Two-scale expansion for periodic conductivity ($\nabla ={\nabla }_x+{ϵ}^{-1}{\nabla }_y$)

A periodic expansion with $y=x/ϵ$ 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.

A bulk cell expansion generally leaves an oscillatory Dirichlet mismatch. A boundary corrector can solve the homogeneous oscillatory conductivity equation with boundary data cancelling that mismatch. It restores the prescribed physical boundary trace while leaving the leading bulk cell formula unchanged. Estimates depend on boundary regularity and on the coefficient assumptions.