On the Energy Scaling Behaviour of Singular Perturbation Models with Prescribed Dirichlet Data Involving Higher Order Laminates

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 \emph{order of lamination of the Dirichlet data}. This follows by deducing (essentially) matching upper and lower scaling bounds. For the \emph{upper} bound we argue by providing iterated branching constructions, and complement this with ansatz-free \emph{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 \cite{RT21}. Relying on these observations, we study models involving laminates of arbitrary order.

51 pages, 10 figures; improved lower bounds to avoid losses in scaling; periodic extension instead of whole space extension (Lemma 4.2); added perimeter contribution in Lemma 4.5; added Corollary 4.9; corrected some typos

References in corpus (1)

• On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

Cited by in corpus (2)

• On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators
• On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions