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A continuous model of transportation in the Heisenberg group

Michele Circelli, Albert Clop

Abstract

We present a minimization problem with a horizontal divergence-type constraint in the Heisenberg group. Our study explores its dual formulation and examines its relationship with the congested optimal transport problem, for $1 < p < +\infty$, as well as the Monge-Kantorovich problem, in the limite case $p=1$.

A continuous model of transportation in the Heisenberg group

Abstract

We present a minimization problem with a horizontal divergence-type constraint in the Heisenberg group. Our study explores its dual formulation and examines its relationship with the congested optimal transport problem, for , as well as the Monge-Kantorovich problem, in the limite case .
Paper Structure (20 sections, 24 theorems, 219 equations)

This paper contains 20 sections, 24 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Heisenberg group $\mathbb{H}^n$
  4. Geodesics in $\mathbb{H}^n$
  5. $H\Omega$-valued finite Radon measures
  6. Mollification in $\mathbb{H}^n$
  7. Riemannian approximation of $\mathbb{H}^n$
  8. Optimal transport theory in $\mathbb{H}^n$
  9. Properties of transport rays and Kantorovich potentials in $\mathbb{H}^n$
  10. Transport rays and transport set
  11. Differentiability of Kantorovich potential
  12. The Beckmann problem for $p=1$
  13. Vector transport density in $\mathbb{H}^n$
  14. The problem
  15. Lagrangian formulation
  16. ...and 5 more sections

Key Result

Theorem 2.1

A non trivial geodesics parametrized on $[0,1]$ and starting from $0$ is the restriction to $[0,1]$ of the curve $\sigma_{\chi,\theta}(t)=\left(x_1(t),\ldots,x_{2n+1}(t)\right)$ either of the form for some $\chi \in \mathbb{R}^{2n}\setminus\{0\}$ and $\theta\in [-2\pi,2\pi]\setminus\left\{0\right\}$, or of the form for some $\chi \in \mathbb{R}^{2n}\setminus\{0\}$ and $\theta=0$. In particular,

Theorems & Definitions (49)

  • Theorem 2.1
  • Theorem 2.2: Kantorovich duality theorem
  • Lemma 2.3
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 39 more