Topological singularities for vector-valued Sobolev maps and applications

Giacomo Canevari; Giandomenico Orlandi

doi:10.5802/afst.1677

Giacomo Canevari ¹; Giandomenico Orlandi ¹

¹ Università di Verona, Dipartimento di Informatica, Strada le Grazie 15, 37134 Verona (Italy)

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 327-351.

Abstract
Abstract

We review the analysis of topological singularities of Sobolev maps into manifolds and their applications to variational problems of Ginzburg–Landau type and to the lifting problem for BV maps into manifolds. We describe in particular recent results obtained in the vector-valued case related to variational models of material science, more precisely the Landau–de Gennes model.

Nous passons en revue certains résultats d’analyse des singularités topologiques des fonctions de Sobolev à valeurs dans des variétés, ainsi que leurs applications aux problèmes variationnels de type Ginzburg–Landau et au problème du relèvement dans l’espace BV. En particulier, nous présentons des résultats récents, portant sur les fonctions à valeurs vectorielles, qui trouvent leur application dans l’étude des modèles variationnels pour la science des matériaux, tels que le modèle de Landau–de Gennes.

Published online: 2021-07-09

DOI: 10.5802/afst.1677

Classification: 58C06, 49Q15, 49Q20
Keywords: Topological singularities, Flat chains, Ginzburg–Landau type functionals, Manifold-valued maps, Lifting problem
Mot clés : Singularités topologiques, Chaînes bémol, Fonctionnelles de type Ginzburg–Landau, Applications à valeurs dans des variétés, Problème du relèvement

Author's affiliations:

Giacomo Canevari ¹; Giandomenico Orlandi ¹

¹ Università di Verona, Dipartimento di Informatica, Strada le Grazie 15, 37134 Verona (Italy)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{AFST_2021_6_30_2_327_0,
     author = {Giacomo Canevari and Giandomenico Orlandi},
     title = {Topological singularities for vector-valued {Sobolev} maps and applications},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {327--351},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 30},
     number = {2},
     year = {2021},
     doi = {10.5802/afst.1677},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1677/}
}

TY  - JOUR
AU  - Giacomo Canevari
AU  - Giandomenico Orlandi
TI  - Topological singularities for vector-valued Sobolev maps and applications
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2021
SP  - 327
EP  - 351
VL  - 30
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1677/
DO  - 10.5802/afst.1677
LA  - en
ID  - AFST_2021_6_30_2_327_0
ER  -

%0 Journal Article
%A Giacomo Canevari
%A Giandomenico Orlandi
%T Topological singularities for vector-valued Sobolev maps and applications
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2021
%P 327-351
%V 30
%N 2
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1677/
%R 10.5802/afst.1677
%G en
%F AFST_2021_6_30_2_327_0

Giacomo Canevari; Giandomenico Orlandi. Topological singularities for vector-valued Sobolev maps and applications. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 327-351. doi : 10.5802/afst.1677. https://afst.centre-mersenne.org/articles/10.5802/afst.1677/

References
Cited by

[1] Giovanni Alberti; Sisto Baldo; Giandomenico Orlandi Functions with prescribed singularities, J. Eur. Math. Soc., Volume 5 (2003) no. 3, pp. 275-311 | DOI | MR | Zbl

[2] Giovanni Alberti; Sisto Baldo; Giandomenico Orlandi Variational convergence for functionals of Ginzburg–Landau type, Indiana Univ. Math. J., Volume 54 (2005) no. 5, pp. 1411-1472 | DOI | MR | Zbl

[3] Roberto Alicandro; Marcello Ponsiglione Ginzburg–Landau functionals and renormalized energy: a revised $Γ$ -convergence approach, J. Funct. Anal., Volume 266 (2014) no. 8, pp. 4890-4907 | DOI | MR | Zbl

[4] Jr. Almgren $Q$ -valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two, Bull. Am. Math. Soc., Volume 8 (1983), pp. 327-328 | DOI | MR | Zbl

[5] Luigi Ambrosio; Halil Mete Soner Level set approach to mean curvature flow in arbitrary codimension, J. Differ. Geom., Volume 43 (1996) no. 4, pp. 693-737 | MR | Zbl

[6] John M. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1976), pp. 337-403 | DOI | MR

[7] John M. Ball; Stefen J. Bedford Discontinuous order parameters in liquid crystal theories, Molecular Crystals and Liquid Crystals, Volume 612 (2015) no. 1, pp. 1-23 | DOI

[8] John M. Ball; Arghir Zarnescu Orientability and energy minimization in liquid crystal models, Arch. Ration. Mech. Anal., Volume 202 (2011) no. 2, pp. 493-535 | DOI | MR | Zbl

[9] Patricia Bauman; Jinhae Park; Daniel Phillips Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., Volume 205 (2012) no. 3, pp. 795-826 | DOI | MR | Zbl

[10] Stephen J. Bedford Function spaces for liquid crystals, Arch. Ration. Mech. Anal., Volume 219 (2016) no. 2, pp. 937-984 | DOI | MR | Zbl

[11] Fabrice Bethuel A characterization of maps in $H^{1} (B^{3}, S^{2})$ which can be approximated by smooth maps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 7 (1990) no. 4, p. 269-186 | DOI | Numdam | MR | Zbl

[12] Fabrice Bethuel The approximation problem for Sobolev maps between two manifolds, Acta Math., Volume 167 (1991) no. 3-4, pp. 153-206 | DOI | MR | Zbl

[13] Fabrice Bethuel; Haïm Brézis; Jean-Michel Coron Relaxed Energies for Harmonic Maps, Variational methods (Paris, 1988) (Progress in Nonlinear Differential Equations and their Applications), Volume 4, Birkhäuser, 1990, pp. 37-52 | DOI | MR | Zbl

[14] Fabrice Bethuel; Haïm Brézis; Frédéric Hélein Ginzburg–Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser, 1994 | MR | Zbl

[15] Fabrice Bethuel; Haïm Brézis; Giandomenico Orlandi Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions, J. Funct. Anal., Volume 186 (2001) no. 2, pp. 432-520 | DOI | MR

[16] Fabrice Bethuel; David Chiron Some questions related to the lifting problem in Sobolev spaces, Perspectives in nonlinear partial differential equations in honor of Haïm Brezis (Contemporary Mathematics), Volume 446, American Mathematical Society, 2007, pp. 125-152 | DOI | MR | Zbl

[17] Fabrice Bethuel; Françoise Demengel Extensions for Sobolev mappings between manifolds, Calc. Var. Partial Differ. Equ., Volume 3 (1995) no. 4, pp. 475-491 | DOI | MR | Zbl

[18] Fabrice Bethuel; Giandomenico Orlandi; Didier Smets Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Ann. Math., Volume 163 (2006) no. 1, pp. 37-163 | DOI | MR

[19] Fabrice Bethuel; Xiaomin Zheng Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., Volume 80 (1988) no. 1, pp. 60-75 | DOI | MR | Zbl

[20] Jean Bourgain; Haïm Brézis; Petru Mironescu On the structure of the Sobolev space $H^{1 / 2}$ with values into the circle, C. R. Math. Acad. Sci. Paris, Volume 331 (2000) no. 2, pp. 119-124 | DOI | MR | Zbl

[21] Jean Bourgain; Haïm Brézis; Petru Mironescu Lifting, degree, and distributional jacobian revisited, Commun. Pure Appl. Math., Volume 58 (2005) no. 4, pp. 529-551 | DOI | MR | Zbl

[22] Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $ℓ$ simply connected manifolds when $s \geq 1$ , Confluentes Math., Volume 5 (2013) no. 2, pp. 3-24 | DOI | Numdam | MR | Zbl

[23] Haïm Brézis; Jean-Michel Coron; Elliott H. Lieb Harmonic maps with defects, Commun. Math. Phys., Volume 107 (1986), pp. 649-705 | DOI | MR | Zbl

[24] Haïm Brézis; Petru Mironescu Sur une conjecture de E. De Giorgi relative à l’énergie de Ginzburg–Landau, C. R. Math. Acad. Sci. Paris, Volume 319 (1994) no. 2, pp. 167-170 | Zbl

[25] Haïm Brezis; Hoai-Minh Nguyen The Jacobian determinant revisited, Invent. Math., Volume 185 (2011) no. 1, pp. 17-54 | DOI | MR | Zbl

[26] Giacomo Canevari Biaxiality in the asymptotic analysis of a 2D Landau–de Gennes model for liquid crystals, ESAIM, Control Optim. Calc. Var., Volume 21 (2015) no. 1, pp. 101-137 | DOI | MR | Zbl

[27] Giacomo Canevari Line defects in the small elastic constant limit of a three-dimensional Landau–de Gennes model, Arch. Ration. Mech. Anal., Volume 223 (2017) no. 2, pp. 591-676 | DOI | MR | Zbl

[28] Giacomo Canevari; Giandomenico Orlandi Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 2, p. 72 | DOI | MR | Zbl

[29] Giacomo Canevari; Giandomenico Orlandi Lifting for manifold-valued maps of bounded variation, J. Funct. Anal., Volume 278 (2020) no. 10, 108453, 17 pages | MR | Zbl

[30] Giacomo Canevari; Giandomenico Orlandi Topological singular set of vector-valued maps, II: $Γ$ -convergence for Ginzburg–Landau type functionals (2020) (https://arxiv.org/abs/2003.01354) | Zbl

[31] Alexandre Chemin; François Henrotte; Jean-François Remacle; Jean Van Schaftingen Representing Three-Dimensional Cross Fields Using Fourth Order Tensors, Proceedings of the 27th International Meshing Roundtable (IMR) (Lecture Notes in Computational Science and Engineering), Volume 127, Springer, 2019, pp. 89-108 | DOI | MR | Zbl

[32] David Chiron Étude mathématique de modèles issus de la physique de la matière condensée, Ph. D. Thesis, Université Pierre et Marie Curie–Paris 6 (France) (2004)

[33] Andres Contreras; Xavier Lamy Singular perturbation of manifold-valued maps with anisotropic energy (2018) (https://arxiv.org/abs/1809.05170)

[34] Juan Dávila; Radu Ignat Lifting of BV functions with values in $S^{1}$ , C. R. Math. Acad. Sci. Paris, Volume 337 (2003) no. 3, pp. 159-164 | DOI | MR | Zbl

[35] Herbert Federer Geometric Measure Theory, Grundlehren der Mathematischen Wissenschaften, 153, Springer, 1969 | MR | Zbl

[36] Herbert Federer The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Am. Math. Soc., Volume 76 (1970), pp. 767-771 | DOI | MR | Zbl

[37] Herbert Federer; Wendell H. Fleming Normal and integral currents, Ann. Math., Volume 72 (1960), pp. 458-520 | DOI | MR

[38] Wendell H. Fleming Flat chains over a finite coefficient group, Trans. Am. Math. Soc., Volume 121 (1966), pp. 160-186 | DOI | MR | Zbl

[39] P. G. de Gennes; J. Prost The Physics of Liquid Crystals, International Series of Monographs on Physics, 83, Clarendon Press, 1993

[40] Mariano Giaquinta; Giuseppe Modica; Jiří Souček Cartesian currents in the calculus of variations I. Cartesian currents, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 37, Springer, 1998 | MR | Zbl

[41] Dmitry Golovaty; José Alberto Montero On minimizers of a Landau-de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal., Volume 213 (2014) no. 2, pp. 447-490 | DOI | MR | Zbl

[42] Robert Hardt; David Kinderlehrer; Fang-Hua Lin Existence and partial regularity of static liquid crystal configurations, Commun. Math. Phys., Volume 105 (1986), pp. 547-570 | DOI | MR | Zbl

[43] Robert Hardt; Fang-Hua Lin Mappings minimizing the $L^{p}$ norm of the gradient, Commun. Pure Appl. Math., Volume 40 (1987) no. 5, pp. 555-588 | DOI | MR | Zbl

[44] Christopher P. Hopper Partial regularity for holonomic minimisers of quasiconvex functionals, Arch. Ration. Mech. Anal., Volume 222 (2016) no. 1, pp. 91-141 | DOI | MR | Zbl

[45] Radu Ignat The space $B V (S^{2}, S^{1})$ : minimal connection and optimal lifting, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005) no. 3, pp. 283-302 | DOI | Numdam | MR | Zbl

[46] Radu Ignat; Robert L. Jerrard Renormalized energy between vortices in some Ginzburg–Landau models on 2-dimensional Riemannian manifolds (1910) (https://arxiv.org/abs/1910.02921) | Zbl

[47] Radu Ignat; Xavier Lamy Lifting of ${ℝℙ}^{d - 1}$ -valued maps in $B V$ and applications to uniaxial $Q$ -tensors, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 2, 68, 26 pages (with an appendix on an intrinsic $B V$ -energy for manifold-valued maps) | Zbl

[48] Radu Ignat; Luc Nguyen; Valeriy Slastikov; Arghir Zarnescu Stability of the melting hedgehog in the Landau–de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal., Volume 215 (2015) no. 2, pp. 633-673 | DOI | MR | Zbl

[49] Radu Ignat; Luc Nguyen; Valeriy Slastikov; Arghir Zarnescu Stability of point defects of degree $\pm \frac{1}{2}$ in a two-dimensional nematic liquid crystal model, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 5, 119, 33 pages | MR | Zbl

[50] Robert L. Jerrard; Halil Mete Soner The Jacobian and the Ginzburg–Landau energy, Calc. Var. Partial Differ. Equ., Volume 14 (2002) no. 2, pp. 151-191 | DOI | MR | Zbl

[51] Robert L. Jerrard; Halil Mete Soner Functions of bounded higher variation, Indiana Univ. Math. J., Volume 51 (2003) no. 3, pp. 645-677 | MR

[52] Fang-Hua Lin; Tristan Rivière Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents, J. Eur. Math. Soc., Volume 1 (1999) no. 3, pp. 237-311 erratum in ibid. 2 (2000), no. 1, p. 87-91 | MR | Zbl

[53] Fang-Hua Lin; Tristan Rivière A quantization property for static Ginzburg–Landau vortices, Commun. Pure Appl. Math., Volume 54 (2001) no. 2, pp. 206-228 | MR | Zbl

[54] Apala Majumdar; Arghir Zarnescu Landau–de Gennes theory of nematic liquid crystals: the Oseen–Frank limit and beyond, Arch. Ration. Mech. Anal., Volume 196 (2010) no. 1, pp. 227-280 | DOI | MR | Zbl

[55] N. David Mermin The topological theory of defects in ordered media, Rev. Mod. Phys., Volume 51 (1979) no. 3, pp. 591-648 | DOI | MR

[56] Petru Mironescu; Emmanuel Russ; Yannick Sire Lifting in Besov spaces, Nonlinear Anal., Theory Methods Appl., Volume 193 (2020), 111489, 44 pages | MR | Zbl

[57] Petru Mironescu; Jean Van Schaftingen Lifting in compact covering spaces for fractional Sobolev mappings (2019) (https://arxiv.org/abs/1907.01373)

[58] A. Monteil; R. Rodiac; Jean Van Schaftingen (in preparation)

[59] Domenico Mucci Maps into projective spaces: liquid crystal and conformal energies, Discrete Contin. Dyn. Syst., Ser. B, Volume 17 (2012) no. 2, pp. 597-635 | MR | Zbl

[60] Olaf Müller A note on closed isometric embeddings, J. Math. Anal. Appl., Volume 349 (2009) no. 1, pp. 297-298 | DOI | MR | Zbl

[61] John Nash The imbedding problem for Riemannian manifolds, Ann. Math., Volume 63 (1956) no. 1, pp. 20-63 | DOI | MR | Zbl

[62] Luc Nguyen; Arghir Zarnescu Refined approximation for minimizers of a Landau–de Gennes energy functional, Calc. Var. Partial Differ. Equ., Volume 47 (2013) no. 1-2, pp. 383-432 | DOI | MR | Zbl

[63] Mohammad R. Pakzad; Tristan Rivière Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 223-257 | DOI | MR | Zbl

[64] Alessandro Pigati; Daniel Stern Minimal submanifolds from the abelian Higgs model (2019) (https://arxiv.org/abs/1905.13726) | Zbl

[65] Tristan Rivière Dense subsets of $H^{1 / 2} (S^{2}, S^{1})$ , Ann. Global Anal. Geom., Volume 18 (2000) no. 5, pp. 517-528 | DOI | MR | Zbl

[66] Étienne Sandier; Sylvia Serfaty Vortices in the magnetic Ginzburg–Landau model, Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser, 2007 | MR | Zbl

[67] Daniel Stern Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds (to appear in J. Diff. Geom.) | MR | Zbl

[68] Brian White The deformation theorem for flat chains, Acta Math., Volume 183 (1999) no. 2, pp. 255-271 | DOI | MR | Zbl

[69] Brian White Rectifiability of flat chains, Ann. Math., Volume 150 (1999) no. 1, pp. 165-184 | DOI | MR | Zbl

[70] Hassler Whitney Geometric integration theory, Princeton Mathematical Series, Princeton University Press, 1957 | Zbl

Cited by Sources: