Microstructures with finite surface energy: the two-well problem

We study solutions of the two-well problem, i.e., maps which satisfy uSO(n)ASO(n)B a.c. in ⁿ, where A and B are n×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 and which do not intersect in . This may not be the case if the assumption on E is dropped. We also discuss applications of this result to magnetostrictive materials.