A local description of 2-dimensional almost minimal sets bounded by a curve

Guy David

doi:10.5802/afst.1697

Guy David ¹

¹ Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 1, pp. 1-382.

Abstract
Résumé

We study the local regularity of sliding almost minimal sets of dimension $2$ in $ℝ^{n}$ , bounded by a smooth curve $L$ . These are a good way to model soap films bounded by a curve, and their definition is similar to Almgren’s, but the results of this paper also hold for other, smaller classes of almost minimal sets. We aim for a local description, in particular near $L$ and modulo $C^{1 + ε}$ diffeomorphisms, of such sets $E$ , but in the present paper we only obtain a full description when $E$ is close enough to a half plane, a plane or a union of two half planes bounded by the same line, or a transverse minimal cone of type $𝕐$ or $𝕋$ . The main tools are adapted near monotonicity formulae for the density, including for balls that are not centered on $L$ , and the same sort of construction of competitors as for the generalization of J. Taylor’s regularity result far from the boundary.

On étudie la régularité locale des ensembles presque minimaux de dimension $2$ dans $ℝ^{n}$ , bordés par une courbe lisse $L$ , et avec une condition glissante de bord semblable à celle d’Almgren. Ces ensembles semblent le meilleur modèle pour les films de savon bordés par une courbe, mais les résultats de ce papier s’appliquent aussi à d’autres classes, plus petites, d’ensembles presque minimaux. Le but est d’obtenir une description locale de ces ensembles, en particulier près de $L$ et modulo un difféomorphisme de classe $C^{1 + ε}$ . Dans ce papier on n’obtient une description complète que lorsque $E$ est assez proche d’un demi plan, un plan ou une union de deux demi plans bordés par la même droite, ou un cône minimal de type $𝕐$ ou $𝕋$ transverse à $L$ . Les outils principaux sont des formules de presque monotonie adaptées pour la densité, y compris pour des boules qui ne sont pas centrées sur $L$ , et la construction du même genre de compétiteurs que pour la généralisation du résultat de J. Taylor sur la régularité loin du bord.

Received: 2019-02-06
Accepted: 2019-07-01
Published online: 2022-03-15

DOI: 10.5802/afst.1697

Classification: 49K99, 49Q20
Keywords: Almost minimal sets, sliding boundary condition, Plateau problem

Author's affiliations:

Guy David ¹

¹ Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Guy David. A local description of 2-dimensional almost minimal sets bounded by a curve. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 1, pp. 1-382. doi : 10.5802/afst.1697. https://afst.centre-mersenne.org/articles/10.5802/afst.1697/

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