Yann Brenier, Marjolaine Puel (2010)
ESAIM: Control, Optimisation and Calculus of Variations
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A multiphase generalization of the Monge–Kantorovich optimal transportation problem is addressed. Existence of optimal solutions is established. The optimality equations are related to classical Electrodynamics.
Guy Bouchitté, Giuseppe Buttazzo (2001)
Journal of the European Mathematical Society
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We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.
Levin, V.L. (2004)
Journal of Mathematical Sciences (New York)
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José Manuel Mazón, Julio Daniel Rossi, Julián Toledo (2014)
Mathematica Bohemica
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We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a -Laplacian system....
Luca Granieri, Francesco Maddalena (2013)
ESAIM: Control, Optimisation and Calculus of Variations
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By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds...
Frisch, Uriel, Sobolevskii, A. (2004)
Journal of Mathematical Sciences (New York)
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Christian Léonard (2011)
ESAIM: Control, Optimisation and Calculus of Variations
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The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to -conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.
W. Szwarc (1965)
Applicationes Mathematicae
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Patrick Bernard, Boris Buffoni (2007)
Journal of the European Mathematical Society
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We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.
Christian Léonard (2011)
ESAIM: Control, Optimisation and Calculus of Variations
Similarity:
The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to -conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.