Optimal Shapes for Tree Roots

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, subjec...

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Bibliographic Details
Published inarXiv.org
Main Authors Bressan, Alberto, Galtung, Sondre T, Sun, Qing
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 03.05.2022
Subjects
Mathematics - Optimization and Control
Moisture content
Nutrients
Optimization
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2108.05254