Microstructures with finite surface energy: The two-well problem. (English) Zbl 0846.73054
Summary: We study solutions of the two-well problem, i.e., maps which satisfy \(\nabla u\in SO(n)A\cup SO(n)B\) a.e. in \(\Omega\subset \mathbb{R}^n\), where \(A\) and \(B\) are \(n\times n\) 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 \(\partial\Omega\) 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.
MSC:
| 74A60 | Micromechanical theories |
| 74M25 | Micromechanics of solids |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74P10 | Optimization of other properties in solid mechanics |
| 82D40 | Statistical mechanics of magnetic materials |
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