  We consider a class of eigenvalue problems for poly-harmonic operators,
including Dirichlet and buckling-type eigenvalue problems. We prove an
analyticity result for the dependence of the symmetric functions of the
eigenvalues upon domain perturbations and compute Hadamard-type formulas for
the Frech\'{e}t differentials. We also consider isovolumetric domain
perturbations and characterize the corresponding critical domains for the
symmetric functions of the eigenvalues. Finally, we prove that balls are
critical domains.
