Sequential laminates in multiple-state optimal design problems

In the study of optimal design related to stationary diffusion problems with multiple-state equations, the description of the set H = { ( Aa 1 , ... , Aa m ) : A ∈ K ( θ ) } for given vectors a 1 , ... , a m ∈ ℠d ( m < d ) is crucial. K ( θ ) denotes all composite materials (in the sense of homogenisation theory) with given local proportion θ of the first material. We prove that the boundary of H is attained by sequential laminates of rank at most m with matrix phase α I and core β I ( α < β ). Examples showing that the information on the rank of optimal laminate cannot be improved, as well as the fact that sequential laminates with matrix phase α I are preferred to those with matrix phase β I , are presented. This result can significantly reduce the complexity of optimality conditions, with obvious impact on numerical treatment, as demonstrated in a simple numerical example.