no. 5
Global regularity for minimal sets near a union of two planes
[Régularité globale pour les ensembles minimaux proche d’une union de deux plans]
Liang, Xiangyu1
1 Institut Camille Jordan Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne cedex (France)
Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 2067-2099.

On traite la régularité globale des ensembles minimaux 2-dimensionnels qui sont proches d’une union de deux plans, et on démontre que tout ensemble minimal proche d’une union de deux plans presque orthogonaux à l’infini dans ℝ 4 est un cône. L’enjeu est de contrôler le comportement d’un ensemble minimal à petite échelle par la topologie à grande échelle.

We discuss the global regularity of 2 dimensional minimal sets that are near a union of two planes, and prove that every global minimal set in ℝ 4 that looks like a union of two almost orthogonal planes at infinity is a cone. The main point is to use the topological properties of a minimal set at a large scale to control its behavior at smaller scales.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3058
Classification : 28A75, 49Q10, 49Q20, 49K99
Keywords: Minimal sets, blow-in limit, existence of singularities, Hausdorff measure, elliptic systems
Mot clés : Ensembles minimaux, limites d’explosion, existence de singularités, mesure de Hausdorff, système elliptiques.

Liang, Xiangyu 1

1 Institut Camille Jordan Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne cedex (France) 
@article{AIF_2016__66_5_2067_0,
     author = {Liang, Xiangyu},
     title = {Global regularity for minimal sets near a union of two planes},
     journal = {Annales de l'Institut Fourier},
     pages = {2067--2099},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     doi = {10.5802/aif.3058},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3058/}
}
TY  - JOUR
AU  - Liang, Xiangyu
TI  - Global regularity for minimal sets near a union of two planes
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 2067
EP  - 2099
VL  - 66
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3058/
DO  - 10.5802/aif.3058
LA  - en
ID  - AIF_2016__66_5_2067_0
ER  - 
%0 Journal Article
%A Liang, Xiangyu
%T Global regularity for minimal sets near a union of two planes
%J Annales de l'Institut Fourier
%D 2016
%P 2067-2099
%V 66
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3058/
%R 10.5802/aif.3058
%G en
%F AIF_2016__66_5_2067_0
Liang, Xiangyu. Global regularity for minimal sets near a union of two planes. Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 2067-2099. doi : 10.5802/aif.3058. https://aif.centre-mersenne.org/articles/10.5802/aif.3058/

[1] Allard, William K. On the first variation of a varifold, Ann. of Math. (2), Volume 95 (1972), pp. 417-491 | DOI

[2] Almgren, F. J. Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc., Volume 4 (1976) no. 165, viii+199 pages

[3] Bernstein, S. Sur un théorème de géométrie et ses applications aux équations aux dérivées partielles du type elliptique, Comm. Soc. Math. de Khardov, Volume 15 (1915–17), pp. 38-45

[4] Carleman, Torsten Zur Theorie der Minimalflächen, Math. Z., Volume 9 (1921) no. 1-2, pp. 154-160 | DOI

[5] David, Guy Singular sets of minimizers for the Mumford-Shah functional, Progress in Mathematics, 233, Birkhäuser Verlag, Basel, 2005, xiv+581 pages

[6] David, Guy Hölder regularity of two-dimensional almost-minimal sets in ℝ n , Ann. Fac. Sci. Toulouse Math. (6), Volume 18 (2009) no. 1, pp. 65-246 http://afst.cedram.org/item?id=AFST_2009_6_18_1_65_0 | DOI

[7] David, Guy C 1+α -regularity for two-dimensional almost-minimal sets in ℝ n , J. Geom. Anal., Volume 20 (2010) no. 4, pp. 837-954 | DOI

[8] David, Guy; Semmes, Stephen Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Mem. Amer. Math. Soc., Volume 144 (2000) no. 687, viii+132 pages | DOI

[9] Federer, Herbert Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969, xiv+676 pages

[10] Liang, Xiangyu Almgren-minimality of unions of two almost orthogonal planes in ℝ 4 , Proc. Lond. Math. Soc. (3), Volume 106 (2013) no. 5, pp. 1005-1059 | DOI

[11] Liang, Xiangyu Almgren and topological minimality for the set Y×Y, J. Funct. Anal., Volume 266 (2014) no. 10, pp. 6007-6054 | DOI

[12] Liang, Xiangyu Global regularity for minimal sets near a 𝕋-set and counterexamples, Rev. Mat. Iberoam., Volume 30 (2014) no. 1, pp. 203-236 | DOI

[13] Morgan, Frank Harnack-type mass bounds and Bernstein theorems for area-minimizing flat chains modulo ν, Comm. Partial Differential Equations, Volume 11 (1986) no. 12, pp. 1257-1283 | DOI

[14] Morgan, Frank Size-minimizing rectifiable currents, Invent. Math., Volume 96 (1989) no. 2, pp. 333-348 | DOI

[15] Morrey, C. B. Jr. Second-order elliptic systems of differential equations, Contributions to the theory of partial differential equations (Annals of Mathematics Studies, no. 33), Princeton University Press, Princeton, N. J., 1954, pp. 101-159

[16] Osserman, Robert A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969, iv+159 pp. (1 plate) pages

[17] Taylor, Jean E. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2), Volume 103 (1976) no. 3, pp. 489-539 | DOI

Cité par Sources :