@article{ccedf078cf7b42d29c22c4420465acc5,
title = "OPTIMAL SHAPES FOR TREE ROOTS",
abstract = "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 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 have 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.",
author = "Alberto Bressan and Galtung, {Sondre T.} and Qing Sun",
note = "Funding Information: \ast Received by the editors August 12, 2021; accepted for publication April 25, 2022; published electronically August 15, 2022. https://doi.org/10.1137/21M1440281 Funding: The work of the first and third authors was partially supported by the National Science Foundation grant DMS-1714237, ``Models of controlled biological growth.{"}{"} The work of the second author was partially supported by the Research Council of Norway project 286822, ``Wave Phenomena and Stability - a Shocking Combination (WaPheS).{"}{"} \dagger Department of Mathematics, Penn State University, University Park, PA 16802 USA (bressan@math.psu.edu, qxs15@psu.edu). \ddagger Department of Mathematical Sciences, NTNU --Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (sondre.galtung@ntnu.no). Publisher Copyright: {\textcopyright} 2022 Society for Industrial and Applied Mathematics.",
year = "2022",
doi = "10.1137/21M1440281",
language = "English (US)",
volume = "54",
pages = "4757--4784",
journal = "SIAM Journal on Mathematical Analysis",
issn = "0036-1410",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",
}