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arXiv · 2506.13311

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

Christos Papadimitriou
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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.

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Christos Papadimitriou. 2025-06-16. Symmetrization of measures and the one-dimensional Poisson equation with Dirichlet boundary conditions. https://arxiv.org/abs/2506.13311

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