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An optimal inequality between nonlinear eigenvalues in convex sets
Abstract
Let λp(Ω) be the first Dirichlet eigenvalue of the p-Laplacian in a bounded convex open set Ω⊂ℝN, N≥2. We prove that p↦λp(Ω)1/p/πp is strictly decreasing, where πpp is the first eigenvalue on the unit interval. Consequently, for 1<q<p<∞, the sharp upper bound for λp(Ω)1/p/λq(Ω)1/q is πp/πq. This bound is not attained and is approached by slabs, resolving the conjecture on the supremum formulated in the plane by Parini (J. Convex Anal. 2017). The proof combines log-concavity of the first eigenfunction with a vector field constructed from the one-dimensional eigenfunctions. The endpoint limits recover the Hersch--Protter and reverse Cheeger (or Buser) inequalities.
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BibTeX
@misc{pietra2026optimal,
title = {{An optimal inequality between nonlinear eigenvalues in convex sets}},
author = {Francesco Della Pietra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25162},
eprint = {2610.25162},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25162},
}
RIS
TY - UNPB
TI - An optimal inequality between nonlinear eigenvalues in convex sets
AU - Francesco Della Pietra
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25162
UR - https://arxiv.org/abs/2610.25162
AB - Let λ_p(Ω) be the first Dirichlet eigenvalue of the p-Laplacian in a bounded convex open set Ω⊂ℝ^N, N≥2. We prove that p↦λ_p(Ω)^1/p/π_p is strictly decreasing, where π_p^p is the first eigenvalue on the unit interval. Consequently, for 1<q<p<∞, the sharp upper bound for λ_p(Ω)^1/p/λ_q(Ω)^1/q is π_p/π_q. This bound is not attained and is approached by slabs, resolving the conjecture on the supremum formulated in the plane by Parini (J. Convex Anal. 2017). The proof combines log-concavity of the first eigenfunction with a vector field constructed from the one-dimensional eigenfunctions. The endpoint limits recover the Hersch--Protter and reverse Cheeger (or Buser) inequalities.
ER -