On the continuity of the second Sobolev best constant

Ezequiel R. Barbosa; Marcos Montenegro

doi:10.1016/j.crma.2007.10.026

Differential Geometry

On the continuity of the second Sobolev best constant

Presented by: Thierry Aubin

Ezequiel R. Barbosa ¹; Marcos Montenegro ¹

¹ Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil

Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 579-582.

Abstract
Résumé

In this Note we prove that the second Riemannian $L^{p}$ -Sobolev best constant $B_{0} (p, g)$ depends continuously on g in the $C^{0}$ -topology when $1 < p < 2$ . The situation changes significantly in the case $p = 2$ . In particular, we prove that $B_{0} (2, g)$ is continuous on g in the $C^{2}$ -topology and is not in the $C^{1, β}$ -topology.

Dans cette Note nous prouvons que la deuxième meilleure constante dans l'inégalité de $L^{p}$ -Sobolev Riemannienne $B_{0} (p, g)$ dépend continûment de g dans la topologie $C^{0}$ quand $1 < p < 2$ . La situation change radicalement lorsque $p = 2$ . En particulier, nous montrons que $B_{0} (2, g)$ est continu en g dans le $C^{2}$ -topologie et ne l'est pas dans le $C^{1, β}$ -topologie.

Received: 2007-07-11
Accepted: 2007-10-12
Published online: 2007-11-07

DOI: 10.1016/j.crma.2007.10.026

Author's affiliations:

Ezequiel R. Barbosa ¹; Marcos Montenegro ¹

¹ Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil

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     title = {On the continuity of the second {Sobolev} best constant},
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     number = {10},
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     language = {en},
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Ezequiel R. Barbosa; Marcos Montenegro. On the continuity of the second Sobolev best constant. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 579-582. doi : 10.1016/j.crma.2007.10.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.026/

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