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|.
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