For a family of second-order elliptic systems of Maxwells type with rapidly oscillating periodic coefficients in a C1,α domain Ω,we establish uniform estimates of solutions up and ▽ × up in Lp(Ω)for 1 ＜ p ≤∞.The proof relies on the uniform W1,p and Lipschitz estimates for solutions of scalar elliptic equations with periodic coefficients.