Symmetrization of measures and the one-dimensional Poisson equation with Dirichlet boundary conditions

TL;DR

This paper investigates the symmetrization of measures and their impact on solutions to the one-dimensional Poisson equation with Dirichlet boundary conditions, establishing uniqueness of solutions that maximize convex integral means.

Contribution

It introduces measure transformations and compares convex integral means of solutions, proving the uniqueness of the maximizer in this context.

Findings

Defined measure transformations related to  measures

Compared convex integral means of original and transformed solutions

Proved uniqueness of the solution maximizing convex integral means

Abstract

Let \(\mu\) be a finite Borel measure on \((-\pi,\pi)\). Consider the one-dimensional Poisson equation \(-u''=\mu\), where equality holds in the sense of distributions, with Dirichlet boundary conditions \(u(\pm\pi)=0\). In this paper, we define measures that are transformations of \(\mu\), we compare the convex integral means of the original solutions \(u_\mu\) and the transformed ones, and we prove the uniqueness of a solution that maximizes the convex integral means.