# American Institute of Mathematical Sciences

March  2012, 17(2): 597-635. doi: 10.3934/dcdsb.2012.17.597

## Maps into projective spaces: Liquid crystal and conformal energies

 1 Dipartimento di Matematica dell’Università di Parma, Parco Area delle Scienze 53/A, I-43100 Parma, Italy

Received  August 2010 Revised  February 2011 Published  December 2011

Variational problems for the liquid crystal energy of mappings from three-dimensional domains into the real projective plane are studied. More generally, we study the dipole problem, the relaxed energy, and density properties concerning the conformal $p$-energy of mappings from $n$-dimensional domains that are constrained to take values into the $p$-dimensional real projective space, for any positive integer $p$. Furthermore, a notion of optimally connecting measure for the singular set of such class of maps is given.
Citation: Domenico Mucci. Maps into projective spaces: Liquid crystal and conformal energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 597-635. doi: 10.3934/dcdsb.2012.17.597
##### References:
 [1] F. J. Almgren, W. Browder and E. H. Lieb, Co-area, liquid crystals, and minimal surfaces, in "Partial Differential Equations" (Tianjin, 1986), Lecture Notes in Math., 1306, Springer, Berlin, (1988), 1-22. [2] F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math., 167 (1991), 153-206. doi: 10.1007/BF02392449. [3] F. Bethuel, H. Brezis and J.-M. Coron, Relaxed energies for harmonic maps, in "Variational Methods" (Paris, 1988) (eds. H. Berestycki, J.-M. Coron and J. Ekeland), Progr. Nonlinear Differential Equations Appl., 4, Birkäuser Boston, Boston, MA, (1990), 37-52. [4] F. Bethuel and X. M. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., 80 (1988), 60-75. doi: 10.1016/0022-1236(88)90065-1. [5] P. Biscari and G. Guidone Peroli, A hierarchy of defects in biaxial nematics, Commun. Math. Phys., 186 (1997), 381-392. doi: 10.1007/s002200050113. [6] H. Brezis, J.-M. Coron and E. H. Lieb, Harmonic maps with defects, Commun. Math. Phys., 107 (1986), 649-705. doi: 10.1007/BF01205490. [7] S. Chandresekhar, "Liquid Crystals," Cambridge Univ. Press, Cambridge, 1977. [8] J.-M. Coron, J.-M. Ghidaglia and F. Helein, eds., "Nematics. Mathematical and Physical Aspects," Proceedings of the NATO Advanced Research Workshop on Defects, Singularities and Patterns in Nematic Liquid Crystals: Mathematical and Physical Aspects held at the Université de Paris XI, Orsay, May 28–June 1, 1990, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 332, Kluwer Academic Publ., Dodrecht, 1991. [9] P.-G. De Gennes, "The Physics of Liquid Crystals," Oxford Univ. Press, Oxford, 1974. [10] L. C. Evans and R. F. Gariepy, Some remarks concerning quasiconvexity and strong convergence, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 53-61. doi: 10.1017/S0308210500018199. [11] J. L. Ericksen, Equilibrium theory of liquid crystals, in "Advances in Liquid Crystals" (ed. G.H. Brown), 2, Academic Press, New York, (1976), 233-299. [12] J. L. Ericksen and D. Kinderlehrer, eds., "Theory and Applications of Liquid Crystals," Papers from the IMA workshop held in Minneapolis, Minn., January 21–25, 1985, The IMA Volumes in Mathematics and its Applications, 5, Springer-Verlag, New York, 1987. [13] H. Federer, "Geometric Measure Theory," Grundlehren Math. Wissen., 153, Springer-Verlag New York, Inc., New York, 1969. [14] H. Federer, Real flat chains, cochains and variational problems,, Indiana Univ. Math. J., 24 (): 351.  doi: 10.1512/iumj.1974.24.24031. [15] H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2), 72 (1960), 458-520. doi: 10.2307/1970227. [16] F. C. Frank, On the theory of liquid crystals, Discuss. Faraday Soc., 28 (1958), 19-28. doi: 10.1039/df9582500019. [17] M. Giaquinta and G. Modica, On sequences of maps with equibounded energies, Calc. Var. Partial Differential Equations, 12 (2001), 213-222. [18] M. Giaquinta, G. Modica and J. Souček, The Dirichlet energy of mappings with values into the sphere, Manuscr. Math., 65 (1989), 489-507. doi: 10.1007/BF01172794. [19] M. Giaquinta, G. Modica and J. Souček, Liquid crystals: Relaxed energies, dipoles, singular lines and singular points, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 17 (1990), 415-437. [20] M. Giaquinta, G. Modica and J. Souček, Variational problems for the conformally invariant integral $\int|du|^n$, in "Progress in Partial Differential Equations: Calculus of Variations, Applications" (eds. C. Bandle, J. M. C. Bemelmans, M. Grüter and J. S. J. Paulin) (Pont-à-Mousson, 1991), Pitman Research Notes in Math. Ser., 267, Longman Sci. Tech., Harlow, (1992), 27-47. [21] M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations," I, II, Ergebnisse Math. Grenzgebiete (III Ser), 37, 38, Springer, Berlin, 1998. [22] M. Giaquinta and D. Mucci, Density results relative to the Dirichlet energy of mappings into a manifold, Commun. Pure Appl. Math., 59 (2006), 1791-1810. doi: 10.1002/cpa.20125. [23] M. Giaquinta and D. Mucci, "Maps into Manifolds and Currents: Area and $W^{1,2}$-, $W^{1/2}$-, $BV$-Energies," CRM Series, 3, Edizioni della Normale, Pisa, 2006. [24] F. Hang and F. Lin, Topology of Sobolev mappings. II, Acta Math., 191 (2003), 55-107. doi: 10.1007/BF02392696. [25] R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystals theory, in "Theory and Applications of Liquid Crystals" (eds. J. L. Ericksen and D. Kinderlehrer) (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., 5, Springer, New York, (1987), 151-184. [26] R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid cristals configurations, Commun. Math. Phys., 105 (1986), 547-570. doi: 10.1007/BF01238933. [27] R. Hardt, D. Kinderlehrer and F. Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 297-322. [28] R. Hardt, D. Kinderlehrer and F. Lin, The variety of configurations of static liquid crystals, in "Variational Methods" (eds. H. Beresticky, J. M. Coron and I. Ekeland) (Paris, 1988), Progr. Nonlinear Differential Equations Appl., 4, Birkäuser Boston, Boston, MA, (1990), 115-131. [29] R. Hardt, D. Kinderlehrer and M. Luskin, Remarks about the variational theory of liquid crystals, in "Calculus of Variations and Partial Differential Equations" (eds. S. Hildebrandt, D. Kinderlehrer and M. Miranda) (Trento, 1986), Lecture Notes in Math., 1340, Springer, Berlin, (1988), 123-138. [30] D. Kinderlehrer, Recent developments in liquid crystals theory, in "Frontiers in Pure and Applied Mathemathics," North-Holland, Amsterdam, (1991), 151-178. [31] F. M. Leslie, Theory of flow phenomena in liquid crystals, in "Advances in Liquid Crystals" (ed. G. H. Brown), 2, Academic Press, New York, (1976), 1-81. [32] F. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), 289-330. [33] D. Mucci, Sobolev maps into the projective line with bounded total variation, Confluentes Mathematici, 2 (2010), 181-216. [34] C. W. Oseen, The theory of liquid crystals, Trans. Faraday Soc., 29 (1933), 883-889. doi: 10.1039/tf9332900883. [35] M. R. Pakzad and T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal., 13 (2003), 223-257. doi: 10.1007/s000390300006. [36] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom., 18 (1983), 253-268. [37] L. Simon, "Lectures on Geometric Measure Theory," Proc. of the Centre for Math. Analysis, Australian National University, 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. [38] E. H. Spanier, "Algebraic Topology," Corrected reprint, Springer-Verlag, New York-Berlin, 1981. [39] Yu. G. Reshetnyak, "Space Mappings with Bounded Distortion," Translation of Mathematical Monographs, 73, American Math. Soc., Providence, RI, 1989. [40] U. Tarp-Ficenc, On the minimizers of the relaxed energy functionals of mappings from higher dimensional balls into $\mathbbS^2$, Calc. Var. Partial Differential Equations, 23 (2005), 451-467. [41] E. G. Virga, "Variational Theories for Liquid Crystals," Applied Mathematics and Mathematical Computation, 8, Chapman & Hall, London, 1994.

show all references

##### References:
 [1] F. J. Almgren, W. Browder and E. H. Lieb, Co-area, liquid crystals, and minimal surfaces, in "Partial Differential Equations" (Tianjin, 1986), Lecture Notes in Math., 1306, Springer, Berlin, (1988), 1-22. [2] F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math., 167 (1991), 153-206. doi: 10.1007/BF02392449. [3] F. Bethuel, H. Brezis and J.-M. Coron, Relaxed energies for harmonic maps, in "Variational Methods" (Paris, 1988) (eds. H. Berestycki, J.-M. Coron and J. Ekeland), Progr. Nonlinear Differential Equations Appl., 4, Birkäuser Boston, Boston, MA, (1990), 37-52. [4] F. Bethuel and X. M. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., 80 (1988), 60-75. doi: 10.1016/0022-1236(88)90065-1. [5] P. Biscari and G. Guidone Peroli, A hierarchy of defects in biaxial nematics, Commun. Math. Phys., 186 (1997), 381-392. doi: 10.1007/s002200050113. [6] H. Brezis, J.-M. Coron and E. H. Lieb, Harmonic maps with defects, Commun. Math. Phys., 107 (1986), 649-705. doi: 10.1007/BF01205490. [7] S. Chandresekhar, "Liquid Crystals," Cambridge Univ. Press, Cambridge, 1977. [8] J.-M. Coron, J.-M. Ghidaglia and F. Helein, eds., "Nematics. Mathematical and Physical Aspects," Proceedings of the NATO Advanced Research Workshop on Defects, Singularities and Patterns in Nematic Liquid Crystals: Mathematical and Physical Aspects held at the Université de Paris XI, Orsay, May 28–June 1, 1990, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 332, Kluwer Academic Publ., Dodrecht, 1991. [9] P.-G. De Gennes, "The Physics of Liquid Crystals," Oxford Univ. Press, Oxford, 1974. [10] L. C. Evans and R. F. Gariepy, Some remarks concerning quasiconvexity and strong convergence, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 53-61. doi: 10.1017/S0308210500018199. [11] J. L. Ericksen, Equilibrium theory of liquid crystals, in "Advances in Liquid Crystals" (ed. G.H. Brown), 2, Academic Press, New York, (1976), 233-299. [12] J. L. Ericksen and D. Kinderlehrer, eds., "Theory and Applications of Liquid Crystals," Papers from the IMA workshop held in Minneapolis, Minn., January 21–25, 1985, The IMA Volumes in Mathematics and its Applications, 5, Springer-Verlag, New York, 1987. [13] H. Federer, "Geometric Measure Theory," Grundlehren Math. Wissen., 153, Springer-Verlag New York, Inc., New York, 1969. [14] H. Federer, Real flat chains, cochains and variational problems,, Indiana Univ. Math. J., 24 (): 351.  doi: 10.1512/iumj.1974.24.24031. [15] H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2), 72 (1960), 458-520. doi: 10.2307/1970227. [16] F. C. Frank, On the theory of liquid crystals, Discuss. Faraday Soc., 28 (1958), 19-28. doi: 10.1039/df9582500019. [17] M. Giaquinta and G. Modica, On sequences of maps with equibounded energies, Calc. Var. Partial Differential Equations, 12 (2001), 213-222. [18] M. Giaquinta, G. Modica and J. Souček, The Dirichlet energy of mappings with values into the sphere, Manuscr. Math., 65 (1989), 489-507. doi: 10.1007/BF01172794. [19] M. Giaquinta, G. Modica and J. Souček, Liquid crystals: Relaxed energies, dipoles, singular lines and singular points, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 17 (1990), 415-437. [20] M. Giaquinta, G. Modica and J. Souček, Variational problems for the conformally invariant integral $\int|du|^n$, in "Progress in Partial Differential Equations: Calculus of Variations, Applications" (eds. C. Bandle, J. M. C. Bemelmans, M. Grüter and J. S. J. Paulin) (Pont-à-Mousson, 1991), Pitman Research Notes in Math. Ser., 267, Longman Sci. Tech., Harlow, (1992), 27-47. [21] M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations," I, II, Ergebnisse Math. Grenzgebiete (III Ser), 37, 38, Springer, Berlin, 1998. [22] M. Giaquinta and D. Mucci, Density results relative to the Dirichlet energy of mappings into a manifold, Commun. Pure Appl. Math., 59 (2006), 1791-1810. doi: 10.1002/cpa.20125. [23] M. Giaquinta and D. Mucci, "Maps into Manifolds and Currents: Area and $W^{1,2}$-, $W^{1/2}$-, $BV$-Energies," CRM Series, 3, Edizioni della Normale, Pisa, 2006. [24] F. Hang and F. Lin, Topology of Sobolev mappings. II, Acta Math., 191 (2003), 55-107. doi: 10.1007/BF02392696. [25] R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystals theory, in "Theory and Applications of Liquid Crystals" (eds. J. L. Ericksen and D. Kinderlehrer) (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., 5, Springer, New York, (1987), 151-184. [26] R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid cristals configurations, Commun. Math. Phys., 105 (1986), 547-570. doi: 10.1007/BF01238933. [27] R. Hardt, D. Kinderlehrer and F. Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 297-322. [28] R. Hardt, D. Kinderlehrer and F. Lin, The variety of configurations of static liquid crystals, in "Variational Methods" (eds. H. Beresticky, J. M. Coron and I. Ekeland) (Paris, 1988), Progr. Nonlinear Differential Equations Appl., 4, Birkäuser Boston, Boston, MA, (1990), 115-131. [29] R. Hardt, D. Kinderlehrer and M. Luskin, Remarks about the variational theory of liquid crystals, in "Calculus of Variations and Partial Differential Equations" (eds. S. Hildebrandt, D. Kinderlehrer and M. Miranda) (Trento, 1986), Lecture Notes in Math., 1340, Springer, Berlin, (1988), 123-138. [30] D. Kinderlehrer, Recent developments in liquid crystals theory, in "Frontiers in Pure and Applied Mathemathics," North-Holland, Amsterdam, (1991), 151-178. [31] F. M. Leslie, Theory of flow phenomena in liquid crystals, in "Advances in Liquid Crystals" (ed. G. H. Brown), 2, Academic Press, New York, (1976), 1-81. [32] F. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), 289-330. [33] D. Mucci, Sobolev maps into the projective line with bounded total variation, Confluentes Mathematici, 2 (2010), 181-216. [34] C. W. Oseen, The theory of liquid crystals, Trans. Faraday Soc., 29 (1933), 883-889. doi: 10.1039/tf9332900883. [35] M. R. Pakzad and T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal., 13 (2003), 223-257. doi: 10.1007/s000390300006. [36] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom., 18 (1983), 253-268. [37] L. Simon, "Lectures on Geometric Measure Theory," Proc. of the Centre for Math. Analysis, Australian National University, 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. [38] E. H. Spanier, "Algebraic Topology," Corrected reprint, Springer-Verlag, New York-Berlin, 1981. [39] Yu. G. Reshetnyak, "Space Mappings with Bounded Distortion," Translation of Mathematical Monographs, 73, American Math. Soc., Providence, RI, 1989. [40] U. Tarp-Ficenc, On the minimizers of the relaxed energy functionals of mappings from higher dimensional balls into $\mathbbS^2$, Calc. Var. Partial Differential Equations, 23 (2005), 451-467. [41] E. G. Virga, "Variational Theories for Liquid Crystals," Applied Mathematics and Mathematical Computation, 8, Chapman & Hall, London, 1994.
 [1] Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $L^p$ type critical Besov spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 [2] Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623 [3] Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681 [4] Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983 [5] Jinhae Park, Feng Chen, Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1419-1440. doi: 10.3934/dcds.2010.26.1419 [6] Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565 [7] J. C. Alvarez Paiva and E. Fernandes. Crofton formulas in projective Finsler spaces. Electronic Research Announcements, 1998, 4: 91-100. [8] Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591 [9] Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106 [10] Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211 [11] Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445 [12] Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757 [13] Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499 [14] Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243 [15] Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475 [16] Xian-Gao Liu, Jianzhong Min, Kui Wang, Xiaotao Zhang. Serrin's regularity results for the incompressible liquid crystals system. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5579-5594. doi: 10.3934/dcds.2016045 [17] Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1 [18] Tiziana Giorgi, Feras Yousef. Analysis of a model for bent-core liquid crystals columnar phases. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2001-2026. doi: 10.3934/dcdsb.2015.20.2001 [19] Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357 [20] Shijin Ding, Junyu Lin, Changyou Wang, Huanyao Wen. Compressible hydrodynamic flow of liquid crystals in 1-D. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 539-563. doi: 10.3934/dcds.2012.32.539

2020 Impact Factor: 1.327