Parabolic Harnack inequality for divergence form second order differential operators. (English) Zbl 0840.31006

Saloff-Coste, L.

Potential Anal. 4, No. 4, 429-467 (1995).

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 \(\psi\) sur \(B\).
On montre assez facilement \(PHP(R) \Rightarrow D(R), P(R)\), ainsi que la continuité hölderienne des solutions de \((*)\) et un encadrement gaussien de la solution fondamentale sur \(M\). La preuve de la réciproque est beaucoup plus difficile et utilise en particulier la technique itérative de Moser.

Reviewer: R.-M.Hervé (Paris)

MSC:

31B35	Connections of harmonic functions with differential equations in higher dimensions
47F05	General theory of partial differential operators
26D10	Inequalities involving derivatives and differential and integral operators

Keywords:

parabolic Harnack principle; Harnack inequality; Poincaré inequality; doubling property; Hölder continuity; fundamental solution; Moser’s iteration

Citations:

Zbl 0769.58054

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