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. [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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.