  In terms of layer potential methods, this paper is devoted to study the $L^2$
boundary value problems for nonhomogeneous elliptic operators with rapidly
oscillating coefficients in a periodic setting. Under a low regularity
assumption on the coefficients, we establish the solvability for Dirichlet,
regular and Neumann problems in a bounded Lipschitz domain, as well as, the
uniform nontangential maximal function estimates and square function estimates.
The main difficulty is reflected in two aspects: (i) we can not treat the lower
order terms as a compact perturbation to the leading term due to the low
regularity assumption; (ii) the nonhomogeneous systems do not possess a
scaling-invariant property in general. Although this work may be regarded as a
follow-up to C. Kenig and Z. Shen's in \cite{SZW24}, we make an effort to find
a clear way of how to handle the nonhomogeneous operators by using the known
results of the homogenous ones. Also, we mention that the periodicity condition
plays a key role in the scaling-invariant estimates.
