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| dμ(x) among the vector-valued maps T with prescribed image measure T # μ) by adding a vanishing Dirichlet energy, namely ε int_Omega |DT |2. We study the Gamma-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|.