Optimal Shapes for Tree Roots

Abstract

The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $\mu$ describing the distribution of root hair cells, we seek to maximize a harvest functional $\mathcal{H}$, computing the total amount of water and nutrients gathered by the roots, subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers had established the existence of an optimal measure, and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension $d=2$, we prove that the support of an optimal measure is nowhere dense.

30 pages, 4 figures

References in corpus (2)

• Optimal micropatterns in transport networks
• Variational Problems for Tree Roots and Branches

Cited by in corpus (1)

• Optimal quantization with branched optimal transport distances