Asymptotic behavior of spectral functions of elliptic operators with Hölder continuous coefficients
type: Item
instance of: scholarly article
title: Asymptotic behavior of spectral functions of elliptic operators with Hölder continuous coefficients
zbMATH Open document ID: 0901.35064
author: Yoichi Miyazaki
published in: Journal of the Mathematical Society of Japan
publication date: 1998-09-21T00:00:00Z
review text: In previous works the author improved the remainder estimate in the asymptotic formula for the counting function \(N(t)\) of a strongly elliptic operator \(A\) of order \(2m\) defined in a bounded domain \(\Omega\) of \(\mathbb{R}^n\), whose coefficients of top order are Hölder continuous of exponent \(\tau \in (0,\infty)\). The purpose of this paper is to obtain a similar remainder estimate for the spectral function \(e(t,x,x)\) when \(2m >n\). Improving Tsujimoto's estimate \[ e(t,x,x) =\mu_A(x) t^{n/2m}+O\bigl( \delta(x)^{-\theta} t^{(n-\theta)/2m}) \quad \text{as }t \to\infty \] with any \(\theta\in (0,\tau/(\tau+2))\), the author proves that the above formula holds with \(\theta= \tau/(\tau+1)\) if we replace \(\delta(x)^{-\theta}\) by \(\delta (x)^{- \theta} +\log_+ (\delta (x)t^{1/2m})\). Here \(\mu_A\) is the function defined by the principal term of \(A\) and \(\delta(x) =\min \{\text{dist} (x,\partial \Omega), 1\}\). Moreover the author also shows that a better remainder estimate holds if we consider the heat kernel \(U(t,x,x)\). Namely, we get \[ U(t,x,x) =c_{n,m} \mu_A(x) t^{-n/2m} +O\bigl(t^{(\tau-n)/2m} +\delta (x)^{-1} t^{(1- n)/2m} \bigr) \] at \(t\to +0\) when \(0< \tau\leq 1\).
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 1172207
zbMATH Keywords: remainder estimate
zbMATH Keywords: heat kernel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04930539
OpenAlex ID: W2078885589
DOI: 10.2969/JMSJ/04930539
description: scientific article; zbMATH DE number 1172207