A spectral asymptotics of an elliptic operator in a bounded domain
type: Item
instance of: scholarly article
title: A spectral asymptotics of an elliptic operator in a bounded domain
zbMATH Open document ID: 0593.47046
DOI: 10.1007/BF01078394
author: A. G. Kostyuchenko
author: K. Kh. Boimatov
publication date: 1985-01-01T00:00:00Z
review text: Let \(P_ 0\) be the symmetric uniformly elliptic operator of 2m-th order with its coefficients in \(C^{\infty}({\bar \Omega})\), (\(\Omega\) a bounded domain in \(R_ n)\), P the Friedrichs extension of \(P_ 0\). In the note the authors prove the asymptotic formula for N(\(\lambda)\) - the number of eigenvalues of the operator not exceeding prescribed value. Namely, for \(\lambda \to +\infty\), \(N(\lambda)=const \lambda^{n/2m}+O(1)\lambda^{(n-1)/2m}\int_{\Omega \lambda}dist^{- 1}(x\), \(\partial \Omega)dx+O(1)\lambda^{n/2m}mes\Omega '_{\lambda}\), \(\Omega_{\lambda}=\{x\in \Omega\); \(dist(x,\partial \Omega)>\lambda^{- 1/2m}\}\), \(\Omega '_{\lambda}=\Omega \setminus {\bar \Omega}_{\lambda}\).
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 3953643
zbMATH Keywords: spectral asymptotics
zbMATH Keywords: bounded domain
zbMATH Keywords: symmetric uniformly elliptic operator
zbMATH Keywords: Friedrichs extension
zbMATH Keywords: asymptotic formula
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf01078394
OpenAlex ID: W1965406655
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 3953643