Solutions to two conjectures in branched transport: stability and regularity of optimal paths

Prof. Antonio De Rosa, Department of Mathematics, University of Maryland

Transport models involving branched structures are employed to describe several biological, natural and supply-demand systems. The transportation cost in these models is proportional to a concave power of the intensity of the flow.

In this talk, we focus on the stability of optimal transports with respect to variations of the source and target measures. Stability was known to hold just for special regimes (supercritical concave powers degenerating with the dimension). We prove that stability holds for every lower semicontinuous cost functional continuous in 0 and we provide counterexamples when these assumptions are not satisfied. Thus we completely solve a conjecture of Bernot, Caselles and Morel. 

To conclude, we prove stability for the mailing problem, too. This was completely open in the literature and allows us to obtain the regularity of the optimal networks.