Variational problems for tree roots and branches

  • Alberto Bressan
  • Michele PalladinoEmail author
  • Qing Sun


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 \(\mu \) describing the distribution of leaves, a sunlight functional \({\mathcal {S}}(\mu )\) computes the total amount of light captured by the leaves. For a measure \(\mu \) describing the distribution of root hair cells, a harvest functional \({\mathcal {H}}(\mu )\) computes the total amount of water and nutrients gathered by the roots. In both cases, we seek a measure \(\mu \) 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 \(\mu \), 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.

Mathematics Subject Classification

35R06 49Q10 92C80 



This research was partially supported by NSF with Grant DMS-1714237, “Models of controlled biological growth”.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA
  2. 2.Gran Sasso Science InstituteL’AquilaItaly

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