Preprint

Inserted: 17 nov 2025

Year: 2025
Doi: https://doi.org/10.48550/arXiv.2511.11316

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.

Keywords: stability, Laplace operator, Robin boundary condition, Talenti comparison