A Variational Approach to the Homogenization of Double Phase ph(x)-Curl Systems in Magnetism

Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

Abstract

We introduce a variational approach to study the homogenization of a class of p h (x)-curl systems arising in Magnetism based on the study of the Γ-convergence of the sequence of associated energies. The explicit characterization of the effective coefficients is obtained by means of a three dimensional minimization problem when p h (x) is a double phase exponent.

Notes

Acknowledgements

This work was supported by projects MTM2013-47053-P from Ministerio de Economía y Competitividad, and PEII-2014-010-P from Junta de Comunidades de Castilla-La Mancha (Spain).

References

  1. 1.
    Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland P.C., Amsterdam (1978)MATHGoogle Scholar
  2. 2.
    Cessenat, M.: Mathematical Methods in Electromagnetism. World Scientific Publishing, Singapore (1996)CrossRefMATHGoogle Scholar
  3. 3.
    Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)MATHGoogle Scholar
  4. 4.
    Dal Maso, G.: An Introduction to Γ-Convergence. Birkhäuser, Basel (1993)CrossRefMATHGoogle Scholar
  5. 5.
    Focardi, M.: Γ-convergence: a tool to investigate physical phenomena across scales. Math. Methods Appl. Sci. 35, 1613–1658 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fonseca, I., Müller, S.: A-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30, 1355–1390 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin/Heidelberg (1994)CrossRefMATHGoogle Scholar
  8. 8.
    Milton, G.: The Theory of Composites. Cambridge University Press, Cambridge (2004)Google Scholar
  9. 9.
    Pedregal, P.: Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997)CrossRefMATHGoogle Scholar
  10. 10.
    Serrano, H.: A variational approach to the homogenization of laminate metamaterials. Nonlinear Anal. Real World Appl. 18, 75–85 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de Castilla-La ManchaCiudad RealSpain

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