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: $${\mu }_s(\Omega ):=\inf \{{\int }_{\Omega }|\nabla u{|}^{2}dx;u\in H_{1,0}^2(\Omega )and{\int }_{\Omega }\frac{|u{|}^{2^{\star }}}{|x{|}^s}dx=1\}$$ 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].

39 pages

References in corpus (1)

• Concentration Estimates for Emden-Fowler Equations with Boundary Singularities and Critical Growth