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

2025/06/16 by Papadimitriou, Christos
#34B05 #34C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.13311

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 \(u(±π)=0\). 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.

Citations

• Integral means of solutions of the one-dimensional Poisson equation with Robin boundary conditions
• Temperature on rods with Robin boundary conditions
• Extremizing Temperature Functions of Rods with Robin Boundary Conditions

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