论文标题
关于两国问题的缩放属性和一个奇异的扰动$ T_3 $结构
On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_3$ Structure
论文作者
论文摘要
在本文中，我们研究了适合$ \ Mathcal {a} $的合适类别的兼容和不兼容的两国问题的定量刚度属性 - 免费运算符，以及用于脱落操作员的奇异扰动$ T_3 $结构。特别是，在两国问题的兼容设置中，我们证明所有具有仿射边界数据的均质的，一阶的线性运算符，它们会产生典型的$ε^{\ frac {\ frac {2} {3} {3}} $ - 较低的缩放范围。如\ cite {cc15}在高阶操作员中所观察到的那样，情况可能不再是这种情况。从\ cite {cc15}重新访问示例时，我们表明这反映在关联符号的结构中，并且可以利用这是针对较低缩放结合的新的基于傅立叶的新证明。此外，在\ cite {rt22，gn04，pp04}上构建，我们讨论了Divergence操作员的$ T_3 $结构的缩放行为。我们证明，如\ cite {rt22}所产生的非代数缩放定律。
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 $ε^{\frac{2}{3}}$-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.
