Uniqueness property for 2-dimensional minimal cones in R3
Liang, Xiangyu (Beihang University. School of Mathematics)

Fecha:	2021
Resumen:	In this article we treat two closely related problems: 1) the upper semicontinuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the 2-dimensional minimal cones in R3 . Given an open set Ω ⊂ Rn, a closed set E ⊂ Ω is said to be Almgren minimal of dimension d in Ω if it minimizes the d-Hausdorff measure among all its Lipschitz deformations in Ω. We say that a d-dimensional minimal set E in an open set Ω admits upper semi-continuity if, whenever {fn(E)}n is a sequence of deformations of E in Ω that converges to a set F, then we have Hd(F) ≥ lim supn Hd(fn(E)). This guarantees in particular that E minimizes the d-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. As proved in [19], when several 2-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known 2-dimensional minimal cones in Rn to generate new families of minimal cones by taking their almost orthogonal unions. The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].
Derechos:	Tots els drets reservats.
Lengua:	Anglès
Documento:	Article ; recerca ; Versió publicada
Materia:	Minimal cones ; Uniqueness ; Hausdorff measure ; Plateau's problem
Publicado en:	Publicacions matemàtiques, Vol. 65 Núm. 1 (2021) , p. 3-59 (Articles) , ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/383612
DOI: 10.5565/PUBLMAT6512101
DOI: 10.5565/publicacionsmatematiques.v65i1.383612

57 p, 664.2 KB

El registro aparece en las colecciones:
Artículos > Artículos publicados > Publicacions matemàtiques
Artículos > Artículos de investigación