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The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles

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arxiv 2305.14063 v6 pith:KZ4KA5GU submitted 2023-05-23 math.SP

Ryoki Endo , Xuefeng Liu This is my paper
classification math.SP
keywords nearlydirichleteigenvalueequilateralsecondtrianglesachievedalgorithm
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This paper solves the open problem of the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a complete solution to Conjecture 6.47 posed by R. Laugesen and B. Siudeja in A. Henrot's book ``Shape Optimization and Spectral Theory." Our proof is achieved by introducing a new difference quotient formula for the behavior of nearly degenerate eigenvalues resulting from domain perturbations, and a novel numerical algorithm that rigorously estimates it using verified computation.

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Cited by 1 Pith paper

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  1. Stable Computation of Laplacian Eigenfunctions Corresponding to Clustered Eigenvalues

    math.SP 2025-06 conditional novelty 4.0 of 10

    A shape-difference-quotient based algorithm stably recovers eigenfunctions for clustered Laplacian eigenvalues caused by small domain perturbations, demonstrated on rectangles and equilateral triangles.

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