Optimal Shapes for Tree Roots

TL;DR

This paper investigates the optimal shapes of tree roots modeled through variational problems, deriving necessary conditions for optimality and analyzing the structure of solutions in two dimensions.

Contribution

It introduces new necessary optimality conditions for the shape of tree roots and characterizes the support of optimal solutions in 2D.

Findings

Support of optimal measure is nowhere dense in 2D

Necessary conditions for optimality derived

Existence and bounds of optimal measure previously established

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.