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

We investigate the approximation of the Monge problem (minimizing \int\_$Ω$ |T (x) -- x| d$μ$(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 $ε$ $\rightarrow$ 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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