  For a class of linear elliptic equations of general type with rapidly
oscillating coefficients, we use the sigma-convergence method to prove the
homogenization result and a corrector-type result. In the case of asymptotic
periodic coefficients we derive the optimal convergence rates for the zero
order approximation of the solution with no smoothness on the coefficients, in
contrast to what has been done up to now in the literature. This follows as a
result of the existence of asymptotic periodic correctors for general nonsmooth
coefficients. The homogenization process is achieved through a compactness
result obtained by proving a Helmholtz-type decomposition theorem in case of
Besicovitch spaces.
