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

TL;DR

This paper investigates how the energy scaling in singular perturbation models with higher-order laminates depends on the lamination order of prescribed boundary data, using branching constructions and Fourier analysis.

Contribution

It establishes a precise relationship between energy scaling and lamination order for complex microstructures, extending understanding of shape-memory alloys modeling.

Findings

Energy scaling is determined by the lamination order of Dirichlet data.

Matching upper and lower bounds are achieved through branching and Fourier analysis.

The results apply to models with arbitrary lamination order.

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.