On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles
type: Item
instance of: scholarly article
title: On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles
zbMATH Open document ID: 0541.35019
DOI: 10.1007/BF01206016
author: Shin Ozawa
published in: Communications in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: We consider a bounded domain \(\Omega\) in \({\mathbb{R}}^ 3\) with smooth boundary \(\Gamma\). We put \(B(\epsilon;w)=\{x\in {\mathbb{R}}^ 3\quad:\quad | x-w|<\epsilon \}.\) Let \(0<\mu_ 1(\epsilon;w(m))\leq \mu_ 2(\epsilon;w(m))\leq...\) be the eigenvalues of \(-\Delta(=-div \text{grad})\) in \(\Omega_{\epsilon,w(m)}=\Omega \backslash \overline{\cup^{m}_{i=1}B(\epsilon;w_ i^{(m)})}\) under the Dirichlet condition on its boundary. Here w(m) denotes the set of m points \(\{w_ i^{(m)}\}^ m_{i=1}\in \Omega^ m\). Let V(x)\(\geq 0\) be a \(C^ 1\)-class function on \({\bar \Omega}\) satisfying \(\int_{\Omega}V(x)\quad dx=1.\) We consider \(\Omega\) as the probability space with the probability density V(x) dx. Let \(\Omega^ m=\prod^{m}_{i=1}\Omega\) be the probability space with the product measure. We prove the Theorem: Fix \(\alpha >0\) and k. Then \[ \lim_{m\to \infty}P(w(m)\in \Omega^ m;\quad m^{\delta}| \mu_ k(\alpha /m;w(m))-\mu^ V_ k|<\epsilon)=1 \] holds for any \(\epsilon>0\) and \(\delta \in [0,1/4)\). Here \(\mu^ V_ k\) denotes the kth eigenvalue of \(-\Delta +4\pi \alpha V(x)\) in \(\Omega\) under the Dirichlet condition on \(\Gamma\). [See also the author's paper in Proc. Jap. Acad., Ser. A 60, 43-45 (1984)].
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 3859587
zbMATH Keywords: randomly distributed small obstacles
zbMATH Keywords: Laplacian
zbMATH Keywords: Kac theorem
zbMATH Keywords: singular variation of domains
zbMATH Keywords: random media
zbMATH Keywords: Dirichlet condition
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3859587