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Variational Analysis of Nematic Shells

  • Giacomo Canevari
  • Antonio SegattiEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 27)

Abstract

In this note we present some recent results on the Mathematical Analysis of Nematic Shells. The type of results we present deal with the analysis of defectless configurations as well as the analysis of defected configurations. The mathematical tools include Topology, Analysis of Partial Differential Equations as well as Variational Techniques like Γ convergence.

Notes

Acknowledgements

G.C.’s research has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement n 291053. A.S. gratefully acknowledges the financial support of the FP7-IDEAS-ERC-StG #256872 (EntroPhase). G.C. and A.S. acknowledge the partial support of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) group of INdAM (Istituto Nazionale di Alta Matematica).

References

  1. 1.
    Alicandro, R., Cicalese, M.: Variational analysis of the asymptotics of the XY model. Arch. Ration. Mech. Anal. 192(3), 501–536 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alicandro, R., De Luca, L., Garroni, A., Ponsiglione, M.: Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach. Arch. Ration. Mech. Anal. 214(1), 269–330 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alicandro, R., Ponsiglione, M.: Ginzburg-Landau functionals and renormalized energy: a revised Γ-convergence approach. J. Funct. Anal. 266(8), 4890–4907 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berezinskii, V.L.: Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. i. Classical systems. J. Exp. Theor. Phys. 61(3), 1144 (1972)Google Scholar
  5. 5.
    Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, vol. 13. Birkhäuser Boston, Inc., Boston, MA (1994)Google Scholar
  6. 6.
    Bowick, M.J., Giomi, L.: Two-dimensional matter: order, curvature and defects. Adv. Phys. 58(5), 449–563 (2009)CrossRefGoogle Scholar
  7. 7.
    Braides, A., Cicalese, M., Solombrino, F.: Q-Tensor continuum energies as limits of head-to-tail symmetric spin systems. SIAM J. Math. Anal. 47(4), 2832–2867 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Canevari, G., Segatti, A.: Defects in nematic shells: a Γ-convergence discrete to continuum approach. Arch. Ration. Mech. Anal. (2018, to appear)Google Scholar
  9. 9.
    Canevari, G., Segatti, A., Veneroni, M.: Morse’s index formula in VMO for compact manifolds with boundary. J. Funct. Anal. 269(10), 3043–3082 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, Y.M.: The weak solutions to the evolution problems of harmonic maps. Math. Z. 201(1), 69–74 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, Y.M., Struwe, M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dal Maso, G.: An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston, Inc., Boston, MA (1993)CrossRefGoogle Scholar
  13. 13.
    do Carmo, M.P.: Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis Flaherty.CrossRefGoogle Scholar
  14. 14.
    Golovaty, D., Montero, A., Sternberg, P.: Dimension reduction for the landau-de gennes model on curved nematic thin films. arXiv, arXiv:1611.03011v1 (2016)Google Scholar
  15. 15.
    Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105(4), 547–570 (1986)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hildebrandt, K., Polthier, K., Wardetzky, M.: On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedicata 123, 89–112 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ignat, R., Jerrard, R.: Interaction energy between vortices of vector fields on Riemannian surfaces. ArXiv: 1701.06546 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ignat, R., Jerrard, R.: Renormalized energy between vortices in some Ginzburg-Landau models on Riemannian surfaces. Preprint (2017)Google Scholar
  19. 19.
    Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Stability of the melting hedgehog in the Landau–de Gennes theory of nematic liquid crystals. Arch. Ration. Mech. Anal. 215(2), 633–673 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Jerrard, R.L.: Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30(4), 721–746 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differ. Equ. 14(2), 151–191 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kosterlitz, J.M., Thouless, D.J.: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C Solid State Phys. 6(7), 1181 (1973)CrossRefGoogle Scholar
  23. 23.
    Kralj, S., Rosso, R., Virga, E.G.: Curvature control of valence on nematic shells. Soft Matter 7, 670–683 (2011)CrossRefGoogle Scholar
  24. 24.
    Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6(1), 59–84 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lin, F., Wang, C.: The Analysis of Harmonic Maps and Their Heat Flows. World Scientific Publishing, Hackensack, NJ (2008)CrossRefGoogle Scholar
  26. 26.
    Lubensky, T.C., Prost, J.: Orientational order and vesicle shape. J. Phys. II France 2(3), 371–382 (1992)CrossRefGoogle Scholar
  27. 27.
    Napoli, G., Vergori, L.: Extrinsic curvature effects on nematic shells. Phys. Rev. Lett. 108(20), 207803 (2012)CrossRefGoogle Scholar
  28. 28.
    Napoli, G., Vergori, L.: Surface free energies for nematic shells. Phys. Rev. E 85(6), 061701 (2012)CrossRefGoogle Scholar
  29. 29.
    Nelson, D.R.: Toward a tetravalent chemistry of colloids. Nano Lett. 2(10), 1125–1129 (2002)CrossRefGoogle Scholar
  30. 30.
    Rosso, R., Virga, E.G., Kralj, S.: Parallel transport and defects on nematic shells. Continum Mech. Thermodyn. 24(4–6), 643–664 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sandier, É.: Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152(2), 379–403 (1998); See Erratum, ibidem 171(1), 233 (2000)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sandier, É., Serfaty, S.: Vortices in the magnetic Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, vol. 70. Birkhäuser Boston, Inc., Boston, MA (2007)Google Scholar
  33. 33.
    Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17(2), 307–335 (1982)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Segatti, A.: Variational models for nematic shells. Lecture Notes for a PhD course at Universidad Autonoma, Madrid (October 2015)Google Scholar
  35. 35.
    Segatti, A., Snarski, M., Veneroni, M.: Equilibrium configurations of nematic liquid crystals on a torus. Phys. Rev. E 90(1), 012501 (2014)CrossRefGoogle Scholar
  36. 36.
    Segatti, A., Snarski, M., Veneroni, M.: Analysis of a variational model for nematic shells. Math. Models Methods Appl. Sci. 26(10), 1865–1918 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Selinger, R.L., Konya, A., Travesset, A., Selinger, J.V.: Monte Carlo studies of the XY model on two-dimensional curved surfaces. J. Phys. Chem B 48, 12989–13993 (2011)Google Scholar
  38. 38.
    Shkoller, S.: Well-posedness and global attractors for liquid crystals on Riemannian manifolds. Comm. Partial Differ. Equ. 27(5–6), 1103–1137 (2002)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Straley, J.P.: Liquid crystals in two dimensions. Phys. Rev. A 4(2), 675–681 (1971)CrossRefGoogle Scholar
  40. 40.
    Virga, E.G.: Variational theories for liquid crystals. Applied Mathematics and Mathematical Computation, vol. 8. Chapman & Hall, London, 1994.CrossRefGoogle Scholar
  41. 41.
    Vitelli, V., Nelson, D.: Nematic textures in spherical shells. Phys. Rev. E 74(2), 021711 (2006)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Vitelli, V., Nelson, D.R.: Defect generation and deconfinement on corrugated topographies. Phys. Rev. E 70, 051105 (2004)CrossRefGoogle Scholar
  43. 43.
    Wang, X., Miller, D.S., Bukusoglu, E., de Pablo, J.J., Abbott, N.L.: Topological defects in liquid crystals as templates for molecular self-assembly. Nat. Mater. 15(1), 106–112 (2016)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Basque Center for Applied MathematicsBilbaoSpain
  2. 2.Dipartimento di Matematica “F. Casorati”Università di PaviaPaviaItaly

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