Abstract
We investigate the approximation of the Monge problem (minimizing ∫Ω|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∫Ω|DT|2. We study the Γ-convergence as ε→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, 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|logε|.
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