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On the energy scaling behaviour of singular perturbation models 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...

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Main Authors: Rüland, Angkana (Author) , Tribuzio, Antonio (Author)
Format: Article (Journal) Chapter/Article
Language:English
Published: 2022
Edition:Version V3
In: Arxiv
Year: 2022, Pages: 1-47
DOI:10.48550/arXiv.2110.15929
Online Access:Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2110.15929
Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2110.15929
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Author Notes:Angkana Rüland and Antonio Tribuzio

MARC

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520 |a 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. 
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