Sharp quantitative Talenti's inequality in particular cases

In this paper, we focus on the famous Talenti’s symmetrization inequality, more precisely its $L^p$ corollary asserting that the $L^p$-norm of the solution to $-\mathrm{\Delta }v=f^{\mathrm{♯}}$ is higher than the $L^p$-norm of the solution to $-\mathrm{\Delta }u=f$ (we are considering Dirichlet boundary conditions, and $f^{\mathrm{♯}}$ denotes the Schwarz symmetrization of $f:\mathrm{\Omega }\to {\mathbb{R}}_{+}$). We focus on the particular case where functions $f$ are defined on the unit ball, and are characteristic functions of a subset of this unit ball. We show in this case that stability occurs for the $L^p$-Talenti inequality with the sharp exponent 2.