum if there exist \(\delta,M,N >0\) such that \[ [x_\varepsilon-\delta, x_\varepsilon + \delta] \cap \{\lambda^\varepsilon_n \}_1^\infty = \emptyset, \quad x_\varepsilon \leq M,\;\varepsilon \in I_0 \] \[ N(x_\varepsilon) = \# \{\lambda^\varepsilon_i: \lambda^\varepsilon_i \leq x_\varepsilon\} \leq N< \infty. \] Let then \(P_{x_\varepsilon}\) be the projection onto \([\varphi^\varepsilon_1, \dots, \varphi^\varepsilon_{N (x_\varepsilon)}]\). Then the principal result of the paper states that if hypothesis \(H\) is satisfied, \(\sigma(A_N^{\Omega_\varepsilon})\) arises from \(\sigma (A_N^\Omega)\) and \(\sigma (A_N^{R_\varepsilon})\) in the sense that: \[ \lim_{\varepsilon \downarrow 0} (\lambda_{n, \Omega_\varepsilon} - \lambda^\varepsilon_n) = 0, \quad |\varphi_{r_\varepsilon, \Omega_\varepsilon} - P_{x_\varepsilon} \varphi_{r_\varepsilon, \Omega_\varepsilon} |_{H^1 (\Omega_\varepsilon \cup R_\varepsilon)} @>\varepsilon \downarrow 0>> 0, \] \(n\in \mathbb{Z}^+\), \(r_\varepsilon = 1, \dots,N(x_\varepsilon)\), for any \(x_\varepsilon\) dividing the spectrum. The proof of this result involves simple but detailed (variational) estimates. A considerable amount of work however is involved in establishing geometrical and variational conditions under which \(H\) holds (conditions \(D_1\) and \(D_2\) of Theorem 2.3 of the paper).
scientific article; zbMATH DE number 759461
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