Solutions to two conjectures in branched transport: stability and regularity of optimal paths
Seminar with Dr. Antonio De Rosa
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. The transportation cost in these models is proportional to a concave power α in (0,1) of the intensity of the flow.
In this seminar, Dr De Rosa focuses on the stability of the optimal transports, for variations of the source and target measures. The stability was known for α bigger than a critical threshold, but in this seminar, stability is proved for every exponent α in (0,1), and a counterexample for α =0 will be provided. Thus, a conjecture of the book "Optimal transportation networks", by Bernot, Caselles and Morel, is completely solved. Moreover, the robustness of the proof allows getting the stability for more general lower semicontinuous cost functionals.
Furthermore, Dr De Rosa will prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the book mentioned above. This latter result is used to show the regularity of the optimal networks.
(Joint works with Maria Colombo and Andrea Marchese)