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On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators

Bogdan Raiţă, Angkana Rüland, Camillo Tissot, Antonio Tribuzio

TL;DR

This work develops a systematic theory for scaling in the compatible two-well problem associated with higher-order homogeneous linear differential operators. By introducing a general lower-bound framework controlled by the maximal vanishing order L of the operator symbol on the unit sphere, it reveals nonstandard scaling laws—specifically epsilon raised to 2L over 2L+1—beyond the classical epsilon^(2/3) regime. The authors provide sharp two-dimensional upper bounds for generalized symmetrized gradients, matching the lower bounds and establishing optimality for model operator classes (curl-like and Saint-Venant generalizations). The results highlight how Fourier-symbol geometry and wave-cone structure govern microstructure formation, with branching and laminate constructions illustrating how to realize the predicted scaling in practice, and they connect to a broad literature on N-well problems and pattern formation in elastic materials.

Abstract

We study the scaling behaviour of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions for highly symmetric boundary data (but arbitrary tensor order $m \in \mathbb{N}$) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from \cite{CC15} which was also discussed in \cite{RRT23} provides an example of this family of new scaling laws.

On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators

TL;DR

This work develops a systematic theory for scaling in the compatible two-well problem associated with higher-order homogeneous linear differential operators. By introducing a general lower-bound framework controlled by the maximal vanishing order L of the operator symbol on the unit sphere, it reveals nonstandard scaling laws—specifically epsilon raised to 2L over 2L+1—beyond the classical epsilon^(2/3) regime. The authors provide sharp two-dimensional upper bounds for generalized symmetrized gradients, matching the lower bounds and establishing optimality for model operator classes (curl-like and Saint-Venant generalizations). The results highlight how Fourier-symbol geometry and wave-cone structure govern microstructure formation, with branching and laminate constructions illustrating how to realize the predicted scaling in practice, and they connect to a broad literature on N-well problems and pattern formation in elastic materials.

Abstract

We study the scaling behaviour of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions for highly symmetric boundary data (but arbitrary tensor order ) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from \cite{CC15} which was also discussed in \cite{RRT23} provides an example of this family of new scaling laws.
Paper Structure (19 sections, 18 theorems, 168 equations, 1 figure)

This paper contains 19 sections, 18 theorems, 168 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The classical two well-problem for the gradient
  3. The compatible two-well problem for constant coefficient, linear differential operators
  4. Outline of the main results
  5. Scaling results for generalized symmetrized gradients
  6. Lower bounds for a class of linear, homogeneous differential operators
  7. Relation to the literature
  8. Outline of the article
  9. Notation and Preliminaries
  10. Tensor notation
  11. Example operators
  12. Computation of the wave cones for symmetrized derivatives and the higher order divergence
  13. Lower Bound Scaling Results
  14. Scaling results
  15. Applications
  16. ...and 4 more sections

Key Result

Theorem 1

Let $d,n \in \mathbb{N}$. Let $\Omega \subset \mathbb{R}^d$ be open, bounded and Lipschitz. Let $\mathcal{A}(D)$ be a homogeneous, constant coefficient, linear, first order differential operator and $A,B \in \mathbb{R}^n$ such that $A-B \in \Lambda_{\mathcal{A}} \setminus I_{\mathcal{A}}$, see eq:wa

Figures (1)

  • Figure 1: An illustration of the branching construction used in \ref{['lem:Branching']}. The individual unit cell constructions from \ref{['lem:UnitCell']} are iteratively combined into a construction refining in the $e_2$ direction. In blue is the region where $\tilde{u}_0 = \tilde{A}_0$ and red corresponds to $\tilde{u}_0 = \tilde{B}_0$. The dashed horizontal lines depict the region in which we have a simple laminate. As in CC15 for $m\geq 2$ we need the curves separating the domains to be of a sufficiently high regularity (see the discussion in \ref{['rmk:UnitCellRemark']}). The unit cell and its copies, are highlighted with a green box. Moreover, for $m \geq 3$, we need to do the reflection-type argument outlined in \ref{['lem:Branching']} to ensure zero boundary values at the left and right.

Theorems & Definitions (41)

  • Theorem 1: RRT23
  • Theorem 2: Symmetrized derivative
  • Definition 1.1: Maximal vanishing order on the unit sphere
  • Theorem 3
  • Example 2.1: $d=2$
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Higher order curl
  • ...and 31 more