Advances in Differential Equations

Asymptotics for sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds

Olivier Druet and Emmanuel Hebey

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Given $(M,g)$ a smooth compact Riemannian manifold of dimension $n \ge 3$, there exist $A, B > 0$ such that for any $u \in H_1^2(M)$, $$\Vert u\Vert_{2^\star}^2 \le A\Vert\nabla u\Vert_2^2 + B\Vert u\Vert_1^2 , $$ where $H_1^2(M)$ is the standard Sobolev space consisting of functions in $L^2(M)$ whose gradient is also in $L^2(M)$. The best possible $A$ in this inequality is $K_n^2$, where $K_n$ is the sharp constant $K$ in the Euclidean Sobolev inequality $\Vert u\Vert_{2^\star} \le K\Vert\nabla u\Vert_2$. Thanks to previous work by Druet-Hebey-Vaugon and Hebey, it turns out that the above inequality with $A = K_n^2$ is always true when $n = 3$, in other words without any assumption on the manifold, and true when $n = 4$ if the scalar curvature is everywhere negative, or the scalar curvature is nonpositive and the manifold is conformally flat, or the sectional curvature is nonpositive and a local isoperimetric inequality as in the Cartan-Hadamard conjecture holds. On the contrary, thanks to previous works by Druet-Hebey-Vaugon, the inequality with $A = K_n^2$ is false when $n \ge 4$ and the scalar curvature is positive somewhere. Independent considerations give that for any $\varepsilon > 0$ there exists $B_\varepsilon$ such that for any $u\in H_1^2(M)$, $$\Vert u\Vert_{2^\star}^2 \le (K_n^2+\varepsilon)\Vert\nabla u\Vert_2^2 + B_\varepsilon\Vert u\Vert_1^2 . $$ Defining $B_\varepsilon(g)$ as the smallest $B_\varepsilon$ in this inequality, the difficult question we are concerned with in this article is to describe the asymptotic behavior of $B_\varepsilon(g)$ as $\varepsilon\to 0$. A complete answer to this question is given.

Article information

Adv. Differential Equations, Volume 7, Number 12 (2002), 1409-1478.

First available in Project Euclid: 27 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J37: Perturbations; asymptotics
Secondary: 35B38: Critical points 35B40: Asymptotic behavior of solutions 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]


Druet, Olivier; Hebey, Emmanuel. Asymptotics for sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds. Adv. Differential Equations 7 (2002), no. 12, 1409--1478.

Export citation