On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_3$ Structure

Abstract

In this article we study quantitative rigidity properties for the compatible and incompatible two-state problems for suitable classes of $\mathcal{A}$-free operators and for a singularly perturbed $T_3$-structure for the divergence operator. In particular, in the compatible setting of the two-state problem we prove that all homogeneous, first order, linear operators with affine boundary data which enforce oscillations yield the typical ${\epsilon }^{\frac{2}{3}}$-lower scaling bounds. As observed in \cite{CC15} for higher order operators this may no longer be the case. Revisiting the example from \cite{CC15}, we show that this is reflected in the structure of the associated symbols and that this can be exploited for a new Fourier based proof of the lower scaling bound. Moreover, building on \cite{RT22, GN04, PP04}, we discuss the scaling behaviour of a $T_3$ structure for the divergence operator. We prove that as in \cite{RT22} this yields a non-algebraic scaling law.

arXiv comment45 pages, comments welcome; contains improvements in Theorem 1, Lemma 3.1 as well as in Section 4.2; further extended Section 3.4 and Appendix B

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Cited by in corpus (2)

• On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators
• On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions