  The paper describes the first exact results in optimal design of three-phase
elastic structures. Two isotropic materials, the "strong" and the "weak" one,
are laid out with void in a given two-dimensional domain so that the compliance
plus weight of a structure is minimized. As in the classical two-phase problem,
the optimal layout of three phases is also determined on two levels: macro- and
microscopic. On the macrolevel, the design domain is divided into several
subdomains. Some are filled with pure phases, and others with their mixtures
(composites). The main aim of the paper is to discuss the non-uniqueness of the
optimal macroscopic multiphase distribution. This phenomenon does not occur in
the two-phase problem, and in the three-phase design it arises only when the
moduli of material isotropy of "strong" and "weak" phases are in certain
relation.
