On the attainability of the singular Wiener bound

Abstract

We characterize the lower and upper attainability of the Wiener bound (also known as the conductive analogue of the Voigt-Reuss-Hill bound in elasticity theory) 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 prove 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.

Figures (7)

• Figure 1: Left: a picture of higher-order veins in a tropical forest tree, Ampelocera ruizii. The picture is reproduced from sack2013leaf, with permission from Wiley; Middle left: a reticulate $\mathbb{Z}^2$-periodic network that has full rank effective tensor; Middle right: a non-reticulate $\mathbb{Z}^2$-periodic network that has zero effective tensor; Right: a non-reticulate $\mathbb{Z}^2$-periodic network that has rank 1 effective tensor.
• Figure 2: For some integer $m\ge 2$, there are $m$ half lines with the unit outward tangent vectors denoted by $T_j$ and densities $w_j>0$ for $1\le j \le m$. In a stationary tangent cone the vectors $T_j$ and the weights $w_j$ satisfy \ref{['eq.mcone']}.
• Figure 3: Both networks are $\mathbb{Z}^2$-periodic stationary 1-varifolds, with one period indicated in the box enclosed by dotted lines. Theorem \ref{['t.forma2']} indicates that these are all appropriate models for leaf vein patterns, as leaves tend to maximize its hydraulic conductance to maintain its functions.
• Figure 4: Backlit lower leaf surface of Arbutus unedo of different times of a day. (Light area: infiltrated area; dark area: non-infiltrated area. A: 9 a.m.; B: 10 a.m.; C: 12p.m.; D: 4 p.m.; E: 5 p.m.) This picture is reproduced from Beyschlag1990, with permission from Springer Nature.
• Figure 5: Instability of maximal valencies of reducible stationary networks.

Theorems & Definitions (121)

• Theorem A: See Theorem \ref{['t.Qtohomotopy']}
• Corollary 1.1: See Theorem \ref{['t.characterizationoflowerboundn=2']}
• Theorem B: See Theorem \ref{['t.equivalence']}
• Corollary 1.2
• Theorem 1.4: See Theorem \ref{['t.countabledecompositionof1dmedium']}
• Theorem 1.5: See Theorem \ref{['t.dimensiononeofnontrivialmedium']}
• Theorem 1.6: See Theorem \ref{['t.realizabledimensionandlocaldimension']}
• Theorem 2.1: Riesz representation
• Lemma 2.2
• proof