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

TL;DR

This paper investigates the scaling laws and rigidity properties of two-state problems and a $T_3$-structure related to divergence operators, revealing new bounds and behaviors through Fourier analysis and structural insights.

Contribution

It establishes the $oxed{ ext{ } extstyle rac{2}{3} ext{-scaling bound for compatible two-state problems and explores non-algebraic scaling laws for a }T_3 ext{-structure, extending prior results.} }$

Findings

Homogeneous, first-order operators with affine boundary data have a $oxed{ ext{ extstyle rac{2}{3}} ext{-scaling lower bound} }$.

Higher order operators may not follow the $oxed{ ext{ extstyle rac{2}{3}} ext{-scaling law}}$ as shown by structural analysis.

The $T_3$-structure exhibits non-algebraic scaling behavior, extending previous theoretical understanding.

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.