树木根、枝的最佳形状
On the Optimal Shape of Tree Roots and Branches
摘要：
本文介绍了两类确定树根和树枝最佳形状的变分问题。给定一个度量$μ$，描述叶片的分布，我们引入了一个太阳光函数$§(μ)$计算叶片捕获的总光量。另一方面，给出一个描述根毛细胞分布的测量$μ$，我们考虑一个计算根收集的水和营养的总量的收获函数$\H(μ)$。在这两种情况下，我们寻求最大限度地提高这些功能的分歧运输成本，从根运输养分的树干和从树干的叶子。主要结果建立了这些泛函的各种性质，并存在最优分布。特别地，我们证明了$§$和$\H$的上连续性，以及最优分布的支持下的先验估计。
英文摘要：
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 $\H(μ)$ 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 $\H$, together with a priori estimates on the support of optimal distributions.
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