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A model of irrigation network, where lower branches must be thicker in order to support the weight of the higher ones, was recently introduced in [7]. This leads to a countable family of ODEs, describing the thickness of every branch, solved by backward induction. The present paper determines what kind of measures can be irrigated with a finite weighted cost. Indeed, the boundedness of the cost depends on the dimension of the support of the irrigated measure, and also on the asymptotic properties of the ODE which determines the thickness of branches.

 

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 functional S(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 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. 

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. 

We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with C^{2,α} boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion. 

The paper studies a PDE model describing the elongation of a plant stem and its bending as a response to gravity. For a suitable range of parameters in the defining equations, it is proved that a feedback response produces stabilization of growth, in the vertical direction.