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 → 1 − and s → 0 + , 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.