Published Paper

Inserted: 31 jan 2022
Last Updated: 28 aug 2023

Journal: ESAIM: COCV
Volume: 29
Number: 68
Year: 2023
Doi: https://doi.org/10.1051/cocv/2023047

Abstract:

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 (essentially) 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 https://cvgmt.sns.it/paper/5099/. Relying on these observations, we study models involving laminates of arbitrary order.