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arXiv · 2610.31063

A unified quantitative approach to Chiti and Kohler--Jobin inequalities

Vincenzo Amato·Luca Barbato·Emanuele Cristoforoni·Alba Lia Masiello·Carlo Nitsch·Cristina Trombetti·Federico Villone
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Abstract

We obtain two quantitative results related to classical rearrangement inequalities for the first Dirichlet eigenvalue. The first one is a quantitative version of Chiti's reverse Hölder inequality. More precisely, for every $1\le q<p<\infty$, we prove that the deficit controls, up to translations, the $L^q$ distance between a first eigenfunction of a bounded domain and the first eigenfunction of the ball having the same first eigenvalue. The second result concerns the Kohler--Jobin inequality and gives a quantitative estimate in terms of the $L^\infty\!\to L^2$ distance between the corresponding Dirichlet resolvents. In both cases, the main idea is to find a suitable superlevel set of the first eigenfunction which is almost optimal for the Faber-Krahn inequality.

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Vincenzo Amato, Luca Barbato, Emanuele Cristoforoni, Alba Lia Masiello, Carlo Nitsch, Cristina Trombetti, Federico Villone. 2026-10-22. A unified quantitative approach to Chiti and Kohler--Jobin inequalities. https://arxiv.org/abs/2610.31063

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