# New P\'olya-Szeg\"o-type inequalities and an alternative approach to comparison results for PDE's

## TL;DR

This paper introduces new Pólya-Szegö-type inequalities involving pairs of functions and their rearrangements, providing a novel approach to comparison results for PDE solutions, simplifying existing proofs and extending classical principles.

## Contribution

The paper presents new inequalities of Pólya-Szegö type involving pairs of functions, offering an alternative proof method for PDE comparison results.

## Key findings

- Established new Pólya-Szegö-type inequalities for function pairs.
- Provided a different proof for PDE comparison results.
- Extended classical rearrangement principles to broader contexts.

## Abstract

We prove some P\'olya-Szeg\"o type inequalities which involve couples of functions and their rearrangements. Our inequalities reduce to the classical P\'olya-Szeg\"o principle when the two functions coincide. As an application, we give a different proof of a comparison result for solutions to Dirichlet boundary value problems for Laplacian equations proved by A. Alvino, G. Trombetti, J. I. Diaz and P. L. Lions.