Optimal Inequalities for Gradients of Solutions of Elliptic Equations Occurring in Two-Phase Heat Conductors

We consider solutions to divergence form partial differential equations that model steady state heat conduction in random two-phase composites. The coefficient representing the conductivity takes two scalar values. Optimal bounds on the L2 norm of the gradient of the solution are found. The optimal upper bound is given in terms of the volume fraction occupied by each conducting phase. The optimal lower bound is independent of the volume fractions of the component conductors. The bounds follow from a Stieltjes integral representation for the L2 norm of the gradient. Maximizing sequences of configurations are found using the corrector theory of homogenization.