Young Researchers Workshop: Kinetic models in biology and social sciences

Stability and regularity of optimal paths in branched transport

Antonio De Rosa

New York University

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. Given a flow (traffic path) that transports a given source onto a given target measure along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power ? ? (0, 1) of the intensity of the flow. The talk introduces the model and focuses on the stability for optimal traffic paths, with respect to variations of the source and target measures. The stability of optimal traffic paths was known when ? is bigger than a critical threshold, but we prove it also for other exponents below the threshold (for instance, for any ? ? (0, 1) in dimension 2). Moreover, we prove the stability in the so called “mailing problem”, which was completely unknown in the literature. We use this result to show the regularity of the optimal networks for the mailing problem. (Joint works with Maria Colombo and Andrea Marchese)