  In this paper we obtain several results concerning the optimization of higher
Steklov eigenvalues both in two and higher dimensional cases. We first show
that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk
is not maximized for a smooth metric on the disk for $k\geq 3$. For $k=1$ the
classical result of [W] shows that $\sigma_1$ is maximized by the standard
metric on the round disk. For $k=2$ it was shown [GP1] that $\sigma_2$ is not
maximized for a smooth metric. We also prove a local rigidity result for the
critical catenoid and the critical M\"obius band as free boundary minimal
surfaces in a ball under $C^2$ deformations. We next show that the first $k$
Steklov eigenvalues are continuous under certain degenerations of Riemannian
manifolds in any dimension. Finally we show that for $k\geq 2$ the supremum of
the $k$-th Steklov eigenvalue on the annulus over all metrics is strictly
larger that that over $S^1$-invariant metrics. We prove this same result for
metrics on the M\"obius band.
