### Abstract

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.

Original language | English (US) |
---|---|

Article number | 7 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 59 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2020 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*59*(1), [7]. https://doi.org/10.1007/s00526-019-1666-1

}

*Calculus of Variations and Partial Differential Equations*, vol. 59, no. 1, 7. https://doi.org/10.1007/s00526-019-1666-1

**Variational problems for tree roots and branches.** / Bressan, Alberto; Palladino, Michele; Sun, Qing.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Variational problems for tree roots and branches

AU - Bressan, Alberto

AU - Palladino, Michele

AU - Sun, Qing

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.

UR - http://www.scopus.com/inward/record.url?scp=85075692237&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075692237&partnerID=8YFLogxK

U2 - 10.1007/s00526-019-1666-1

DO - 10.1007/s00526-019-1666-1

M3 - Article

AN - SCOPUS:85075692237

VL - 59

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 1

M1 - 7

ER -