The sector of analyticity of nonsymmetric submarkovian semigroups generated by elliptic operators

Chill, Ralph; Fašangová, Eva; Metafune, Giorgio; Pallara, Diego

doi:10.1016/j.crma.2006.04.003

Partial Differential Equations

The sector of analyticity of nonsymmetric submarkovian semigroups generated by elliptic operators
[Le secteur d'analyticité de semi-groupes sous-markoviens non-symétriques engendrés par des opérateurs elliptiques]

Présenté par : Paul Malliavin

Chill, Ralph ¹ ; Fašangová, Eva ² ; Metafune, Giorgio ³ ; Pallara, Diego ³

¹ Université Paul-Verlaine – Metz, LMAM et CNRS, UMR 7122, bâtiment A, île du Saulcy, 57045 Metz cedex 1, France
² Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
³ Dipartimento di Matematica “Ennio De Giorgi”, P.O.B. 193, 73100 Lecce, Italy

Comptes Rendus. Mathématique, Tome 342 (2006) no. 12, pp. 909-914.

Résumé
Abstract

Nous prouvons qu'une borne inférieure de l'angle $θ_{p}$ du secteur d'analyticité de semi-groupes sous-markoviens non nécessairement symétriques qui sont engendrés par des opérateurs elliptiques sous forme divergencielle ou par des opérateurs de Ornstein–Uhlenbeck dans $L_{μ}^{p}$ est donnée par la formule $\cot θ_{p} = \sqrt{{(p - 2)}^{2} + p^{2} {(\cot θ_{2})}^{2}} / (2 \sqrt{p - 1})$ . Si le semi-groupe est symétrique on retrouve alors des résultats connus. En général, cette borne inférieure est optimale.

We prove that a lower bound for the angle $θ_{p}$ of the sector of analyticity of not necessarily symmetric submarkovian semigroups generated by second order elliptic operators in divergence form or by Ornstein–Uhlenbeck in $L_{μ}^{p}$ is given by $\cot θ_{p} = \sqrt{{(p - 2)}^{2} + p^{2} {(\cot θ_{2})}^{2}} / (2 \sqrt{p - 1})$ . If the semigroup is symmetric then we recover known results. In general, this lower bound is optimal.

Reçu le : 2006-01-08
Accepté le : 2006-03-30
Publié le : 2006-05-15

DOI : 10.1016/j.crma.2006.04.003

Affiliations des auteurs :

Chill, Ralph ¹ ; Fašangová, Eva ² ; Metafune, Giorgio ³ ; Pallara, Diego ³

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Chill, Ralph; Fašangová, Eva; Metafune, Giorgio; Pallara, Diego. The sector of analyticity of nonsymmetric submarkovian semigroups generated by elliptic operators. Comptes Rendus. Mathématique, Tome 342 (2006) no. 12, pp. 909-914. doi : 10.1016/j.crma.2006.04.003. http://www.numdam.org/articles/10.1016/j.crma.2006.04.003/

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