Φ^*[I] 的正则性与非退化性蕴含固定边界 ∂Ω 的正则性
Regularity and non-degeneracy of $Φ^*[I]$ implies regularity of the fixed boundary $\partialΩ$
浏览论文内容
中文总结 AI 辅助
研究满足特定非退化条件的微分同胚 Φ，证明其推前 Φ^*[I] 的局部正则性可推出固定边界 ∂Ω 的局部正则性，补充了非散射非均匀介质的正则性结果。
中文摘要 AI 辅助
对于将区域 \u03a9 的闭包映射到自身且在 ∂Ω 上为恒等映射的微分同胚 Φ，我们证明：在满足特定非退化条件时，推前 Φ^*[I] 的局部正则性蕴含 ∂Ω 的局部正则性。具体而言，若 ν 是 ∂Ω 上点 P 处的法向量，且满足条件 (Φ^*[I](P)-I)ν ≠ 0，同时假设 Φ^*[I] 在 P 附近为 C^{k+1,α} 类，则 ∂Ω 在 P 附近也为 C^{k+1,α} 类。该结果自然补充了非散射非均匀介质的近期正则性结果。
英文摘要
For a diffeomorphism $Φ$ of a domain $\overlineΩ$ onto itself, which is the identity on $\partialΩ$, we prove that local regularity of the push-forward $Φ^*[I]$ implies local regularity of $\partialΩ$, provided a certain non-degeneracy condition is satisfied. To be precise, if $ν$ is a normal to $\partialΩ$ at a point $P$, then the condition $(Φ^*[I](P)-I)ν\neq 0$ and the assumption that $Φ^*[I]$ is of class $C^{k+1,α}$ near $P$ imply that $\partialΩ$ is also of class $C^{k+1,α}$ near $P$. This result naturally complements recent regularity results for non-scattering inhomogeneities.
