 Polarization field of a conductivity inclusion

Relative to a uniform reference conductivity $a_0$, this polarization is the excess positive conductivity flux. The equation becomes $a_0\Delta u+\nabla \cdot P=0$. For a prescribed mean gradient $E$, averaging gives $a^{\ast }E=a_{0}E+⟨P⟩$. This sign convention differs from the physical heat-flux sign.