Calculus of Variations and Geometric Measure Theory

A. Marchese - A. Massaccesi

An optimal irrigation network with infinitely many branching points

created by massaccesi on 31 Jul 2014
modified on 12 Jan 2016


ESAIM Control Optim. Calc. Var.

Inserted: 31 jul 2014
Last Updated: 12 jan 2016

Journal: ESAIM: COCV
Year: 2014


The Gilbert-Steiner problem is a mass transportation problem, where the cost to transport a measure $\mu_-$ onto a measure $\mu_+$ depends on the network used to move the mass and it is proportional to a certain power of the ``flow''. In this paper, we introduce a new formulation of the problem which turns it into the minimization of a convex functional. By associating to $\mu_-$ and $\mu_+$ a group $G$ and a 0-dimensional current $B$ with coefficients in $G$, we prove that the Gilbert-Steiner problem is equivalent to the problem of finding a mass minimizer, among all 1-dimensional currents $Z$ with coefficients in $G$ having boundary $\partial Z = B$. This framework allows us to define calibrations, which can be used to prove the optimality of concrete configurations. We apply this technique to prove the optimality of a certain irrigation network, having the topological property mentioned in the title.