TY - JOUR
T1 - On the optimal shape of tree roots and branches
AU - Bressan, Alberto
AU - Sun, Qing
N1 - Funding Information:
This research was partially supported by NSF with grant DMS-1714237, “Models of controlled biological growth”.
Publisher Copyright:
© 2018 World Scientific Publishing Company.
PY - 2018/12/30
Y1 - 2018/12/30
N2 - This paper introduces two classes of variational problems, determining optimal shapes for tree roots and branches. Given a measure μ, describing the distribution of leaves, we introduce a sunlight functional (μ) computing the total amount of light captured by the leaves. On the other hand, given a measure μ describing the distribution of root hair cells, we consider a harvest functional (μ) computing the total amount of water and nutrients gathered by the roots. In both cases, we seek to maximize these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk and from the trunk to the leaves. The main results establish various properties of these functionals, and the existence of optimal distributions. In particular, we prove the upper semicontinuity of and, together with a priori estimates on the support of optimal distributions.
AB - This paper introduces two classes of variational problems, determining optimal shapes for tree roots and branches. Given a measure μ, describing the distribution of leaves, we introduce a sunlight functional (μ) computing the total amount of light captured by the leaves. On the other hand, given a measure μ describing the distribution of root hair cells, we consider a harvest functional (μ) computing the total amount of water and nutrients gathered by the roots. In both cases, we seek to maximize these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk and from the trunk to the leaves. The main results establish various properties of these functionals, and the existence of optimal distributions. In particular, we prove the upper semicontinuity of and, together with a priori estimates on the support of optimal distributions.
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U2 - 10.1142/S0218202518500604
DO - 10.1142/S0218202518500604
M3 - Article
AN - SCOPUS:85056221519
SN - 0218-2025
VL - 28
SP - 2719
EP - 2762
JO - Mathematical Models and Methods in Applied Sciences
JF - Mathematical Models and Methods in Applied Sciences
IS - 14
ER -