Year: 2026
Type: preprint
Abstract: We prove two conjectures of Laugesen and Siudeja: the equilateral triangle uniquely maximizes the harmonic mean of the first two positive Neumann eigenvalues at fixed perimeter and their geometric mean at fixed area. The proof uses the two equilateral eigenfunctions and four projected polynomial corrections. Triangle symmetry reduces the associated variational matrices to two-by-two operations and two scalar invariants. Angular comparison and horizontal trial functions then yield the global inequalities. The same argument gives explicit quadratic stability bounds in terms of normalized side-length asymmetry.
Source: arXiv (Cornell University)
Authors: Dai, Guowei, Sun, Yingxin
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G4772512679
OpenAlex ID: W7215079437