  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.
