Bounds and estimates on the effective properties for nonlinear composites (Q700939)

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 \textit{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 \textit{D. Lukkassen, L. E. Persson} and \textit{P. Wall} [Appl. Anal. 58, No. 1-2, 123-135 (1995; Zbl 0832.35009)].