Laminated Microstructure in a Variational Problem with a Non-Rank-One Connected Double Well Potential*

A variational problem with a non-rank-one connected double well potential is considered. It is proved that the problem has a laminated microstructure solution which is uniquely determined by the two potential wells yet certainly not characterized by the fine oscillations between them because of the lack of rank-one connection. The laminated microstructure is explicitly worked out. It is also shown that an application of a nonconforming finite element method can cause overrelaxation and fail to approximate the right microstructure.