Some isoperimetric inequalities on ${\mathbb{R}}^N$ with respect to weights $|x{|}^{\alpha }$

Abstract

We solve a class of isoperimetric problems on ${\mathbb{R}}^N$ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in ${\mathbb{R}}^N$ with fixed Lebesgue measure, ${\int }_{\partial \Omega }|x{|}^k{\mathcal{H}}_{N-1}(dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Szëgo principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.