Continuity and Estimates for Multimarginal Optimal Transportation Problems with Singular Costs
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We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence and some basic regularity of a maximizer for the dual problem (Kantorovich potential). This is then applied to obtain some estimates of the cost and to the study of continuity properties.
KeywordsMultimarginal optimal transportation Monge–Kantorovich problem Duality theory Coulomb cost
Mathematics Subject Classification49J45 49N15 49K30
This paper has been written during some visits of the authors at the Department of Mathematics of University of Pisa and at the Laboratoire IMATH of University of Toulon. The authors gratefully acknowledge the warm hospitality of both institutions. The research of the first and third authors is part of the project 2010A2TFX2 Calcolo delle Variazioni funded by the Italian Ministry of Research and is partially financed by the “Fondi di ricerca di ateneo” of the University of Pisa. The authors also would like to thank the anonymous referees for their valuable comments and suggestions to improve the paper.
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