A Talenti comparison result for a class of Neumann boundary value problems

TL;DR

This paper develops a comparison principle for solutions to Neumann boundary value problems, relating solutions of the original problem to those of a symmetrized version using Lorentz norms and point-wise inequalities.

Contribution

It introduces a novel comparison principle connecting solutions of Neumann problems with their symmetrized counterparts through Lorentz norms.

Findings

Established a comparison principle for Neumann problems

Linked solutions via Lorentz norms and point-wise inequalities

Enhanced understanding of solution behavior in boundary value problems

Abstract

In this paper, we establish a comparison principle in terms of Lorentz norms and point-wise inequalities between a positive solution $u$ to the Poisson equation with non-homogeneous Neumann boundary conditions and a specific positive solution $v$ to the Schwartz symmetrized problem, which is related to $u$ through an additional boundary condition.