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

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.