Boundary regularity of weakly anchored harmonic maps

Andres Contreras; Xavier Lamy; Rémy Rodiac

doi:10.1016/j.crma.2015.09.014

Partial differential equations/Mathematical physics

Boundary regularity of weakly anchored harmonic maps

Presented by: Haïm Brézis

Andres Contreras ¹; Xavier Lamy ^2,³; Rémy Rodiac ^4,⁵

¹ Science Hall 224, New Mexico State University, Department of Mathematical Sciences, USA
² Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
³ Institut Camille Jordan, Université Lyon-1, Villeurbanne, France
⁴ Département de mathématiques, Université Paris-Est Créteil, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France
⁵ Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuna Mackenna 4860, Macul, Santiago, Chile

Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1093-1097.

Abstract
Résumé

In this note, we study the boundary regularity of the minimizers of a family of weak anchoring energies that model the states of liquid crystals. We establish optimal boundary regularity in all dimensions $n \geq 3$ . In dimension $n = 3$ , this yields full regularity at the boundary, which stands in sharp contrast with the observation of boundary defects in physics works. We also show that, in the cases of weak and strong anchoring, the regularity of the minimizers is inherited from that of their corresponding limit problems.

Dans cette note, nous étudions la régularité au bord des minimiseurs d'une famille d'énergies avec ancrage faible utilisée dans la modélisation des cristaux liquides. Nous établissons la régularité au bord optimale en toute dimension supérieure à 3. En dimension $n = 3$ , de tels minimiseurs sont lisses près du bord, ce qui va à l'encontre des observations de défauts sur le bord dans les travaux physiques. Nous montrons également que, dans les cas de faible et de fort ancrage, la régularité des minimiseurs est héritée de la régularité des minimiseurs des problèmes limites correspondants.

Received: 2015-09-14
Accepted: 2015-09-18
Published online: 2015-10-23

DOI: 10.1016/j.crma.2015.09.014

Author's affiliations:

Andres Contreras ¹; Xavier Lamy ^{2, 3}; Rémy Rodiac ^{4, 5}

¹ Science Hall 224, New Mexico State University, Department of Mathematical Sciences, USA
² Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
³ Institut Camille Jordan, Université Lyon-1, Villeurbanne, France
⁴ Département de mathématiques, Université Paris-Est Créteil, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France
⁵ Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuna Mackenna 4860, Macul, Santiago, Chile

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     author = {Andres Contreras and Xavier Lamy and R\'emy Rodiac},
     title = {Boundary regularity of weakly anchored harmonic maps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1093--1097},
     publisher = {Elsevier},
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     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.014},
     language = {en},
}

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Andres Contreras; Xavier Lamy; Rémy Rodiac. Boundary regularity of weakly anchored harmonic maps. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1093-1097. doi : 10.1016/j.crma.2015.09.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.014/

References
Cited by

[1] M. Giaquinta; L. Martinazzi An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Edizioni della Normale, Pisa, Italy, 2012

[2] R. Hardt; F.H. Lin Partially constrained boundary conditions with energy minimizing mappings, Commun. Pure Appl. Math., Volume 42 (1989) no. 3, pp. 309-334

[3] M. Kléman; O.D. Lavrentovich Soft Matter Physics: An Introduction, Springer, 2007

[4] S. Luckhaus Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J., Volume 37 (1988) no. 2, pp. 349-367

[5] A. Majumdar; A. Zarnescu Landau–De Gennes theory of nematic liquid crystals: the Oseen–Frank limit and beyond, Arch. Ration. Mech. Anal., Volume 196 (2010) no. 1, pp. 227-280

[6] R. Schoen; K. Uhlenbeck A regularity theory for harmonic maps, J. Differ. Geom., Volume 17 (1982) no. 2, pp. 307-335

[7] R. Schoen; K. Uhlenbeck Boundary regularity and the Dirichlet problem for harmonic maps, J. Differ. Geom., Volume 18 (1983) no. 2, pp. 253-268

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