Singularities of divergence-free vector fields with values into S 1 or S 2 : Applications to micromagnetics
Confluentes Mathematici, Tome 4 (2012) no. 3, article no. 1230001.
Publié le :
DOI : 10.1142/S1793744212300012
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Ignat, Radu. Singularities of divergence-free vector fields with values into $S^1$ or $S^2$: Applications to micromagnetics. Confluentes Mathematici, Tome 4 (2012) no. 3, article no. 1230001. doi : 10.1142/S1793744212300012. http://archive.numdam.org/articles/10.1142/S1793744212300012/

[1] F. Alouges, T. Rivière and S. Serfaty, Néel and cross-tie wall energies for planar micromagnetic configurations, ESAIM Control Optim. Calc. Var. 8 (2002) 31–68.

[2] L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane, Calc. Var. Partial Differential Equations 9 (1999) 327–255.

[3] L. Ambrosio, B. Kirchheim, M. Lecumberry and T. Rivière, On the rectifiability of defect measures arising in a micromagnetics model, in Nonlinear Problems in Mathe- matical Physics and Related Topics, II, Int. Math. Ser., Vol. 2 (Kluwer/Plenum, 2002), pp. 29–60.

[4] P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations, in Miniconference on Geometry and Partial Differential Equations, 2 (Canberra, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 12 (Austral. Nat. Univ., 1987), pp. 1–16.

[5] P. Aviles and Y. Giga, The distance function and defect energy, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 923–938.

[6] P. Aviles and Y. Giga, On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg–Landau type energy for gradient fields, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 1–17.

[7] F. Béthuel, H. Brezis and F. Hélein, Ginzburg–Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, Vol. 13 (Birkhäuser, 1994).

[8] F. Bethuel and X. M. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80 (1988) 60–75.

[9] S. Bianchini, C. De Lellis and R. Robyr, SBV regularity for Hamilton–Jacobi equations in Rn , Arch. Rational Mech. Anal. 200 (2011) 1003–1021.

[10] J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86.

[11] J. Bourgain, H. Brezis and P. Mironescu, H 1 2 maps with values into the circle: Minimal connections, lifting, and the Ginzburg–Landau equation, Publ. Math. Inst. Hautes Études Sci. 99 (2004) 1–115.

[12] H. Brezis, P. Mironescu and A. C. Ponce, W 1,1 -maps with values into S1 , in Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., Vol. 368 (Amer. Math. Soc., 2005), pp. 69–100.

[13] W. F. Brown, Micromagnetics (Wiley Interscience, 1963).

[14] L. A. Caffarelli and M. G. Crandall, Distance functions and almost global solutions of eikonal equations, Comm. Partial Differential Equations 35 (2010) 391–414.

[15] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial energy, J. Chem. Phys. 28 (1958) 258–267.

[16] G. Carbou, Thin layers in micromagnetism, Math. Models Methods Appl. Sci. 11 (2001) 1529–1546.

[17] S. Conti and C. De Lellis, Sharp upper bounds for a variational problem with singular perturbation, Math. Ann. 338 (2007) 119–146.

[18] J. D´avila and R. Ignat, Lifting of BV functions with values in S1 , C. R. Math. Acad. Sci. Paris 337 (2003) 159–164.

[19] C. De Lellis, An example in the gradient theory of phase transitions, ESAIM Control Optim. Calc. Var. 7 (2002) 285–289.

[20] C. De Lellis and F. Otto, Structure of entropy solutions to the eikonal equation, J. Eur. Math. Soc. 5 (2003) 107–145.

[21] A. DeSimone, H. Knüpfer and F. Otto, 2d stability of the Néel wall, Calc. Var. Partial Differential Equations 27 (2006) 233–253.

[22] A. DeSimone, R. V. Kohn, S. Müller and F. Otto, Magnetic microstructures — a paradigm of multiscale problems, in ICIAM 99 (Edinburgh), (Oxford Univ. Press, 2000), pp. 175–190.

[23] A. Desimone, R. V. Kohn, S. Müller and F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1408–1460.

[24] A. Desimone, R. V. Kohn, S. Müller and F. Otto, Repulsive interaction of Néel walls, and the internal length scale of the cross-tie wall, Multiscale Model. Simul. 1 (2003) 57–104.

[25] A. DeSimone, R. V. Kohn, S. Müller, F. Otto and R. Schäfer, Two-dimensional mod- elling of soft ferromagnetic films, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001) 2983–2991.

[26] A. DeSimone, S. Müller, R. V. Kohn and F. Otto, A compactness result in the gradient theory of phase transitions, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 833–844.

[27] A. DeSimone, S. Müller, R. V. Kohn and F. Otto, Recent analytical developments in micromagnetics, in The Science of Hysteresis, Vol. 2 (Elsevier, 2005), pp. 269–381.

[28] L. Döring, R. Ignat and F. Otto, Asymmetric domain walls of small angle in micro- magnetics, preprint.

[29] L. Döring, R. Ignat and F. Otto, Cross-over from symmetric to asymmetric transition layers in micromagnetics, preprint.

[30] M. Giaquinta, G. Modica and J. Souˇcek, Cartesian Currents in the Calculus of Vari- ations, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 38 (Springer-Verlag, 1998).

[31] Y. Giga, M. Kubo and Y. Tonegawa, Magnetic clusters and fold energies, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 23–40.

[32] F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988) 110–125.

[33] A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstruc- tures (Springer-Verlag, 1998).

[34] R. Ignat, On an open problem about how to recognize constant functions, Houston J. Math. 31 (2005) 285–304.

[35] R. Ignat, The space BV(S2 , S1 ): Minimal connection and optimal lifting, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 283–302.

[36] R. Ignat, A Γ-convergence result for Néel walls in micromagnetics, Calc. Var. Partial Differential Equations 36 (2009) 285–316.

[37] R. Ignat, A survey of some new results in ferromagnetic thin films, in Séminaire: Équations aux Dérivées Partielles. 2007–2008, Sémin. Équ. Dériv. Partielles, pages Exp. No. VI, 21 (École Polytech., 2009).

[38] R. Ignat, Gradient vector fields with values into S1 , C. R. Math. Acad. Sci. Paris 349 (2011) 883–887.

[39] R. Ignat, Two-dimensional unit-length vector fields of vanishing divergence, J. Funct. Anal. 262 (2012) 3465–3494.

[40] R. Ignat and H. Knüpfer, Vortex energy and 360◦ Néel walls in thin-film micromag- netics, Comm. Pure Appl. Math. 63 (2010) 1677–1724.

[41] R. Ignat and M. Kurzke, An effective model for boundary vortices in thin-film micro- magnetics, in preparation.

[42] R. Ignat and B. Merlet, Lower bound for the energy of Bloch walls in micromagnetics, Arch. Rational Mech. Anal. 199 (2011) 369–406.

[43] R. Ignat and B. Merlet, Entropy method for line-energies, Calc. Var. Partial Differ- ential Equations 44 (2012) 375–418.

[44] R. Ignat and R. Moser, A zigzag pattern in micromagnetics, J. Math. Pures Appl. 98 (2012) 139–159.

[45] R. Ignat and F. Otto, A compactness result in thin-film micromagnetics and the optimality of the Néel wall, J. Eur. Math. Soc. 10 (2008) 909–956.

[46] R. Ignat and F. Otto, A compactness result of Landau state in thin-film micromag- netics, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011) 247–282.

[47] P.-E. Jabin, F. Otto and B. Perthame, Line-energy Ginzburg–Landau models: Zero- energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202.

[48] P.-E. Jabin and B. Perthame, Compactness in Ginzburg–Landau energy by kinetic averaging, Comm. Pure Appl. Math. 54 (2001) 1096–1109.

[49] R. L. Jerrard, Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal. 30 (1999) 721–746.

[50] W. Jin and R. V. Kohn, Singular perturbation and the energy of folds, J. Nonlinear Sci. 10 (2000) 355–390.

[51] R. V. Kohn and V. V. Slastikov, Another thin-film limit of micromagnetics, Arch. Rational Mech. Anal. 178 (2005) 227–245.

[52] S. N. Kruˇzkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81 (1970) 228–255.

[53] M. Kurzke, Boundary vortices in thin magnetic films, Calc. Var. Partial Differential Equations 26 (2006) 1–28.

[54] M. Kurzke, A nonlocal singular perturbation problem with periodic well potential, ESAIM Control Optim. Calc. Var. 12 (2006) 52–63.

[55] F. H. Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math. 52 (1999) 737–761.

[56] C. Melcher, The logarithmic tail of Néel walls, Arch. Rational Mech. Anal. 168 (2003) 83–113.

[57] C. Melcher, Logarithmic lower bounds for Néel walls, Calc. Var. Partial Differential Equations 21 (2004) 209–219.

[58] P. Mironescu, Lifting of S1 -valued maps in sums of Sobolev spaces, preprint.

[59] P. Mironescu, S1 -valued Sobolev mappings, Sovrem. Mat. Fundam. Napravl. 35 (2010) 86–100.

[60] R. Moser, Ginzburg–Landau vortices for thin ferromagnetic films, AMRX Appl. Math. Res. Express 1 (2003) 1–32.

[61] R. Moser, Boundary vortices for thin ferromagnetic films, Arch. Rational Mech. Anal. 174 (2004) 267–300.

[62] F. Murat, Compacité par compensation: Condition nécessaire et suffisante de conti- nuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) 69–102.

[63] C. B. Muratov and V. V. Osipov, Theory of 360◦ domain walls in thin ferromagnetic films, J. Appl. Phys. 104 (2008) 053908.

[64] M. Ortiz and G. Gioia, The morphology and folding patterns of buckling-driven thin- film blisters, J. Mech. Phys. Solids 42 (1994) 531–559.

[65] F. Otto, Cross-over in scaling laws: A simple example from micromagnetics, in Proc. of the Int. Congress of Mathematicians, Vol. III (Beijing, 2002) (Higher Ed. Press, 2002), pp. 829–838.

[66] A. Poliakovsky, Upper bounds for singular perturbation problems involving gradient fields, J. Eur. Math. Soc. 9 (2007) 1–43.

[67] T. Rivière, Dense subsets of H 1 2 (S 2 ,S 1 ), Ann. Global Anal. Geom. 18 (2000) 517–528.

[68] T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 (2001) 294–338.

[69] T. Rivière and S. Serfaty, Compactness, kinetic formulation, and entropies for a prob- lem related to micromagnetics, Comm. Partial Differential Equations 28 (2003) 249– 269.

[70] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998) 379–403.

[71] E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg–Landau Model, Progress in Nonlinear Differential Equations and their Applications, Vol. 70 (Birkhäuser, 2007).

[72] L. Tartar. Compensated compactness and applications to partial differential equa- tions, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., Vol. 39 (Pitman, 1979), pp. 136–212.

[73] H. A. M. van den Berg, Self-consistent domain theory in soft-ferromagnetic media. II, Basic domain structures in thin film objects, J. Appl. Phys. 60 (1986) 1104–1113.

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