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PreprintCC BY 4.0arXiv

The fractional Faber--Krahn inequality in sharp quantitative form

  • Michael Goldman
  • Matteo Novaga
  • Berardo Ruffini
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Abstract

We prove that the first Dirichlet eigenvalue of the fractional Laplacian satisfies the sharp quantitative Faber--Krahn inequality. This was conjectured by Brasco, Cinti and Vita, who proved the inequality with a sub-optimal exponent. The proof adapts to the non-local setting the level-set argument recently introduced by Guerra, Machado and Ramos for the Laplacian. It is carried out on the horizontal level sets of the Caffarelli--Silvestre extension of the torsion function, where the asymmetry appears only as a boundary value at the corner of the extension variables.

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Cite

BibTeXRIS
BibTeX
@misc{goldman2026fractional,
  title = {{The fractional Faber--Krahn inequality in sharp quantitative form}},
  author = {Michael Goldman and Matteo Novaga and Berardo Ruffini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40232},
  eprint = {2610.40232},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40232},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The fractional Faber--Krahn inequality in sharp quantitative form
AU  - Michael Goldman
AU  - Matteo Novaga
AU  - Berardo Ruffini
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40232
UR  - https://arxiv.org/abs/2610.40232
AB  - We prove that the first Dirichlet eigenvalue of the fractional Laplacian satisfies the sharp quantitative Faber--Krahn inequality. This was conjectured by Brasco, Cinti and Vita, who proved the inequality with a sub-optimal exponent. The proof adapts to the non-local setting the level-set argument recently introduced by Guerra, Machado and Ramos for the Laplacian. It is carried out on the horizontal level sets of the Caffarelli--Silvestre extension of the torsion function, where the asymmetry appears only as a boundary value at the corner of the extension variables.
ER  -