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.