Volume 29, 2023
|Number of page(s)||54|
|Published online||11 August 2023|
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: email@example.com
Accepted: 15 June 2023
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.
Mathematics Subject Classification: 35B25 / 74N15 / 35B36
Key words: Higher order laminates / scaling law / singular perturbation / branching construction
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through SPP 2256, project ID 441068247. At the time of writing A.R. was a member of the Heidelberg STRUCTURES Excellence Cluster, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948.
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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