Microstructures with finite surface energy: The two-well problem

We study solutions of the two-well problem, i.e., maps which satisfy Vu is an element of SO(n)A boolean OR SO(n)B a.e. in Omega subset of R '', where A and B are n x n matrices with positive determinants. This problem arises in the study of microstructure in solid-solid phase transitions. Under the additional hypothesis that the set E where the gradient lies in SO(n)A has finite perimeter, we show that u is locally only a function of one variable and that the boundary of E consists of (subsets of) hyperplanes which extend to partial derivative Omega and which do not intersect in Omega. This may not be the case if the assumption on E is dropped. We also discuss applications of this result to magnetostrictive materials.