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For a family of second-order elliptic systems of Maxwell's type with rapidlyoscillating periodic coefficients in a $C^{1, \alpha}$ domain $\Omega$, weestablish uniform estimates of solutions $u_\varep$ and $\nabla \timesu_\varep$ 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 equationswith periodic coefficients.

On Lp Estimates in Homogenization of Elliptic Equations of Maxwell's Type