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PreprintCC BY 4.0arXiv

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
Open at sourcePDFDOICite

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≤ q<p<∞, we prove that the deficit controls, up to translations, the Lq 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∞ → L2 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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Cite

BibTeXRIS
BibTeX
@misc{amato2026unified,
  title = {{A unified quantitative approach to Chiti and Kohler--Jobin inequalities}},
  author = {Vincenzo Amato and Luca Barbato and Emanuele Cristoforoni and Alba Lia Masiello and Carlo Nitsch and Cristina Trombetti and Federico Villone},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31063},
  eprint = {2610.31063},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31063},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A unified quantitative approach to Chiti and Kohler--Jobin inequalities
AU  - Vincenzo Amato
AU  - Luca Barbato
AU  - Emanuele Cristoforoni
AU  - Alba Lia Masiello
AU  - Carlo Nitsch
AU  - Cristina Trombetti
AU  - Federico Villone
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31063
UR  - https://arxiv.org/abs/2610.31063
AB  - 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≤ q<p<∞, 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^∞ → 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.
ER  -