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

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

2005-03-01

arxiv.org/abs/math/0503025v1

Mathematics

Analysis of PDEs

39 pages

Scientific paper

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= 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].

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