Abstract

Branched optimal transport is a variant of optimal transport in which an economy of scale principle is present. Grouped transportation is favoured, leading to mass moving on networks, instead of following geodesics.

This talk is concerned with several conjectures concerning fractal measures, which are related to two different branched transport type models. The first model is the standard branched optimal transport problem developed by Xia and by Bernot, Caselles and Morel as a generalization of the classical model by Gilbert for communication networks. The second one is a model for pattern formation of the magnetic field in type-I superconductors, which was introduced as a reduced model for the Ginzburg-Landau functional by Conti, Goldman, Otto and Serfaty. To both models we can associate an irrigation energy, which is the cost of transporting a Dirac delta to a target measure.

In recent years, several conjectured were formulated concerning problems in which the irrigation energy, which favours mass concentration, is paired with a non local energy or constraint, which favours a diffused behaviour. In all cases, the optimal measure seems to exhibit fractal behaviour. I this talk I present some of these conjectures and recent advances in solving them. In particular, concerning the first model, I introduce the conjecture proposed by Xia, Santambrogio and Pegon for the “unit” ball for branched optimal transport, and the conjecture on the optimal shape of tree roots proposed by Bressan. For the second model, I present a conjecture by Conti, Otto and Serfaty on the behaviour of the magnetic field on the boundary of a superconductor, when the adherence to the external magnetic field is imposed weakly. While the models and the techniques used are different, the spirit of the results is similar. Hence, analogies between the two models are an interesting source of ideas for future research and will be underlined.

This talk is based on some works with Michael Goldman, Melanie Koser, Felix Otto and Paul Pegon.

Seminari di Equazioni Differenziali e Applicazioni