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2018 | OriginalPaper | Chapter

Variational Analysis of Nematic Shells

Authors : Giacomo Canevari, Antonio Segatti

Published in: Trends in Applications of Mathematics to Mechanics

Publisher: Springer International Publishing

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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.

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Literature
1.
go back to reference Alicandro, R., Cicalese, M.: Variational analysis of the asymptotics of the XY model. Arch. Ration. Mech. Anal. 192(3), 501–536 (2009)MathSciNetCrossRef Alicandro, R., Cicalese, M.: Variational analysis of the asymptotics of the XY model. Arch. Ration. Mech. Anal. 192(3), 501–536 (2009)MathSciNetCrossRef
2.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
3.
go back to reference Alicandro, R., Ponsiglione, M.: Ginzburg-Landau functionals and renormalized energy: a revised Γ-convergence approach. J. Funct. Anal. 266(8), 4890–4907 (2014)MathSciNetCrossRef Alicandro, R., Ponsiglione, M.: Ginzburg-Landau functionals and renormalized energy: a revised Γ-convergence approach. J. Funct. Anal. 266(8), 4890–4907 (2014)MathSciNetCrossRef
4.
go back to reference 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) 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)
5.
go back to reference 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) 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)
6.
go back to reference Bowick, M.J., Giomi, L.: Two-dimensional matter: order, curvature and defects. Adv. Phys. 58(5), 449–563 (2009)CrossRef Bowick, M.J., Giomi, L.: Two-dimensional matter: order, curvature and defects. Adv. Phys. 58(5), 449–563 (2009)CrossRef
7.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
8.
go back to reference Canevari, G., Segatti, A.: Defects in nematic shells: a Γ-convergence discrete to continuum approach. Arch. Ration. Mech. Anal. (2018, to appear) Canevari, G., Segatti, A.: Defects in nematic shells: a Γ-convergence discrete to continuum approach. Arch. Ration. Mech. Anal. (2018, to appear)
9.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
10.
11.
go back to reference Chen, Y.M., Struwe, M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989)MathSciNetCrossRef Chen, Y.M., Struwe, M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989)MathSciNetCrossRef
12.
go back to reference Dal Maso, G.: An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston, Inc., Boston, MA (1993)CrossRef Dal Maso, G.: An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston, Inc., Boston, MA (1993)CrossRef
13.
go back to reference do Carmo, M.P.: Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis Flaherty.CrossRef do Carmo, M.P.: Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis Flaherty.CrossRef
14.
go back to reference Golovaty, D., Montero, A., Sternberg, P.: Dimension reduction for the landau-de gennes model on curved nematic thin films. arXiv, arXiv:1611.03011v1 (2016) Golovaty, D., Montero, A., Sternberg, P.: Dimension reduction for the landau-de gennes model on curved nematic thin films. arXiv, arXiv:1611.03011v1 (2016)
15.
go back to reference Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105(4), 547–570 (1986)MathSciNetCrossRef Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105(4), 547–570 (1986)MathSciNetCrossRef
16.
go back to reference Hildebrandt, K., Polthier, K., Wardetzky, M.: On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedicata 123, 89–112 (2006)MathSciNetCrossRef Hildebrandt, K., Polthier, K., Wardetzky, M.: On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedicata 123, 89–112 (2006)MathSciNetCrossRef
17.
go back to reference Ignat, R., Jerrard, R.: Interaction energy between vortices of vector fields on Riemannian surfaces. ArXiv: 1701.06546 (2017)MathSciNetCrossRef Ignat, R., Jerrard, R.: Interaction energy between vortices of vector fields on Riemannian surfaces. ArXiv: 1701.06546 (2017)MathSciNetCrossRef
18.
go back to reference Ignat, R., Jerrard, R.: Renormalized energy between vortices in some Ginzburg-Landau models on Riemannian surfaces. Preprint (2017) Ignat, R., Jerrard, R.: Renormalized energy between vortices in some Ginzburg-Landau models on Riemannian surfaces. Preprint (2017)
19.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
20.
go back to reference Jerrard, R.L.: Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30(4), 721–746 (1999)MathSciNetCrossRef Jerrard, R.L.: Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30(4), 721–746 (1999)MathSciNetCrossRef
21.
go back to reference Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differ. Equ. 14(2), 151–191 (2002)MathSciNetCrossRef Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differ. Equ. 14(2), 151–191 (2002)MathSciNetCrossRef
22.
go back to reference 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)CrossRef 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)CrossRef
23.
go back to reference Kralj, S., Rosso, R., Virga, E.G.: Curvature control of valence on nematic shells. Soft Matter 7, 670–683 (2011)CrossRef Kralj, S., Rosso, R., Virga, E.G.: Curvature control of valence on nematic shells. Soft Matter 7, 670–683 (2011)CrossRef
24.
go back to reference Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6(1), 59–84 (1996)MathSciNetCrossRef Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6(1), 59–84 (1996)MathSciNetCrossRef
25.
go back to reference Lin, F., Wang, C.: The Analysis of Harmonic Maps and Their Heat Flows. World Scientific Publishing, Hackensack, NJ (2008)CrossRef Lin, F., Wang, C.: The Analysis of Harmonic Maps and Their Heat Flows. World Scientific Publishing, Hackensack, NJ (2008)CrossRef
26.
go back to reference Lubensky, T.C., Prost, J.: Orientational order and vesicle shape. J. Phys. II France 2(3), 371–382 (1992)CrossRef Lubensky, T.C., Prost, J.: Orientational order and vesicle shape. J. Phys. II France 2(3), 371–382 (1992)CrossRef
27.
go back to reference Napoli, G., Vergori, L.: Extrinsic curvature effects on nematic shells. Phys. Rev. Lett. 108(20), 207803 (2012)CrossRef Napoli, G., Vergori, L.: Extrinsic curvature effects on nematic shells. Phys. Rev. Lett. 108(20), 207803 (2012)CrossRef
28.
go back to reference Napoli, G., Vergori, L.: Surface free energies for nematic shells. Phys. Rev. E 85(6), 061701 (2012)CrossRef Napoli, G., Vergori, L.: Surface free energies for nematic shells. Phys. Rev. E 85(6), 061701 (2012)CrossRef
29.
go back to reference Nelson, D.R.: Toward a tetravalent chemistry of colloids. Nano Lett. 2(10), 1125–1129 (2002)CrossRef Nelson, D.R.: Toward a tetravalent chemistry of colloids. Nano Lett. 2(10), 1125–1129 (2002)CrossRef
30.
go back to reference Rosso, R., Virga, E.G., Kralj, S.: Parallel transport and defects on nematic shells. Continum Mech. Thermodyn. 24(4–6), 643–664 (2012)MathSciNetCrossRef Rosso, R., Virga, E.G., Kralj, S.: Parallel transport and defects on nematic shells. Continum Mech. Thermodyn. 24(4–6), 643–664 (2012)MathSciNetCrossRef
31.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
32.
go back to reference 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) 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)
33.
34.
go back to reference Segatti, A.: Variational models for nematic shells. Lecture Notes for a PhD course at Universidad Autonoma, Madrid (October 2015) Segatti, A.: Variational models for nematic shells. Lecture Notes for a PhD course at Universidad Autonoma, Madrid (October 2015)
35.
go back to reference Segatti, A., Snarski, M., Veneroni, M.: Equilibrium configurations of nematic liquid crystals on a torus. Phys. Rev. E 90(1), 012501 (2014)CrossRef Segatti, A., Snarski, M., Veneroni, M.: Equilibrium configurations of nematic liquid crystals on a torus. Phys. Rev. E 90(1), 012501 (2014)CrossRef
36.
go back to reference Segatti, A., Snarski, M., Veneroni, M.: Analysis of a variational model for nematic shells. Math. Models Methods Appl. Sci. 26(10), 1865–1918 (2016)MathSciNetCrossRef Segatti, A., Snarski, M., Veneroni, M.: Analysis of a variational model for nematic shells. Math. Models Methods Appl. Sci. 26(10), 1865–1918 (2016)MathSciNetCrossRef
37.
go back to reference 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) 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)
38.
go back to reference Shkoller, S.: Well-posedness and global attractors for liquid crystals on Riemannian manifolds. Comm. Partial Differ. Equ. 27(5–6), 1103–1137 (2002)MathSciNetCrossRef Shkoller, S.: Well-posedness and global attractors for liquid crystals on Riemannian manifolds. Comm. Partial Differ. Equ. 27(5–6), 1103–1137 (2002)MathSciNetCrossRef
39.
go back to reference Straley, J.P.: Liquid crystals in two dimensions. Phys. Rev. A 4(2), 675–681 (1971)CrossRef Straley, J.P.: Liquid crystals in two dimensions. Phys. Rev. A 4(2), 675–681 (1971)CrossRef
40.
go back to reference Virga, E.G.: Variational theories for liquid crystals. Applied Mathematics and Mathematical Computation, vol. 8. Chapman & Hall, London, 1994.CrossRef Virga, E.G.: Variational theories for liquid crystals. Applied Mathematics and Mathematical Computation, vol. 8. Chapman & Hall, London, 1994.CrossRef
42.
go back to reference Vitelli, V., Nelson, D.R.: Defect generation and deconfinement on corrugated topographies. Phys. Rev. E 70, 051105 (2004)CrossRef Vitelli, V., Nelson, D.R.: Defect generation and deconfinement on corrugated topographies. Phys. Rev. E 70, 051105 (2004)CrossRef
43.
go back to reference 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)CrossRef 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)CrossRef
Metadata
Title
Variational Analysis of Nematic Shells
Authors
Giacomo Canevari
Antonio Segatti
Copyright Year
2018
DOI
https://doi.org/10.1007/978-3-319-75940-1_5

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