Calculus of Variations and Geometric Measure Theory

M. Colombo - A. De Rosa - A. Marchese

Improved stability of optimal traffic paths

created by derosa on 03 Jan 2017
modified on 11 Jan 2022


Published Paper

Inserted: 3 jan 2017
Last Updated: 11 jan 2022

Journal: Calc. Var. Partial Differential Equations
Year: 2018


Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure $\mu^-$ onto a target measure $\mu^+$, along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow.

In this paper we address an open problem in the book "Optimal transportation networks" by Bernot, Caselles and Morel and we improve the stability for optimal traffic paths with respect to variations of the given measures $(\mu^-,\mu^+)$, which was known up to now only for $\alpha>1-\frac1d$.

We prove it for exponents $\alpha>1-\frac1{d-1}$ (in particular, for every $\alpha \in (0,1)$ when $d=2$), for a fairly large class of measures $\mu^+$ and $\mu^-$.