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Optimal Shapes for Tree Roots

Alberto Bressan, Sondre T. Galtung, Qing Sun

Abstract

The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $μ$ 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.

Optimal Shapes for Tree Roots

Abstract

The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure describing the distribution of root hair cells, we seek to maximize a harvest functional , 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 , we prove that the support of an optimal measure is nowhere dense.

Paper Structure

This paper contains 9 sections, 11 theorems, 184 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of the basic functionals
  3. Harvest functionals
  4. Optimal irrigation plans
  5. The landscape function
  6. Necessary conditions for optimal tree roots
  7. The support of the optimal measure
  8. A lower Hölder estimate for the landscape function
  9. Proof of Lemma \ref{['l:43']}

Key Result

Lemma \oldthetheorem

For any positive Radon measure $\mu$ on $I\!\!R^d$ and any $\alpha\in [0,1]$, one has In particular, for every $r>0$ one has

Figures (4)

  • Figure 1: Proving that $\Phi(x)<c Z(x)$, at several points $x$ near $x_0$.
  • Figure 2: As shown in the proof of Lemma \ref{['l:42']}, in the shaded region inside $Q_\varepsilon$ the landscape function grows at least at a Hölder rate.
  • Figure 3: Taking $p = 1+{2\delta\over\pi}$, the conformal map $z\mapsto z^p$ transforms the half disc $D'$ into the domain $D_\delta$.
  • Figure 4: Left: the interval $\Gamma(r)$, where the average value for the solution $u$ of (\ref{['PPu']}) can be estimated. Right: the domain ${\cal D}^\gamma$ considered at (\ref{['Ph12']}).

Theorems & Definitions (18)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 8 more