A Compactness and Structure Result for a Discrete Multi-well Problem with SO(n) Symmetry in Arbitrary Dimension

© 2018, The Author(s). In this note we combine the “spin-argument” from Kitavtsev et al. (Proc R Soc Edinb Sect A Mater 147(5):1041–1089, 2017) and the n-dimensional incompatible, one-well rigidity result from Lauteri and Luckhaus (An energy estimate for dislocation configurations and the emergence of Cosserat-type structures in metal plasticity, 2016), in order to infer a new proof for the compactness of discrete multi-well energies associated with the modelling of surface energies in certain phase transitions. Mathematically, a main novelty here is the reduction of the problem to an incompatible one-well problem. The presented argument is very robust and applies to a number of different physically interesting models, including for instance phase transformations in shape-memory materials but also anti-ferromagnetic transformations or related transitions with an “internal” microstructure on smaller scales.