Abstract
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.
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Acerbi, E., Fusco, N.: Regularity for minimizers of nonquadratic functionals: the case \(1<p<2\). J. Math. Anal. Appl.140(1), 115–135 (1989)
Ball, J.M.:Liquid Crystals and Their Defects, vol. 2200, pp. 1–46. Springer, Berlin 2017
Ball, J.M., Bedford, S.J.: Discontinuous order parameters in liquid crystal theories. Mol. Cryst. Liq. Cryst.612(1), 1–23 (2015). https://doi.org/10.1080/15421406.2015.1030571
Brezis, H.,Coron, J.M.,Lieb, E.H.: Harmonic maps with defects.Commun. Math. Phys. 107(4), 649–705 1986. http://projecteuclid.org/euclid.cmp/1104116234
Canevari, G.: Biaxiality in the asymptotic analysis of a 2D Landau-de Gennes model for liquid crystals. ESAIM Control Optim. Calc. Var.21(1), 101–137 (2015). https://doi.org/10.1051/cocv/2014025
Canevari, G.: Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model. Arch. Ration. Mech. Anal.223(2), 591–676 (2017). https://doi.org/10.1007/s00205-016-1040-9
Contreras, A., Lamy, X.: Biaxial escape in nematics at low temperature. J. Funct. Anal.272(10), 3987–3997 (2017)
De Gennes, P.G.,Prost, J.:The Physics of Liquid Crystals. International Series of Monographs on Physics. Clarendon Press, Oxford 1993. http://books.google.fr/books?id=o1cmngEACAAJ
Di Fratta, G., Robbins, J., Slastikov, V., Zarnescu, A.: Half-integer point defects in the Q-tensor theory of nematic liquid crystals. J. Nonlinear Sci.26(1), 121–140 (2016). https://doi.org/10.1007/s00332-015-9271-8
Diening, L., Ettwein, F.: Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math.20(3), 523556 (2008)
Diening, L., Stroffolini, B., Verde, A.: Everywhere regularity of functionals with \(\phi \)-growth. Manuscr. Math.129(4), 449–481 (2009). https://doi.org/10.1007/s00229-009-0277-0
Diening, L., Stroffolini, B., Verde, A.: The \(\varphi \)-harmonic approximation and the regularity of \(\varphi \)-harmonic maps. J. Differ. Equ.253, 1943–1958 (2012)
Duzaar, F., Mingione, G.: The \(p\)-harmonic approximation and the regularity of \(p\)-harmonic maps. Calc. Var. Partial Differ.20, 235–256 (2004)
Evans, L.C.:Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence, RI 2010
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences, vol. 224. Springer, Berlin (1983)
Giusti, E.: Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Bologna (1994)
Hardt, R.,Kinderlehrer, D.,Lin, F.H.: Existence and partial regularity of static liquid crystal configurations.Commun. Math. Phys. 105(4), 547–570 1986. http://projecteuclid.org/getRecord?id=euclid.cmp/1104115500
Hardt, R., Lin, F.H.: Mappings minimizing the \(L^p\) norm of the gradient. Commun. Pure Appl. Math.40(5), 555–588 (1987). https://doi.org/10.1002/cpa.3160400503
Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne.C. R. Acad. Sci. Paris Sér. I Math.312(8), 591–596 1991
Henao, D., Majumdar, A., Pisante, A.: Uniaxial versus biaxial character of nematic equilibria in three dimensions. Calc. Var. Partial Differ. Equ.56(2), 55 (2017). https://doi.org/10.1007/s00526-017-1142-8
Luckhaus, S.: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J.37(2), 349–367 (1988). https://doi.org/10.1512/iumj.1988.37.37017
Majumdar, A.: Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math.21(2), 181–203 (2010). https://doi.org/10.1017/S0956792509990210
Majumdar, A., Zarnescu, A.: Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal.196(1), 227–280 (2010). https://doi.org/10.1007/s00205-009-0249-2
Marcellini, P., Papi, G.: Nonlinear elliptic systems with general growth. J. Differ. Equ.221(2), 412443 (2006)
Mottram, N.J.,Newton, C.:Introduction to Q-tensor theory. Technical Report 10, Department of Mathematics, University of Strathclyde 2004
Nguyen, L., Zarnescu, A.: Refined approximation for minimizers of a Landau-de Gennes energy functional. Calc. Var. Partial Differ. Equ.47(1–2), 383–432 (2013). https://doi.org/10.1007/s00526-012-0522-3
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc, New York (1991)
Schoen, R.,Uhlenbeck, K.: A regularity theory for harmonic maps.J. Differ. Geom. 17(2), 307–335 1982 http://projecteuclid.org/getRecord?id=euclid.jdg/1214436923
Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math.138, 219–240 (1977). https://doi.org/10.1007/BF02392316
Wang, C.: Limits of solutions to the generalized Ginzburg-Landau functional. Commun. Partial Differ. Equ.27(5–6), 877–906 (2002). https://doi.org/10.1081/PDE-120004888
Acknowledgements
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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Canevari, G., Majumdar, A. & Stroffolini, B. Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy. Arch Rational Mech Anal 233, 1169–1210 (2019). https://doi.org/10.1007/s00205-019-01376-7
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DOI: https://doi.org/10.1007/s00205-019-01376-7