The Effect of Curvature on the Best Constatnt in the Hardy-Sobolev Inequalities
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: $$ μ_s (Ω) := \inf \{\int_Ω| \nabla u|^2 dx; u \in {H_{1,0}^2(Ω)} \hbox{and} \int_Ω \frac {|u|^{2^{\star}}}{|x|^s} dx =1\}$$ when 0 = 4, the negativity of the mean curvature of $\partial Ω$ 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 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].