  We consider a classical shape optimization problem for the eigenvalues of
elliptic operators with homogeneous boundary conditions on domains in the
$N$-dimensional Euclidean space. We survey recent results concerning the
analytic dependence of the elementary symmetric functions of the eigenvalues
upon domain perturbation and the role of balls as critical points of such
functions subject to volume constraint. Our discussion concerns Dirichlet and
buckling-type problems for polyharmonic operators, the Neumann and the
intermediate problems for the biharmonic operator, the Lam\'{e} and the
Reissner-Mindlin systems.
