= Two-scale expansion for periodic conductivity
{title2=$\nabla=\nabla_x+\epsilon^{-1}\nabla_y$}
A periodic expansion with $y=x/\epsilon$ separates the macroscopic variable from the cell variable. Its leading cell equation forces $u_0$ to be independent of $y$, the next equation gives $u_1=w_j(y)\partial_{x_j}u_0$, and averaging the order-one flux equation yields the homogenized divergence equation. Bulk correctors generally require boundary-layer adjustments at a physical Dirichlet boundary.
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