PDE Comparison Principles for Robin Problems

TL;DR

This paper establishes comparison principles for solutions of Poisson problems with Robin boundary conditions, demonstrating how symmetrization affects solution convex means and extending results to various geometries.

Contribution

It introduces new comparison principles for Robin problems, including symmetrization effects and results on generalized cylinders with mixed boundary conditions.

Findings

Symmetrized solutions have larger convex means when Robin parameters are nonnegative.

Comparison principles hold for Poisson problems on balls and generalized cylinders.

Results extend classical symmetrization inequalities to Robin and mixed boundary conditions.

Abstract

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data has been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger (increasing) convex means. We prove similar results on balls. We also prove a comparison principle on generalized cylinders with mixed boundary conditions (Neumann and Robin).