A reduced model for the polarization in a ferroelectric thin wire

  • Antonio Gaudiello
  • Kamel Hamdache


In this paper, starting from a non-convex and nonlocal 3D-variational model for the electric polarization in a ferroelectric material, via an asymptotic process we obtain a rigorous 1D-variational model for a thin wire.


Electric polarization Thin wire Nonlocal problems 

Mathematics Subject Classification

35Q61 78A25 


  1. 1.
    Alicandro R., Leone C.: 3D-2D asymptotic analysis for micromagnetic thin films. ESAIM Control Optim. Calc. Var. 6, 489–498 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ammari H., Halpern L., Hamdache K.: Asymptotic behavior of thin ferromagnetic films. Asymptot. Anal. 24, 277–294 (2000)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Baía M., Zappale E.: A note on the 3D-2D dimensional reduction of a micromagnetic thin film with nonhomogeneous profile. Appl. Anal. 86(5), 555–575 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Carbou P.G., Labbé S., Trélat E.: Control of travelling walls in a ferromagnetic nanowire. Discrete Contin. Dyn. Syst. Ser. S 1(1), 51–59 (2008)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Chandra, P.; Littlewood, P.B.: A Landau primer for ferroelectrics, The Physics of ferroelectrics: a modern perspective. In Rabe, K., Ahn, C.H., Triscone, J.-M. (eds.) Topics Applied Physics, vol. 105 (2007), pp. 69–116Google Scholar
  6. 6.
    Costabel M., Dauge M., Nicaise S.: Singularities of Maxwell interface problems. Math. Model. Numer. Anal. 33(3), 627–649 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cross, L.E.; Newnham, R.E.: History of Ferroelectrics. Reprinted from the Ceramics and Civilization, Volume III High-Technology Ceramics-Past, Present, and Future. The American Ceramic Society. Inc. (1987)Google Scholar
  8. 8.
    De Maio, U.; Faella, L.; Perugia, C.: Quasy-stationary ferromagnetic thin films in degenerated cases. Ricerche Mat. (2014). doi: 10.1007/s11587-014-0197-5
  9. 9.
    DeSimone A., Kohn R.V., Muller S., Otto F.: A reduced theory for thin-film micromagnetics. Commun. Pure Appl. Math. 55(11), 1408–1460 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Gaudiello A., Hadiji R.: Junction of one-dimensional minimization problems involving S 2 valued maps. Adv. Differ. Equ. 13(9–10), 935–958 (2008)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Gaudiello A., Hadiji R.: Asymptotic analysis, in a thin multidomain, of minimizing maps with values in S 2. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 26(1), 59–80 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gaudiello A., Hadiji R.: Junction of ferromagnetic thin films. Calc. Var. Partial Differ. Equ. 39(3), 593–619 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gaudiello A., Hadiji R.: Ferromagnetic thin multi-structures. J. Differ. Equ. 257, 1591–1622 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gaudiello A., Hamdache K.: The polarization in a ferroelectric thin film: local and nonlocal limit problems. ESAIM Control Optim. Calc. Var. 19, 657–667 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Gaudiello, A.; Panasenko, G.; Piatnitski, A.: Asymptotic analysis and domain decomposition for a biharmonic problem in a thin multi-structure. Commun. Contemp. Math. doi: 10.1142/S0219199715500571(2015)
  16. 16.
    Gioia G., James R.D.: Micromagnetism of very thin films. Proc. R. Lond. A 453, 213–223 (1997)CrossRefGoogle Scholar
  17. 17.
    Hadiji R., Shirakawa K.: Asymptotic analysis of micromagnetics on thin films governed by indefinite material coefficients. Commun. Pure Appl. Anal. 9(5), 1345–1361 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kohn R.V., Slastikov V.V.: Another thin-film limit of micromagnetics. Arch. Rational Mech. Anal. 178, 227–245 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Sanchez D.: Behaviour of the Landau-Lifschitz equation in a ferromagnetic wire. Math. Methods Appl. Sci. 32(2), 167–205 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Slastikov V., Sonnenberg C.: Reduced models for ferromagnetic nanowires. IMA J. Appl. Math. 77(2), 220–235 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Su Y., Landis C.M.: Continuum thermodynamics of ferroelectric domain evolution: theory, finite element implementation, and application to domain wall pinning. J. Mech. Phys. Solids. 55(2), 280–305 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Ballato, J., Gupta, M.C (Eds.).: The handbook of photonics, Second Edition, CRC Press 2006. In Gopalan, V., Schepler, K.L., Dierolf, V., Biaggio, I., Chapter 6. Ferroelectric MaterialsGoogle Scholar
  23. 23.
    Zhang W., Bhattacharya K.: A computational model of ferroelectric domains. Part I. Model formulation and domain switching. Acta Mater. 53, 185–198 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.DIEIUniversità degli Studi di Cassino e del Lazio MeridionaleCassinoItalia
  2. 2.Technology LabLeonard de Vinci Pôle UniversitaireParis La Défense CedexFrance

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