The eigenvalue distribution of elliptic operators with Hölder continuous coefficients
type: Item
instance of: scholarly article
title: The eigenvalue distribution of elliptic operators with Hölder continuous coefficients
zbMATH Open document ID: 0754.35101
published in: Osaka Journal of Mathematics
publication date: 1993-01-16T00:00:00Z
review text: Improvements are studied for the estimates of the remainder term in the asymptotic formula for the eigenvalue distribution of an elliptic operator \(A\) of order \(2m\) whose principal part has Hölder continuous coefficients with exponent \(\tau\), \(0<\tau\leq 1\). In some cases, mainly when \(n=1\), a better estimate may be obtained for the remainder term of the counting function \(N(T)\): \[ N(t)=\mu_ A(\Omega)t^{n/2m}+O(t^{(n-\tau)/2m})\quad \text{as} \quad t\to\infty. \] Conditions are determined under which this estimate is valid when the following asymptotic formula for the partition function (the trace of the heat kernel) holds: \[ U(t)=\Gamma({n \over 2m}+1)\mu_ A(\Omega)t^{-n/2m}+O(t^{(\tau-n)/2m}) \quad\text{as}\quad t\to 0, \] where \(\Gamma(x)\) is the Gamma function.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 91631
zbMATH Keywords: estimates of the remainder term
zbMATH Keywords: asymptotic formula for the eigenvalue distribution
zbMATH Keywords: trace of the heat kernel
author: Yoichi Miyazaki
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 91631