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 & 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, 1306 (1988), 1.   Google Scholar

[2]

F. Bethuel, The approximation problem for Sobolev maps between two manifolds,, Acta Math., 167 (1991), 153.  doi: 10.1007/BF02392449.  Google Scholar

[3]

F. Bethuel, H. Brezis and J.-M. Coron, Relaxed energies for harmonic maps,, in, 4 (1990), 37.   Google Scholar

[4]

F. Bethuel and X. M. Zheng, Density of smooth functions between two manifolds in Sobolev spaces,, J. Funct. Anal., 80 (1988), 60.  doi: 10.1016/0022-1236(88)90065-1.  Google Scholar

[5]

P. Biscari and G. Guidone Peroli, A hierarchy of defects in biaxial nematics,, Commun. Math. Phys., 186 (1997), 381.  doi: 10.1007/s002200050113.  Google Scholar

[6]

H. Brezis, J.-M. Coron and E. H. Lieb, Harmonic maps with defects,, Commun. Math. Phys., 107 (1986), 649.  doi: 10.1007/BF01205490.  Google Scholar

[7]

S. Chandresekhar, "Liquid Crystals,", Cambridge Univ. Press, (1977).   Google Scholar

[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, 332 (1990).   Google Scholar

[9]

P.-G. De Gennes, "The Physics of Liquid Crystals,", Oxford Univ. Press, (1974).   Google Scholar

[10]

L. C. Evans and R. F. Gariepy, Some remarks concerning quasiconvexity and strong convergence,, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 53.  doi: 10.1017/S0308210500018199.  Google Scholar

[11]

J. L. Ericksen, Equilibrium theory of liquid crystals,, in, 2 (1976), 233.   Google Scholar

[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 (1987).   Google Scholar

[13]

H. Federer, "Geometric Measure Theory,", Grundlehren Math. Wissen., 153 (1969).   Google Scholar

[14]

H. Federer, Real flat chains, cochains and variational problems,, Indiana Univ. Math. J., 24 (): 351.  doi: 10.1512/iumj.1974.24.24031.  Google Scholar

[15]

H. Federer and W. H. Fleming, Normal and integral currents,, Ann. of Math. (2), 72 (1960), 458.  doi: 10.2307/1970227.  Google Scholar

[16]

F. C. Frank, On the theory of liquid crystals,, Discuss. Faraday Soc., 28 (1958), 19.  doi: 10.1039/df9582500019.  Google Scholar

[17]

M. Giaquinta and G. Modica, On sequences of maps with equibounded energies,, Calc. Var. Partial Differential Equations, 12 (2001), 213.   Google Scholar

[18]

M. Giaquinta, G. Modica and J. Souček, The Dirichlet energy of mappings with values into the sphere,, Manuscr. Math., 65 (1989), 489.  doi: 10.1007/BF01172794.  Google Scholar

[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.   Google Scholar

[20]

M. Giaquinta, G. Modica and J. Souček, Variational problems for the conformally invariant integral $\int|du|^n$,, in, 267 (1992), 27.   Google Scholar

[21]

M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations," I, II,, Ergebnisse Math. Grenzgebiete (III Ser), 37 (1998).   Google Scholar

[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.  doi: 10.1002/cpa.20125.  Google Scholar

[23]

M. Giaquinta and D. Mucci, "Maps into Manifolds and Currents: Area and $W^{1,2}$-, $W^{1/2}$-, $BV$-Energies,", CRM Series, 3 (2006).   Google Scholar

[24]

F. Hang and F. Lin, Topology of Sobolev mappings. II,, Acta Math., 191 (2003), 55.  doi: 10.1007/BF02392696.  Google Scholar

[25]

R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystals theory,, in, 5 (1987), 151.   Google Scholar

[26]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid cristals configurations,, Commun. Math. Phys., 105 (1986), 547.  doi: 10.1007/BF01238933.  Google Scholar

[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.   Google Scholar

[28]

R. Hardt, D. Kinderlehrer and F. Lin, The variety of configurations of static liquid crystals,, in, 4 (1990), 115.   Google Scholar

[29]

R. Hardt, D. Kinderlehrer and M. Luskin, Remarks about the variational theory of liquid crystals,, in, 1340 (1988), 123.   Google Scholar

[30]

D. Kinderlehrer, Recent developments in liquid crystals theory,, in, (1991), 151.   Google Scholar

[31]

F. M. Leslie, Theory of flow phenomena in liquid crystals,, in, 2 (1976), 1.   Google Scholar

[32]

F. Lin and C. Liu, Static and dynamic theories of liquid crystals,, J. Partial Differential Equations, 14 (2001), 289.   Google Scholar

[33]

D. Mucci, Sobolev maps into the projective line with bounded total variation,, Confluentes Mathematici, 2 (2010), 181.   Google Scholar

[34]

C. W. Oseen, The theory of liquid crystals,, Trans. Faraday Soc., 29 (1933), 883.  doi: 10.1039/tf9332900883.  Google Scholar

[35]

M. R. Pakzad and T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds,, Geom. Funct. Anal., 13 (2003), 223.  doi: 10.1007/s000390300006.  Google Scholar

[36]

R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps,, J. Diff. Geom., 18 (1983), 253.   Google Scholar

[37]

L. Simon, "Lectures on Geometric Measure Theory,", Proc. of the Centre for Math. Analysis, 3 (1983).   Google Scholar

[38]

E. H. Spanier, "Algebraic Topology,", Corrected reprint, (1981).   Google Scholar

[39]

Yu. G. Reshetnyak, "Space Mappings with Bounded Distortion,", Translation of Mathematical Monographs, 73 (1989).   Google Scholar

[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.   Google Scholar

[41]

E. G. Virga, "Variational Theories for Liquid Crystals,", Applied Mathematics and Mathematical Computation, 8 (1994).   Google Scholar

show all references

References:
[1]

F. J. Almgren, W. Browder and E. H. Lieb, Co-area, liquid crystals, and minimal surfaces,, in, 1306 (1988), 1.   Google Scholar

[2]

F. Bethuel, The approximation problem for Sobolev maps between two manifolds,, Acta Math., 167 (1991), 153.  doi: 10.1007/BF02392449.  Google Scholar

[3]

F. Bethuel, H. Brezis and J.-M. Coron, Relaxed energies for harmonic maps,, in, 4 (1990), 37.   Google Scholar

[4]

F. Bethuel and X. M. Zheng, Density of smooth functions between two manifolds in Sobolev spaces,, J. Funct. Anal., 80 (1988), 60.  doi: 10.1016/0022-1236(88)90065-1.  Google Scholar

[5]

P. Biscari and G. Guidone Peroli, A hierarchy of defects in biaxial nematics,, Commun. Math. Phys., 186 (1997), 381.  doi: 10.1007/s002200050113.  Google Scholar

[6]

H. Brezis, J.-M. Coron and E. H. Lieb, Harmonic maps with defects,, Commun. Math. Phys., 107 (1986), 649.  doi: 10.1007/BF01205490.  Google Scholar

[7]

S. Chandresekhar, "Liquid Crystals,", Cambridge Univ. Press, (1977).   Google Scholar

[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, 332 (1990).   Google Scholar

[9]

P.-G. De Gennes, "The Physics of Liquid Crystals,", Oxford Univ. Press, (1974).   Google Scholar

[10]

L. C. Evans and R. F. Gariepy, Some remarks concerning quasiconvexity and strong convergence,, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 53.  doi: 10.1017/S0308210500018199.  Google Scholar

[11]

J. L. Ericksen, Equilibrium theory of liquid crystals,, in, 2 (1976), 233.   Google Scholar

[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 (1987).   Google Scholar

[13]

H. Federer, "Geometric Measure Theory,", Grundlehren Math. Wissen., 153 (1969).   Google Scholar

[14]

H. Federer, Real flat chains, cochains and variational problems,, Indiana Univ. Math. J., 24 (): 351.  doi: 10.1512/iumj.1974.24.24031.  Google Scholar

[15]

H. Federer and W. H. Fleming, Normal and integral currents,, Ann. of Math. (2), 72 (1960), 458.  doi: 10.2307/1970227.  Google Scholar

[16]

F. C. Frank, On the theory of liquid crystals,, Discuss. Faraday Soc., 28 (1958), 19.  doi: 10.1039/df9582500019.  Google Scholar

[17]

M. Giaquinta and G. Modica, On sequences of maps with equibounded energies,, Calc. Var. Partial Differential Equations, 12 (2001), 213.   Google Scholar

[18]

M. Giaquinta, G. Modica and J. Souček, The Dirichlet energy of mappings with values into the sphere,, Manuscr. Math., 65 (1989), 489.  doi: 10.1007/BF01172794.  Google Scholar

[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.   Google Scholar

[20]

M. Giaquinta, G. Modica and J. Souček, Variational problems for the conformally invariant integral $\int|du|^n$,, in, 267 (1992), 27.   Google Scholar

[21]

M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations," I, II,, Ergebnisse Math. Grenzgebiete (III Ser), 37 (1998).   Google Scholar

[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.  doi: 10.1002/cpa.20125.  Google Scholar

[23]

M. Giaquinta and D. Mucci, "Maps into Manifolds and Currents: Area and $W^{1,2}$-, $W^{1/2}$-, $BV$-Energies,", CRM Series, 3 (2006).   Google Scholar

[24]

F. Hang and F. Lin, Topology of Sobolev mappings. II,, Acta Math., 191 (2003), 55.  doi: 10.1007/BF02392696.  Google Scholar

[25]

R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystals theory,, in, 5 (1987), 151.   Google Scholar

[26]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid cristals configurations,, Commun. Math. Phys., 105 (1986), 547.  doi: 10.1007/BF01238933.  Google Scholar

[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.   Google Scholar

[28]

R. Hardt, D. Kinderlehrer and F. Lin, The variety of configurations of static liquid crystals,, in, 4 (1990), 115.   Google Scholar

[29]

R. Hardt, D. Kinderlehrer and M. Luskin, Remarks about the variational theory of liquid crystals,, in, 1340 (1988), 123.   Google Scholar

[30]

D. Kinderlehrer, Recent developments in liquid crystals theory,, in, (1991), 151.   Google Scholar

[31]

F. M. Leslie, Theory of flow phenomena in liquid crystals,, in, 2 (1976), 1.   Google Scholar

[32]

F. Lin and C. Liu, Static and dynamic theories of liquid crystals,, J. Partial Differential Equations, 14 (2001), 289.   Google Scholar

[33]

D. Mucci, Sobolev maps into the projective line with bounded total variation,, Confluentes Mathematici, 2 (2010), 181.   Google Scholar

[34]

C. W. Oseen, The theory of liquid crystals,, Trans. Faraday Soc., 29 (1933), 883.  doi: 10.1039/tf9332900883.  Google Scholar

[35]

M. R. Pakzad and T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds,, Geom. Funct. Anal., 13 (2003), 223.  doi: 10.1007/s000390300006.  Google Scholar

[36]

R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps,, J. Diff. Geom., 18 (1983), 253.   Google Scholar

[37]

L. Simon, "Lectures on Geometric Measure Theory,", Proc. of the Centre for Math. Analysis, 3 (1983).   Google Scholar

[38]

E. H. Spanier, "Algebraic Topology,", Corrected reprint, (1981).   Google Scholar

[39]

Yu. G. Reshetnyak, "Space Mappings with Bounded Distortion,", Translation of Mathematical Monographs, 73 (1989).   Google Scholar

[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.   Google Scholar

[41]

E. G. Virga, "Variational Theories for Liquid Crystals,", Applied Mathematics and Mathematical Computation, 8 (1994).   Google Scholar

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