On some properties of three-dimensional minimal sets in ℝ 4

Tien Duc Luu

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 3, page 465-493
  • ISSN: 0240-2963

Abstract

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in ℝ 4 around a 𝕐 -point and the existence of a point of particular type of a Mumford-Shah minimal set in ℝ 4 , which is very close to a 𝕋 . This will give a local description of minimal sets of dimension 3 in ℝ 4 around a singular point and a property of Mumford-Shah minimal sets in ℝ 4 .

How to cite

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Luu, Tien Duc. "On some properties of three-dimensional minimal sets in ${\mathbb{R}}^4$." Annales de la faculté des sciences de Toulouse Mathématiques 22.3 (2013): 465-493. <http://eudml.org/doc/275284>.

@article{Luu2013,
abstract = {We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $\{\mathbb\{R\}\}^4$ around a $\{\mathbb\{Y\}\}$-point and the existence of a point of particular type of a Mumford-Shah minimal set in $\{\mathbb\{R\}\}^4$, which is very close to a $\{\mathbb\{T\}\}$. This will give a local description of minimal sets of dimension 3 in $\{\mathbb\{R\}\}^4$ around a singular point and a property of Mumford-Shah minimal sets in $\{\mathbb\{R\}\}^4$.},
author = {Luu, Tien Duc},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Almgren minimal sets; Hölder regularity; Mumford-Shah minimal sets},
language = {eng},
month = {6},
number = {3},
pages = {465-493},
publisher = {Université Paul Sabatier, Toulouse},
title = {On some properties of three-dimensional minimal sets in $\{\mathbb\{R\}\}^4$},
url = {http://eudml.org/doc/275284},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Luu, Tien Duc
TI - On some properties of three-dimensional minimal sets in ${\mathbb{R}}^4$
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 3
SP - 465
EP - 493
AB - We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in ${\mathbb{R}}^4$ around a ${\mathbb{Y}}$-point and the existence of a point of particular type of a Mumford-Shah minimal set in ${\mathbb{R}}^4$, which is very close to a ${\mathbb{T}}$. This will give a local description of minimal sets of dimension 3 in ${\mathbb{R}}^4$ around a singular point and a property of Mumford-Shah minimal sets in ${\mathbb{R}}^4$.
LA - eng
KW - Almgren minimal sets; Hölder regularity; Mumford-Shah minimal sets
UR - http://eudml.org/doc/275284
ER -

References

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  4. . David (G.).— C 1 + α -regularity for two dimensional almost-minimal sets in ℝ n , Journal of Geometric Analysis, Vol 20, Number 4, p. 837-954. Zbl1225.49044MR2683770
  5. . David (G.).— Singular sets of minimizers for the Mumford-Shah functional, Progress in Mathematics 233 (581p.), Birkhäuser (2005). Zbl1086.49030MR2129693
  6. . David (G.), De Pauw (T.), and Toro (T.).— A generalization of Reifenberg’s theorem in ℝ 3 , Geom. Funct. Anal. Vol. 18, p. 1168-1235 (2008). Zbl1169.49040MR2465688
  7. . David (G.) and Semmes (S.).— Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Memoirs of the A.M.S. Number 687, Vol 144 (2000). Zbl0966.49024MR1683164
  8. . Dugundji (J.).— Topology, Allyn and Bacon, Boston (1966). Zbl0397.54003MR193606
  9. . Federer (H.).— Geometric measure theory, Grundlehren der Mathematishen Wissenschaften 153, Springer Verlag (1969). Zbl0176.00801MR257325
  10. . Simons (J.).— Minimal varieties in riemannian manifolds, Ann. of Math, (2), Vol. 88, p. 62-105 (1968). Zbl0181.49702MR233295
  11. . Taylor (J.).— The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103, no. 3, p. 489-539 (1976). Zbl0335.49032MR428181

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