On weighted isoperimetric inequalities with non-radial densities

TL;DR

This paper investigates weighted isoperimetric inequalities in the upper half-space with non-radial weights, providing solutions in a special case and implications for weighted inequalities and elliptic PDE estimates.

Contribution

It introduces new weighted isoperimetric inequalities with non-radial densities and explores their applications to PDE boundary value problems.

Findings

Solved a specific case of weighted isoperimetric problems

Derived a weighted Polya-Szegö principle

Established a priori estimates for degenerate elliptic equations

Abstract

We consider a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the volume and the area element carry two different weights of the type $|x|^lx_N^\alpha$. We solve them in a special case while a more detailed study is contained in \cite{ABCMP2}. Our results imply a weighted Polya-Sz\"ego principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations