Neumann eigenvalue problems on exterior perturbations of the domain
type: Item
instance of: scholarly article
title: Neumann eigenvalue problems on exterior perturbations of the domain
zbMATH Open document ID: 0860.35086
published in: Journal of Differential Equations
publication date: 1997-04-21T00:00:00Z
review text: 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 \(\varepsilon\)…
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 759461
zbMATH Keywords: semiclassical approximations
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: magnetic field
zbMATH Keywords: singular geometric perturbation
author: Jose M. Arrieta
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1067
OpenAlex ID: W2017279692
DOI: 10.1006/JDEQ.1995.1067
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description: scientific article; zbMATH DE number 759461