Mathematische Zeitschrift

, Volume 237, Issue 4, pp 669–695 | Cite as

A Poincaré-Sobolev type inequality on compact Riemannian manifolds with boundary

  • David Holcman
  • Emmanuel Humbert
Original article


In this paper we study a Poincaré-Sobolev type inequality on compact Riemannian n-manifolds with boundary where the exponent growth is critical. Two constants have to be determined. We show that, contrary to the classical Sobolev inequality, the first best constant in this inequality does not depend on the dimension only, but depends on the geometry. It can be represented as the minimum \(\mu\) of a given energy functional. We study the nonlinear PDE associated to this functional which involves the geometry of the boundary. For a star-shaped domain D in \({\mathbb R}^n\) whose boundary has positive Ricci curvature, we give explicitly two Sobolev constants corresponding to the embedding \(H^1(D)\) in \(L^{\frac{2(n-1)}{n-2}}(\partial D)\). This result is used to obtain an explicit geometrical lower bound for \(\mu\).


Riemannian Manifold Type Inequality Sobolev Inequality Ricci Curvature Compact Riemannian Manifold 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • David Holcman
    • 1
  • Emmanuel Humbert
    • 2
  1. 1.Scuola Normale Superiore di Pisa, 7, Piazza dei Cavalieri, 56126 Pisa, Italy IT
  2. 2.Université de Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France (e-mail: FR

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