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

2023/05/23 by Ryoki Endo, Xuefeng Liu, Endo, Ryoki +1 · 1 citation
Engineering · Medicine · Pharmacology, Toxicology and Pharmaceutics · #35P15 #65N25 #Advanced Measurement and Metrology Techniques #Cytokine Signaling Pathways and Interactions #FOS: Mathematics #Medicinal Plant Pharmacodynamics Research #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2305.14063

openalex publication_date 2023/05/23 · openalex created_date 2023/05/27 · openalex updated_date 2026/07/28

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.

Cited by

• Stable Computation of Laplacian Eigenfunctions Corresponding to Clustered Eigenvalues

Related

• The second Dirichlet eigenvalue is simple on every non-equilateral triangle
• Any three eigenvalues do not determine a triangle
• Dirichlet eigenvalue sums on triangles are minimal for equilaterals
• Sharp bounds for eigenvalues of triangles
• Isoperimetric inequalities for eigenvalues of triangles