我们建立了基于 $\mathrm{L}^{1}$ 的函数空间中无散（或螺线管）元素的第一个延拓结果。这里的关键点是在保持无散约束的同时，保持底层的 $\mathrm{L}^{1}$ 有界性。虽然先前的结果，如 Kato 等人 [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] 针对基于 $\mathrm{L}^{p}$ 的函数空间（$1<p<\infty$）的结果依赖于偏微分方程方法，但调和分析的基本原理排除了在 $\mathrm{L}^{1}$ 情形下使用此类策略。通过一种借助微分形式适应无散约束的新方法，我们建立了在基于 $\mathrm{L}^{1}$ 的情形下此类延拓算子的存在性。这既适用于凸区域的情形，此时可以实现全局延拓，也适用于 Lipschitz 情形，此时可以实现局部延拓。由于该方法同样适用于 $1<p<\infty$，我们的方法为 $p\in\{1,\infty\}$ 和 $1<p<\infty$ 的情形提供了统一的处理途径。具体而言，覆盖指数 $p\in\{1,\infty\}$，这以肯定的方式回答了 Kato 等人 [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] 留下的一个边界情形。通过使用显式例子，表明关于底层区域的假设几乎是最优的。
英文摘要：
We establish the first extension results for divergence-free (or solenoidal) elements of $\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying $\mathrm{L}^{1}$-boundedness. While previous results as in Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] for $\mathrm{L}^{p}$-based function spaces, $1<p<\infty$, rely on PDE approaches, basic principles from harmonic analysis rule out such strategies in the $\mathrm{L}^{1}$-context. By means of a novel method adapted to the divergence-free constraint via differential forms, we establish the existence of such extension operators in the $\mathrm{L}^{1}$-based situation. This applies both to the case of convex domains, where a global extensions can be achieved, as well as to the Lipschitz case, where a local extension can be achieved. Being applicable to $1<p<\infty$ too, our method provides a unifying approach to the cases $p\in\{1,\infty\}$ and $1<p<\infty$. Specifically, covering the exponents $p\in\{1,\infty\}$, this answers a borderline case left open by Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] in the affirmative. By use of explicit examples, the assumptions on the underlying domains are shown to be almost optimal.
Curl and gradient of divergence operators in Spaces $ \mathbf{W}^{m}$ and $\mathbf{A}^{2k}$ vortex and potential fields and in the classes $\mathbf{C}(2k, m)$
A two-dimensional structural local-defect theory for scalar non-divergence advection-diffusion homogenization
SKLMS, 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）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
