Deep estimates for higher eigenvalues of the poly-Laplacian

2025/08/06 by Ji, Zhengchao, Xu, Hongwei
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.04069

Abstract

We investigate the lower bound for higher eigenvalues λ_i of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the Pólya conjecture, the clamped plate problem and the eigenvalue problem of the poly-Laplacian and deliver a series of deep eigenvalue inequalities for these problems respectively. Secondly, we establish a sharp lower bound for the eigenvalues of the poly-Laplacia in arbitrary dimension under some certain restrictive conditions. Finally, we provide an improved inequality for λ_i in arbitrary dimension without any restrictive conditions. Our results also yield the improvement of the lower bounds for the Stokes eigenvalue problems and the Generalized Pólya conjecture.

Citations

• Existence of an optimal shape for the first eigenvalue of polyharmonic operators
• Lord Rayleigh's Conjecture for Vibrating Clamped Plates in Positively Curved Spaces
• Lower bounds for eigenvalues of Laplacian operator and the clamped plate problem
• Fundamental tones of clamped plates in nonpositively curved spaces
• The Weyl formula for planar annuli
• Upper and lower bounds for eigenvalues of the clamped plate problem
• Estimates for lower bounds of eigenvalues of the poly-Laplacian and quadratic polynomial operator of the Laplacian
• A lower bound for eigenvalues of the poly-Laplacian with arbitrary order
• Lower bounds for the spectrum of the Laplace and Stokes operators
• A lower bound for eigenvalues of a clamped plate problem

Related

• Estimates for Eigenvalues of Poly-harmonic Operators
• Lower bounds for eigenvalues of Laplacian operator and the clamped plate problem
• A lower bound for eigenvalues of the poly-Laplacian with arbitrary order
• Proof of the Pólya conjecture
• Estimates for lower bounds of eigenvalues of the poly-Laplacian and quadratic polynomial operator of the Laplacian