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On the collapsing sandpile problem
1. | LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex |
2. | LAMFA, CNRS UMR 6140, Universite de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex, France |
References:
[1] |
G. Aronson, L. C. Evans and Y. Wu, Fast/Slow diffusion and growing sandpiles, J. Differential Equations, 131 (1996), 304-335.
doi: doi:10.1006/jdeq.1996.0166. |
[2] |
P. Bak, C. Tang and K. Weisenfeld, Self-organized criticality, Phys. Rev. A, 38 (1988), 364-378.
doi: doi:10.1103/PhysRevA.38.364. |
[3] |
J. W. Barrett and L. Prigozhin, Dual formulation in critical state problems, Interfaces and Free Boundaries, 8 (2006), 349-370.
doi: doi:10.4171/IFB/147. |
[4] |
Ph. Bénilan, M. G. Crandall and A. Pazy, "Evolution Equations Governed by Accretive Operators," Preprint book. |
[5] |
Ph. Bénilan, L. C. Evans and R. F. Gariepy, On some singular limits of homogeneous semigroups, J. Evol. Equ., 3 (2003), 203-214. |
[6] |
J. P. Bouchaud, M. E. Cates, J. Ravi Prakash and S. F. Edwards, A model for the Dynamic of Sandpile Surfaces, J. Phys. I France, 4 (1994), 1383-1410. |
[7] |
G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a Measure and Applications to Low Dimensional Structures, Calc. Var. Partial Differential Equations, 5 (1997), 37-54. |
[8] |
H. Brézis, Opérateurs maximaux monotones et semi-groupes 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. |
[9] |
S. Dumont and N. Igbida, On a Dual Formulation for the Growing Sandpile Problem, European Journal Applied Math., 20 (2009), 169-185.
doi: doi:10.1017/S0956792508007754. |
[10] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. |
[11] |
L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), pp. 163-188, |
[12] |
L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997 (Cambridge, MA), 65-126, Int. Press, Boston, MA, 1999. |
[13] |
L. C. Evans, M. Feldman and R. F. Gariepy, Fast/Slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997), 166-209.
doi: doi:10.1006/jdeq.1997.3243. |
[14] |
L. C. Evans and F. Rezakhanlou, A stochastic model for sandpiles and its continum limit, Comm. Math. Phys., 197 (1998), 325-345.
doi: doi:10.1007/s002200050453. |
[15] |
N. Igbida, Equivalent formulations for Monge-Kantorovich equation, Submitted. |
[16] |
L. Prigozhin, Variational model of sandpile growth, Euro. J. Appl. Math., 7 (1996), 225-236.
doi: doi:10.1017/S0956792500002321. |
[17] |
J. E. Roberts and J.-M. Thomas, "Mixed and Hybrid Methods," (P. G. Ciarlet and J. L. Lions eds.), Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991. |
[18] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. |
show all references
References:
[1] |
G. Aronson, L. C. Evans and Y. Wu, Fast/Slow diffusion and growing sandpiles, J. Differential Equations, 131 (1996), 304-335.
doi: doi:10.1006/jdeq.1996.0166. |
[2] |
P. Bak, C. Tang and K. Weisenfeld, Self-organized criticality, Phys. Rev. A, 38 (1988), 364-378.
doi: doi:10.1103/PhysRevA.38.364. |
[3] |
J. W. Barrett and L. Prigozhin, Dual formulation in critical state problems, Interfaces and Free Boundaries, 8 (2006), 349-370.
doi: doi:10.4171/IFB/147. |
[4] |
Ph. Bénilan, M. G. Crandall and A. Pazy, "Evolution Equations Governed by Accretive Operators," Preprint book. |
[5] |
Ph. Bénilan, L. C. Evans and R. F. Gariepy, On some singular limits of homogeneous semigroups, J. Evol. Equ., 3 (2003), 203-214. |
[6] |
J. P. Bouchaud, M. E. Cates, J. Ravi Prakash and S. F. Edwards, A model for the Dynamic of Sandpile Surfaces, J. Phys. I France, 4 (1994), 1383-1410. |
[7] |
G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a Measure and Applications to Low Dimensional Structures, Calc. Var. Partial Differential Equations, 5 (1997), 37-54. |
[8] |
H. Brézis, Opérateurs maximaux monotones et semi-groupes 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. |
[9] |
S. Dumont and N. Igbida, On a Dual Formulation for the Growing Sandpile Problem, European Journal Applied Math., 20 (2009), 169-185.
doi: doi:10.1017/S0956792508007754. |
[10] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. |
[11] |
L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), pp. 163-188, |
[12] |
L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997 (Cambridge, MA), 65-126, Int. Press, Boston, MA, 1999. |
[13] |
L. C. Evans, M. Feldman and R. F. Gariepy, Fast/Slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997), 166-209.
doi: doi:10.1006/jdeq.1997.3243. |
[14] |
L. C. Evans and F. Rezakhanlou, A stochastic model for sandpiles and its continum limit, Comm. Math. Phys., 197 (1998), 325-345.
doi: doi:10.1007/s002200050453. |
[15] |
N. Igbida, Equivalent formulations for Monge-Kantorovich equation, Submitted. |
[16] |
L. Prigozhin, Variational model of sandpile growth, Euro. J. Appl. Math., 7 (1996), 225-236.
doi: doi:10.1017/S0956792500002321. |
[17] |
J. E. Roberts and J.-M. Thomas, "Mixed and Hybrid Methods," (P. G. Ciarlet and J. L. Lions eds.), Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991. |
[18] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. |
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