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Electrostatic capacity and eigenvalues of the Laplacian. (English) Zbl 0531.35061
Consider a bounded domain \(\Omega\) in \({\mathbb{R}}^ 3\) with \(C^{\infty}\) boundary \(\gamma\) and \(\omega\) a fixed point in \(\Omega\). Let D be a bounded open neighbourhood of the origin with \(C^ 2\) boundary. Assume that \(R^ 3\backslash \bar D\) is connected. Put \(\epsilon D=\{x\in {\mathbb{R}}^ 3\), \(\epsilon^{-1}(x-\omega)\in D\}\) and \(\Omega_{\epsilon}=\Omega \backslash \overline{\epsilon.D}\). Let \(m_ j(\epsilon)\) (resp. \(m_ j)\) be the j-th eigenvalue of the Laplacian in \(\Omega_{\epsilon}\) (resp. \(\Omega)\) under the Dirichlet condition on \(\partial \Omega_{\epsilon}\) (resp. \(\gamma)\). The author proves the following nice result: For fixed j suppose \(m_ j\) is simple. Then \[ m_ j(\epsilon)=m_ j-4\pi \epsilon.cap(D)\phi_ j(\omega)^ 2+0(\epsilon^{2-s}) \] as \(\epsilon\to 0\), s being any fixed positive constant. Here \(\phi_ j\) is the normalized eigenfunction associated with \(m_ j\) and cap(D) is the electrostatic capacity of D.
Reviewer: D.Robert

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31B15 Potentials and capacities, extremal length and related notions in higher dimensions