 Periodic conductivity cell problem

The periodic zero-mean corrector $w_j$ adjusts a unit imposed gradient so microscopic flux is divergence-free. For bounded uniformly elliptic coefficients, the weak problem has a unique solution modulo constants. The averaged flux is $a^{\ast }{e}_j=⟨a(e_j+\nabla w_j)⟩$. For symmetric coefficients this also gives a symmetric positive-definite effective tensor.