Multiphase interchange inequalities

Phase interchange inequalities have been studied since the early work of Keller [J. Math. Phys. 5, 548 (1964)]. They constrain the effective conductivity of composite materials which are obtained from each other, for fixed configuration, by interchanging the position of the phases. Optimal results in this direction for the case of a two-phase composite are due to Keller in spatial dimension d = 2 and to Avellaneda et al. [J. Appl. Phys. 63, 4989 (1988)] in dimension d = 3. In this paper new inequalities in spatial dimension d = 2 and d = 3, which are valid when an arbitrary number of phases is present, are proven. When specialized to two-phase composites, they agree with those of Keller in d = 2 and of Avellaneda et al. in d = 3, respectively. © 1991 American Institute of Physics.