Calculus of Variations and Geometric Measure Theory

L. Ambrosio - A. Pratelli

Existence and stability results in the $L^1$ theory of optimal transportation

created on 29 May 2002
modified on 22 Dec 2003


Published Paper

Inserted: 29 may 2002
Last Updated: 22 dec 2003

Year: 2002

Lecture Notes in Mathematics ``Optimal transportation and applications'' (CIME Series, Martina Franca, 2001) {\bf 1813}, L.A. Caffarelli and S. Salsa Eds., 123--160, 2003.


This is conceived both as a survey paper and a research paper on the theory of optimal transportation of mass, with special emphasis on the case cost=distance. Concerning the general theory, we show a duality formula under quite general growth conditions on the cost, providing also a counterexample. Conserning the case cost=distance, we revisit several recent papers on this argument (starting from Sudakov's one) and using some tools from the theory of $\Gamma$-convergence we provide general existence and stability results for the problem.