Sharp stability for the Talenti inequality

We prove a sharp quantitative version of the Talenti rearrangement inequality. Specifically, we show that, for any $p\in [1;+\mathrm{\infty }]$, for any $f\in L^{\mathrm{\infty }}(\mathrm{\Omega }),f\ge 0,f\equiv ̸0$, for any open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with finite volume, the solution $u_{f,\mathrm{\Omega }}$ to $-\mathrm{\Delta }{u}_{f,\mathrm{\Omega }}=f,u_{f,\mathrm{\Omega }}\in W_0^{1,2}(\mathrm{\Omega })$, satisfies, in dimension $d\ge 1$, $\mathrm{\parallel }{u}_{f^{\mathrm{♯}},{\mathrm{\Omega }}^{\mathrm{♯}}}{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{♯}})}-\mathrm{\parallel }{u}_{f,\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\gtrsim \alpha (\mathrm{\Omega }{)}^2+{\inf }_{z\in {\mathbb{R}}^d}\mathcal{A}(f,z{)}^2$ and, in dimension 1, $\mathrm{\parallel }{u}_{f^{\mathrm{♯}},{\mathrm{\Omega }}^{\mathrm{♯}}}{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{♯}})}-\mathrm{\parallel }{u}_{f,\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\gtrsim \alpha (\mathrm{\Omega })+{\inf }_{z\in \mathbb{R}}\mathcal{A}(f,z{)}^2$ where $\alpha (\mathrm{\Omega })$ denotes the Fraenkel asymmetry of $\mathrm{\Omega }$, $\mathcal{A}(f,z)$ a weighted asymmetry of $f$ with respect to some center $z$ and ${\mathrm{\Omega }}^{\mathrm{♯}}$ (resp. $f^{\mathrm{♯}}$) is the Schwarz rearrangement of $\mathrm{\Omega }$ (resp. of $f$). In dimension $d\ge 2$, this estimate implies $\mathrm{\parallel }{u}_{f^{\mathrm{♯}},{\mathrm{\Omega }}^{\mathrm{♯}}}{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{♯}})}-\mathrm{\parallel }{u}_{f,\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\gtrsim \alpha (\mathrm{\Omega }{)}^2+{\inf }_{z\in {\mathbb{R}}^d}\mathrm{\parallel }f-f^{\mathrm{♯}}(\cdot -z){\mathrm{\parallel }}_{L^1}^2\mathrm{.}$ 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.