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

David, Guy

Journées équations aux dérivées partielles (2008), article no. 1, 27 p.

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].

[Al] 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. | MR | Zbl

[Am] L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal. 111 (1990), 291-322. | MR | Zbl

[AFP1] 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. | Numdam | MR | Zbl

[AFP2] L. Ambrosio, N. Fusco and D. Pallara, Higher regularity of solutions of free discontinuity problems. Differential Integral Equations 12 (1999), no. 4, 499-520. | MR | Zbl

[CL] 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. | Numdam | MR | Zbl

[DCL] 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. | MR | Zbl

[DMS] 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. | MR | Zbl

[D1] 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. | MR | Zbl

[D2] G. David, Singular sets of minimizers for the Mumford-Shah functional, Progress in Mathematics 233 (581p.), Birkhäuser 2005. | MR | Zbl

[D3] 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. | MR | Zbl

[D4] 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/ $\tilde{}$ gdavid/

[D5] G. David, $C^{1+\alpha }$-regularity for two-dimensional almost-minimal sets in ${ℝ}^n$, to be found at the same web addresses.

[DDT] G. David, T. De Pauw, and T. Toro, A generalization of Reifenberg’s theorem in ${ℝ}^3$, to appear, Geometric And Functional Analysis. | Zbl

[DS1] 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. | Numdam | MR | Zbl

[DS2] G. David and S. Semmes, Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Memoirs of the A.M.S. Number 687, volume 144, 2000. | MR | Zbl

[Fe] H. Federer, Geometric measure theory, Grundlehren der Mathematishen Wissenschaften 153, Springer Verlag 1969. | Zbl

[Feu] 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.

[He] A Heppes, Isogonal sphärischen netze, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 7 (1964), 41-48. | MR | Zbl

[La] E. Lamarle, Sur la stabilité des systèmes liquides en lames minces, Mém. Acad. R. Belg. 35 (1864), 3-104.

[LM] 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. | MR | Zbl

[Le] 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.

[Ma] P. Mattila, Geometry of sets and measures in Euclidean space, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press l995. | MR | Zbl

[Mo1] F. Morgan, Size-minimizing rectifiable currents, Invent. Math. 96 (1989), no. 2, 333-348. | MR | Zbl

[Mo2] 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. | MR | Zbl

[Ne] M. H. A. Newman, Elements of the topology of plane sets of points, Second edition, reprinted, Cambridge University Press, New York 1961. | MR | Zbl

[R1] E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104, 1960, 1–92. | MR | Zbl

[R2] E. R. Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Ann. of Math. (2) 80, 1964, 1–14. | MR | Zbl

[Ri] S. Rigot, Big Pieces of $C^{1,\alpha }$-Graphs for Minimizers of the Mumford-Shah Functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 2, 329-349. | Numdam | MR | Zbl

[St] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton university press 1970. | MR | Zbl

[Ta] 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. | MR | Zbl