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

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.