Abstract
We observe that for a unit tangent vector field u ∈ TM on a 3-dimensional Riemannian manifold M, there is a unique unit cotangent vector field A ∈ T∗M associated to u such that we can define the curl of u by dA. Through a unit cotangent vector field A ∈ T∗M, we define the Oseen–Frank energy functional on 3-dimensional Riemannian manifolds. Moreover, we prove partial regularity of minimizers of the Oseen–Frank energy on 3-dimensional Riemannian manifolds.
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References
Almgren, F.J., Lieb, E.H.: Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds. Ann. Math. 128, 483–530 (1988)
Bethuel, F., Brezis, H.: Regularity of minimizers of relaxed problems for harmonic maps. J. Funct. Anal. 101, 145–161 (1991)
Bethuel, F., Brezis, H., Coron, J.M.: Relaxed energies for harmonic maps. In: Berestycki, H., Coron, J.-M., Ekeland, I (eds.) Variational Methods. Progress in Nonlinear Differential Equations and Their Applications, vol. 4, pp 37–52. Birkhäuser, Basel (1990)
Brezis, H., Coron, J.-M., Lieb, E.H.: Harmonic maps with defects. Commun. Math. Phys. 107, 649–705 (1986)
Chen, Y., Struwe, M.: Existence and partial regular results for the heat flow for harmonic maps. Math. Z. 201, 83–103 (1989)
Chern, S.S., Chen, W.H., Lam, K.S.: Lectures on Differential Geometry. Series on University Mathematics, vol. 1. World Scientific, Singapore (1998)
Eells, J., Lemaire, L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)
Eells, J., Sampson, J.H.: Harmonic map**s of Riemannian manifolds. Amer. J. Math. 86, 109–160 (1964)
Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)
Feng, Z., Hong, M.-C., Mei, Y.: Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system. SIAM J. Math. Anal. 52, 481–523 (2020)
Frank, F.C.: On theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)
Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)
Giaquinta, M., Giusti, E.: The singular set of the minima of certain quadratic functionals. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11, 45–55 (1984)
Giaquinta, M., Hong, M. -C., Yin, H.: A new approximation of relaxed energies for harmonic maps and the Faddeev model. Calc. Var. Partial Differ. Equ. 41, 45–69 (2011)
Giaquinta, M., Modica, G., Souček, J.: The Dirichlet energy of map**s with values into the sphere. Manuscripta Math. 65, 489–507 (1989)
Giaquinta, M., Modica, G., Souček, J.: Liquid crystals: Relaxed energies, dipoles, singular lines and singular points. Ann. Scuola Norm. Sup. Pisa 17, 415–437 (1990)
Giaquinta, M., Modica, G., Souček, J.: Cartesian Currents in the Calculus of Variations, Part II: Variational Integrals. A Series of Modern Surveys in Mathematics, vol. 38. Springer, Berlin (1998)
Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003)
Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105, 547–570 (1986)
Hardt, R., Kinderlehrer, D., Lin, F.-H.: Stable defects of minimizers of constrained variational principles. Ann. Inst. Henri Poincaré, Non Linear Anal. 5, 297–322 (1988)
Hong, M.-C.: Partial regularity of weak solutions of the liquid crystal equilibrium system. Indiana Univ. Math. J. 53, 1401–1414 (2004)
Hong, M.-C.: Existence of infinitely many equilibrium configurations of the liquid crystal system prescribing the same nonconstant boundary value. Pac. J. Math. 232, 177–206 (2007)
Hong, M.-C.: Some results on harmonic maps. Bull. Inst. Math. Acad. Sin. (N.S.) 9, 187–221 (2014)
Hong, M. -C., Jost, J., Struwe, M.: Asymptotic limits of a Ginzburg-Landau type functional. In: Jost, J. (ed.) Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt, pp 99–123. International Press (1996)
Hong, M.-C., Li, J., **n, Z.: Blow-up criteria of strong solutions to the Ericksen-Leslie system in \(\mathbb {R}^{3}\). Commun. Partial Differ. Equ. 39, 1284–1328 (2014)
Hong, M. -C., Mei, Y.: Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in \(L^{3}_{uloc}(\mathbb {R}^{3})\). Calc. Var. Partial Differ. Equ. 58, 3 (2019)
Hong, M. -C., **n, Z.: Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in \(\mathbb {R}^{2}\). Adv. Math. 231, 1364–1400 (2012)
Jost, J.: Harmonic Maps Between Surfaces. Lecture Notes in Mathematics, vol. 1062. Springer, Berlin (1984)
Jost, J.: Nonlinear Methods in Riemannian and Kählerian Geometry: Delivered at the German Mathematical Society Seminar in Düsseldorf in June, vol. 1986. Birkhäuser, Basel (1988)
Jost, J.: Riemannian Geometry and Geometric Analysis, 4th edn. Springer, Cham (2005)
Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28, 265–283 (1968)
Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113, 1–24 (1981)
Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17, 307–335 (1982)
Stewart, I.W.: The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction. Taylor and Francis, London (2004)
Oseen, C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29, 833–899 (1933)
Struwe, M.: On the evolution of harmonic maps of Riemannian surfaces. Commun. Math. Helv. 60, 558–581 (1985)
Uhlenbeck, K.: Morse theory by perturbation methods with applications to harmonic maps. Trans. Amer. Math. Soc. 267, 569–583 (1981)
Acknowledgements
I would take this opportunity to thank Professors Mariano Giaquinta and Enrico Giusti for their strong influence and support; in particular, the main idea of this paper was taught by them. Part of the research was supported by the Australian Research Council grant DP150101275.
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Dedicated to Professor Jürgen Jost on the occasion of his 65th birthday.
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Hong, MC. The Oseen–Frank Energy Functional on Manifolds. Vietnam J. Math. 49, 597–613 (2021). https://doi.org/10.1007/s10013-020-00468-2
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DOI: https://doi.org/10.1007/s10013-020-00468-2