Equivalence between regularity theorems and heat kernel estimates for higher-order elliptic operators and systems under divergence form
type: Item
instance of: scholarly article
title: Equivalence between regularity theorems and heat kernel estimates for higher-order elliptic operators and systems under divergence form
zbMATH Open document ID: 0979.35044
published in: Journal of Functional Analysis
publication date: 2002-02-13T00:00:00Z
review text: The authors study the heat kernel of higher-order elliptic operators under divergence form of the type \[ L=\sum_{|\alpha|\leq m,|\beta|\leq m}(-1)^{|\alpha|}D^\alpha(a_{\alpha\beta}D^\beta) \] whose principal part satisfies the Gårding inequality. It is shown that for homogeneous operators Gaussian upper bounds and Hölder regularity of the heat kernel is equivalent to local regularity of weak solutions. In addition, it is studied the stability of such bounds under \(L^\infty\)-perturbations of the coefficients or under perturbation with bounded coefficients lower-order terms. There are obtained heat kernel bounds for operators with uniformly continuous or \(VMO\) coefficients. Analogous questions are discussed also for elliptic, in sense of the Gårding inequality, systems.
Mathematics Subject Classification ID: 35J35
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1556911
zbMATH Keywords: Gårding inequality
zbMATH Keywords: local elliptic regularity
zbMATH Keywords: Morrey-Campanato spaces
zbMATH Keywords: Gaussian upper bounds
author: Pascal Auscher
author: Mahmoud Qafsaoui
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: jfan.2000.3643
OpenAlex ID: W1965709042
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DOI: 10.1006/JFAN.2000.3643
description: scientific article; zbMATH DE number 1556911