Harnack's inequality for parabolic operators with singular low order terms
type: Item
instance of: scholarly article
title: Harnack's inequality for parabolic operators with singular low order terms
zbMATH Open document ID: 0808.35044
DOI: 10.1007/BF02572341
published in: Mathematische Zeitschrift
publication date: 1995-03-05T00:00:00Z
full work available at URL: https://eudml.org/doc/174670
review text: We prove a uniform Harnack inequality for parabolic operators \(L= L_ 0+ b\cdot\nabla +c\) with time-dependent, singular low order terms \(b\) and \(c\). The unperturbed operator \(L_ 0\) may be a general time-dependent second order parabolic operator (either of divergence form \(L_ 0= {1\over 2}\nabla a\nabla+ \alpha\nabla\) or of non-divergence form \(L_ 0= {1\over 2}a \nabla^ 2+ \alpha\nabla\)) which satisfies a uniform Harnack inequality. The Harnack constant for the operator \(L\) depends only on the Harnack constant for the unperturbed operator \(L_ 0\) and the Kato norms of \(\langle b,a^{-1} b\rangle\) and \(c\) with respect to \(L_ 0\). Moreover, we prove that solutions of \(Lu=0\) are continuous (if \(c=0\) even Hölder continuous) and that boundary points are regular for \(L\) if and only if they are regular for \(L_ 0\). In particular, the sheaf of solutions defines a harmonic space.
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 703848
zbMATH Keywords: uniform Harnack inequality
zbMATH Keywords: time-dependent, singular low order terms
author: Karl-Theodor Sturm
MaRDI profile type: Publication
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OpenAlex ID: W1991130911
description: scientific article; zbMATH DE number 703848