 Hashin-Shtrikman conductivity variational principle

For a scalar reference $b$, choose a trial polarization $q$ and solve $b\Delta v+\nabla \cdot q=0$ with zero Dirichlet data. The trial functional isA reference above all phase conductivities gives $J(u)\le {\mathcal{F}}_b(q)$; one below all of them gives $J(u)\ge {\mathcal{F}}_b(q)$. If the reference equals a phase conductivity, admissible polarization vanishes in that phase and the inverse is taken only on the contrast subspace.