We examine the composition of the $L^{\infty}$ norm with weakly convergent
sequences of gradient fields associated with the homogenization of second order
divergence form partial differential equations with measurable coefficients.
Here the sequences of coefficients are chosen to model heterogeneous media and
are piecewise constant and highly oscillatory. We identify local representation
formulas that in the fine phase limit provide upper bounds on the limit
superior of the $L^{\infty}$ norms of gradient fields.

We obtain a convergent power series expansion for the first branch of the
dispersion relation for subwavelength plasmonic crystals consisting of
plasmonic rods with frequency-dependent dielectric permittivity embedded in a
host medium with unit permittivity. The expansion parameter is $\eta=2\pi
d/\lambda$, where $\lambda$ is a fixed wavelength and $d$ is the period of the
crystal, and the plasma frequency scales inversely to $d$, making the
dielectric permittivity in the rods large and negative.