Published Paper

Inserted: 12 feb 2016
Last Updated: 28 jun 2016

Journal: J. Math. Pures Appl.
Volume: 106
Number: 2
Pages: 237-279
Year: 2016
Doi: 10.1016/j.matpur.2016.02.009

Abstract:

We investigate the approximation of the Monge problem (minimizing $\int_\Omega
T(x)-x
\,\text{d}\mu(x)$ among the vector-valued maps $T$ with prescribed image measure $T_\#\mu$) by adding a vanishing Dirichlet energy, namely $\varepsilon\int_\Omega
DT
^2$. We study the $\Gamma$-convergence as $\varepsilon\to 0$, 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, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on $\varepsilon$, where the leading term is of order $\varepsilon
\log\varepsilon
$

Keywords: Monge problem, Optimal transport, $\Gamma$-convergence, Monotone transport, Density of smooth maps

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