Isoperimetric inequalities for eigenvalues of triangles

2007/07/24 by Bartłomiej Siudeja, Siudeja, Bartłomiej · 1 citation
Computer Science · Engineering · Mathematics · #35P15 #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.0707.3631

openalex publication_date 2007/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of Pólya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-Pólya-Weinberger conjecture proved by Ashbaugh and Benguria.

Cited by

• Flow approach on Riesz type nonlocal energies

Related

• Sharp bounds for eigenvalues of triangles
• Spectral isoperimetric inequalities for a class of mixed eigenvalue problems of the Laplacian on triangles and trapezoids
• On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian
• Geometric inequalities between Dirichlet and Neumann eigenvalues
• Isoperimetric relations between Dirichlet and Neumann eigenvalues