## Topological Methods in Nonlinear Analysis

### On the existence of skyrmions in planar liquid crystals

Carlo Greco

#### Abstract

The study of topologically nontrivial field configurations is an important topic in many branches of physics and applied sciences. In this paper we are interested to the existence of such structures, the so-called skyrmions, in the context of liquid crystals. More precisely, we consider a two-dimensional nematic or cholesteric liquid crystal. In the nematic case we use a Bogomol'nyi type decomposition in order to get a topological lower bound on the configurations with a given degree for the full Oseen-Frank energy functional, and so we can find a global minimum of degree $\pm 1$ for the energy. Then we consider the cholesteric case in presence of an electric field under the one constant approximation assumption, and, by using the concentration-compactness method, we prove the existence of a minimum again on the configurations of degree $\pm 1$, for sufficiently large electric fields.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 54, Number 2A (2019), 567-586.

Dates
First available in Project Euclid: 7 October 2019

https://projecteuclid.org/euclid.tmna/1570413618

Digital Object Identifier
doi:10.12775/TMNA.2019.053

#### Citation

Greco, Carlo. On the existence of skyrmions in planar liquid crystals. Topol. Methods Nonlinear Anal. 54 (2019), no. 2A, 567--586. doi:10.12775/TMNA.2019.053. https://projecteuclid.org/euclid.tmna/1570413618

#### References

• A. Belavin and A. Polyakov, Metastable states of two-dimensional isotropic ferromagnets, JETP Lett. 22 (1975), 245–248.
• A.N. Bogdanov, Theory of spherulitic domains in cholesteric liquid crystals with positive dielectric anisotropy, JETP Lett. 71 (2000), 85–88.
• A.N. Bogdanov and A. Hubert, Thermodynamically stable magnetic vortex states in magnetic crystals, J. Mag. Mag. Mater. 138 (1994), 255–269.
• A.N. Bogdanov and A. Hubert, The stability of vortex-like structures in uniaxial ferromagnets, J. Mag. Mag. Mater. 195 (1999), 182–192.
• A.N. Bogdanov, U.K. Rö$\ss$ler and A.A. Shestakov, Skyrmions in nematic liquid crystals, Phys. Rev. E 67 (2003), 016602.
• H. Brezis and J.-M. Coron, Large solutions for harmonic maps in two dimensions, Commun. Math. Phys. 92 (1983), 203–215.
• G. De Matteis, L. Martina and V. Turco, Skyrmion states in chiral liquid crystals, Theor. Math. Phys. 196 (2018), 1150–1163.
• L. Döring and C. Melcher, Compactness results for static and dynamic chiral skyrmions near the conformal limit, Calc. Var. 56 (2017), 60.
• M.J. Esteban, A direct variational approach to Skyrme's model for meson fields, Commun. Math. Phys. 105 (1986), 571–591.
• M.J. Esteban, A new setting for Skyrme's problem, Variational Methods (Paris, 1988), Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 77–93.
• M.J. Esteban, Existence of \rom3D skyrmions, Erratum to: “A direct variational approach to Skyrme's model for meson field” (Comm. Math. Phys. 105 (1986), no. 4, 571–591); A new setting for Skyrme's problem, Variational Methods (Paris, 1988), Birkhäuser Boston, Boston, MA, 1990, pp. 77–93; Commun. Math. Phys. 251 (2004), 209–210.
• A.O. Leonov, I.E. Dragunov, U.K. Rö$\ss$ler and A.N. Bogdanov, Theory of skyrmion states in liquid crystals, Phys. Rev. E 90 (2014), 042502.
• A.O. Leonov, T.L. Monchesky, N. Romming, A. Kubetzka, A.N. Bogdanov and R. Wiesendanger, The properties of isolated chiral skyrmions in thin magnetic films, New J. Phys. 18 (2016), 065003.
• J. Li and X. Zhu, Existence of \rom2D skyrmions, Math. Z. 268 (2011), 305–315.
• F. Lin, Y. Yang, Existence of two-dimensional skyrmions via the concentration-compactness method, Comm. Pure Appl. Math. 57 (2004), 1332–1351.
• P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145.
• C. Melcher, Chiral skyrmions in the plane, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), 20140394.
• L. Nirenberg, Topics in nonlinear functional analysis, Courant Lect. Notes Math., vol. 6, New York University Courant Institute of Mathematical Sciences, New York (2001).
• R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom. 18 (1983), 253–268.
• T.H.R. Skyrme, A non-linear field theory, Proc. Roy. Soc. London Ser. A 260 (1961), 127–138.
• T.H.R. Skyrme, A unified field theory of mesons and baryons, Nuclear Phys. 31 (1962), 556–569.
• I.W. Stewart, The static and dynamic continuum theory of liquid crystals, Taylor & Francis, London, 2004.
• T. Weidig, The baby Skyrme models and their multi-skyrmions, Nonlinearity 12 (1999), 1489–1503.
• Y. Yang, Solitons in field theory and nonlinear analysis, Springer Monographs in Mathematics, Springer–Verlag, New York, 2001.