Sequential laminates in multiple-state optimal design problems (Q610047)

Summary: In the study of optimal design related to stationary diffusion problems with multiple-state equations, the description of the set \(\mathcal H=\{(Aa_1,\cdots ,Aa_m):A \in \mathcal K(\theta)\}\) for given vectors \(a_1,\cdots ,a_m \in \mathbb R^d (m<d)\) is crucial. \(\mathcal K(\theta)\) denotes all composite materials (in the sense of homogenisation theory) with given local proportion \(\theta\) of the first material. We prove that the boundary of \(\mathcal H\) is attained by sequential laminates of rank at most \(m\) with matrix phase \(\alpha \mathbf I\) and core \(\beta \mathbf I (\alpha < \beta)\). 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 \(\alpha\mathbf I\) are preferred to those with matrix phase \(\beta \mathbf I\), 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.