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**Bounds and estimates on the effective properties for nonlinear composites.**
*(English)*
Zbl 0996.74062

The paper deals with the mathematical description of physical behaviour of a composite material (that is, a material which is a mix of two or more distinct materials). The main task is to establish lower and upper bounds on effective properties of nonlinear heterogeneous systems which govern the physical process. In an earlier work of P. Ponte Castaneda [Phil. Trans. R. Soc. Lond., Ser. A 340, No. 1659, 531-567 (1992; Zbl 0776.73062)], a key method for obtaining the bounds was developed based on certain variational principle. However, only lower or only upper bounds were obtained in that way. In the paper under review the author overcomes this problem by developing a new variational principle which uses ideas from an earlier paper by D. Lukkassen, L. E. Persson and P. Wall [Appl. Anal. 58, No. 1-2, 123-135 (1995; Zbl 0832.35009)].

Reviewer: Luboš Pick (Praha)

### MSC:

74Q20 | Bounds on effective properties in solid mechanics |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

### References:

[1] | A. Braides, D. Lukkassen: Reiterated homogenization of integral functionals. Math. Models Methods Appl. Sci, to appear. · Zbl 1010.49011 |

[2] | P. Ponte Castaneda: Bounds and estimates for the properties of nonlinear heterogeneous systems. Philos. Trans. Roy. Soc. London Ser. A. 340 (1992), 531-567. · Zbl 0776.73062 |

[3] | P. Ponte Castaneda: A new variational principle and its application to nonlinear heterogeneous systems. SIAM J. Appl. Math. 52 (1992), 1321-1341. · Zbl 0759.73064 |

[4] | G. Dal Maso: An Introduction to \(\Gamma \)-convergence. Birkhäuser, Boston, 1993. · Zbl 0816.49001 |

[5] | I. Ekeland, R. Temam: Convex analysis and variational problems. Studies in Mathematics and Its Applications, Vol. 1. North-Holland, Amsterdam, 1976. |

[6] | V. V. Jikov, S. M. Kozlov and O. A. Oleinik: Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin-Heidelberg-New York, 1994. |

[7] | D. Lukkassen: Formulae and bounds connected to optimal design and homogenization of partial differential operators and integral functionals. (1996), Ph. D. Thesis, Dept. of Math., Tromsö University, Norway. |

[8] | D. Lukkassen: Some sharp estimates connected to the homogenized \(p\)-Laplacian equation. Z. Angew. Math. Mech. 76 (S2) (1996), 603-604. · Zbl 1126.35303 |

[9] | D. Lukkassen, L. E. Persson and P. Wall: On some sharp bounds for the \(p\)-Poisson equation. Appl. Anal. 58 (1995), 123-135. · Zbl 0832.35009 |

[10] | D. R .S. Talbot, J. R. Willis: Variational principles for nonlinear inhomogeneous media. IMA J. Appl. Math. 35 (1985), 39-54. · Zbl 0588.73025 |

[11] | D. R. S. Talbot, J. R. Willis: Bounds and self-consistent estimates for the overall properties of nonlinear composites. IMA J. Appl. Math. 39 (1987), 215-240. · Zbl 0649.73012 |

[12] | J. van Tiel: Convex Analysis. John Wiley and Sons Ltd., New York, 1984. · Zbl 0565.49001 |

[13] | P. Wall: Optimal bounds on the effective shear moduli for some nonlinear and reiterated problems. Acta Sci. Math. 65 (1999), 553-566. · Zbl 0987.35027 |

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