On the energy scaling behaviour of singular perturbation models involving higher order laminates

• Verfasst von: Rüland, Angkana [VerfasserIn]
• Verfasst von: Tribuzio, Antonio [VerfasserIn]
• Titel: On the energy scaling behaviour of singular perturbation models involving higher order laminates
• Verf.angabe: Angkana Rüland and Antonio Tribuzio
• Ausgabe: Version V3
• Jahr: 2022
• Umfang: 47 S.
• Fussnoten: V1 29. Oktober 2021, V2 17. Januar 2022, V3 25. November 2022 (this version, v3) ; Gesehen am 14.02.2023
• Titel Quelle: Enthalten in: De.arxiv.org
• Ort Quelle: [S.l.] : Arxiv.org, 1991
• Jahr Quelle: 2022
• Band/Heft Quelle: (2022), Artikel-ID 2110.15929, Seite 1-47
• DOI: doi:10.48550/arXiv.2110.15929
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: Mathematics - Analysis of PDEs
• K10plus-PPN: 1816982164

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.