= Hashin-Shtrikman conductivity variational principle
{c}
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 is
$$
\mathcal F_b(q)=b|\Omega||\lambda|^2+2\lambda\cdot\int_\Omega q+\int_\Omega q\cdot\nabla v-\int_\Omega q\cdot(a-bI)^{-1}q.
$$
A 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.
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