Calculus of Variations
& Geometric Analysis

Nov 7, 2025

Invited Speakers

Jimmy Lamboley, Sorbonne université

Title: First examples of sharp quantitative Talenti’s inequality

Abstract: A classical result by Talenti provides a comparison for solutions of Elliptic PDE with Dirichlet boundary conditions. In particular, it implies that the $L^p$-norm of the solution to $-\Delta u = f$ in a domain $\Omega$ is lower to the $L^p$-norm of the solution to $-\Delta v=f^\sharp$ where $f^\sharp$ is the symmetric decreasing rearrangement of $f$.

Recently, [Amato, Barbato, Lia Masiello, Paoli 2023] proved a quantitative version of this result, but only in the case $p=\infty$, and with a non-sharp exponent.

In this talk, we aim at obtaining a similar result, but for any $p$ and with a sharp exponent. To that end we make the extra-assumption that $\Omega$ is a ball $B$ (therefore $\Omega$ is already symmetric and only $f$ is symmetrized). We start by analyzing the case where $f$ is the characteristic function of a set $E\subset B$ and show that in this case, one can reach a sharp quantitative estimate, using a shape derivative approach. We will then explain how one can tackle the more general case of a general bounded function $f$.