Open Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 77
Number of page(s) 32
DOI https://doi.org/10.1051/cocv/2021075
Published online 22 July 2021
  1. F. Almgren, Deformations and multiple-valued functions, Geometric measure theory and the calculus of variations. Proc. Sympos. Pure Math, American Math. Soc. 44 (1986) 29–130. [Google Scholar]
  2. F. Almgren, Plateau’s problem. Mathematics monograph series. W. A. Benjamin, Inc. (1966). [Google Scholar]
  3. F. Almgren and J. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differ. Geometry 42 (1995) 1–22. [Google Scholar]
  4. F. Almgren, J. Taylor and L. Wang, Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387–483. [Google Scholar]
  5. M. Bernot, V. Caselles and J.-M. Morel, Optimal transportation networks. Lecture Notes in Mathematics 1955. Springer (2009). [Google Scholar]
  6. M. Bernot, Irrigation and optimal transport. Ph.D. Thesis, Ecole Normale Superieure de Cachan (2005). [Google Scholar]
  7. K.A. Brakke, The motion of a surface by its mean curvature. Princeton University Press (1978). [Google Scholar]
  8. X. Cheng, A mass reducing flow for integral currents, Indiana Univ. Math. J. 42 (1993) 425–44. [Google Scholar]
  9. O. Chodosh, Brian White – Topics in GMT (MATH 258) Lecture Notes, Stanford University. Available from: http://web.stanford.edu/ochodosh/GMTnotes.pdf (Spring 2012). [Google Scholar]
  10. T. Colding, W. Minicozzi and E. Pedersen, Mean curvature flow. Bull. Am. Math. Soc. 52 (2015) 297–333. [Google Scholar]
  11. J. Douglas, Minimal surfaces of higher topological structure. Ann. Math. (1939) 205–298. [Google Scholar]
  12. T. De Pauw and R. Hardt, Rectifiable and flat g chains in a metric space. Am. J. Math. 134 (2012) 1–69. [Google Scholar]
  13. L. Evans and J. Spruck, Motion of level sets by mean curvature I. J. Differ. Geometry 33 (1991) 635–681. [Google Scholar]
  14. L. Evans and J. Spruck, Motion of level sets by mean curvature II. Trans. Am. Math. Soc. 330 (1992) 32-1-332. [Google Scholar]
  15. L. Evans and J. Spruck, Motion of level sets by mean curvature III. J. Geometric Anal. 2 (1992) 121–150. [Google Scholar]
  16. L. Evans and J. Spruck, Motion of level sets by mean curvature IV. J. Geometric Anal. 5 (1995) 77–114. [Google Scholar]
  17. H. Federer, Geometric measure theory. Springer-Verlag (1969). [Google Scholar]
  18. H. Federer and W.H. Fleming, Normal and integral currents. Ann. Math. (1960) 458–520. [Google Scholar]
  19. H. Federer, Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24 (1974) 351–407. [CrossRef] [MathSciNet] [Google Scholar]
  20. W.H. Fleming, Flat chains over a finite coefficient group. Trans. Am. Math. Soc. 121 (1966) 160–186. [Google Scholar]
  21. M. Gage and R. Hamilton, The heat equation shrinking convex plane curves. J. Differ. Geometry 23 (1986) 69–96. [Google Scholar]
  22. E.N. Gilbert, Minimum cost communication networks. Bell Labs Tech. J. 46 (1967) 2209–2227. [Google Scholar]
  23. M. Grayson, The heat equation shrinks embedded plane curves to round points. J. Differ. Geometry 26 (1987) 285–314. [Google Scholar]
  24. J. Haga, K. Hoshino and N. Kikuchi, Construction of harmonic map flows through the method of discrete Morse flows. Comput. Visual. Sci. 7 (2004) 53–59. [Google Scholar]
  25. G. Huisken, Flow by mean curvature of convex surfaces into spheres. J. Differ. Geometry 20 (1984) 237–266. [Google Scholar]
  26. L.V. Kantorovich, On the translocation of masses. Dokl. Akad. Nauk SSSR 37 (1942) 199–201. [Google Scholar]
  27. F. Maddalena, J.-M. Morel and S. Solimini, A variational model of irrigation patterns. Interfaces Free Boundaries 5 (2003) 391–415. [Google Scholar]
  28. G. Monge, Mémoire sur la théorie des déblais et de remblais. Histoire de l’Académie Royale des Sciences de Paris (1781) 666–704. [Google Scholar]
  29. F. Morgan, Geometric measure theory. Academic Press (2009). [Google Scholar]
  30. E. Oudet and F. Santambrogio, A Modica-Mortola approximation for brached transport and applications. Arch. Ratl. Mech. Anal. 201 (2011) 115–142. [Google Scholar]
  31. P. Pegon, F. Santambrogio and Q. Xia, A fractal shape optimization problem in branched transport. J. Math. Pures Appl. (2017). [Google Scholar]
  32. J.A.F. Plateau, Statique expérimentale et théoreique des liquides soumis aux seules forces molécaires 2, Gauthier-Villars (1873). [Google Scholar]
  33. T. Radó, The problem of the least area and the problem of Plateau. Math. Zeitschrift 32 (1930) 763–796. [Google Scholar]
  34. E.R. Reifenberg, Solution of the Plateau problem form-dimensional surfaces of varying topological type. Acta Math. 104 (1960) 1–92. [Google Scholar]
  35. F. Santambrogio, Optimal channel networks, landscape function and branched transport. Interfaces Free Bound 9 (2007) 149–169. [Google Scholar]
  36. H. Whitney, Geometric integration theory. Princeton University Press (1957). [Google Scholar]
  37. B. White, The deformation theorem for flat chains. Acta Math. 183 (1999) 255–271. [Google Scholar]
  38. B. White, Rectifiability of flat chains. Ann. Math. 150 (1999) 165–184. [Google Scholar]
  39. B. White, Evolution of curves and surfaces by mean curvature. Proc. ICM (2002). [Google Scholar]
  40. B. White, Currents and flat chains associated to varifolds, with an application to mean curvature flow. Duke Math J. 148 (2009) 41–62. [Google Scholar]
  41. Q. Xia, Optimal paths related to transport problems. Commun. Contemp. Math. 5 (2003) 251–279. [Google Scholar]
  42. Q. Xia, Interior regularity of optimal transport paths. Calc. Variat. Partial Differ. Equ. 20 (2004) 283–299. [Google Scholar]
  43. Q. Xia, An application of optimal transport paths to urban transport networks. Discrete Continu. Dyn. Syst. (2005) 904–910. [Google Scholar]
  44. Q. Xia, The formation of a tree leaf. ESAIM: COCV 13 (2007) 359–377. [CrossRef] [EDP Sciences] [Google Scholar]
  45. Q. Xia, Boundary regularity of optimal transport paths. Adv. Calc. Variat. 4 (2011) 153–174. [Google Scholar]
  46. Q. Xia, Ramified optimal transportation in geodesic metric spaces. Adv. Calc. Variat. 4 (2011) 277–307. [Google Scholar]
  47. Q. Xia, On landscape functions associated with transport paths. Discrete Continu. Dyn. Syst. 34 (2014) 1683–1700. [Google Scholar]
  48. Q. Xia, Motivations, ideas, and applications of ramified optimal transportation. ESAIM: M2AN 49 (2015) 1791–1832. [EDP Sciences] [Google Scholar]
  49. Q. Xia and A. Vershynina, On the transport dimension of measures. SIAM J. Math. Anal. 41 (2010) 2407–2430. [Google Scholar]
  50. Q. Xia and S. Xu, The exchange value embedded in a transport system. Appl. Math. Optim. 62 (2010) 229–252. [Google Scholar]

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