The second Dirichlet eigenvalue is simple on every non-equilateral triangle
Spectral Theory
2025-04-01 v2
Abstract
The Dirichlet eigenvalues of the Laplacian on a triangle that collapses into a line segment diverge to infinity. In this paper, to track the behavior of the eigenvalues during the collapsing process of a triangle, we establish a quantitative error estimate for the Dirichlet eigenvalues on collapsing triangles. As an application, we solve the open problem concerning the simplicity of the second Dirichlet eigenvalue for nearly degenerate 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".
Cite
@article{arxiv.2503.06786,
title = {The second Dirichlet eigenvalue is simple on every non-equilateral triangle},
author = {Ryoki Endo and Xuefeng Liu},
journal= {arXiv preprint arXiv:2503.06786},
year = {2025}
}