## Abstract

In this chapter we connect the Monge-Kantorovich problem, which is mainly “Lagrangian” in spirit, with other models which could describe optimal transport issues, in Eulerian form. This primarily concerns the original Monge cost | *x* − *y* | , rather than other power costs. We see the equivalence with Beckmann’s *continuous transportation model*, which consists in \(\min \int \vert \mathbf{w}\vert \mathrm{d}x\,:\, \nabla \cdot \mathbf{w} =\mu -\nu \}\). We also analyze properties and summability of the optimal **w**, in relation to the notion of transport density and we develop a framework of transport problems via measures on paths. These ideas are then applied, in the discussion section, to two variants models, which provide a useful modeling for other transport-related phenomena: traffic congestion and branched structures.

### Keywords

- Transporter Density
- Continuous Transportation Model
- Minimum Flow Problem
- Absolute Continuity
- Monge Problem

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

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- 1.
- 2.
The strange notation

**w**_{[γ]}is chosen so as to distinguish from the object**w**_{ Q }that we will introduce in Section 4.2.3. - 3.
Pay attention to the use of the gradient of the Kantorovich potential

*u*: we are using the result of Lemma 3.6 which provides differentiability of*u*on transport rays. - 4.
- 5.
Note that this is just an example of Nash equilibrium with a continuum of players, as we will see in Section 7.4.3 and, more particularly, in Box 7.3.

- 6.
For a survey on the continuous framework, see also 111.

- 7.
Note that, even for

*Γ*= {*γ*} (which is the most restrictive case), assuming \(\mu,\nu \in L^{\infty }\), considerations from incompressible fluid mechanics in [83] allow to build a*Q*such that \(i_{Q} \in L^{\infty }\). - 8.
Note that the same duality trick is also used in the discrete problem over networks, where solving the dual problem is much more convenient than the original one.

- 9.
This procedure is just a particular case of what is usually called

*forward automatic differentiation*. - 10.
- 11.
Observe that the same result can also be directly obtained from duality arguments, as it is done in Theorem 2.1 in [80].

- 12.
To have an exact equivalence between the Steiner minimal connection problem and Gilbert problem with

*α*= 0, one needs to consider a measure*μ*composed of a unique Dirac mass, so that every point in \(\mathop{\mathrm{spt}}\nolimits (\nu )\) must be connected to it, hence getting connectedness of the graph. - 13.
Unfortunately, to cope with the language usually adopted in branched transport, we cannot be completely coherent with the rest of the chapter, where we called “cycles” what we call here “strong cycles.”

- 14.
For triple junctions, which are the most common ones, this gives interesting results: when

*α*= 0, we get the well-known condition about Steiner trees, with three 120^{∘}angles, and for*α*= 0. 5, we have a 90^{∘}angle (see**Ex**(30)); yet, in general, the angles depend on the masses \(\theta _{k}\). - 15.
We already met rectifiable sets when studying the differentiability of convex functions, in Chapter 1: 1-rectifiable sets are defined as those sets which are covered, \(\mathcal{H}^{1}\)-a.e., by a countable union of Lipschitz curves. Anyway, the reader can easily pretend that “1-rectifiable” is a synonym of “1D” and it should be enough to follow the rest of the discussion.

- 16.
We define rectifiable vector measures as those which can be expressed in the form \([E,\tau,\theta ]\) with

*E*rectifiable. This language is borrowed from that of rectifiable currents, but currents can be considered vector measures. The reader can look at [161] for more details. - 17.
Indeed, even if not completely evident, it can be proven that, at least in the case where \(\mathop{\mathrm{spt}}\nolimits (\mu ) \cap \mathop{\mathrm{spt}}\nolimits (\nu ) =\emptyset\), the condition \(d_{\alpha }(\mu,\nu ) < +\infty \) is actually equivalent to \(d_{\alpha }(\mu,\delta _{0}),d_{\alpha }(\nu,\delta _{0}) < +\infty \).

- 18.
- 19.
The terminology has been introduced for this very purpose by the authors of [48].

- 20.
*We can give a precise definition in the following way: for x,y ∈Ω, define*\([x,y] =\{\omega \in \mathcal{C}: \exists \,s < t\text{ such that }\omega (s) = x\text{ and }\omega (t) = y\}\)*; we say that Q is cycle-free if there are not x*_{1}*,…,x*_{n}*with x*_{1}*= x*_{n}*such that Q([x*_{i}*,x*_{i+1}*]) > 0 or Q([x*_{i+1}*,x*_{i}*]) > 0 for all i = 1,…,n − 1.* - 21.
But it does not seem to come from a dual problem.

- 22.
*Other Holder results exist under different assumptions, if ν admits different dimensional lower bounds, i.e., it is Ahlfors regular of another exponent k < d, thus obtaining*\(\beta = d(\alpha -(1 - \frac{1} {k}))\)*; see*[78].

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Santambrogio, F. (2015). Minimal flows, divergence constraints, and transport density. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_4

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