Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy

  • Giacomo Canevari
  • Apala MajumdarEmail author
  • Bianca Stroffolini
Open Access


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.


  1. 1.
    Acerbi, E., Fusco, N.: Regularity for minimizers of nonquadratic functionals: the case \(1<p<2\). J. Math. Anal. Appl.140(1), 115–135 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ball, J.M.:Liquid Crystals and Their Defects, vol. 2200, pp. 1–46. Springer, Berlin 2017Google Scholar
  3. 3.
    Ball, J.M., Bedford, S.J.: Discontinuous order parameters in liquid crystal theories. Mol. Cryst. Liq. Cryst.612(1), 1–23 (2015). CrossRefGoogle Scholar
  4. 4.
    Brezis, H.,Coron, J.M.,Lieb, E.H.: Harmonic maps with defects.Commun. Math. Phys. 107(4), 649–705 1986.
  5. 5.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Contreras, A., Lamy, X.: Biaxial escape in nematics at low temperature. J. Funct. Anal.272(10), 3987–3997 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    De Gennes, P.G.,Prost, J.:The Physics of Liquid Crystals. International Series of Monographs on Physics. Clarendon Press, Oxford 1993.
  9. 9.
    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). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Diening, L., Ettwein, F.: Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math.20(3), 523556 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diening, L., Stroffolini, B., Verde, A.: Everywhere regularity of functionals with \(\phi \)-growth. Manuscr. Math.129(4), 449–481 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Diening, L., Stroffolini, B., Verde, A.: The \(\varphi \)-harmonic approximation and the regularity of \(\varphi \)-harmonic maps. J. Differ. Equ.253, 1943–1958 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duzaar, F., Mingione, G.: The \(p\)-harmonic approximation and the regularity of \(p\)-harmonic maps. Calc. Var. Partial Differ.20, 235–256 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Evans, L.C.:Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence, RI 2010Google Scholar
  15. 15.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)zbMATHGoogle Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences, vol. 224. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  17. 17.
    Giusti, E.: Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Bologna (1994)zbMATHGoogle Scholar
  18. 18.
    Hardt, R.,Kinderlehrer, D.,Lin, F.H.: Existence and partial regularity of static liquid crystal configurations.Commun. Math. Phys. 105(4), 547–570 1986.
  19. 19.
    Hardt, R., Lin, F.H.: Mappings minimizing the \(L^p\) norm of the gradient. Commun. Pure Appl. Math.40(5), 555–588 (1987). CrossRefzbMATHGoogle Scholar
  20. 20.
    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 1991Google Scholar
  21. 21.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Majumdar, A.: Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math.21(2), 181–203 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Marcellini, P., Papi, G.: Nonlinear elliptic systems with general growth. J. Differ. Equ.221(2), 412443 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mottram, N.J.,Newton, C.:Introduction to Q-tensor theory. Technical Report 10, Department of Mathematics, University of Strathclyde 2004Google Scholar
  27. 27.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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)Google Scholar
  29. 29.
    Schoen, R.,Uhlenbeck, K.: A regularity theory for harmonic maps.J. Differ. Geom. 17(2), 307–335 1982
  30. 30.
    Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math.138, 219–240 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wang, C.: Limits of solutions to the generalized Ginzburg-Landau functional. Commun. Partial Differ. Equ.27(5–6), 877–906 (2002). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Giacomo Canevari
    • 1
    • 2
  • Apala Majumdar
    • 3
    Email author
  • Bianca Stroffolini
    • 4
  1. 1.Basque Center for Applied MathematicsBilbaoSpain
  2. 2.Dipartimento di InformaticaUniversità degli Studi di VeronaVeronaItaly
  3. 3.Mathematical SciencesUniversity of BathBathUK
  4. 4.Dipartimento di Matematica e ApplicazioniUniversità Federico IINapoliItaly

Personalised recommendations