We consider a class of eigenvalue problems for polyharmonic operators, includingDirichlet and buckling-type eigenvalue problems. We prove an analyticity result for thedependence of the symmetric functions of the eigenvalues upon domain perturbations andcompute Hadamard-type formulas for the Frechét differentials. We also considerisovolumetric domain perturbations and characterize the corresponding critical domains forthe symmetric functions of the eigenvalues. Finally, we prove that balls are criticaldomains.
