The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2016

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

Résumé

We investigate the approximation of the Monge problem (minimizing \int_Ω |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_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|.
Fichier principal
Vignette du fichier
deplousan_cvgmt.pdf (456.46 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01052294 , version 1 (25-07-2014)
hal-01052294 , version 2 (30-12-2016)

Identifiants

Citer

Luigi de Pascale, Jean Louet, Filippo Santambrogio. The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile. Journal de Mathématiques Pures et Appliquées, 2016, 106, pp.237 - 279. ⟨10.1016/j.matpur.2016.02.009⟩. ⟨hal-01052294v2⟩
612 Consultations
210 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More