Eigenvalue inequalities and three-term asymptotic formulas of the heat traces for the Lamé operator and Stokes operator

Abstract

This paper is devoted to establish the most essential connections of the eigenvalue problems for the Laplace operator, Lamé 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 Lamé eigenvalue for the Dirichlet or traction boundary condition as the Lamé coefficient $λ$ tends to $+\infty$ (respectively, to $-μ$). Furthermore, we establish the eigenvalue inequalities and three-term asymptotic formulas of the heat traces for the Laplace operator, the Lamé operator, the Stokes operator and buckling operator with the Dirichlet and traction boundary conditions.