PreprintCC BY 4.0arXiv
On the effective sharp stability of the Faber-Krahn inequality
Abstract
We prove the sharp quantitative Faber--Krahn inequality with a computable dimensional constant, resolving a problem posed by Brasco and De Philippis~\cite[Open Problem~2]{brasco2017spectral}. In contrast with the known proof of the sharp estimate, our argument avoids indirect compactness arguments and selection principles. It runs through a new level-set proof of the Saint--Venant inequality, using that the torsional deficit controls both the average isoperimetric deficit and the oscillation of the gradient magnitude along the boundaries of level sets. An effective local strong isoperimetric estimate for confined sets then yields the result.
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BibTeX
@misc{guerra2026effective,
title = {{On the effective sharp stability of the Faber-Krahn inequality}},
author = {André Guerra and João Miguel Machado and João P. G. Ramos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07772},
eprint = {2609.07772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07772},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the effective sharp stability of the Faber-Krahn inequality
AU - André Guerra
AU - João Miguel Machado
AU - João P. G. Ramos
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07772
UR - https://arxiv.org/abs/2609.07772
AB - We prove the sharp quantitative Faber--Krahn inequality with a computable dimensional constant, resolving a problem posed by Brasco and De Philippis~\cite[Open Problem~2]{brasco2017spectral}. In contrast with the known proof of the sharp estimate, our argument avoids indirect compactness arguments and selection principles. It runs through a new level-set proof of the Saint--Venant inequality, using that the torsional deficit controls both the average isoperimetric deficit and the oscillation of the gradient magnitude along the boundaries of level sets. An effective local strong isoperimetric estimate for confined sets then yields the result.
ER -