Periodic conductivity cell problem
= Periodic conductivity cell problem
{title2=$\nabla_y\cdot[a(y)(e_j+\nabla_yw_j)]=0$}
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 $a^*e_j=\langle a(e_j+\nabla w_j)\rangle$. For symmetric coefficients this also gives a symmetric positive-definite effective tensor.