Eigenvalue inequalities and three-term asymptotic formulas of the heat traces for the Lam\'{e} operator and Stokes operator

TL;DR

This paper explores the relationships between eigenvalue problems for various operators like Laplace, Lamé, and Stokes, establishing limits, inequalities, and asymptotic formulas for heat traces to deepen understanding of their spectral properties.

Contribution

It introduces new connections between eigenvalues of different operators and derives asymptotic formulas and inequalities for their heat traces under various boundary conditions.

Findings

Stokes eigenvalues are limits of Lamé eigenvalues as Lamé coefficient tends to infinity.

Eigenvalue inequalities are established for multiple operators.

Three-term asymptotic formulas for heat traces are derived for several operators.

Abstract

This paper is devoted to establish the most essential connections of the eigenvalue problems for the Laplace operator, Lam\'{e} operator, Stokes operator, buckling operator and clamped plate operator. We show that the $k$-th Stokes (respectively, Laplace) eigenvalue is the limit of the $k$-th Lam\'{e} eigenvalue for the Dirichlet or traction boundary condition as the Lam\'{e} coefficient $\lambda$ tends to $+\infty$ (respectively, to $-\mu$). Furthermore, we establish the eigenvalue inequalities and three-term asymptotic formulas of the heat traces for the Laplace operator, the Lam\'{e} operator, the Stokes operator and buckling operator with the Dirichlet and traction boundary conditions.