Tamped functions: A rearrangement in dimension 1

TL;DR

This paper introduces a new rearrangement called tamping for non-negative functions on R+ that shares properties with Schwarz rearrangement but uniquely preserves boundary conditions.

Contribution

The paper defines and analyzes a novel rearrangement method called tamping, which maintains boundary conditions unlike traditional rearrangements.

Findings

Tamping preserves boundary conditions.

Tamping shares properties with Schwarz rearrangement.

Establishes inequalities similar to Pólya-Szegő.

Abstract

We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the P{\'o}lya-Szeg{\"o} inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.