Title : First example 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 in the case $p=infty$.
In this talk, we focus on a similar question in the particular case where $Omega$ is a ball $B$, and $f$ is the characteristic function of a set $Esubset B$: in this case,  we show that one can reach a quantitative estimate for the $L^p$-norm, with a sharp exponent. This is a joint work with Paolo ACAMPORA.

Paolo Acampora, Carlo Nitsch, Cristina Trombetti