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Title: On the optimal shape of tree roots and branches
This paper introduces two classes of variational problems, determining optimal shapes for tree roots and branches. Given a measure [Formula: see text], describing the distribution of leaves, we introduce a sunlight functional [Formula: see text] computing the total amount of light captured by the leaves. On the other hand, given a measure [Formula: see text] describing the distribution of root hair cells, we consider a harvest functional [Formula: see text] 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 [Formula: see text] and [Formula: see text], together with a priori estimates on the support of optimal distributions.
Award ID(s):
Publication Date:
Journal Name:
Mathematical Models and Methods in Applied Sciences
Page Range or eLocation-ID:
2763 to 2801
Sponsoring Org:
National Science Foundation
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