The Effect of Curvature on the Best Constatnt in the Hardy-Sobolev Inequalities

Abstract

We address the question of attainability of the best constant in the following Hardy-Sobolev inequality on a smooth domain $\Omega$ of \mathbb{R}^n: $$ \mu_s (\Omega) := \inf \{\int_{\Omega}| \nabla u|^2 dx; u \in {H_{1,0}^2(\Omega)} \hbox{and} \int_{\Omega} \frac {|u|^{2^{\star}}}{|x|^s} dx =1\}$$ when 0<s<2, 2^*:=2^*(s)=\frac{2(n-s)}{n-2}, and when 0 is on the boundary $\partial \Omega$. This question is closely related to the geometry of $\partial\Omega$, as we extend here the main result obtained in [15] by proving that at least in dimension n >= 4, the negativity of the mean curvature of $\partial \Omega $ at 0 is sufficient to ensure the attainability of $\mu_{s}(\Omega)$. Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions correrresponding 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 [17].