We want to perform a variational treatment of the mechanisms governing network formation in biology (leaf venation, vascular system. . . ). The existing literature suggests taking into account two distinct energy effects: energy dissipation via Joule effect and a material cost taking into account diffusion effects, among other things. The Joule effect favours a high capacity of the network, while the material cost prefers a smaller network. The conductance, that represents the network, then realizes the balance between the two effects. We expect concentration on a 1D network as the diffusion intensity, represented by a parameter D > 0 tends to 0. After two previous meetings between the project members, we realized that the limiting contribution of the material cost should be given by a variant of the branched transport energy (a non-convex variant of optimal transport), but it was still unclear how the Joule effect is relaxed in the limit and whether it is coupled with this limiting material cost.
Our aim is to clarify this question by rigorous mathematical analysis and numerical simulations. The CIRM offers the perfect environement for collaborative coding, and is easily  reachable by the three members. A duration of one week is a reasonable balance between schedule constraints of the members, and the intended objectives of the meeting.