Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals
Abstract
We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace--Neumann eigenvalue. This is the quadrilateral case of the Neumann analogue of the long-standing Pólya--Szegő conjecture, which asserts that the regular n-gon minimizes the first Laplace--Dirichlet eigenvalue among n-gons of equal area. The proof uses a Rayleigh--Ritz lower bound, obtained from the span of the first five nonconstant Neumann modes of the square, on a smooth auxiliary functional whose minimization implies the original maximization. Local maximality at the square follows from a symmetry-reduced Hessian, computed in closed form, together with a certified second-order difference-quotient test that establishes the local inequality on an explicit ball around the square. Global maximality is then established by a certified box covering of the parameter space, certifying on each box a direct upper bound on the first Rayleigh--Ritz eigenvalue. Both certifications operate entirely on integrals over a fixed reference square and avoid a posteriori finite-element bounds on the perturbed quadrilateral.
Related
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- Sharp Neumann eigenvalue means on triangles
- Maximisers for the second Neumann eigenvalue of annuli in two-dimensional space forms
- Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains
- A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound
BibTeX
@misc{endo2026maximizing,
title = {{Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals}},
author = {Ryoki Endo and Braxton Osting},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02784},
eprint = {2609.02784},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02784},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals
AU - Ryoki Endo
AU - Braxton Osting
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02784
UR - https://arxiv.org/abs/2609.02784
AB - We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace--Neumann eigenvalue. This is the quadrilateral case of the Neumann analogue of the long-standing Pólya--Szegő conjecture, which asserts that the regular n-gon minimizes the first Laplace--Dirichlet eigenvalue among n-gons of equal area. The proof uses a Rayleigh--Ritz lower bound, obtained from the span of the first five nonconstant Neumann modes of the square, on a smooth auxiliary functional whose minimization implies the original maximization. Local maximality at the square follows from a symmetry-reduced Hessian, computed in closed form, together with a certified second-order difference-quotient test that establishes the local inequality on an explicit ball around the square. Global maximality is then established by a certified box covering of the parameter space, certifying on each box a direct upper bound on the first Rayleigh--Ritz eigenvalue. Both certifications operate entirely on integrals over a fixed reference square and avoid a posteriori finite-element bounds on the perturbed quadrilateral.
ER -