TY - JOUR
T1 - Variational problems for tree roots and branches
AU - Bressan, Alberto
AU - Palladino, Michele
AU - Sun, Qing
N1 - Funding Information:
This research was partially supported by NSF with Grant DMS-1714237, “Models of controlled biological growth”. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Funding Information:
This research was partially supported by NSF with Grant DMS-1714237, ?Models of controlled biological growth?.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/2/1
Y1 - 2020/2/1
N2 - This paper studies two classes of variational problems introduced in Bressan and Sun (On the optimal shape of tree roots and branches. arXiv:1803.01042), related to the optimal shapes of tree roots and branches. Given a measure μ describing the distribution of leaves, a sunlight functionalS(μ) computes the total amount of light captured by the leaves. For a measure μ describing the distribution of root hair cells, a harvest functionalH(μ) computes the total amount of water and nutrients gathered by the roots. In both cases, we seek a measure μ that maximizes these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk or from the trunk to the leaves. Compared with Bressan and Sun, here we do not impose any a priori bound on the total mass of the optimal measure μ, and more careful a priori estimates are thus required. In the unconstrained optimization problem for branches, we prove that an optimal measure exists, with bounded support and bounded total mass. In the unconstrained problem for tree roots, we prove that an optimal measure exists, with bounded support but possibly unbounded total mass. The last section of the paper analyzes how the size of the optimal tree depends on the parameters defining the various functionals.
AB - This paper studies two classes of variational problems introduced in Bressan and Sun (On the optimal shape of tree roots and branches. arXiv:1803.01042), related to the optimal shapes of tree roots and branches. Given a measure μ describing the distribution of leaves, a sunlight functionalS(μ) computes the total amount of light captured by the leaves. For a measure μ describing the distribution of root hair cells, a harvest functionalH(μ) computes the total amount of water and nutrients gathered by the roots. In both cases, we seek a measure μ that maximizes these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk or from the trunk to the leaves. Compared with Bressan and Sun, here we do not impose any a priori bound on the total mass of the optimal measure μ, and more careful a priori estimates are thus required. In the unconstrained optimization problem for branches, we prove that an optimal measure exists, with bounded support and bounded total mass. In the unconstrained problem for tree roots, we prove that an optimal measure exists, with bounded support but possibly unbounded total mass. The last section of the paper analyzes how the size of the optimal tree depends on the parameters defining the various functionals.
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U2 - 10.1007/s00526-019-1666-1
DO - 10.1007/s00526-019-1666-1
M3 - Article
AN - SCOPUS:85075692237
SN - 0944-2669
VL - 59
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 1
M1 - 7
ER -