Calculus of Variations and Geometric Measure Theory
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T. Champion - L. De Pascale - P. Juutinen

The $\infty$-Wasserstein distance: local solutions and existence of optimal transport maps

created by depascal on 07 May 2008


Published Paper

Inserted: 7 may 2008

Journal: SIAM Journal of Mathematical Analysis
Volume: 40
Number: 1
Pages: 1-20
Year: 2008


We consider the non-nonlinear optimal transportation problem of minimizing the cost functional $\C_\infty(\lambda)= \lambda\text{-}\esssup_{(x,y) \in \Omega^2}
$ in the set of probability measures on $\Omega^2$ having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of ``local'' solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

Keywords: Monge problem, infinite Wasserstein distance, restrictable solutions, infinite cyclical monotonicity


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