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

TL;DR

This work analyzes energy scaling in singularly perturbed multi-well elastic energies without gauge invariance, focusing on Dirichlet data that induce laminates of finite order. By combining iterated branching constructions (upper bounds) with Fourier-multiplier and bootstrap-based lower bounds, the authors establish a precise correspondence between the order of lamination of boundary data and the minimal energy scaling across two, three, and four wells, as well as for arbitrary orders in $n$ dimensions. The results reveal a hierarchical spectrum of scalings, with top-order laminates contributing distinct rates $\epsilon^{2/(m+2)}$ (and $\epsilon^{2/3}$ for the first order) that depend only on the lamination order of the boundary data. This advances the understanding of microstructure-energy coupling in shape-memory and martensitic models by explicitly linking boundary lamination structure to the energetics of admissible microstructures and providing scalable constructions and bounds for higher-order laminates. The techniques integrate $L^p$-growth analyses, $L^2$ Fourier methods, and cone-based localization to yield essentially sharp scaling laws with explicit constructions across a hierarchical laminate framework, offering a clear pathway to explore even more complex laminates and gauge-free models in elasticity.

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.

Figures (10)

• Figure 1: Representation of the set $K_2$ on the plane of diagonal matrices (see Section \ref{['sec:Lpintro']}). The red hashed segment depicts the set $K_2^{(lc)}$.
• Figure 2: An illustration of the microstructures used in the heuristic explanation on the differences between $p=1$ and $p\in (1,\infty)$. On the left a branching construction. The hashed lines mark the building blocks of different refinement generations. On the right a simple laminate. The darker regions represent the boundary layer.
• Figure 3: The set of matrices $K_3$ from Section \ref{['sec:three_wellsinto']}. In blue $K^{(1)}_3$, in red $K^{(lc)}_3\setminus K^{(1)}_3 = K^{(2)}_3\setminus K_3^{(1)}$. $J_1$ is rank-$1$-connected to all the matrices of $K_3$.
• Figure 4: The set of matrices $K_4$ in the space of diagonal matrices (see Section \ref{['sec:fourthintro']}). In blue $K^{(1)}_4$, in red $K^{(2)}_4\setminus K^{(1)}_4$, in green $K^{(lc)}_4\setminus K_4^{(2)} = K^{(3)}_4\setminus K^{(2)}_4$.
• Figure 5: Example of a building block of a (vertical) branching construction as given in Lemma \ref{['lem:loc-vert']}.

Theorems & Definitions (60)

• Theorem 1
• Theorem 2: Scaling of the three-well problem
• Theorem 3: Scaling of the four-well problem
• Theorem 4: Scaling of the $(n+1)$-well problem
• Proposition 2.1
• Definition 2.2: The lamination convex hull
• Lemma 3.1
• proof
• Lemma 3.2
• proof