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Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals
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- OpenAlex
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- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2609.02784
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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.
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