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

Abstract

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

30 pages, 2 figures