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

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. We study the $\Gamma$-convergence as $\epsilon$ $\rightarrow$ 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 $\epsilon$, where the leading term is of order $\epsilon$| log $\epsilon$|.