Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and uniqueness of Monge minimizers for a Multi-marginal Optimal Transport problem with intermolecular interactions cost

Augusto Gerolin, Mircea Petrache, Adolfo Vargas-Jimenez

Abstract

We investigate a new multi-marginal optimal transport problem arising from a dissociation model in the Strong Interaction Limit of Density Functional Theory. In this short note, we introduce such dissociation model, the corresponding optimal transport problem as well as show preliminary results on the existence and uniqueness of Monge solutions assuming absolute continuity of at least two of the marginals. Finally, we show that such marginal regularity conditions are necessary for the existence of an unique Monge solution.

Existence and uniqueness of Monge minimizers for a Multi-marginal Optimal Transport problem with intermolecular interactions cost

Abstract

We investigate a new multi-marginal optimal transport problem arising from a dissociation model in the Strong Interaction Limit of Density Functional Theory. In this short note, we introduce such dissociation model, the corresponding optimal transport problem as well as show preliminary results on the existence and uniqueness of Monge solutions assuming absolute continuity of at least two of the marginals. Finally, we show that such marginal regularity conditions are necessary for the existence of an unique Monge solution.
Paper Structure (5 sections, 5 theorems, 63 equations)

This paper contains 5 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Dissociation energy of many-electrons quantum systems
  3. A Dissociation model in the Strong Interaction limit of DFT
  4. Multi-marginal Optimal Transport problem
  5. Existence and uniqueness of Monge solutions

Key Result

Lemma 1

Let $\eta>0$, $r^{(\eta)}_\alpha,r^{(\eta)}_\beta\in\mathbb R^d$, $\rho_\alpha,\rho_\beta\in\mathcal{P}(\mathbb R^d)$ and $\rho_\eta$ as defined in rhoR1. Then, $\Pi_{{N_\alpha},{N_\beta}}^\eta(\rho_\alpha,\rho_\beta)\subset \Pi(\rho_\eta)$.

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 4
  • Remark 5
  • Definition 6
  • Remark 7
  • ...and 4 more