  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. The local representation
formulas are expressed in terms of the weak limit of the gradient fields and
local corrector problems. The upper bounds may diverge according to the
presence of rough interfaces. We also consider the fine phase limits for
layered microstructures and for sufficiently smooth periodic microsturctures.
For these cases we are able to provide explicit local formulas for the limit of
the $L^\infty$ norms of the associated sequence of gradient fields. Local
representation formulas for lower bounds are obtained for fields corresponding
to continuously graded periodic microstructures as well as for general
sequences of oscillatory coefficients. The representation formulas are applied
to problems of optimal material design.
