The isoperimetric problem for a class of non-radial weights and applications

TL;DR

This paper investigates a class of isoperimetric problems with non-radial weights in the upper half-space, leading to new sharp functional inequalities such as Caffarelli-Kohn-Nirenberg type inequalities.

Contribution

It introduces a novel analysis of isoperimetric problems with non-radial weights of specific form, deriving sharp inequalities in this setting.

Findings

Establishment of isoperimetric inequalities with non-radial weights

Derivation of sharp functional inequalities including Caffarelli-Kohn-Nirenberg type

Extension of classical inequalities to weighted, non-radial contexts

Abstract

We study a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type $|x|^k x_N^\alpha$. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities.