Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals

Ryan Arbon

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arXiv · 2207.05814

Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals

Ryan Arbon

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Abstract

We present a new, computer-assisted, proof that for all triangles in the plane, the equilateral triangle uniquely maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This proves an independent proof the triangular Ashbaugh-Benguria-Payne-Polya-Weinberger conjecture first proved in arXiv:0707.3631 [math.SP] and arXiv:2009.00927 [math.SP]. Inspired by arXiv:1109.4117 [math.SP], the primary method is to use a perturbative estimate to determine a local optimum, and to then use a continuity estimate for the fundamental ratio to perform a rigorous computational search of parameter space. We repeat a portion of this proof to show that the square is a strict local optimizer of the fundamental ratio among quadrilaterals in the plane

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Ryan Arbon. 2022-07-12. Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals. https://arxiv.org/abs/2207.05814

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