Geometric & Functional Analysis GAFA

, Volume 16, Issue 6, pp 1201–1245

The effect of curvature on the best constant in the Hardy–Sobolev inequalities

Authors

    • Department of MathematicsUniversity of British Columbia
  • F. Robert
    • Laboratoire J.A. DieudonnéUniversité de Nice Sophia-Antipolis
Article

DOI: 10.1007/s00039-006-0579-2

Cite this article as:
Ghoussoub, N. & Robert, F. GAFA, Geom. funct. anal. (2006) 16: 1201. doi:10.1007/s00039-006-0579-2

Abstract.

We address the question of attainability of the best constant in the following Hardy–Sobolev inequality on a smooth domain Ω of \(\mathbb{R}^n\):
$$ \mu_s(\Omega):=\text{inf}\left\{\int_\Omega |\nabla u|^2 dx; u\in H_{1,0}^2 (\Omega)\;\text{and}\;\int_{\Omega}\frac{|u|^{2^{\star}}}{|x|^s}dx=1\right\}$$
when \(0 < s < 2,\; 2^{\star}:= 2^{\ast}(s) = \frac{2(n-s)}{n-2},\) and when 0 is on the boundary ∂Ω. This question is closely related to the geometry of ∂Ω, as we extend here the main result obtained in [GhK] by proving that at least in dimension n  ≥  4, the negativity of the mean curvature of ∂Ω at 0 is sufficient to ensure the attainability of μs(Ω). Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions corresponding to the best constant in half-space, as well as a fine analysis of the asymptotic behaviour of appropriate minimizing sequences. The result holds true also in dimension 3 but the more involved proof will be dealt with in a forthcoming paper [GhR2].

Keywords and phrases.

Asymptotic behaviorSobolev inequalities

AMS Mathematics Subject Classification.

35B4035J60

Copyright information

© Birkhäuser Verlag, Basel 2006