Parabolic Harnack inequality for divergence form second order differential operators
type: Item
instance of: scholarly article
title: Parabolic Harnack inequality for divergence form second order differential operators
zbMATH Open document ID: 0840.31006
DOI: 10.1007/BF01053457
author: Laurent Saloff-Coste
published in: Potential Analysis
publication date: 1996-07-04T00:00:00Z
review text: Cet article est une analyse approfondie des liens entre: (1) le principe de Harnack parabolique pour les opérateurs différentiels du deuxième ordre sous forme divergence; et (2) deux propriétés géométriques, à savoir inégalité de Poincaré et propriété de doublement. Il donne en particulier une preuve complète de l'équivalence entre (1) et (2), annoncée par l'Auteur [Int. Math. Res. Not. 1992, No. 2, 27-38 (1992; Zbl 0769.58054)], sous les hypothèses suivantes: \(M\) étant une variété \({\mathcal C}^\infty\) connexe, \(\lll\) un opérateur différentiel du deuxième ordre à coefficients \({\mathcal C}^\infty\), sans terme d'ordre 0, autoadjoint et positif sur \(L^2 (M, \mu)\), \(\mu\) une mesure \({\mathcal C}^\infty \) positive sur \(M\), on suppose que la ``distance'' \(\rho\) associée à \(\lll\) est finie sur \(M\), qu'elle est continue et induit la topologie donnée de \(M\), enfin que l'espace métrique \((M, \rho)\) est complet; on note \(B(x,r) = \{y \in M : \rho (x,y) < r\}\). Alors, \(R > 0\) étant fixé, il y a équivalence entre (1) \(\lll\) satisfait \([PHP(R)]\): Il existe \(C > 0\) tel que, \(\forall x \in M\), \( \forall s \in \mathbb{R}\), \(\forall r \in ]0,R[\), toute solution \(u \geq 0\) de \[ \left( {\partial \over \partial t} + \lll \right) u = 0 \quad \text{sur} \quad Q = ]s,\;s + r^2[ \times B(x,r)\tag \(*\) \] vérifie \(\sup_{Q_-} u \leq C \sup_{Q_+} u\), où \(Q_- = ]s + r^2/6\), \(s + r^2/3 [\times B (x,r/2)\) et \(Q_+ = ]s + 2r^2/3\), \(s + r^2 [\times B (x,r/2)\) (2) \(\lll\) satisfait \([D(R)]\): Il existe \(D > 0\) tel que, \(\forall x \in M\), \(\forall r \in ]0,R[\), on ait \(\mu [B(x,2r)] \leq D\) \(\mu [B(x,r)]\); \(\lll\) satisfait \([P(R)]\): Il existe \(P > 0\) tel que, \(\forall x \in M\), \(\forall r \in ]0,R[\), \(\forall \psi \in {\mathcal C}^\infty [B(x,2r)]\), on ait \[ \int_B |\psi - \psi_B |^2 d \mu \leq Pr^2 \int_{2B} |\nabla \psi |^2 d \mu \] en notant \(B = B(x,r)\), \(2B = B(x,2r)\), \(\psi_B = \mu\)-Valeur moyenne de \(\psi\…
Mathematics Subject Classification ID: 31B35
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 819164
zbMATH Keywords: parabolic Harnack principle
zbMATH Keywords: Harnack inequality
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: doubling property
zbMATH Keywords: Hölder continuity
zbMATH Keywords: fundamental solution
zbMATH Keywords: Moser's iteration
reviewed by: Rose-Marie Hervé
MaRDI profile type: Publication
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Parabolic Harnack inequality for divergence form second order differential operators (Wikidata QID)
Wikidata QID: Q126189853

Parabolic Harnack inequality for divergence form second order differential operators (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf01053457

Parabolic Harnack inequality for divergence form second order differential operators (OpenAlex ID)
OpenAlex ID: W4247653541