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

Ryoki Endo; Xuefeng Liu

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arXiv · 2305.14063

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

Ryoki Endo·Xuefeng Liu

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Abstract

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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BibTeX RIS

Ryoki Endo, Xuefeng Liu. 2023-05-23. The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles. https://arxiv.org/abs/2305.14063

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