  In this paper we extend the ideas presented in Onofrei and Vernescu
[\textit{Asymptotic Analysis, 54, 2007, 103-123}] and introduce suitable second
order boundary layer correctors, to study the $H^1$-norm error estimate for the
classical problem in homogenization. Previous second order boundary layer
results assume either smooth enough coefficients (which is equivalent to
assuming smooth enough correctors $\chi_j,\chi_{ij}\in W^{1,\infty}$), or
smooth homogenized solution $u_0$, to obtain an estimate of order
$\displaystyle O(\epsilon^{\frac{3}{2}})$. For this we use the periodic
unfolding method developed by Cioranescu, Damlamian and Griso [\textit{C. R.
Acad. Sci. Paris, Ser. I 335, 2002, 99-104}]. We prove that in two dimensions,
for nonsmooth coefficients and general data, one obtains an estimate of order
$\displaystyle O(\epsilon^\frac{3}{2})$. In three dimenssions the same estimate
is obtained assuming $\chi_j,\chi_{ij}\in W^{1,p}$, with $p>3$. We also discuss
how our results extend, in the case of nonsmooth coefficients, the convergence
proof for the finite element multiscale method proposed by T.Hou et al.
[\textit{ J. of Comp. Phys., 134, 1997, 169-189}] and the first order
correctoranalysis for the first eigenvalue of a composite media obtained by
Vogelius et al.[\textit{Proc. Royal Soc. Edinburgh, 127A, 1997, 1263-1299}].
