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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds


Author: Emmanuel Hebey
Journal: Trans. Amer. Math. Soc. 354 (2002), 1193-1213
MSC (2000): Primary 58E35
DOI: https://doi.org/10.1090/S0002-9947-01-02913-0
Published electronically: November 1, 2001
MathSciNet review: 1867378
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Abstract: Given $(M,g)$ a smooth compact Riemannian $n$-manifold, $n \ge 3$, we return in this article to the study of the sharp Sobolev-Poincaré type inequality

\begin{displaymath}\Vert u\Vert_{2^\star}^2 \le K_n^2\Vert\nabla u\Vert_2^2 + B\Vert u\Vert_1^2\tag*{(0.1)}\end{displaymath}

where $2^\star = 2n/(n-2)$ is the critical Sobolev exponent, and $K_n$ is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that $(0.1)$ is true if $n = 3$, that $(0.1)$is true if $n \ge 4$ and the sectional curvature of $g$ is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension $n$ is true and the sectional curvature of $g$ is nonpositive, but that $(0.1)$ is false if $n \ge 4$ and the scalar curvature of $g$ is positive somewhere. When $(0.1)$ is true, we define $B(g)$ as the smallest $B$ in $(0.1)$. The saturated form of $(0.1)$ reads as

\begin{displaymath}\Vert u\Vert_{2^\star}^2 \le K_n^2\Vert\nabla u\Vert_2^2+B(g)\Vert u\Vert_1^2. \tag*{(0.2)}\end{displaymath}

We assume in this article that $n \ge 4$, and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality $(0.1)$. We prove that $(0.1)$ is true, and that $(0.2)$ possesses extremal functions when the scalar curvature of $g$ is negative. A fairly complete answer to the question of the validity of $(0.1)$ under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given.


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Additional Information

Emmanuel Hebey
Affiliation: Département de Mathématiques, Site de Saint-Martin, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
Email: Emmanuel.Hebey@math.u-cergy.fr

DOI: https://doi.org/10.1090/S0002-9947-01-02913-0
Received by editor(s): November 18, 2000
Published electronically: November 1, 2001
Article copyright: © Copyright 2001 American Mathematical Society