Geometric & Functional Analysis GAFA

, Volume 16, Issue 6, pp 1201–1245

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

Article

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 behavior Sobolev inequalities 

AMS Mathematics Subject Classification.

35B40 35J60 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaCanada
  2. 2.Laboratoire J.A. DieudonnéUniversité de Nice Sophia-AntipolisNice cedex 2France

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