Weighted isoperimetric inequalities in cones and applications

Abstract

This paper deals with weighted isoperimetric inequalities relative to cones of ${\mathbb{R}}^N$. We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls 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 $d\mu =a{x}_N^k\exp (c|x{|}^2)dx$, for some $a>0$, $k,c\ge 0$. 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.