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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

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