Analysis Seminar

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

Speaker: Antonio De Rosa, CIMS

Location: Warren Weaver Hall 1302

Date: Thursday, February 7, 2019, 11 a.m.

Synopsis:

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 source onto a 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 $\alpha \in
(0,1)$ of the intensity of the flow.
We introduce the model and focus 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 $\alpha$ is bigger than
a critical threshold, but we prove it for every exponent $\alpha \in
(0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely
solve a conjecture of the book Optimal transportation networks by Bernot,
Caselles and Morel.
Moreover, we prove the stability also for the mailing problem, which was
completely open in the literature, solving another conjecture of the
aforementioned book. We use this latter result to show the regularity of
the optimal networks for the mailing problem.
(Joint works with Maria Colombo and Andrea Marchese)