Asymptotic behavior of the W^1∕q,q-norm of mollified BV functions and applications to singular perturbation problems

Department of Mathematics, Ben Gurion University of the Negev, PO Box 653, Be’er Sheva 84105, Israel.

^* Corresponding author: poliakov@math.bgu.ac.il

Received: 16 December 2018
Accepted: 25 August 2019

Abstract

Motivated by results of Figalli and Jerison [J. Funct. Anal. 266 (2014) 1685–1701] and Hernández [Pure Appl. Funct. Anal., Preprint https://arxiv.org/abs/1709.08262 (2017)], we prove the following formula:

where Ω ⊂ ℝ^N is a regular domain, u ∈ BV (Ω) ∩ L^∞(Ω), q > 1 and η_ε(z) = ε^−Nη(z∕ε) is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.