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

Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

^** Current address: Institute for Applied Mathematics and Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
^*** Current address: Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
^**** Corresponding author: tribuzio@iam.uni-bonn.de

Received: 30 November 2021
Accepted: 15 June 2023

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 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.