A quantitative Talenti-type comparison result with Robin boundary conditions

TL;DR

This paper develops a quantitative comparison principle for solutions to the Poisson equation with Robin boundary conditions, linking domain asymmetry to solution behavior, and provides new proofs for related inequalities.

Contribution

It introduces a quantitative Talenti comparison result for Robin boundary conditions, enhancing understanding of domain asymmetry effects on PDE solutions.

Findings

Quantitative Talenti comparison for Robin boundary conditions.

Alternative proof of the quantitative Saint-Venant inequality.

Extension of rigidity results for Talenti inequalities with Robin conditions.

Abstract

The purpose of this paper is to establish a quantitative version of the Talenti comparison principle for solutions to the Poisson equation with Robin boundary conditions. This quantitative enhancement is proved in terms of the asymmetry of domain. The key role is played by a careful analysis of the propagation of asymmetry for the level sets of the solutions of a PDE. As a byproduct, we obtain an alternative proof of the quantitative Saint-Venant inequality for the Robin torsion and, in the planar case, of the quantitative Faber-Krahn inequality for the first Robin eigenvalue. In addition, we complete the framework of the rigidity result of the Talenti inequalities with Robin boundary conditions.