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∈ [1;+∞], for any f∈ L∞(Ω), f≥ 0, fnot≡ 0, for any open set Ω⊂ ℝd with finite volume, the solution uf,Ω to −Δuf,Ω=f, uf,Ω ∈ W1,20(Ω), satisfies, in dimension d≥ 1, ‖ uf♯,Ω♯‖Lp(Ω♯)−‖ uf,Ω‖Lp(Ω)≳ α(Ω)2+infz∈ ℝd𝒜(f,z)2 and, in dimension 1, $ \Vert u_{f^\sharp,Ω^\sharp}\Vert_{L^p(Ω^\sharp)}-\Vert u_{f,Ω}\Vert_{L^p(Ω)}\gtrsim α(Ω)+\inf_{z\in \mathbb{R}}\mathscr A(f,z)^2$ where α(Ω) denotes the Fraenkel asymmetry of Ω, 𝒜(f,z) a weighted asymmetry of f with respect to some center z and Ω♯ (resp. f♯) is the Schwarz rearrangement of Ω (resp. of f). In dimension d≥ 2, this estimate implies $ \Vert u_{f^\sharp,Ω^\sharp}\Vert_{L^p(Ω^\sharp)}-\Vert u_{f,Ω}\Vert_{L^p(Ω)}\gtrsim α(Ω)^2+\inf_{z\in \mathbb{R}^d}\Vert f-f^\sharp(\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.
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BibTeX
@misc{acampora2026sharp,
title = {{Sharp stability for the Talenti inequality}},
author = {Paolo Acampora and Jimmy Lamboley and Idriss Mazari},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31369},
eprint = {2610.31369},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05836411v1},
note = {CC BY-ND 4.0},
}
RIS
TY - UNPB
TI - Sharp stability for the Talenti inequality
AU - Paolo Acampora
AU - Jimmy Lamboley
AU - Idriss Mazari
PY - 2026
DA - 2026/10/22
PB - HAL
DO - 10.48550/arxiv.2610.31369
UR - https://hal.science/hal-05836411v1
AB - We prove a sharp quantitative version of the Talenti rearrangement inequality. Specifically, we show that, for any p∈ [1;+∞], for any f∈ L^∞(Ω), f≥ 0, fnot≡ 0, for any open set Ω⊂ ℝ^d with finite volume, the solution u_f,Ω to −Δu_f,Ω=f, u_f,Ω ∈ W^1,2_0(Ω), satisfies, in dimension d≥ 1, ‖ u_f^♯,Ω^♯‖_L^p(Ω^♯)−‖ u_f,Ω‖_L^p(Ω)≳ α(Ω)^2+inf_z∈ ℝ^d𝒜(f,z)^2 and, in dimension 1, $ \Vert u_{f^\sharp,Ω^\sharp}\Vert_{L^p(Ω^\sharp)}-\Vert u_{f,Ω}\Vert_{L^p(Ω)}\gtrsim α(Ω)+\inf_{z\in \mathbb{R}}\mathscr A(f,z)^2$ where α(Ω) denotes the Fraenkel asymmetry of Ω, 𝒜(f,z) a weighted asymmetry of f with respect to some center z and Ω^♯ (resp. f^♯) is the Schwarz rearrangement of Ω (resp. of f). In dimension d≥ 2, this estimate implies $ \Vert u_{f^\sharp,Ω^\sharp}\Vert_{L^p(Ω^\sharp)}-\Vert u_{f,Ω}\Vert_{L^p(Ω)}\gtrsim α(Ω)^2+\inf_{z\in \mathbb{R}^d}\Vert f-f^\sharp(\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.
ER -