Another approach to the thin-film $\Gamma$-limit of the micromagnetic free energy in the regime of small samples
Author:
Carolin Kreisbeck
Journal:
Quart. Appl. Math. 71 (2013), 201-213
MSC (2010):
Primary 49J45, 35E99, 35Q61, 74F15, 74K35
DOI:
https://doi.org/10.1090/S0033-569X-2012-01323-5
Published electronically:
August 28, 2012
MathSciNet review:
3087419
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The asymptotic behavior of the micromagnetic free energy governing a ferromagnetic film is studied as its thickness gets smaller and smaller compared to its cross section. Here the static Maxwell equations are treated as a Murat’s constant-rank PDE constraint on the energy functional. In contrast to previous work, this approach allows us to keep track of the induced magnetic field without solving the magnetostatic equations. In particular, the mathematical results of Gioia and James [Proc. R. Soc. Lond. A 453 (1997), pp. 213–223] regarding convergence of minimizers are recovered by giving a characterization of the corresponding $\Gamma$-limit.
References
- G. Anzellotti, S. Baldo, and A. Visintin, Asymptotic behavior of the Landau-Lifshitz model of ferromagnetism, Appl. Math. Optim. 23 (1991), no. 2, 171–192. MR 1086467, DOI https://doi.org/10.1007/BF01442396
- Andrea Braides, Irene Fonseca, and Giovanni Leoni, $\scr A$-quasiconvexity: relaxation and homogenization, ESAIM Control Optim. Calc. Var. 5 (2000), 539–577. MR 1799330, DOI https://doi.org/10.1051/cocv%3A2000121
- Brown, W. Micromagnetics. John Wiley and Sons, New York, 1963.
- Brown, W. Magnetostatic principles in ferromagnetism. In Selected Topics in Solid State Physics. North-Holland Publishing Company, Amsterdam, 1962.
- Bernard Dacorogna, Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Mathematics, vol. 922, Springer-Verlag, Berlin-New York, 1982. MR 658130
- Antonio De Simone, Energy minimizers for large ferromagnetic bodies, Arch. Rational Mech. Anal. 125 (1993), no. 2, 99–143. MR 1245068, DOI https://doi.org/10.1007/BF00376811
- Antonio Desimone, Robert V. Kohn, Stefan Müller, and Felix Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002), no. 11, 1408–1460. MR 1916988, DOI https://doi.org/10.1002/cpa.3028
- Desimone, A., Kohn, R. V., Müller, S., and Otto, F. Recent analytical developments in micromagnetics. In The Science of Hysteresis II: Physical Modeling, Micromagnetics, and Magnetization Dynamics. G. Bertotti and I. Mayergoyz eds., Elsevier, 2006, pp. 269–381.
- Irene Fonseca, Gilles Francfort, and Giovanni Leoni, Thin elastic films: the impact of higher order perturbations, Quart. Appl. Math. 65 (2007), no. 1, 69–98. MR 2313149, DOI https://doi.org/10.1090/S0033-569X-06-01035-7
- Irene Fonseca and Stefan Krömer, Multiple integrals under differential constraints: two-scale convergence and homogenization, Indiana Univ. Math. J. 59 (2010), no. 2, 427–457. MR 2648074, DOI https://doi.org/10.1512/iumj.2010.59.4249
- Irene Fonseca, Giovanni Leoni, and Stefan Müller, $\scr A$-quasiconvexity: weak-star convergence and the gap, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 2, 209–236 (English, with English and French summaries). MR 2021666, DOI https://doi.org/10.1016/S0294-1449%2803%2900036-2
- Irene Fonseca and Stefan Müller, $\scr A$-quasiconvexity, lower semicontinuity, and Young measures, SIAM J. Math. Anal. 30 (1999), no. 6, 1355–1390. MR 1718306, DOI https://doi.org/10.1137/S0036141098339885
- Gioia, G., and James, R. D. Micromagnetics of very thin films. Proc. R. Soc. Lond. A 453 (1997), 213–223.
- Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
- Hubert, A. and Schäfer, R. Magnetic Domains. The Analysis of Magnetic Microstructures. Springer, Berlin-Heidelberg-New York, 1998.
- R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Contin. Mech. Thermodyn. 2 (1990), no. 3, 215–239. MR 1069400, DOI https://doi.org/10.1007/BF01129598
- R. D. James and Stefan Müller, Internal variables and fine-scale oscillations in micromagnetics, Contin. Mech. Thermodyn. 6 (1994), no. 4, 291–336. MR 1308877, DOI https://doi.org/10.1007/BF01140633
- Kreisbeck, C., and Rindler, F. Thin-film limits of functionals on $\mathcal {A}$-free vector fields. arXiv:1105.3848 (2012).
- Krömer, S. Dimension reduction for functionals on solenoidal vector fields. ESAIM Control Optim. Calc. Var. 18 (2012), 259–276.
- L. D. Landau and E. M. Lifshitz, Course of theoretical physics. Vol. 8, Pergamon International Library of Science, Technology, Engineering and Social Studies, Pergamon Press, Oxford, 1984. Electrodynamics of continuous media; Translated from the second Russian edition by J. B. Sykes, J. S. Bell and M. J. Kearsley; Second Russian edition revised by Lifshits and L. P. Pitaevskiĭ. MR 766230
- Hervé Le Dret and Annie Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9) 74 (1995), no. 6, 549–578 (English, with English and French summaries). MR 1365259
- Hervé Le Dret and Annie Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results, Arch. Ration. Mech. Anal. 154 (2000), no. 2, 101–134. MR 1784962, DOI https://doi.org/10.1007/s002050000100
- François Murat, Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 69–102 (French). MR 616901
- Ohring, M. Materials Science of Thin Films: Deposition and Structure, second ed. Elsevier, 2002.
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
- A. Visintin, On Landau-Lifshitz’ equations for ferromagnetism, Japan J. Appl. Math. 2 (1985), no. 1, 69–84. MR 839320, DOI https://doi.org/10.1007/BF03167039
References
- Anzellotti, G., Baldo, S., and Visintin, A. Asymptotic behavior of the Landau-Lifshitz model of ferromagnetism. Appl. Math. Optim. 23, 2 (1991), 171–192. MR 1086467 (92a:82131)
- Braides, A., Fonseca, I., and Leoni, G. $\mathcal {A}$-quasiconvexity: relaxation and homogenization. ESAIM Control Optim. Calc. Var. 5 (2000), 539–577. MR 1799330 (2001k:49034)
- Brown, W. Micromagnetics. John Wiley and Sons, New York, 1963.
- Brown, W. Magnetostatic principles in ferromagnetism. In Selected Topics in Solid State Physics. North-Holland Publishing Company, Amsterdam, 1962.
- Dacorogna, B. Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals, vol. 922 of Lecture Notes in Mathematics. Springer, Berlin - New York, 1982. MR 658130 (84f:49020)
- De Simone, A. Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125, 2 (1993), 99–143. MR 1245068 (94j:82084)
- Desimone, A., Kohn, R. V., Müller, S., and Otto, F. A reduced theory for thin-film micromagnetics. Comm. Pure Appl. Math. 55, 11 (2002), 1408–1460. MR 1916988 (2004c:78031)
- Desimone, A., Kohn, R. V., Müller, S., and Otto, F. Recent analytical developments in micromagnetics. In The Science of Hysteresis II: Physical Modeling, Micromagnetics, and Magnetization Dynamics. G. Bertotti and I. Mayergoyz eds., Elsevier, 2006, pp. 269–381.
- Fonseca, I., Francfort, G., and Leoni, G. Thin elastic films: the impact of higher order perturbations. Quart. Appl. Math. 65, 1 (2007), 69–98. MR 2313149 (2009b:74073)
- Fonseca, I., and Krömer, S. Multiple integrals under differential constraints: two-scale convergence and homogenization. Indiana Univ. Math. J. 59, 2 (2010), 427–457. MR 2648074 (2011j:49016)
- Fonseca, I., Leoni, G., and Müller, S. $\mathcal {A}$-quasiconvexity: weak-star convergence and the gap. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 2 (2004), 209–236. MR 2021666 (2005c:49031)
- Fonseca, I., and Müller, S. $\mathcal {A}$-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30, 6 (1999), 1355–1390. MR 1718306 (2000j:49020)
- Gioia, G., and James, R. D. Micromagnetics of very thin films. Proc. R. Soc. Lond. A 453 (1997), 213–223.
- Grafakos, L. Classical Fourier analysis, second ed., vol. 249 of Graduate Texts in Mathematics. Springer, New York, 2008. MR 2445437 (2011c:42001)
- Hubert, A. and Schäfer, R. Magnetic Domains. The Analysis of Magnetic Microstructures. Springer, Berlin-Heidelberg-New York, 1998.
- James, R. D., and Kinderlehrer, D. Frustration in ferromagnetic materials. Contin. Mech. Thermodyn. 2, 3 (1990), 215–239. MR 1069400 (92a:82132)
- James, R. D., and Müller, S. Internal variables and fine-scale oscillations in micromagnetics. Contin. Mech. Thermodyn. 6, 4 (1994), 291–336. MR 1308877 (96a:82045)
- Kreisbeck, C., and Rindler, F. Thin-film limits of functionals on $\mathcal {A}$-free vector fields. arXiv:1105.3848 (2012).
- Krömer, S. Dimension reduction for functionals on solenoidal vector fields. ESAIM Control Optim. Calc. Var. 18 (2012), 259–276.
- Landau, L. D., Lifshitz, E. M., and Pitaevskii, L. P. Electrodynamics of continuous media. In Course of Theoretical Physics, vol. 8. Pergamon Press, New York, 1960. MR 766230 (86b:78001)
- Le Dret, H., and Raoult, A. The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. (9) 74, 6 (1995), 549–578. MR 1365259 (97d:73009)
- Le Dret, H., and Raoult, A. Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Ration. Mech. Anal. 154, 2 (2000), 101–134. MR 1784962 (2002b:74038)
- Murat, F. Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8, 1 (1981), 69–102. MR 616901 (82h:46051)
- Ohring, M. Materials Science of Thin Films: Deposition and Structure, second ed. Elsevier, 2002.
- Tartar, L. Compensated compactness and applications to partial differential equations. In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Res. Notes in Math. Pitman, Boston, Mass., 1979, pp. 136–212. MR 584398 (81m:35014)
- Visintin, A. On Landau-Lifshitz’ equations for ferromagnetism. Japan J. Appl. Math. 2, 1 (1985), 69–84. MR 839320 (87j:78013)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
49J45,
35E99,
35Q61,
74F15,
74K35
Retrieve articles in all journals
with MSC (2010):
49J45,
35E99,
35Q61,
74F15,
74K35
Additional Information
Carolin Kreisbeck
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Address at time of publication:
Departamento de Matemática and Centro de Matemática e Aplicaçoẽs, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
Email:
carolink@andrew.cmu.edu
Received by editor(s):
May 12, 2011
Published electronically:
August 28, 2012
Additional Notes:
The author is grateful to Irene Fonseca for pointing her to this topic, for valuable conversations on the subject and for reading carefully a first draft of the manuscript. Also, the author thanks Filip Rindler, who contributed with useful ideas on closely related issues. This research was carried out during a one-year stay at Carnegie Mellon University funded by the Fundação para a Ciência e a Tecnologia (FCT) through the ICTI CMU–Portugal program and UTA-CMU/MAT/0005/2009.
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.