Weighted isoperimetric inequalities in cones and applications

TL;DR

This paper investigates weighted isoperimetric inequalities within cones, characterizes measures for which cone-balls are isoperimetric sets, and applies these results to eigenvalue estimates, Hardy inequalities, and PDE comparisons.

Contribution

It provides a complete characterization of measures with cone-based isoperimetric sets and applies these findings to various problems in analysis and PDEs.

Findings

Characterization of measures with cone-based isoperimetric sets.

Derivation of isoperimetric estimates for weighted Laplacian eigenvalues.

Establishment of sharp Hardy inequalities and PDE comparison results.

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=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.