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Comptes Rendus Mathématique
Volume 353, n° 12
pages 1093-1097 (décembre 2015)
Doi : 10.1016/j.crma.2015.09.014
Received : 14 September 2015 ;  accepted : 18 September 2015
Boundary regularity of weakly anchored harmonic maps
Régularité au bord des applications harmoniques avec ancrage faible
 

Andres Contreras a , Xavier Lamy b, c , Rémy Rodiac d, e
a Science Hall 224, New Mexico State University, Department of Mathematical Sciences, USA 
b Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany 
c Institut Camille Jordan, Université Lyon-1, Villeurbanne, France 
d Département de mathématiques, Université Paris-Est Créteil, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France 
e Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuna Mackenna 4860, Macul, Santiago, Chile 

Abstract

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  . In dimension  , 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.

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Résumé

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  , 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.

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