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

2022/01/02 by Gu, Qingsong, Huang, Qingzhong
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 46E35 #Secondary 52A20

paper · doi:10.48550/arxiv.2201.00353

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.

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