Title: Improved stability of optimal traffic paths,
Abstract: 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 measure $\mu^-$ onto a target measure
$\mu^+$, 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.
In this paper we address an open problem in the book "Optimal transportation
networks" by Bernot, Caselles and Morel and we improve the stability for
optimal traffic paths in the Euclidean space $\mathbb{R}^d$, with respect to
variations of the given measures $(\mu^-,\mu^+)$, which was known up to now
only for $\alpha>1-\frac1d$. We prove it for exponents $\alpha>1-\frac1{d-1}$
(in particular, for every $\alpha \in (0,1)$ when $d=2$), for a fairly large
class of measures $\mu^+$ and $\mu^-$.