Estimates for the Homogenized Matrix
The bounds specified by the Voigt — Reiss inequality established in Section 1.6 are usually too wide and give little information about the homogenized matrix. The problem of tighter bounds has been the subject of intensive research in physics and continuum mechanics, especially in the theory of dispersion of electromagnetic waves on small particles and the theory of elasticity for microscopically non-homogeneous media. After the classical works of Maxwell  and Rayleigh , an enormous amount of facts accumulated in this direction. For a long time, preference has been given to the potential theory methods, and it was only the case of two-phase media that the analysts were concerned with; an important role in the previous studies belongs to the geometric properties of the inclusions.
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