We investigate the approximation of the Monge problem (minimizing ∫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∫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 H1 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|logε|.
We investigate the approximation of the Monge problem (minimizing ?????|T(x)???x|d??(x) among the vector-valued maps T with prescribed image measure $T_\\#\mu$) by adding a vanishing Dirichlet energy, namely ???????|DT|2, where ?????0. We study the ??-convergence as ?????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 H1 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 ??, where the leading term is of order ??|log??|
The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile / De Pascale, Luigi; Louet, Jean; Santambrogio, Filippo;. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 1776-3371. - STAMPA. - 106:(2016), pp. 237-279. [10.1016/j.matpur.2016.02.009]
The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile
DE PASCALE, LUIGI;
2016
Abstract
We investigate the approximation of the Monge problem (minimizing ?????|T(x)???x|d??(x) among the vector-valued maps T with prescribed image measure $T_\\#\mu$) by adding a vanishing Dirichlet energy, namely ???????|DT|2, where ?????0. We study the ??-convergence as ?????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 H1 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 ??, where the leading term is of order ??|log??|File | Dimensione | Formato | |
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