General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary

Abstract

A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a final distribution. The cost of the scheme encodes a higher transport efficiency the more mass is moved together, which automatically leads to optimal transportation networks with a hierarchical branching structure. The two major existing model formulations, either using mass fluxes (vector-valued measures) or patterns (probabilities on the space of particle paths), are rather different. Once their equivalence was established, the analysis of optimal networks could rest on both.The transportation cost of classical branched transport is a fractional power of the transported mass, and several model properties and proof techniques build on its strict concavity. We generalize the model and its analysis to the most general class of reasonable transportation costs, essentially increasing, subadditive functions. This requires several modifications or new approaches. In particular, for the equivalence between mass flux and pattern formulation it turns out advantageous to resort to a description via 1-currents, an intuition which already Xia exploited. In addition, some already existing arguments are given a more concise and perhaps simpler form. The analysis includes the well-posedness, a metrization and a length space property of the model cost, the equivalence between the different model formulations, as well as a few network properties.

Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass

“…The existence of minimizers and their properties are discussed in [3]. Note that mass fluxes are also known as divergence measure vector fields [13] or vector charges [12] or 1-dimensional normal currents [7].…”

“…Note that flat m-chains with finite mass and boundary mass are also known as normal m-currents [7, 4.1.23 & 4.2.23], [3,Rem. 2.29(2)].…”

“…The existence of minimizers for d J h (µ + , µ − ) is shown in [3,Cor. 2.20], and since each minimizer ρ for Let us first restrict ourselves to the case where h ′ (0) < ∞.…”

“…Following [17] or its generalization [3], the generalized branched transport model can be introduced as follows (where our notation is chosen slightly differently to emphasize the correspondence to the h-mass later). Throughout the article we consider Ω ⊂ R d to be the closure of an open bounded connected domain, and we denote by M(Ω) the set of Radon measures, by M + (Ω) ⊂ M(Ω) the subset of nonnegative measures, and by M(Ω; R d ) the set of R d -valued Radon measures on Ω.…”

“…">A transportation cost is a subadditive, nondecreasing, lower semi-continuous function6. Given a transportation cost h, the corresponding generalized branched transport cost of a polyhedralThe generalized branched transport cost of a mass flux ρ isthe relaxation of the generalized branched transport cost on polyhedral fluxes with respect to weak convergence of mass fluxes.The variational problem of finding optimal mass transportation schemes between a given mass source µ + ∈ M + (Ω) and a sink µ − ∈ M + (Ω) then isThe existence of minimizers and their properties are discussed in [3]. Note that mass fluxes are also known as divergence measure vector fields [13] or vector charges [12] or 1-dimensional normal currents [7].The definition of the h-mass of a flat chain follows the same strategy.where γ # µ denotes the pushforward of a measure µ under a function f and γ : [0, 1] → Ω is an injective Lipschitz curve.…”