On the geometric dependence of Riemannian Sobolev best constants

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November 2009, 8(6): 1759-1777. doi: 10.3934/cpaa.2009.8.1759

On the geometric dependence of Riemannian Sobolev best constants

Ezequiel R. Barbosa ^1, and Marcos Montenegro ^1,

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

Received September 2008 Revised February 2009 Published August 2009

Abstract
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We concerns here with the continuity on the geometry of the second Riemannian $L^p$-Sobolev best constant $B_0(p,g)$ associated to the AB program. Precisely, for $1 \leq p \leq 2$, we prove that $B_0(p,g)$ depends continuously on $g$ in the $C^2$-topology. Moreover, this topology is sharp for $p = 2$. From this discussion, we deduce some existence and $C^0$-compactness results on extremal functions.

Keywords: extremal functions, uniformity problem., sharp Sobolev inequalities, compactness.

Mathematics Subject Classification: 32Q10, 53C2.

Citation: Ezequiel R. Barbosa, Marcos Montenegro. On the geometric dependence of Riemannian Sobolev best constants. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1759-1777. doi: 10.3934/cpaa.2009.8.1759

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