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 Lp norm in the Gagliardo s-seminorm by the weak Lp 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.