  We consider a variant of the classic Steklov eigenvalue problem, which arises
in the study of the best trace constant for functions in Sobolev space. We
prove that the elementary symmetric functions of the eigenvalues depend
real-analytically upon variation of the underlying domain and we compute the
corresponding Hadamard-type formulas for the shape derivatives. We also
consider isovolumetric and isoperimetric domain perturbations and we
characterize the corresponding critical domains in terms of appropriate
overdetermined systems. Finally, we prove that balls are critical domains for
the elementary symmetric functions of the eigenvalues subject to volume or
perimeter constraint.
