Geometric measure theory and differential inclusions

Camillo De Lellis; Guido De Philippis; Bernd Kirchheim; Riccardo Tione

doi:10.5802/afst.1691

Camillo De Lellis ¹ ; Guido De Philippis ² ; Bernd Kirchheim ³ ; Riccardo Tione ⁴

¹ School of Mathematics, Institute for Advanced Study and Universität Zürich, 1 Einstein Dr., Princeton NJ 05840, USA
² Courant Institute for Mathematical Sciences, Mercer St. 251, New York NY 10012, USA
³ Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany
⁴ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 4, pp. 899-960.

Résumé
Abstract

Dans cet article on considère des graphes de fonctionnes lipschitziennes qui sont points stationnaires d’énergies strictement polyconvexes. Ces graphes peuvent être pensés soit comme des courants entièrs soit comme des varifolds, qui sont stationnaires pour des intégrandes elliptiques. La théorie de la régularité pour ce genre d’intégrandes est un problème ouvert, en particulier il n’existe aucun version du classique théorème d’Allard. On étudie ce problème en adoptant le point de vue des inclusions différentielles, et on démontre que l’inclusion différentielle associée avec le problème de la stationnarité ne contient pas la classe des laminés utilisés en [23] et [26] pour construire des solutions qui ne sont pas régulières. Notre résultat suggère que un théorème de régularité à la Allard peut rester valide pour intégrandes elliptiques générales. On conclut ce travail avec des questions concernant le problème de régularité des points stationnaires des intégrandes elliptiques.

In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard’s theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the class of laminates which are used in [23] and [26] to construct nonregular solutions. Our result is thus an indication that an Allard’s type result might be valid for general elliptic integrands. We conclude the paper by listing a series of open questions concerning the regularity of stationary points for elliptic integrands.

Reçu le : 2019-10-25
Accepté le : 2020-02-05
Publié le : 2021-12-06

DOI : 10.5802/afst.1691

Affiliations des auteurs :

Camillo De Lellis ¹ ; Guido De Philippis ² ; Bernd Kirchheim ³ ; Riccardo Tione ⁴

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {899--960},
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Camillo De Lellis; Guido De Philippis; Bernd Kirchheim; Riccardo Tione. Geometric measure theory and differential inclusions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 4, pp. 899-960. doi : 10.5802/afst.1691. https://afst.centre-mersenne.org/articles/10.5802/afst.1691/

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