The Monge problem with vanishing gradient penalization: vortices and asymptotic profile
Résumé
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 $\epsilon\int_\Omega |DT|^2$, where $\epsilon \to 0$. We study the $\Gamma$-convergence as $\epsilon\to 0$, 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, 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 $\epsilon|\log\epsilon|$
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