, Volume 16, Issue 6, pp 1201-1245
Date: 21 Sep 2006

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

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 μ s (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].

N.G.’s research partially supported by the Natural Sciences and Engineering Research Council of Canada. The first named author gratefully acknowledges the hospitality and support of the Université de Nice where this work was initiated. F.R. gratefully acknowledges the hospitality and support of the University of British Columbia where this work was completed.
Received: February 2005; Accepted: May 2005