Hölder regularity of two-dimensional almost-minimal sets in n
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 18 (2009) no. 1, pp. 65-246.

On donne une démonstration différente et sans doute plus élémentaire d’une bonne partie du résultat de régularité de Jean Taylor sur les ensembles presque-minimaux d’Almgren. On en profite pour donner des précisions sur les ensembles presque minimaux, généraliser une partie du théorème de Taylor aux ensembles presque minimaux de dimension 2 dans n , et donner la caractérisation attendue des ensembles fermés E de dimension 2 dans 3 qui sont minimaux, au sens où H 2 (EF)H 2 (FE) pour tout fermé F tel qu’il existe une partie bornée B telle que F=E hors de B et F sépare les points de 3 B qui sont séparés par E.

We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension 2 in 3 . We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension 2 in n , and give the expected characterization of the closed sets E of dimension 2 in 3 that are minimal, in the sense that H 2 (EF)H 2 (FE) for every closed set F such that there is a bounded set B so that F=E out of B and F separates points of 3 B that E separates.

DOI : 10.5802/afst.1205
David, Guy 1

1 Mathématiques, Bâtiment 425, Université de Paris-Sud 11, 91 405 Orsay Cedex, France
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David, Guy. Hölder regularity of two-dimensional almost-minimal sets in $\mathbb{R}^n$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 18 (2009) no. 1, pp. 65-246. doi : 10.5802/afst.1205. http://www.numdam.org/articles/10.5802/afst.1205/

[Al1] Almgren (F. J.).— Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87, p. 321-391 (1968). | MR | Zbl

[Al2] Almgren (F. J.).— Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs of the Amer. Math. Soc. 165, volume 4, p. i-199 (1976). | MR | Zbl

[Bo] Bombieri (E.).— Regularity theory for almost minimal currents, Arch. Rational Mech. Anal. 78, n 2, p. 99-130 (1982). | MR | Zbl

[DMS] Dal Maso (G.), Morel (J.-M.) and Solimini (S.).— A variational method in image segmentation: Existence and approximation results, Acta Math. 168, n 1-2, p. 89-151 (1992). | MR | Zbl

[D1] David (G.).— Limits of Almgren-quasiminimal sets, Proceedings of the conference on Harmonic Analysis, Mount Holyoke, A.M.S. Contemporary Mathematics series, Vol. 320, p. 119-145 (2003). | MR | Zbl

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

[D3] David (G.).— Quasiminimal sets for Hausdorff measures, Recent developments in nonlinear partial differential equations, p. 81-99, Contemp. Math., 439, Amer. Math. Soc., Providence, RI (2007). | MR | Zbl

[D4] David (G.).— C 1+α -regularity for two-dimensional almost-minimal sets in n , preprint, available on arXiv, Hal, or at http://mahery.math.u-psud.fr/ gdavid | Zbl

[DDT] David (G.), De Pauw (T.) and Toro (T.).— A generalization of Reifenberg’s theorem in 3 , to appear, Geom. Funct. Anal. | Zbl

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

[Do] Dold (A.).— Lectures on algebraic topology, Second edition, Grundlehren der Mathematishen Wissenschaften 200, Springer Verlag (1980). | MR | Zbl

[Du] Dugundji (J.).— Topology, Allyn and Bacon, Boston (1966). | MR | Zbl

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

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

[Fv] Feuvrier (V.).— Un résultat d’existence pour les ensembles minimaux par optimisation sur des grilles polyédrales, Thèse de l’université de Paris-Sud 11, Orsay, Septembre 2008.

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

[LM] Lawlor (G.) and Morgan (F.).— Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math. 166, n 1, p. 55-83 (1994). | MR | Zbl

[Le] Lemenant (A.).— Sur la régularité des minimiseurs de Mumford-Shah en dimension 3 et supérieure, Thèse de l’université de Paris-Sud 11, Orsay, Juin 2008.

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

[Mo1] Morgan (F.).— Area-minimizing currents bounded by higher multiples of curves, Rend. Circ. Mat. Palermo 33, p. 37-46 (1984). | MR | Zbl

[Mo2] Morgan (F.).— Size-minimizing rectifiable currents, Invent. Math. 96, n 2, p. 333-348 (1989). | Zbl

[Mo3] Morgan (F.).— (M,ε,δ)-minimal curve regularity, Proc. Amer. Math. Soc. 120, n 3, p. 677-686 (1994). | MR | Zbl

[Mo4] Morgan (F.).— Geometric measure theory. A beginner’s guide, Second edition. Academic Press, Inc., San Diego, CA, x+175 pp (1995). | MR | Zbl

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

[R2] Reifenberg (E. R.).— Epiperimetric Inequality related to the analyticity of minimal surfaces, Annals Math., 80, p. 1-14 (1964). | MR | Zbl

[R3] Reifenberg (E. R.).— On the analyticity of minimal surfaces, Annals of Math., 80, p. 15-21 (1964). | MR | Zbl

[SS] Schoen (R.) and Simon (L.).— A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals, Indiana Univ. Math. J. 31, n 3, p. 415-434 (1982). | MR | Zbl

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

[Tam] Tamanini (I.).— Regularity results for almost minimal oriented hypersurfaces in n , Quaderni del dipartimento di matematica dell’universitá di Lecce (1984).

[Tay] Taylor (J.).— The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103, n 3, p. 489-539 (1976). | MR | Zbl

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