树根和树枝的变分问题
Variational Problems for Tree Roots and Branches
摘要：
本文研究了文献[7]中提出的两类与树根和树枝的最优形状有关的变分问题。给定描述叶子分布的度量$μ$，阳光函数$§(μ)$计算叶子捕获的光的总量。对于描述根毛细胞分布的测量$μ$，收获函数$\H(μ)$计算根收集的水和营养物的总量。在这两种情况下，我们寻求一种措施$μ$，最大限度地提高这些功能受到支？运输成本，用于将营养物质从根运输到树干或从树干运输到叶子。与文献[7]相比，本文没有对最优测度$μ$的总质量施加任何先验界，因而需要更仔细的先验估计.在分枝的无约束最优化问题中，我们证明了存在一个最优测度，它具有有界的支撑和有界的总质量。在树根的无约束问题中，我们证明了存在一个最优的措施，有界的支持，但可能无界的总质量。本文的最后一节分析了最优树的大小如何取决于定义各种泛函的参数。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
