Extremizing Temperature Functions of Rods with Robin Boundary Conditions

TL;DR

This paper compares solutions of Poisson problems with Robin boundary conditions under symmetrization, showing that symmetrized solutions have larger convex means depending on boundary conditions and rearrangement type.

Contribution

It provides new inequalities for solutions of Poisson problems with Robin and Neumann boundary conditions under symmetrization.

Findings

Symmetrized solutions have larger increasing convex means for Robin conditions.

Symmetrized solutions have larger convex means for Neumann conditions.

Results extend symmetrization inequalities to Robin boundary problems.

Abstract

We compare the solutions of two one-dimensional Poisson problems on an interval with Robin boundary conditions, one with given data, and one where the data has been symmetrized. When the Robin parameter is positive and the symmetrization is symmetric decreasing rearrangement, we prove that the solution to the symmetrized problem has larger increasing convex means. When the Robin parameter equals zero (so that we have Neumann boundary conditions) and the symmetrization is decreasing rearrangement, we similarly show that the solution to the symmetrized problem has larger convex means.