Cauchy problem for effectively hyperbolic operators with triple characteristics

Bernardi, Enrico; Bove, Antonio; Petkov, Vesselin

doi:10.1016/j.crma.2013.10.009

Partial differential equations

Cauchy problem for effectively hyperbolic operators with triple characteristics
[Problème de Cauchy pour des opérateurs effectivement hyperboliques ayant des caractéristiques triples]

Présenté par : Jean-Michel Bony

Bernardi, Enrico ¹ ; Bove, Antonio ² ; Petkov, Vesselin ³

¹ Dipartimento di Scienze Statistiche, Università di Bologna, Viale Filopanti 5, 40126 Bologna, Italy
² Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy
³ Université Bordeaux I, Institut de Mathématiques, 351, cours de la Libération, 33405 Talence, France

Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 109-112.

Résumé
Abstract

On étudie une classe dʼopérateurs effectivement hyperboliques P dans $G = {(t, x) : 0 ⩽ t ⩽ T, x \in U ⋐ R^{n}}$ ayant des caractéristiques triples pour $ρ = (0, x_{0}, ξ), ξ \in R^{n} ∖ {0}$ . V. Ivrii a introduit la conjecture selon laquelle chaque opérateur effectivement hyperbolique est fortement hyperbolique, cʼest-à-dire telle que le problème de Cauchy pour $P + Q$ soit localement bien posé pour tout opérateur Q dʼordre inférieur à celui de P. Pour des opérateurs ayant des caractéristiques triples, cette conjecture a été démontrée [3] pour le cas où le symbole principal de P admet une factorisation comme produit de deux symboles du type principal. Un opérateur fortement hyperbolique pourrait avoir des caractéristiques triples seulement pour $t = 0$ ou pour $t = T$ . Les opérateurs que nous examinons ont en général un symbole principal qui nʼest pas factorisable, et nous prouvons quʼils sont fortement hyperboliques si T est suffisamment petit.

We study a class of third-order effectively hyperbolic operators P in $G = {(t, x) : 0 ⩽ t ⩽ T, x \in U ⋐ R^{n}}$ with triple characteristics at $ρ = (0, x_{0}, ξ), ξ \in R^{n} ∖ {0}$ . V. Ivrii introduced the conjecture that every effectively hyperbolic operator is strongly hyperbolic, that is the Cauchy problem for $P + Q$ is locally well posed for any lower-order terms Q. For operators with triple characteristics, this conjecture was established [3] in the case when the principal symbol of P admits a factorization as a product of two symbols of principal type. A strongly hyperbolic operator in G could have triple characteristics in G only for $t = 0$ or for $t = T$ . The operators that we investigate have a principal symbol which in general is not factorizable and we prove that these operators are strongly hyperbolic if T is small enough.

Reçu le : 2013-08-22
Accepté le : 2013-10-11
Publié le : 2013-12-30

DOI : 10.1016/j.crma.2013.10.009

Affiliations des auteurs :

Bernardi, Enrico ¹ ; Bove, Antonio ² ; Petkov, Vesselin ³

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Bernardi, Enrico; Bove, Antonio; Petkov, Vesselin. Cauchy problem for effectively hyperbolic operators with triple characteristics. Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 109-112. doi : 10.1016/j.crma.2013.10.009. http://www.numdam.org/articles/10.1016/j.crma.2013.10.009/

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Cité par

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