Lecture notes : The local regularity of soap films after Jean Taylor

Guy David[1]

  • [1] Département de Mathématiques d’Orsay, Université de Paris Sud 11, F-91405 Orsay.

Journées Équations aux dérivées partielles (2008)

  • page 1-27
  • ISSN: 0752-0360

Abstract

top
The following text is a minor modification of the transparencies that were used in the conference; please excuse the often telegraphic style.The main goal of the series of lectures is a presentation (with some proofs) of Jean Taylor’s celebrated theorem on the regularity of almost minimal sets of dimension 2 in ℝ 3 , and a few more recent extensions or perspectives. Some of the results presented below are work of, or with T. De Pauw, V. Feuvrier A. Lemenant, and T. Toro.The main references for these lectures are [D4] and [D5] (for the proofs), [D3] (for some of the questions), and the theses [Feu] and [Le].

How to cite

top

David, Guy. "Lecture notes : The local regularity of soap films after Jean Taylor." Journées Équations aux dérivées partielles (2008): 1-27. <http://eudml.org/doc/10633>.

@article{David2008,
abstract = {The following text is a minor modification of the transparencies that were used in the conference; please excuse the often telegraphic style.The main goal of the series of lectures is a presentation (with some proofs) of Jean Taylor’s celebrated theorem on the regularity of almost minimal sets of dimension $2$ in $\mathbb\{R\}^3$, and a few more recent extensions or perspectives. Some of the results presented below are work of, or with T. De Pauw, V. Feuvrier A. Lemenant, and T. Toro.The main references for these lectures are [D4] and [D5] (for the proofs), [D3] (for some of the questions), and the theses [Feu] and [Le].},
affiliation = {Département de Mathématiques d’Orsay, Université de Paris Sud 11, F-91405 Orsay.},
author = {David, Guy},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-27},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Lecture notes : The local regularity of soap films after Jean Taylor},
url = {http://eudml.org/doc/10633},
year = {2008},
}

TY - JOUR
AU - David, Guy
TI - Lecture notes : The local regularity of soap films after Jean Taylor
JO - Journées Équations aux dérivées partielles
DA - 2008/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 27
AB - The following text is a minor modification of the transparencies that were used in the conference; please excuse the often telegraphic style.The main goal of the series of lectures is a presentation (with some proofs) of Jean Taylor’s celebrated theorem on the regularity of almost minimal sets of dimension $2$ in $\mathbb{R}^3$, and a few more recent extensions or perspectives. Some of the results presented below are work of, or with T. De Pauw, V. Feuvrier A. Lemenant, and T. Toro.The main references for these lectures are [D4] and [D5] (for the proofs), [D3] (for some of the questions), and the theses [Feu] and [Le].
LA - eng
UR - http://eudml.org/doc/10633
ER -

References

top
  1. F. J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs of the Amer. Math. Soc. 165, volume 4 (1976), i-199. Zbl0327.49043MR420406
  2. L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal. 111 (1990), 291-322. Zbl0711.49064MR1068374
  3. L. Ambrosio, N. Fusco, and D. Pallara, Partial regularity of free discontinuity sets II., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 39-62. Zbl0896.49024MR1475772
  4. L. Ambrosio, N. Fusco and D. Pallara, Higher regularity of solutions of free discontinuity problems. Differential Integral Equations 12 (1999), no. 4, 499-520. Zbl1007.49025MR1697242
  5. M. Carriero and A. Leaci, S k -valued maps minimizing the L p -norm of the gradient with free discontinuities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 321-352. Zbl0753.49018MR1145314
  6. E. De Giorgi, M. Carriero, and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal. 108 (1989), 195-218. Zbl0682.49002MR1012174
  7. G. Dal Maso, J.-M. Morel, and S. Solimini, A variational method in image segmentation: Existence and approximation results, Acta Math. 168 (1992), no. 1-2, 89–151. Zbl0772.49006MR1149865
  8. G. David, Limits of Almgren-quasiminimal sets, Proceedings of the conference on Harmonic Analysis, Mount Holyoke, A.M.S. Contemporary Mathematics series, Vol. 320 (2003), 119-145. Zbl1090.49025MR1979936
  9. G. David, Singular sets of minimizers for the Mumford-Shah functional, Progress in Mathematics 233 (581p.), Birkhäuser 2005. Zbl1086.49030MR2129693
  10. G. David, Quasiminimal sets for Hausdorff measures, in Recent Developments in Nonlinear PDEs, Proceeding of the second symposium on analysis and PDEs (June 7-10, 2004), Purdue University, D. Danielli editor, 81–99, Contemp. Math. 439, Amer. Math. Soc., Providence, RI, 2007. Zbl1137.49038MR2359022
  11. G. David, Low regularity for almost-minimal sets in ℝ 3 , submitted and to be found at HAL, ArXiv, or http://math.u-psud.fr/ ˜ gdavid/ 
  12. G. David, C 1 + α -regularity for two-dimensional almost-minimal sets in ℝ n , to be found at the same web addresses. 
  13. G. David, T. De Pauw, and T. Toro, A generalization of Reifenberg’s theorem in ℝ 3 , to appear, Geometric And Functional Analysis. Zbl1169.49040
  14. G. David and S. Semmes, Uniform rectifiability and Singular sets, Annales de l’Inst. Henri Poincaré, Analyse non linéaire, Vol 13, N¡ 4 (1996), p. 383-443. Zbl0908.49030MR1404317
  15. G. David and S. Semmes, Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Memoirs of the A.M.S. Number 687, volume 144, 2000. Zbl0966.49024MR1683164
  16. H. Federer, Geometric measure theory, Grundlehren der Mathematishen Wissenschaften 153, Springer Verlag 1969. Zbl0176.00801
  17. V. Feuvrier, Un résultat d’existence pour les ensembles minimaux par optimisation sur des grilles polyédrales, thèse de l’université de Paris-Sud 11 (Orsay), 2008. 
  18. A Heppes, Isogonal sphärischen netze, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 7 (1964), 41-48. Zbl0127.37601MR173193
  19. E. Lamarle, Sur la stabilité des systèmes liquides en lames minces, Mém. Acad. R. Belg. 35 (1864), 3-104. 
  20. Gary Lawlor and Frank Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math. 166 (1994), no. 1, 55–83. Zbl0830.49028MR1306034
  21. A. Lemenant, Sur la régularité des minimiseurs de Mumford-Shah en dimension 3 et supérieure, Thèse de l’Université de Paris-sud 11 (Orsay), 2008. 
  22. P. Mattila, Geometry of sets and measures in Euclidean space, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press l995. Zbl0911.28005MR1333890
  23. F. Morgan, Size-minimizing rectifiable currents, Invent. Math. 96 (1989), no. 2, 333-348. Zbl0645.49024MR989700
  24. F. Morgan, Minimal surfaces, crystals, shortest networks, and undergraduate research, Math. Intelligencer 14 (1992), no. 3, 37–44. Morgan bis avec la calibration pour le 4eme minimiseur. Zbl0765.52015MR1184317
  25. M. H. A. Newman, Elements of the topology of plane sets of points, Second edition, reprinted, Cambridge University Press, New York 1961. Zbl0123.39301MR132534
  26. E. R. Reifenberg, Solution of the Plateau Problem for m -dimensional surfaces of varying topological type, Acta Math. 104, 1960, 1–92. Zbl0099.08503MR114145
  27. E. R. Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Ann. of Math. (2) 80, 1964, 1–14. Zbl0151.16701MR171197
  28. S. Rigot, Big Pieces of C 1 , α -Graphs for Minimizers of the Mumford-Shah Functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 2, 329-349. Zbl0960.49024MR1784178
  29. E. M. Stein, Singular integrals and differentiability propertiesof functions, Princeton university press 1970. Zbl0207.13501MR290095
  30. J. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. Zbl0335.49032MR428181

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.