Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

TL;DR

This paper establishes comparison results for solutions of Poisson equations with Robin boundary conditions on complete Riemannian manifolds using rearrangement and isoperimetric inequalities, extending previous results and deriving new inequalities.

Contribution

It extends comparison results for Poisson equations on Riemannian manifolds with non-negative Ricci curvature and applies these to derive Saint-Venant and Bossel-Daners inequalities.

Findings

Comparison results for Poisson solutions on Riemannian manifolds.

Extension of previous inequalities to Robin Laplacian.

Derivation of Saint-Venant and Bossel-Daners inequalities.

Abstract

In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with $Ric\geq (n-1)\kappa$, $\, \kappa\geq 0$, which extends the results in \cite{ANT-Talenti-a}. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian.