On the energy scaling behaviour of singular perturbation models with prescribed dirichlet data involving higher order laminates

Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the energy scaling behaviour of a simplified m -well problem without gauge invariances. Considering wells for which the lamination convex hull consists of one-dimensional line segments of increasing order of lamination, we prove that for prescribed Dirichlet data the energy scaling is determined by the order of lamination of the Dirichlet data . This follows by deducing matching upper and lower scaling bounds. For the upper bound we argue by providing iterated branching constructions, and complement this with ansatz-free lower bounds. These are deduced by a careful analysis of the Fourier multipliers of the associated energies and iterated “bootstrap arguments” based on the ideas from [A. Rüland and A. Tribuzio, Arch. Rational Mech. Anal. 243 (2022) 401–431]. Relying on these observations, we study models involving laminates of arbitrary order.