A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

  • P. Cannarsa
  • P. Cardaliaguet
  • G. Crasta
  • E. Giorgieri
Article

Abstract

The system of partial differential equations
$$ \left\{ \begin{array}{l}-{\rm div}(vDu)=f\quad {\rm in}\;{\rm \Omega}\\ |Du|-1=0\quad {\rm in }\;\{v > 0 \} \end{array} \right. $$
arises in the analysis of mathematical models for sandpile growth and in the context of the Monge–Kantorovich optimal mass transport theory. A representation formula for the solutions of a related boundary value problem is here obtained, extending the previous two-dimensional result of the first two authors to arbitrary space dimension. An application to the minimization of integral functionals of the form
$$\int_{{\rm \Omega}} [h(|Du|)-f(x) u]{\rm d}x, $$
with f≥ 0, and h≥ 0 possibly non-convex, is also included.

Keywords

Granular matter Eikonal equation Singularities Semiconcave functions Viscosity solutions Optimal mass transfer Existence of minimizers Distance function Calculus of variations Nonconvex integrands 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosio, L.: Lecture notes on optimal transport problems. Mathematical aspects of evolving interfaces (Funchal, 2000), Lecture Notes in Math. 1812, pp. 1–52. Springer, Berlin, (2003)Google Scholar
  2. 2.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems and Control: Foundations and Applications. Boston, Birkhäuser (1997)Google Scholar
  3. 3.
    Boutreux, T., de Gennes, P.-G.: Surface flows of granular mixtures. I. General principles and minimal model. J. Phys. I France 6, 1295–1304 (1996)CrossRefGoogle Scholar
  4. 4.
    Cannarsa, P., Cardaliaguet, P.: Representation of equilibrium solutions to the table problem for growing sandpile. J. Eur. Math. Soc. 6, 1–30 (2004)MathSciNetGoogle Scholar
  5. 5.
    Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and Theyr Applications. Boston, Birkhäuser (2004)Google Scholar
  6. 6.
    Celada, P., Cellina, A.: Existence and non existence of solutions to a variational problem on a square. Houston J. Math. 24, 345–375 (1998)MathSciNetGoogle Scholar
  7. 7.
    Celada, P., Perrotta, S., Treu, G.: Existence of solutions for a class of non convex minimum problems. Math. Z. 228, 177–199 (1997)MathSciNetGoogle Scholar
  8. 8.
    Cellina, A.: Minimizing a functional depending on ∇ u and on u. Ann. Inst. H. Poincaré, Anal. Non Linéaire 14, 339–352 (1997)MathSciNetMATHGoogle Scholar
  9. 9.
    Clarke, F.H., Ledyaev, Yu. S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. New York, Springer (1998)Google Scholar
  10. 10.
    Crasta, G., Malusa, A.: Geometric constraints on the domain for a class of minimum problems. ESAIM Control Optim. Calc. Var. 9, 125–133 (2003)MathSciNetGoogle Scholar
  11. 11.
    Evans, L.C., Feldman, M., Gariepy, R.: Fast/slow diffusion and collapsing sandpiles. J. Differential Equations 137(1), 166–209 (1997)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137(653), (1999)Google Scholar
  13. 13.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)Google Scholar
  14. 14.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)MathSciNetMATHGoogle Scholar
  15. 15.
    Feldman, M.: Variational evolution problems and nonlocal geometric motion. Arch. Rational Mech. Anal. 146, 221–274 (1999)CrossRefMATHGoogle Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, 224. Berlin, Springer-Verlag (1983).Google Scholar
  17. 17.
    Hadeler, K.P., Kuttler, C.: Dynamical Models for Granular Matter. Granular Matter 2, 9–18 (1999)CrossRefGoogle Scholar
  18. 18.
    Itoh, J., Tanaka, M.: The Lipschitz continuity of the distance function to the cut locus. Trans. Am. Math. Soc. 353(1), 21–40 (2001)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Li, Y.Y., Nirenberg, L.: The distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton–Jacobi equations, Commun. Pure Appl. Math. 58, 85–146 (2005)MathSciNetGoogle Scholar
  20. 20.
    Lions, P.L.: Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics, vol. 69. Boston London Melbourne, Pitman (1982)Google Scholar
  21. 21.
    Prigozhin, L.: Variational model of sandpile growth. European J. Appl. Math. 7(3), 225–235 (1996)MathSciNetMATHGoogle Scholar
  22. 22.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski theory. Cambridge University Press, Cambridge (1993)Google Scholar
  23. 23.
    Treu, G.: An existence result for a class of non convex problems of the Calculus of Variations. J. Convex Anal. 5, 31–44 (1998)MathSciNetMATHGoogle Scholar
  24. 24.
    Vornicescu, M.: A variational problem on subsets of ℝn. Proc. Roy. Soc. Edinburgh Sect. A 127, 1089–1101 (1997)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • P. Cannarsa
    • 1
  • P. Cardaliaguet
    • 2
  • G. Crasta
    • 3
  • E. Giorgieri
    • 4
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomaItaly
  2. 2.UFR des Sciences et TechniquesUniversité de Bretagne OccidentaleBrestFrance
  3. 3.Dipartimento di MatematicaUniversità di Roma La SapienzaRomaItaly
  4. 4.Dipartimento di MatematicaUniversità di Roma Tor VergataRomaItaly

Personalised recommendations