Skip to main content
SpringerLink
Log in

A Modica-Mortola Approximation for Branched Transport and Applications

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

The M α energy which is usually minimized in branched transport problems among singular one-dimensional rectifiable vector measures is approximated by means of a sequence of elliptic energies defined on more regular vector fields. The procedure recalls the one of Modica-Mortola related to the approximation of the perimeter. In our context, the double-well potential is replaced by a concave term. The paper contains a proof of Γ−convergence and numerical simulations of optimal networks based on that previous result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ambrosio L., De Lellis C., Mantegazza C.: Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9(4), 327–355 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio L., Tortorelli V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6(1), 105–123 (1992)

    MathSciNet  MATH  Google Scholar 

  3. Ambrosio L., Tortorelli V.M.: Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43(8), 999–1036 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aviles, P., Giga, Y.: A mathematical problem related to the physical theory of liquid crystal configurations. In: Miniconference on geometry and partial differential equations, 2 (Canberra, 1986), volume 12 of Proceedings of the Centre Mathematical Analysis. Australian National University, pp. 1–16. Australian National University, Canberra, 1987

  5. Aviles P., Giga Y.: Singularities and rank one properties of Hessian measures. Duke Math. J. 58(2), 441–467 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bernot M., Caselles V., Morel J.-M.: Traffic plans. Publ. Mat. 49(2), 417–451 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Bernot, M., Caselles, V., Morel, J.-M.: Optimal transportation networks, volume 1955 of Lecture Notes in Mathematics. Models and theory. Springer, Berlin, 2009

  8. Bethuel F., Brezis H., Hélein F.: Asymptotics for the minimization of a Ginzburg-Landau functional. Calc. Var. Partial Differential Equations 1(2), 123–148 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, 13. Birkhäuser Boston Inc., Boston, 1994

  10. Bouchitté G., Dubs C., Seppecher P.: Transitions de phases avec un potentiel dégénéré à l’infini, application à l’équilibre de petites gouttes. C. R. Acad. Sci. Paris Sér. I Math. 323(9), 1103–1108 (1996)

    MATH  Google Scholar 

  11. Bouchitté G., Jimenez C., Rajesh M.: Asymptotique d’un problème de positionnement optimal. C. R. Math. Acad. Sci. Paris, 335(10), 853–858 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Braides, A.: Approximation of free-discontinuity problems volume 1694 of Lecture Notes in Mathematics. Springer, Berlin, 1998

  13. Dal Maso, G.: An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston, 1993

  14. De Giorgi E., Franzoni T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6), 842–850 (1975)

    MathSciNet  MATH  Google Scholar 

  15. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York Inc., New York, 1969

  16. Gilbarg, D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Grundlehren der Mathematischen Wissenschaften, Vol. 224, 1977

  17. Gilbert E.N.: Minimum cost communication networks. Bell System Tech. J. 46, 2209–2227 (1967)

    Google Scholar 

  18. Kelley, C.T.: Iterative methods for optimization, volume 18 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999)

  19. Maddalena F., Solimini S., Morel J.-M.: A variational model of irrigation patterns. Interfaces Free Bound. 5(4), 391–415 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Modica L., Mortola S.: Un esempio di Γ−-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

  21. Mosconi S.J.N., Tilli P.: Γ-convergence for the irrigation problem. J. Convex Anal. 12(1), 145–158 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Oudet, É.: Approximation of partitions of least perimeter by Γ-convergence: around kelvin’s conjecture. http://www.lama.univ-savoie.fr/~oudet, 2009

  23. Rodriguez-Iturbe I., Rinaldo A.: Fractal River Basins. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  24. Santambrogio F.: Optimal channel networks, landscape function and branched transport. Interfaces Free Bound 9(1), 149–169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Santambrogio F.: A modica-mortola approximation for branched transport. Comptes Rendus Acad. Sci. 348, 941–945 (2010)

    MathSciNet  MATH  Google Scholar 

  26. White B.: Rectifiability of flat chains. Ann. of Math. (2) 150(1), 165–184 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xia Q.: Optimal paths related to transport problems. Commun. Contemp. Math. 5(2), 251–279 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xia Q.: Interior regularity of optimal transport paths. Calc. Var. Partial Differential Equations 20(3), 283–299 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Xue G., Lillys T.P., Dougherty D.E.: Computing the minimum cost pipe network interconnecting one sink and many sources. SIAM J. Optim 10(1), 22–42 (1999) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAMA, Universit de Savoie, Campus scientifique, 73376, Le Bourget du Lac Cedex, France

    Edouard Oudet

  2. Departement de Mathmatiques, Bt. 425, Facult des Sciences, Universit Paris-Sud XI, 91405, Orsay Cedex, France

    Filippo Santambrogio

Authors
  1. Edouard Oudet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filippo Santambrogio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edouard Oudet.

Additional information

Communicated by Y. Brenier

Rights and permissions

Reprints and permissions

About this article

Cite this article

Oudet, E., Santambrogio, F. A Modica-Mortola Approximation for Branched Transport and Applications. Arch Rational Mech Anal 201, 115–142 (2011). https://doi.org/10.1007/s00205-011-0402-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-011-0402-6

Keywords

Search

Navigation