Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T08:46:40.809Z Has data issue: false hasContentIssue false
  • English
  • Français

On a dual formulation for the growing sandpile problem

Published online by Cambridge University Press:  01 April 2009

S. DUMONT  and
N. IGBIDA
S. DUMONT
Affiliation:
LAMFA, Université de Picardie Jules Verne, CNRS UMR 6140, 33, rue Saint-Leu, 80 039 Amiens, France email: [email protected], [email protected]
N. IGBIDA
Affiliation:
LAMFA, Université de Picardie Jules Verne, CNRS UMR 6140, 33, rue Saint-Leu, 80 039 Amiens, France email: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we are interested in the mathematical and numerical study of the Prigozhin model for a growing sandpile. Based on implicit Euler discretization in time, we give a simple improvement of theoretical and numerical analyses of the dual formulation for the problem. By using this model, we also give some application to the Monge–Kantorovich problem for optimal mass transportation.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 2 , April 2009 , pp. 169 - 185
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ambrosio, L. (2003) Lecture notes on optimal transport Problems, in Mathematical Aspects of Evolving Interfaces. In: Lecture Notes in Mathematics, LNM 1812, Springer, Berlin.Google Scholar
[2]Aronson, G., Evans, L. C. & Wu, Y. (1996) Fast/Slow diffusion and growing sandpiles. J. Differential Equations 131, 304–335.CrossRefGoogle Scholar
[3]Barrett, J. W. & Prigozhin, L. (2006) Dual formulation in critical state problems. Interface Free Bound. 8, 349–370.CrossRefGoogle Scholar
[4]Barrett, J. W. & Prigozhin, L. (2007) A mixed formulation of the Monge–Kantorovich equations. M2AN, 41 (6), 1041–1060.CrossRefGoogle Scholar
[5]Benilan, Ph., Crandall, M. G. & Pazy, A.Evolution Equations Governed by Accretive Operators, Book to appear.Google Scholar
[6]Bouchitté, G. & Buttazzo, G. (2001) Characterization of optimal shapes and masses through Monge–Kantorovich equation. J. Eur. Math. Soc. 3 (2), 139–168.CrossRefGoogle Scholar
[7]Bouchitté, G., Buttazzo, G. & Seppecher, P. (1997) Energies with respect to a measure and applications to low dimensional structures. Calc. Var. Partial Differential Equations 5, 37–54.CrossRefGoogle Scholar
[8]Bouchitté, G., Buttazzo, G. & Seppecher, P. (1997) Shape optimization solutions via Monge–Kantorovich. C.R. Acad. Sci. Paris, t.324, Série I, 1185–1191.CrossRefGoogle Scholar
[9]Brézis, H. (1973) Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert. (French). North-Holland Mathematics Studies, No. 5. Notas de Matemàtica (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.Google Scholar
[10]De Pascale, L., & Pratelli, A. (2002) Regularity properties for Monge transport density and for solutions of some shape optimization problem. Calc. Var. Partial Differential Equations 14 (3), 249–274.CrossRefGoogle Scholar
[11]De Pascale, L., & Pratelli, A. (2004) Sharp summability for Monge transport density via interpolation. ESAIM Contrôle Optim. Calc. Var. 10, 549–552.CrossRefGoogle Scholar
[12]De Pascale, L., Evans, L. C. & Pratelli, A. (2004) Integral estimates for transport densities. Bull. London Math. Soc., 36 (3), 383–395.CrossRefGoogle Scholar
[13]Ekeland, I. & Témam, R. (1999) Convex analysis and variational problems. In Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.CrossRefGoogle Scholar
[14]Evans, L. C. (1997) Partial differential equations and Monge–Kantorovich mass transfer. In Current Developments in Mathematics, International Press, Boston, MA, pp. 65–126.Google Scholar
[15]Evans, L. C. & Gangbo, W. (1999) Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc., 137, 653.Google Scholar
[16]Evans, L. C. & Rezakhanlou, F. (1998) A stochastic model for sandpiles and its continum limit. Comm. Math. Phys. 197 (2), 325–345.CrossRefGoogle Scholar
[17]Glowinski, R., Lions, & Trémolières, (1976) Analyse numérique des inéquations variationalles. Méthodes Mathématiques de l'Informatique, 5. Dunod, Paris.Google Scholar
[18]Igbida, N. On Equivalent Formulations for Monge–Kantorovich Equation. submitted.Google Scholar
[19]Igbida, N. Evolution Monge–Kantorovich equation. preprint.Google Scholar
[20]Prigozhin, L.Variational model of sandpile growth. Euro. J. Appl. Math. 7, 225–236.CrossRefGoogle Scholar
[21]Roberts, J. E. & Thomas, J.-M. (1991) Mixed and hybrid methods. In: Ciarlet, P. G. & Lions, J. L. (editors), Handbook of Numerical Analysis Finite Element Methods (Part 1), Vol. II, North-Holland, Amsterdam.Google Scholar
[22]Showalter, R. E. (1997) Monotone operators in Banach space and nonlinear partial differential equations. In Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI.Google Scholar