= Solution
Choose the affine trial function $v(x)=\lambda\cdot x$. It has the required trace and constant <gradient>, so the <Dirichlet principle> gives
$$
J(u)\le J(v)=\int_\Omega a(x)\lambda\cdot\lambda\,dx
=|\Omega|(p_1a_1+p_2a_2)|\lambda|^2.
$$
For the macroscopic <isotropic> effective medium, the corresponding affine-boundary energy is $|\Omega|a^*|\lambda|^2$. Dividing by $|\Omega||\lambda|^2$ for a nonzero imposed <gradient> yields the <Voigt bound>
$$
\boxed{a^*\le p_1a_1+p_2a_2.}
$$
More generally the same calculation is the <tensor> inequality in the <Loewner order> $a^*\preceq\langle a\rangle$. For a finite unresolved specimen the calculation bounds its apparent affine-boundary response; identification with the bulk <effective conductivity> uses the macroscopic scale separation.
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