Hölder regularity of two-dimensional almost-minimal sets in ${ℝ}^n$

David, Guy

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 ${ℝ}^n$, et donner la caractérisation attendue des ensembles fermés $E$ de dimension $2$ dans ${ℝ}^3$ qui sont minimaux, au sens où $H^2(E\setminus F)\le H^2(F\setminus E)$ 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 ${ℝ}^3\setminus B$ 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 ${ℝ}^3$. 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 ${ℝ}^n$, and give the expected characterization of the closed sets $E$ of dimension $2$ in ${ℝ}^3$ that are minimal, in the sense that $H^2(E\setminus F)\le H^2(F\setminus E)$ 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 ${ℝ}^3\setminus B$ that $E$ separates.

[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${}^{\circ }$ 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${}^{\circ }$ 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.).— $C^{1+\alpha }$-regularity for two-dimensional almost-minimal sets in ${ℝ}^n$, 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 ${ℝ}^3$, 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${}^{\circ }$ 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${}^{\circ }$ 2, p. 333-348 (1989). | Zbl

[Mo3] Morgan (F.).— $(M,\epsilon ,\delta )$-minimal curve regularity, Proc. Amer. Math. Soc. 120, n${}^{\circ }$ 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${}^{\circ }$ 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 ${ℝ}^n$, 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${}^{\circ }$ 3, p. 489-539 (1976). | MR | Zbl