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 setH={(Aa1,...,Aam):A∈K(θ)}for given vectorsa1,...,am∈ℝd(m<d) is crucial.K(θ)denotes all composite materials (in the sense of homogenisation theory) with given local proportionθof the first material. We prove that the boundary ofHis attained by sequential laminates of rank at mostmwith matrix phaseαIand coreβI(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 phaseαIare preferred to those with matrix phaseβI, 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.