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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 $\int_\Omega |T(x)-x| d\mu(x)$ among the vector-valued maps $T$ with prescribed image measure $T_\#\mu$) by adding a vanishing Dirichlet energy, namely $\epsilon\int_\Omega |DT|^2$, where $\epsilon \to 0$. We study the $\Gamma$-convergence as $\epsilon\to 0$, 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, different from the monotone one, and we find the precise asymptotics of the optimal cost depending on $\epsilon$, where the leading term is of order $\epsilon|\log\epsilon|$
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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 (2), pp.237-279 ⟨10.1016/j.matpur.2016.02.009⟩. ⟨hal-01052294v1⟩
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