Analysis of a PDE Model for Sandpile Growth

  • P. Cannarsa
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 199)


In the dynamical theory of granular matter, the so-called table problem consists in studying the evolution of a heap of matter poured continuously onto a bounded domain Ω ⊂ R2. The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of Ω.


granular matter eikonal equation singularities viscosity solutions optimal mass tranfer 


  1. [1]
    L. Ambrosio. Optimal transport maps in Monge-Kantorovich problem. In Proceedings of the International Congress of Mathematicians Higher Ed. Press, 2002.Google Scholar
  2. [2]
    G. Aronsson, L. C. Evans, Y. Wu. Fast/slow diffusion and growing sandpiles. J. Differential Equations 131:304–335, 1996.MathSciNetCrossRefGoogle Scholar
  3. [3]
    T. Boutreux, P.-G. de Gennes. Surface flows of granular mixtures. I. General principles and minimal model. J. Phys. 1 France 6: 1295–1304, 1996.CrossRefGoogle Scholar
  4. [4]
    G. Bouchitté, G. Buttazzo, P. Seppechere. Shape optimization solutions via Monge-Kantorovich equation. C. R. Acad. Sci. Paris Ser. 1 Math. 324 10: 1185–1191, 1997.Google Scholar
  5. [5]
    P. Cannarsa, P. Cardaliaguet. Representation of equilibrium solutions to the table problem for growing sandpile. J. Eur. Math. Soc. 6: 1–30, 2004.MathSciNetGoogle Scholar
  6. [6]
    P. Cannarsa, P. Cardaliaguet, G. Crasta, E. Giorgieri. A boundary value problem for a PDE model in mass transfer theory: representation of solutions and applications. Calc. Var. DOI 10.1007/s00526-005-0328-7, 2005.Google Scholar
  7. [7]
    P. Cannarsa, P. Cardaliaguet, E. Giorgieri. Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles. Pre-print.Google Scholar
  8. [8]
    P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Birkhäuser progress in nonlinear differential equations and their applications. Boston, 2004.Google Scholar
  9. [9]
    P. Celada, A. Cellina. Existence and non existence of solutions to a variational problem on a square. Houston J. Math. 24: 345–375, 1998.MathSciNetGoogle Scholar
  10. [10]
    P. Celada, S. Perrotta, G. Treu. Existence of solutions for a class of non convex minimum problems. Math. Z. 228: 177–199, 1997.MathSciNetCrossRefGoogle Scholar
  11. [11]
    A. Cellina. Minimizing a functional depending on ∇u and on u. Ann. Inst. H. Poincaré, Anal Non Linéaire 14: 339–352, 1997.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Cellina, S. Perrotta. On the validity of the maximum principle and of the Euler-Lagrange equation for a minimum problem depending on the gradient. SIAM J. Control Optim. 36: 1987–1998, 1997.MathSciNetCrossRefGoogle Scholar
  13. [13]
    G. Crasta, A. Malusa. Geometric constraints on the domain for a class of minimum problems. ESAIM Control Optim. Calc. Var. 9:125–133, 2003.MathSciNetGoogle Scholar
  14. [14]
    L.C. Evans, M. Feldman, R. Gariepy. Fast/slow diffusion and growing sandpiles. J. Differential Equations 137:166–209, 1997.MathSciNetCrossRefGoogle Scholar
  15. [15]
    L.C. Evans, W. Gangbo. Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137, no. 653, 1999.Google Scholar
  16. [16]
    M. Feldman, R. J. McCann. Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. 15:81–113, 2002.MathSciNetCrossRefGoogle Scholar
  17. [17]
    K. P. Hadeler, C. Kuttler. Dynamical models for granular matter. Granular Matter 2:9–18, 1999.CrossRefGoogle Scholar
  18. [18]
    J. Itoh, M. Tanaka. The Lipschitz continuity of the distance function to the cut locus. Trans. Am. Math. Soc. 353:21–40, 2001.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Y. Y. Li, L. Nirenberg. The distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math. 58:85–146, 2005.MathSciNetCrossRefGoogle Scholar
  20. [20]
    L. Prigozhin. Variational model of sandpile growth. European J. Appl. Math. 7:225–235, 1996.MATHMathSciNetGoogle Scholar
  21. [21]
    G. Treu. An existence result for a class of non convex problems of the Calculus of Variations. J. Convex Anal. 5:31–44, 1998.MATHMathSciNetGoogle Scholar
  22. [22]
    M. Vornicescu. A variational problem on subsets of Rn. Proc. Roy. Soc. Edinburgh Sect. A 127:1089–1101, 1997.MATHMathSciNetGoogle Scholar

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© International Federation for Information Processing 2006

Authors and Affiliations

  • P. Cannarsa
    • 1
  1. 1.Department of MathematicsUniversity of Rome “Tor Vergata”RomeItaly

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