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

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1 Dipartimento di Matematica Pisa 2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision 3 LM-Orsay - Laboratoire de Mathématiques d-Orsay

Abstract : We investigate the approximation of the Monge problem minimizing \int Ω |T x − x| dµx among the vector-valued maps T with prescribed image measure T # µ by adding a vanishing Dirichlet energy, namely ε \int Ω |DT |^2. We study the Γ-convergence as ε → 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|.

Keywords : 49J45 Optimal transport Monge problem monotone transport Γ-convergence density of smooth maps AMS subject classification 49J30

Autor: Luigi De Pascale - Jean Louet - Filippo Santambrogio -

Fuente: https://hal.archives-ouvertes.fr/

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