Eigenvalues of the incompressibile elasticity problem

Dear all,

    I have a question about the eigenvalue of the incompressible elasticity problem. Assume that we have a incompressible elasticity continuum body whose constitutive response holds normality. Obviously, we can treat the incompressiblity limit as a Lagrangian constraint and express the governing equation via the Lagrange-Multiplier method. If mixed finite element method is used, one can use Courant-Fischer theorem to show that the tangential stiffness matrix of  constrained variational form would have its maximum eigenvalue larger than its unconstrained counterpart and its minimum eigenvalue smaller than its uncsontrained counterpart.

On the other hand, if penalty method is used, then Wely's monotonicity theorem shows that the smallest eigenvale of the stiffness matrix K_ij + alpha I_i _j would be always larger than that of K_ij.

Now, suppose we choose not to discretize the problem. Instead, assume that the elasticity problem can be expressed with a compact, self-adjointed linear operators A on a Hibert space such that

A(u) = f

where A(u) = div (C: epsilon(u))  

Would the incompressible constraint impose a simiar effect on the linear operator A? Is there any published results that show how the eigenvalue of the operator A changes to the constraint which is somehow "equvalient" to the monotonicity therorem/Courant-Fischer theorem applied to the discrete Hessian matrix? Any comment/suggestion is appreciated. Thank you.

Regards,

WaiChing Sun