Sharp stability for the Talenti inequality

We prove a sharp quantitative version of the Talenti rearrangement inequality. Specifically, we show that, for any $p\in [1;+\infty]$, for any $f\in L^\infty(Ω),\, f\geq 0,\, f\not\equiv 0$, for any open set $Ω\subset \mathbb{R}^d$ with finite volume, the solution $u_{f,Ω}$ to $-Δu_{f,Ω}=f,\, u_{f,Ω