  For a family of second-order elliptic systems of Maxwell's type with rapidly
oscillating periodic coefficients in a $C^{1, \alpha}$ domain $\Omega$, we
establish uniform estimates of solutions $u_\varep$ and $\nabla \times
u_\varep$ in $L^p(\Omega)$ for $1<p\le \infty$. The proof relies on the uniform
$W^{1,p}$ and Lipschitz estimates for solutions of scalar elliptic equations
with periodic coefficients.
