Weighted isoperimetric inequalities in cones and applications

Details Weighted isoperimetric inequalities in cones and applications Weighted isoperimetric inequalities in cones and applications

2011-07-27

arxiv.org/abs/1107.5406v3

Mathematics

Analysis of PDEs

Scientific paper

This paper deals with weighted isoperimetric inequalities relative to cones of $\mathbb{R}^{N}$. We study the structure of measures that admit as an isoperimetric set the intersection of a cone with the ball centered at the vertex of the cone. For instance, in case that the cone is the half-space $\mathbb{R}_{+}^{N}=\{x \in \mathbb{R}^{N} : x_{N}>0\}$ and the measure is factorized, we prove that this phenomenon occurs if and only if the measure has the form $d\mu=ax_{N}^{k}\exp (c|x|^{2})dx $, for some $a>0$, $k,c\geq 0$. Our results are then used to obtain isoperimetric estimates for Neumann eigenvalues of a weighted Laplace-Beltrami operator on the sphere, sharp Hardy-type inequalities for functions defined in a quarter space and, finally, via symmetrization arguments, a comparison result for a class of degenerate PDE's.

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