论文标题
树根和树枝的变分问题
Variational Problems for Tree Roots and Branches
论文作者
论文摘要
本文研究了[7]中引入的两类变异问题，与树根和分支的最佳形状有关。给定描述叶子分布的度量$μ$，阳光功能性$§（μ）$计算叶子捕获的光总数。对于描述根毛细胞分布的度量$μ$，收获功能性$ \ h（μ）$计算根部收集的水和营养总量。在这两种情况下，我们都会寻求一种$μ$，以最大化这些功能，但要承担拉米的运输成本，用于将养分从根部运输到树干或从树干到叶子到叶子。与[7]相比，在这里，我们不对最佳度量的总质量$μ$施加任何先验结合，因此需要更加谨慎的先验估计。在分支机构的不受约束的优化问题中，我们证明存在一个最佳度量，并具有有限的支持和有限的总质量。在树根的不受约束的问题中，我们证明存在一个最佳度量，并具有有限的支撑，但可能是无限的总质量。本文的最后一部分分析了最佳树的大小如何取决于定义各种功能的参数。
This paper studies two classes of variational problems introduced in [7], related to the optimal shapes of tree roots and branches. Given a measure $μ$ describing the distribution of leaves, a sunlight functional $§(μ)$ computes the total amount of light captured by the leaves. For a measure $μ$ describing the distribution of root hair cells, a harvest functional $\H(μ)$ 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 rami?ed transportation cost, for transporting nutrients from the roots to the trunk or from the trunk to the leaves. Compared with [7], 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.
