 Phase polarization comparison bound

Take a positive comparison elastic stiffness tensor $C^0$, excess energies $F_r(e)=W_r(e)-e:C^{0}e/2$, and a phasewise constant elastic polarization. The lower conjugate inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible strains gives the displayed bound, where $A_{rs}=\int \int ({\chi }_r-c_r)\Gamma ({\chi }_s-c_s)$. Stationarity in ${\tau }^r$ gives $c_r{e}^r+{\sum }_s{A}_{rs}{\tau }^s=c_{r}E$, with $e^r=D{F}_{r\ast }({\tau }^r)$. Without this stationarity, dual strains must not be identified with the comparison solution's phase means.