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

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??|