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Domain branching near the boundary appears in many singularly-perturbedmodels for microstructure in materials and was first demonstratedmathematically by Kohn and M\"uller for a scalar problem modeling the elasticbehavior of shape-memory alloys. We study here a model for shape-memory alloysbased on the full vectorial problem of nonlinear elasticity, includinginvariance under rotations, in the case of two wells in two dimensions. We showthat, for two wells with two rank-one connections, the energy scalesproportional to the power $2/3$ of the surface energy, in agreement with thescalar model. In a case where only one rank-one connection is present, we showthat the energy exhibits a different behavior, proportional to the power $4/5$of the surface energy. This lower energy is achieved by a suitable interactionof the two components of the deformations and hence cannot be reproduced by thescalar model. Both scalings are proven by explicit constructions and matchinglower bounds.

Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance