有界域上拟正则映射的Liouville定理的一个尖锐稳定性不等式
A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains
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中文总结 AI 辅助
该研究针对有界域上外失真接近1的拟正则映射，建立了Liouville定理在W^{1,p}中的尖锐定量稳定性结果，通过Whitney链与Lorentzian表示给出两种全局化方案，并推广到一般有界连通开集。
中文摘要 AI 辅助
设$n \ge 3$并固定$p>n$。我们针对外失真$K$充分接近1的非恒定、保向拟正则映射，建立了Liouville定理在$W^{1,p}$中的尖锐定量稳定性结果。我们首先重新组织Reshetnyak的经典局部论证\cite{R1976}，系统地借鉴了如\cite{FZ2022}中发展的现代技术。然后，我们在有界连通域上给出两种全局化方案。对于有界John域，我们不再使用Reshetnyak的原始构造\cite{R19762}（该构造在相邻Whitney立方体上粘合Möbius变换），而是利用Whitney链和Bojarski的放大立方体估计，获得定量尖锐的欧几里得稳定性结果。最后，我们通过Möbius群的Lorentzian表示将此思想推广到$\mathbb{R}^n$的一般有界连通开子集，得到一个紧化的加权稳定性定理，其底层测度关于Lebesgue测度绝对连续，且密度上界为1。
英文摘要
Let $n \ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \cite{R1976}, drawing systematically on modern techniques such as those developed in \cite{FZ2022}. We then give two globalization schemes on bounded, connected domains. For bounded John domains, instead of Reshetnyak's original construction \cite{R19762}, which glues Möbius transformations on adjacent Whitney cubes, we use Whitney chains and Bojarski's enlarged-cube estimate to obtain a quantitatively sharp Euclidean stability result. Finally, we extend this idea to general bounded, connected open subsets of $\mathbb{R}^n$ via the Lorentzian representation of the Möbius group, yielding a compactified, weighted stability theorem whose underlying measure is absolutely continuous with respect to Lebesgue measure and has density bounded above by one.
发表机构
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
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