On theWell‐Posednessof Branched Transportation

Abstract

We show in full generality the stability of optimal transport paths in branched transport: namely, we prove that any limit of optimal transport paths is optimal as well. This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks by Bernot, Caselles, and Morel), which has been addressed up to now only under restrictive assumptions. © 2020 Wiley Periodicals LLC

Stability of optimal traffic plans in the irrigation problem

“…In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [7], extending it to the Lagrangian framework. 2020 Mathematics Subject Classification.…”

“…The positive answer is already known above the critical threshold α > 1 − 1/d both for the Lagrangian and the Eulerian formulation. A positive answer for every α ∈ (0, 1) has been recently given for the Eulerian formulation in [7]. Although the Eulerian and Lagrangian problems have the same minimizers (see [14,16]), the Eulerian viewpoint carries less information than the Lagrangian one.…”

“…Strategy of the proof. Our proof relies on the general stability result proved for the Eulerian setting in [7,Theorem 1.1]. The classical way to associate to a Lagrangian traffic plan P an Eulerian transport path T = T P consists in integrating (w.r.t.…”

“…Taking P, {P n } n∈N as in Theorem 1.1, we consider the induced transport paths T , {T n } n∈N . One can easily show that T and {T n } n∈N satisfy the hypotheses of [7,Theorem 1.1], so that T is an optimal transport path for the marginals (µ − , µ + ). Nevertheless in principle it could happen that the cost of T as a transport path and the cost of P as a traffic plan do not coincide.…”

Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks

“…In the last decade, the communities of Calculus of Variations and Geometric Measure Theory made some efforts to study (Gilbert-)Steiner problems in many aspects, such as existence, regularity, stability and numerical feasibility (see for example [30,28,23,24,15,16,27,10,26,6,8,7] and references therein). Among all the significant results, we would like to mention recent works in [23,24] and [6,7], which are closely related to the present paper.…”