  We study the dependence of the eigenvalues of time-harmonic Maxwell's
equations in a cavity upon variation of its shape. The analysis concerns all
eigenvalues both simple and multiple. We provide analyticity results for the
dependence of the elementary symmetric functions of the eigenvalues splitting a
multiple eigenvalue, as well as a Rellich-Nagy-type result describing the
corresponding bifurcation phenomenon. We also address an isoperimetric problem
and characterize the critical cavities for the symmetric functions of the
eigenvalues subject to isovolumetric or isoperimetric domain perturbations and
prove that balls are critical. We include known formulas for the eigenpairs in
a ball and calculate the first one.
