Sharp inequalities and asymptotics for polyharmonic eigenvalues

2025/06/15 by Davide Buoso, Pedro Freitas, Buoso, Davide +1 · 2 citations
Computer Science · Mathematics · #35J30 #35P15 #49R05 #74K20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2506.12791

openalex publication_date 2025/06/15 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

We study eigenvalues of general scalar Dirichlet polyharmonic problems in domains in \mathbb R^d. We first prove a number of inequalities satisfied by the eigenvalues on general domains, depending on the relations between the orders of the operators involved. We then obtain several estimates for these eigenvalues, yielding their growth as a function of these orders. For the problem in the ball we derive the general form of eigenfunctions together with the equations satisfied by the corresponding eigenvalues, and obtain several bounds for the first eigenvalue. In the case of the polyharmonic operator of order 2m we derive precise bounds yielding the first two terms in the asymptotic expansion for the first normalised eigenvalue as m grows to infinity. These results allow us to obtain the order of growth for the k^{\rm th} polyharmonic eigenvalue on general domains.

Citations

• Existence of an optimal shape for the first eigenvalue of polyharmonic operators
• Bulk-Boundary eigenvalues for Bilaplacian problems
• On the paper "The Payne conjecture for Dirichlet and buckling eigenvalues", by Genquian Liu
• The Payne conjecture for Dirichlet and Buckling eigenvalues
• The determinant of one-dimensional polyharmonic operators of arbitrary\n order
• Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles
• Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators
• Eigenvalues of poly-harmonic operators on variable domains
• A lower bound for eigenvalues of the poly-Laplacian with arbitrary order
• Universal Bounds for Eigenvalues of the Polyharmonic Operators
• Universal Inequalities for Eigenvalues of the Buckling Problem of Arbitrary Order
• Some inequalities and asymptotic formulas for eigenvalues on Riemannian manifolds

Cited by

• On the eigenvalues of the biharmonic operator on annuli
• Extremal problems for clamped plates under tension

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• Universal Bounds for Eigenvalues of the Polyharmonic Operators
• The sharp estimates of eigenvalues of Polyharmonic operator and higher order Stokes operator
• Perturbed eigenvalues of polyharmonic operators in domains with small holes
• Estimates for eigenvalues of the Neumann and Steklov problems
• Estimates for Eigenvalues of Poly-harmonic Operators