On the attainability of singular Wiener bound

We characterize the lower and upper attainability of the Wiener bound (also known as Voigt-Reuss bound) for singularly distributed conductive material mixtures. For the lower attainability we consider mixtures in which high-conductance materials support on sets having finite one-dimensional Hausdorff measures. We show that, under a mild coercivity condition, the kernel of the effective tensor of the mixture is equal to the orthogonal complement of the homotopy classes of closed paths in the supporting set. This shows that a periodic planar network has positive definite effective tensor, i.e., it is resilient to fluctuations, if and only if the network is reticulate. We provide a geometric characterization of the upper attainability by applying a transformation from varifolds to matrix-valued measures. We show that this transformation leads to an equivalence between two distinct notions from material science and geometric measure theory respectively: conductance maximality and area criticality. Based on this relation we show a pointwise dimension bound for mixtures that attain the upper Wiener bound by applying a fractional version of the monotonicity formula for stationary varifolds. This dimension bound illustrates how the maximality condition constrains the local anisotropy and the local distribution of conductance magnitudes. Both the lower and upper attainability results have potential novel applications in modeling leaf venation patterns.