Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$

Abstract

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.