Archive for Rational Mechanics and Analysis

, Volume 230, Issue 2, pp 665–700 | Cite as

The Dispersion Tensor and Its Unique Minimizer in Hashin–Shtrikman Micro-structures

  • Loredana BălilescuEmail author
  • Carlos Conca
  • Tuhin Ghosh
  • Jorge San Martín
  • Muthusamy Vanninathan


In this paper, we introduce a macroscopic quantity, namely the dispersion tensor or the Burnett coefficients in the class of generalized Hashin–Shtrikman micro-structures (Tartar in The general theory of homogenization, volume 7 of Lecture notes of the Unione Matematica Italiana, Springer, Berlin, p 281, 2009). In the case of two-phase materials associated with the periodic Hashin–Shtrikman structures, we settle the issue that the dispersion tensor has a unique minimizer, which is the so called Apollonian–Hashin–Shtrikman micro-structure.


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  1. 1.
    Allaire, G.: Shape Optimization by the Homogenization Method, volume 146 of Applied Mathematical Sciences. Springer-Verlag, New York (2002)CrossRefGoogle Scholar
  2. 2.
    Allaire, G., Briane, M., Vanninathan, M.: A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures. SeMA J. 73(3), 237–259 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allaire, G., Yamada, T.: Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures. Submitted to Numerische Mathematik, HAL preprint: hal-01341082, 2016Google Scholar
  4. 4.
    Bă lilescu, L., Conca, C., Ghosh, T., San Martín, J., Vanninathan, M. Bloch Wave Spectral Analysis in the Class of Generalized Hashin–Shtrikman Micro-structures. Submitted, arXiv preprint: arXiv:1608.07540, 2016
  5. 5.
    Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and Its Applications. North-Holland Publishing Co., Amsterdam (1978)zbMATHGoogle Scholar
  6. 6.
    Conca, C., Orive, R., Vanninathan, M.: Bloch approximation in homogenization and applications. SIAM J. Math. Anal. 33(5), 1166–1198 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conca, C., Orive, R., Vanninathan, M.: On Burnett coefficients in periodic media. J. Math. Phys., 47(3), 032902, 11 (2006)Google Scholar
  8. 8.
    Conca, C., San Martín, J., Smaranda, L., Vanninathan, M.: On Burnett coefficients in periodic media in low contrast regime. J. Math. Phys., 49(5), 053514, 23 (2008)Google Scholar
  9. 9.
    Conca, C., San Martín, J., Smaranda, L., Vanninathan, M.: Optimal bounds on dispersion coefficient in one-dimensional periodic media. Math. Models Methods Appl. Sci. 19(9), 1743–1764 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conca, C., San Martín, J., Smaranda, L., Vanninathan, M.: Burnett coefficients and laminates. Appl. Anal. 91(6), 1155–1176 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Conca, C., Vanninathan, M.: Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57(6), 1639–1659 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dohnal, T., Lamacz, A., Schweizer, B.: Bloch-wave homogenization on large time scales and dispersive effective wave equations. Multiscale Model. Simul. 12(2), 488–513 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125–3131 (1962)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994. Translated from the Russian by G.A. Yosifian [G.A. Iosifʹyan]Google Scholar
  15. 15.
    Kesavan, S.: Topics in Functional Analysis and Applications. Wiley, New York (1989)zbMATHGoogle Scholar
  16. 16.
    Meyer, P.A.: Probability and Potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, MA, 1966Google Scholar
  17. 17.
    Murat, F., Tartar, L.: Calculus of variations and homogenization [ MR0844873 (87i:73059)]. In Topics in the Mathematical Modelling of Composite Materials, volume 31 of Progress in Nonlinear Differential Equations Applications, pp. 139–173. Birkhäuser, Boston, MA, 1997.Google Scholar
  18. 18.
    Murat, F., Tartar, L.: \(H\)-convergence. In Topics in the Mathematical Modelling of Composite Materials, volume 31 of Progress in Nonlinear Differential Equations Applications, pp. 21–43. Birkhäuser, Boston, MA, 1997.Google Scholar
  19. 19.
    Tartar, L.: Estimations fines des coefficients homogénéisés. In Ennio De Giorgi colloquium (Paris. 1983), volume 125 of Research Notes in Mathematics, pp. 168–187. Pitman, Boston, MA, 1985.Google Scholar
  20. 20.
    Tartar, L.: The General Theory of Homogenization, volume 7 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin; UMI, Bologna, 2009. A personalized introduction.Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Loredana Bălilescu
    • 1
    • 2
    Email author
  • Carlos Conca
    • 3
  • Tuhin Ghosh
    • 4
  • Jorge San Martín
    • 3
  • Muthusamy Vanninathan
    • 5
  1. 1.Department of Mathematics and Computer ScienceUniversity of PiteştiPiteştiRomania
  2. 2.Department of MathematicsFederal University of Santa CatarinaFlorianópolisBrazil
  3. 3.Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático UMR 2071/UMI 2807 CNRS-UChile, and Centro de Biotecnología y BioingenieríaFacultad de Ciencias Físicas y Matemáticas, Universidad de ChileSantiagoChile
  4. 4.Centre For Applicable MathematicsTata Institute of Fundamental ResearchBangaloreIndia
  5. 5.Department of MathematicsIIT-BombayMumbaiIndia

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