Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles

In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying a...

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Autor principal: Rossi, Julio Daniel
Publicado: 2014
Materias:
Mass transport
Monge-Kantorovich problems
P-Laplacian equation
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v256_n9_p3208_Mazon
http://hdl.handle.net/20.500.12110/paper_00220396_v256_n9_p3208_Mazon
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_00220396_v256_n9_p3208_Mazon2023-06-08T14:45:12Z Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles Rossi, Julio Daniel Mass transport Monge-Kantorovich problems P-Laplacian equation In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of -g1(x) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at y. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p→∞ to a double obstacle problem (with obstacles g1, g2) for the p-Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved. We also show that this problem can be interpreted as an optimal mass transport problem in which one can make the transport directly (paying a cost given by the Euclidean distance) or may hire a courier that cost g2(y)-g1(x) to pick up a unit of mass at y and deliver it to x. For this different interpretation we provide examples and a decomposition of the optimal transport plan that shows when we have to use the courier. © 2014 Elsevier Inc. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2014 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v256_n9_p3208_Mazon http://hdl.handle.net/20.500.12110/paper_00220396_v256_n9_p3208_Mazon
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Mass transport
Monge-Kantorovich problems
P-Laplacian equation
spellingShingle Mass transport
Monge-Kantorovich problems
P-Laplacian equation
Rossi, Julio Daniel
Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles
topic_facet Mass transport
Monge-Kantorovich problems
P-Laplacian equation
description In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of -g1(x) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at y. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p→∞ to a double obstacle problem (with obstacles g1, g2) for the p-Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved. We also show that this problem can be interpreted as an optimal mass transport problem in which one can make the transport directly (paying a cost given by the Euclidean distance) or may hire a courier that cost g2(y)-g1(x) to pick up a unit of mass at y and deliver it to x. For this different interpretation we provide examples and a decomposition of the optimal transport plan that shows when we have to use the courier. © 2014 Elsevier Inc.
author Rossi, Julio Daniel
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles
title_short Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles
title_full Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles
title_fullStr Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles
title_full_unstemmed Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles
title_sort mass transport problems for the euclidean distance obtained as limits of p-laplacian type problems with obstacles
publishDate 2014
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v256_n9_p3208_Mazon
http://hdl.handle.net/20.500.12110/paper_00220396_v256_n9_p3208_Mazon
work_keys_str_mv AT rossijuliodaniel masstransportproblemsfortheeuclideandistanceobtainedaslimitsofplaplaciantypeproblemswithobstacles
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