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.
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.
Accepted:
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Camillo De Lellis 1; Guido De Philippis 2; Bernd Kirchheim 3; Riccardo Tione 4
CC-BY 4.0
@article{AFST_2021_6_30_4_899_0,
author = {Camillo De Lellis and Guido De Philippis and Bernd Kirchheim and Riccardo Tione},
title = {Geometric measure theory and differential inclusions},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {899--960},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 30},
number = {4},
year = {2021},
doi = {10.5802/afst.1691},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1691/}
}
TY - JOUR AU - Camillo De Lellis AU - Guido De Philippis AU - Bernd Kirchheim AU - Riccardo Tione TI - Geometric measure theory and differential inclusions JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2021 SP - 899 EP - 960 VL - 30 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1691/ DO - 10.5802/afst.1691 LA - en ID - AFST_2021_6_30_4_899_0 ER -
%0 Journal Article %A Camillo De Lellis %A Guido De Philippis %A Bernd Kirchheim %A Riccardo Tione %T Geometric measure theory and differential inclusions %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2021 %P 899-960 %V 30 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1691/ %R 10.5802/afst.1691 %G en %F AFST_2021_6_30_4_899_0
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, Serie 6, Volume 30 (2021) no. 4, pp. 899-960. doi : 10.5802/afst.1691. https://afst.centre-mersenne.org/articles/10.5802/afst.1691/
[1] On the first variation of a varifold, Ann. Math., Volume 95 (1972), pp. 417-491 | DOI | MR | Zbl
[2] A characterization of the Area Integrand, Convegni del Gennaio e del Maggio del 1973 (Symposia Mathematica), Volume XIV, Academic Press Inc., 1974, pp. 429-444 | MR | Zbl
[3] An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, Geometric Measure Theory and the Calculus of Variations (Proceedings of Symposia in Pure Mathematics), Volume 44 (1986), pp. 1-28 | DOI | MR | Zbl
[4] The homotopy groups of the integral cycle groups, Topology, Volume 1 (1962) no. 4, pp. 257-299 | DOI | MR
[5] Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. Math., Volume 87 (1968), pp. 321-391 | DOI | MR | Zbl
[6] Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, 2000
[7] Allard’s interior regularity theorem: an invitation to stationary Varifolds, 2017 (http://cvgmt.sns.it/paper/3454/) | Zbl
[8] The Euler equations as a differential inclusion, Ann. Math., Volume 170 (2009) no. 3, pp. 1417-1436 | DOI | MR | Zbl
[9] Rectifiability of Varifolds with Locally Bounded First Variation with Respect to Anisotropic Surface Energies, Commun. Pure Appl. Math., Volume 71 (2017) no. 6, pp. 1123-1148 | DOI | MR | Zbl
[10] Equivalence of the ellipticity conditions for geometric variational problems, Commun. Pure Appl. Anal., Volume 73 (2020) no. 11, pp. 2473-2515 | MR | Zbl
[11] On the atomic condition and compactness (2019) (in preparation)
[12] Quasiconvexity and Partial Regularity in the Calculus of Variations, Arch. Ration. Mech. Anal., Volume 95 (1986) no. 3, pp. 227-252 | DOI | MR | Zbl
[13] Tartar’s conjecture and localization of the quasiconvex hull in , Acta Math., Volume 200 (2008), pp. 279-305 | DOI | MR | Zbl
[14] Geometric measure theory, Grundlehren der Mathematischen Wissenschaften, 153, Springer, 1969
[15] -Configurations and non-rigid sets of matrices, Calc. Var. Partial Differ. Equ., Volume 57 (2018) no. 1, 19, 12 pages | DOI | MR | Zbl
[16] Cartesian currents in the calculus of variations I. Cartesian currents, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 37, Springer, 1998
[17] Cartesian currents in the calculus of variations II. Variational integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 38, Springer, 1998 | DOI
[18] Characterizations of young measures generated by gradients, Arch. Ration. Mech. Anal., Volume 115 (1991) no. 4, pp. 329-365 | DOI | MR | Zbl
[19] Studying Nonlinear pde by Geometry in Matrix Space, Geometric analysis and nonlinear partial differential equations, Springer, 2003, pp. 347-395 | DOI | Zbl
[20] Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations, Arch. Ration. Mech. Anal., Volume 170 (2003) no. 1, pp. 63-89 | DOI | MR | Zbl
[21] Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system, Acta Math., Volume 139 (1977), pp. 1-17 | DOI | MR
[22] Variational models for microstructure and phase transitions, Calculus of variations and geometric evolution problems (Lecture Notes in Mathematics), Volume 1713, Springer, 1999, pp. 85-210 | DOI | MR | Zbl
[23] Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. Math., Volume 157 (2003) no. 3, pp. 715-742 | DOI | MR | Zbl
[24] Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities, Ph. D. Thesis, Princeton University, USA (1974) | MR
[25] Lectures on Geometric Measure Theory, Australian National University, 2008
[26] The Regularity of Critical Points of Polyconvex Functionals, Arch. Ration. Mech. Anal., Volume 172 (2004) no. 1, pp. 133-152 | DOI | MR | Zbl
[27] Rank-one convex hulls in , Calc. Var. Partial Differ. Equ., Volume 28 (2007) no. 4, pp. 545-546 | DOI | Zbl
[28] Private communication, 2018
[29] Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations, Proc. Am. Math. Soc., Volume 131 (2003) no. 10, pp. 3101-3107 | DOI | MR | Zbl
[30] Minimal graphs and differential inclusions (2020) (https://arxiv.org/abs/2002.02157)
[31] Surfaces paramétriques généralisées, Bull. Soc. Math. Fr., Volume 79 (1951), pp. 59-84 | DOI | Numdam | MR | Zbl
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