# The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile

arXiv: https://arxiv.org/abs/1407.7022 · HTML: https://mlsanity.com/paper/1407.7022/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2014.

## Built on by (most cited first, 1 of 1)

- [ICON: Learning Regular Maps Through Inverse Consistency](https://mlsanity.com/paper/2105.04459/) (2021)

## Abstract (arXiv)

We investigate the approximation of the Monge problem (minimizing int_Omega |T (x) -- x| dμ(x) among the vector-valued maps T with prescribed image measure T # μ) by adding a vanishing Dirichlet energy, namely ε int_Omega |DT |2. We study the Gamma-convergence as ε rightarrow 0, proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we analyze in details, when no optimal plan is induced by an H 1 map, we study the selected limit map, which is a new "special" Monge transport, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on ε, where the leading term is of order ε| log ε|.