Boundary corrector in periodic homogenization ($b^{ϵ}{|}_{\partial \Omega }=-ϵw_j(x/ϵ){\partial }_j{u}_0$)

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.