Artículo
Abstract:
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.
Registro:
| Documento: | Artículo |
| Título: | Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles |
| Autor: | Mazón, J.M.; Rossi, J.D.; Toledo, J. |
| Filiación: | Departament d'Anàlisi Matemàtica, Universitat de València, Valencia, Spain Departamento de Análisis Matemático, Universidad de Alicante, Ap. correos 99, 03080, Alicante, Spain Departamento de Matemática, FCEyN UBA, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina |
| Palabras clave: | Mass transport; Monge-Kantorovich problems; P-Laplacian equation |
| Año: | 2014 |
| Volumen: | 256 |
| Número: | 9 |
| Página de inicio: | 3208 |
| Página de fin: | 3244 |
| DOI: | http://dx.doi.org/10.1016/j.jde.2014.01.039 |
| Título revista: | Journal of Differential Equations |
| Título revista abreviado: | J. Differ. Equ. |
| ISSN: | 00220396 |
| CODEN: | JDEQA |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v256_n9_p3208_Mazon |
Referencias:
- Ambrosio, L., Lecture notes on optimal transport problems (2003) Lecture Notes in Math., 1812, pp. 1-52. , Springer, Berlin, Mathematical Aspects of Evolving Interfaces
- Ambrosio, L., Fusco, N., Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (2000) Oxford Math. Monogr.
- Ambrosio, L., Pratelli, A., Existence and stability results in the L1 theory of optimal transportation (2003) Lecture Notes in Math., 1813, pp. 123-160. , Springer, Berlin, Optimal Transportation and Applications
- Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011) Universitext, , Springer, New York
- Dellacherie, C., Meyer, P.A., Probabilities and Potential (1978) North-Holland Math. Stud., , North-Holland Publishing Co., Amsterdam
- Evans, L.C., Partial differential equations and Monge-Kantorovich mass transfer (1999) Current Developments in Mathematics, pp. 65-126. , Int. Press, Boston
- Evans, L.C., Gangbo, W., Differential equations methods for the Monge-Kantorovich mass transfer problem (1999) Mem. Amer. Math. Soc., 137 (653)
- Mazón, J.M., Rossi, J.D., Toledo, J., An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary (2014) Rev. Mat. Iberoamericana., , in press
- Mazón, J.M., Rossi, J.D., Toledo, J., An optimal matching problem for the Euclidean distance (2014) SIAM J. Math. Anal., 46, pp. 233-255
- Talenti, G., Inequalities in rearrangement invariant function spaces (1994) Nonlinear Analysis, Function Spaces and Applications, vol. 5, pp. 177-230. , Prometheus, Prague
- Villani, C., Topics in Optimal Transportation (2003) Grad. Stud. Math., 58
- Villani, C., Optimal Transport, Old and New (2009) Grundlehren Math. Wiss., 338. , Springer-Verlag, Berlin-New York
Citas:
---------- APA ----------
Mazón, J.M., Rossi, J.D. & Toledo, J. (2014) . Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles. Journal of Differential Equations, 256(9), 3208-3244.http://dx.doi.org/10.1016/j.jde.2014.01.039
---------- CHICAGO ----------
Mazón, J.M., Rossi, J.D., Toledo, J. "Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles" . Journal of Differential Equations 256, no. 9 (2014) : 3208-3244.http://dx.doi.org/10.1016/j.jde.2014.01.039
---------- MLA ----------
Mazón, J.M., Rossi, J.D., Toledo, J. "Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles" . Journal of Differential Equations, vol. 256, no. 9, 2014, pp. 3208-3244.http://dx.doi.org/10.1016/j.jde.2014.01.039
---------- VANCOUVER ----------
Mazón, J.M., Rossi, J.D., Toledo, J. Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles. J. Differ. Equ. 2014;256(9):3208-3244.http://dx.doi.org/10.1016/j.jde.2014.01.039
