Abstract. 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, or the nervous and the cardiovascular systems.
Given a flow that transports a given mass distribution $\mu^-$ onto a target distribution $\mu^+$, along a 1-dimensional network, the transportation cost per unit length is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. This favors grouping of the particles' trajectories and prevents diffusion.
After a general description of the main features of the model, I will discuss in particular the issue of stability of minimizers under variations of the given mass distributions $\mu^-$ and $\mu^+$. I will also explain how one can exploit currents with coefficients in a normed group to describe a "multi-commodity" version of the problem, where the interaction between different types of transported goods is taken into account. Based on joint works with M. Colombo, A. De Rosa, A. Massaccesi, and R. Tione.