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Optimal transport problems with gradient penalization

Abstract : The optimal transportation problem was originally introduced by Monge in the 18th century; it consists in minimizing the total energy of the displacement of a given repartition of mass onto another given repartition of mass. This is mathematically expressed by: find the minimizer of the integral of c(x,T(x)) (where c(x,T(x)) is the cost to send x onto T(x)) among the maps T with prescribed image measure.This thesis is devoted to similar variational problems, which involve the Jacobian matrix of the transport map, meaning that the cost depends on three variables c(x,T(x),DT(x)); we typically add the Dirichlet energy to the transport functional in view to obtain a Sobolev-type penalization. This kind of constraints finds its motivations in continuum mechanics, incompressible elasticity or shape analysis, and a quite different mathematical approach than in the usual theory of optimal transportation is needed.We consider the following questions:- proper definition of the problem, in particular of the Dirichlet energy, thanks to the theory of Sobolev spaces with respect to a measure, and existence results;- characterizations of these minimizers: optimality of the monotone transport map on the real line, and Euler-Lagrange-like approach in any dimension;- selection of a minimizer via a Gamma-convergence-like penalization procedure (we multiply the Dirihlet energy with a vanishing positive parameter) where the transport cost is the Monge cost given by the distance (for which the optimal transport map is not unique);- other related problems and perspectives: dynamic Benamou-Brenier-like formulation, and dual Kantorovich-like formulation.
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Jean Louet. Optimal transport problems with gradient penalization. General Mathematics [math.GM]. Université Paris Sud - Paris XI, 2014. English. ⟨NNT : 2014PA112132⟩. ⟨tel-01070163⟩

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