Sharp stability for the Talenti inequality

Abstract

We prove a sharp quantitative version of the Talenti rearrangement inequality. Specifically, we show that, for any $p\in [1;+\infty ]$, for any $f\in L^{\infty }(\Omega ),f\ge 0,f\not\equiv 0$, for any open set $\Omega \subset {\mathbb{R}}^d$ with finite volume, the solution $u_{f,\Omega }$ to $-\Delta u_{f,\Omega }=f,u_{f,\Omega }\in W_0^{1,2}(\Omega )$, satisfies, in dimension $d\ge 1$, $\Vert u_{f^{♯},{\Omega }^{♯}}{\Vert }_{L^p({\Omega }^{♯})}-\Vert u_{f,\Omega }{\Vert }_{L^p(\Omega )}\gtrsim \alpha (\Omega {)}^2+{\inf }_{z\in {\mathbb{R}}^d}\mathcal{A}(f,z{)}^2$ and, in dimension 1, $\Vert u_{f^{♯},{\Omega }^{♯}}{\Vert }_{L^p({\Omega }^{♯})}-\Vert u_{f,\Omega }{\Vert }_{L^p(\Omega )}\gtrsim \alpha (\Omega )+{\inf }_{z\in \mathbb{R}}\mathcal{A}(f,z{)}^2$ where $\alpha (\Omega )$ denotes the Fraenkel asymmetry of $\Omega$, $\mathcal{A}(f,z)$ a weighted asymmetry of $f$ with respect to some center $z$ and ${\Omega }^{♯}$ (resp. $f^{♯}$) is the Schwarz rearrangement of $\Omega$ (resp. of $f$). In dimension $d\ge 2$, this estimate implies $\Vert u_{f^{♯},{\Omega }^{♯}}{\Vert }_{L^p({\Omega }^{♯})}-\Vert u_{f,\Omega }{\Vert }_{L^p(\Omega )}\gtrsim \alpha (\Omega {)}^2+{\inf }_{z\in {\mathbb{R}}^d}\Vert f-f^{♯}(\cdot -z){\Vert }_{L^1}^2.$ Our result sharpens and extends recent quantitative versions, and does not rely on shape differentiation arguments, but on a generalisation of the quantitative Saint-Venant inequality.