SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3158-3187, January 2021.
This paper provides an extension to fractional order Sobolev spaces of the classical result of Murat and Brezis which states that the cone of positive elements in $H^{-1}(\Omega)$ compactly embeds in $W^{-1,q}(\Omega)$ for every $q < 2$ and for any open and bounded set $\Omega$ with Lipschitz boundary. In particular, our proof contains the classical result. Several new analysis tools are developed during the course of the proof to our main result which are of wider interest. Subsequently, we apply our results to the convergence of convex sets and establish a fractional version of the Mosco convergence result of Boccardo and Murat. We conclude with an application of this result to quasi-variational inequalities.

中文翻译：

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关于 Murat 紧凑性结果的分数版本及其应用

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3158-3187 页，2021 年 1 月。
本文提供了对 Murat 和 Brezis 经典结果的分数阶 Sobolev 空间的扩展，其中指出 $H^ 中的正元素锥{-1}(\Omega)$ 紧凑地嵌入在 $W^{-1,q}(\Omega)$ 中，对于每一个 $q < 2$ 以及任何具有 Lipschitz 边界的开集和有界集 $\Omega$。特别是，我们的证明包含经典结果。在证明我们的主要结果的过程中开发了几种新的分析工具，这些工具引起了更广泛的兴趣。随后，我们将我们的结果应用于凸集的收敛，并建立 Boccardo 和 Murat 的 Mosco 收敛结果的分数版本。最后，我们将此结果应用于拟变分不等式。