Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy
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We study a modified Landau–de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two- and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.
A. M. would like to thank John Ball for suggesting this problem to her when she was a postdoctoral researcher at OxPDE. Part of this work was carried out when the authors were visiting the International Centre for Mathematical Sciences (ICMS) in Edinburgh (UK), supported by the Research-in-Groups program. The authors would like to thank the ICMS for its hospitality. G.C.’s research was supported by the Basque Government through the BERC 2018-2021 program and by the Spanish Ministry of Economy and Competitiveness: MTM2017-82184-R. A.M. is supported by an EPSRC Career Acceleration Fellowship EP/J001686/1 and EP/J001686/2 and an OCIAM Visiting Fellowship, the Keble Advanced Studies Centre. B.S.’s research was supported by the Project: Variational Advanced TEchniques for compleX MATErials (VATEXMATE) of University Federico II of Naples. B.S. would like to thank the OxPDE center whose hospitality in Michaelmas term 2015 and 2016 made it possible to interact with G.C. and A.M. and with the research group on Liquid Crystals.
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