PreprintCC BY 4.0arXiv
A Quantitative Robin Faber-Krahn Inequality in Terms of Eigenfunction Distance
Abstract
We establish a quantitative stability estimate for the first eigenvalue of the Laplacian with positive Robin boundary conditions. For every open set Ω⊂ℝn of prescribed volume, we prove that λ1,β(Ω)−λ1,β(B) ≥ c|uΩ−uB|L2(ℝn)4, where B is a ball achieving the Fraenkel asymmetry of Ω, and uΩ and uB are nonnegative, L2-normalized first Robin eigenfunctions, extended by zero outside their respective domains. The constant c>0 depends only on the dimension, the Robin parameter, and the prescribed volume. The exponent $4$ is optimal.
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BibTeX
@misc{nitsch2026quantitative,
title = {{A Quantitative Robin Faber-Krahn Inequality in Terms of Eigenfunction Distance}},
author = {Carlo Nitsch and Cristina Trombetti and Federico Villone},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28032},
eprint = {2610.28032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28032},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Quantitative Robin Faber-Krahn Inequality in Terms of Eigenfunction Distance
AU - Carlo Nitsch
AU - Cristina Trombetti
AU - Federico Villone
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28032
UR - https://arxiv.org/abs/2610.28032
AB - We establish a quantitative stability estimate for the first eigenvalue of the Laplacian with positive Robin boundary conditions. For every open set Ω⊂ℝ^n of prescribed volume, we prove that λ_1,β(Ω)−λ_1,β(B) ≥ c|u_Ω−u_B|_L^2(ℝ^n)^4, where B is a ball achieving the Fraenkel asymmetry of Ω, and u_Ω and u_B are nonnegative, L^2-normalized first Robin eigenfunctions, extended by zero outside their respective domains. The constant c>0 depends only on the dimension, the Robin parameter, and the prescribed volume. The exponent $4$ is optimal.
ER -