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

Abstract

Motivated by results of Figalli and Jerison and Hernández, we prove the following formula: \begin{equation*} \lim_{ε\to 0^+}\frac{1}{|\lnε|}\big\|η_ε*u\big\|^q_{W^{1/q,q}(Ω)}= C_0\int_{J_u}\Big|u^+(x)-u^-(x)\Big|^qd\mathcal{H}^{N-1}(x), \end{equation*} where $Ω\subset\mathbb{R}^N$ is a regular domain, $u\in BV(Ω)\cap L^\infty$, $q>1$ and $η_ε(z)=ε^{-N}η(z/ε)$ is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.