Stability for the mailing problem

TL;DR

This paper proves the stability of optimal traffic plans in the mailing problem for certain exponents, resolving an open question and analyzing the regularity and structure of minimizers in optimal transportation networks.

Contribution

It establishes the stability of optimal traffic plans for the mailing problem above a critical exponent, advancing understanding of their structure and regularity.

Findings

Optimal traffic plans are stable under coupling variations for bove the critical exponent.

Finitely many connected components meet at any branching point in an optimal plan.

The results solve an open problem from the book 'Optimal transportation networks'.

Abstract

We prove that optimal traffic plans for the mailing problem in $\mathbb{R}^d$ are stable with respect to variations of the given coupling, above the critical exponent $\alpha=1-1/d$, thus solving an open problem stated in the book "Optimal transportation networks", by Bernot, Caselles and Morel. We apply our novel result to study some regularity properties of the minimizers of the mailing problem. In particular, we show that only finitely many connected components of an optimal traffic plan meet together at any branching point.