论文标题
Brezis-Van Schaftingen-Yung方法的各向异性版本$ s = 1 $和$ s = 0 $
Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$
论文作者
论文摘要
2014年，路德维格（Ludwig）展示了各向异性gagliardo $ s $ s $ s $ f $的eminorm as $ f $ as $ s \ rightArrow 1^ - $和$ s \ rightArrow0^+$，这扩展了由于Bourgain-Bregain-Brezis-Brezis-Mironescu（bbm）和maz'ya-shaposhaposhaposhapshaphikova（Masshapshikova）而扩大了结果。最近，布雷兹斯（Brezis），范·沙夫丁根（Van Schaftingen）和杨（Yung）通过替换了弱$ l^p $ quasinorm的gagliardo $ s $ seminorm中强的$ l^p $ norm，提供了另一种方法。他们表征了与BBM公式相辅相成的$ s = 1 $的情况。 Yung和第一作者后来建立了$ S = 0 $的相应MS公式。在本文中，我们遵循Brezis-Van Schaftingen-Yung的方法，并显示$ s = 1 $和$ s = 0 $的各向异性版本。我们的结果概括了Brezis，Van Schaftingen，Yung和第一作者的作品，并补充了Ludwig的作品。
In 2014, Ludwig showed the limiting behavior of the anisotropic Gagliardo $s$-seminorm of a function $f$ as $s\rightarrow 1^-$ and $s\rightarrow0^+$, which extend the results due to Bourgain-Brezis-Mironescu(BBM) and Maz'ya-Shaposhnikova(MS) respectively. Recently, Brezis, Van Schaftingen and Yung provided a different approach by replacing the strong $L^p$ norm in the Gagliardo $s$-seminorm by the weak $L^p$ quasinorm. They characterized the case for $s=1$ that complements the BBM formula. The corresponding MS formula for $s=0$ was later established by Yung and the first author. In this paper, we follow the approach of Brezis-Van Schaftingen-Yung and show the anisotropic versions of $s=1$ and $s=0$. Our result generalizes the work by Brezis, Van Schaftingen, Yung and the first author and complements the work by Ludwig.
