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 Hern\'andez, we prove the following formula: \begin{equation*} \lim_{\epsilon\to 0^+}\frac{1}{|\ln{\epsilon}|}\big|\eta_\epsilon*u\big|^q_{W^{1/q,q}(\Omega)}= C_0\int_{J_u}\Big|u^+(x)-u^-(x)\Big|^qd\mathcal{H}^{N-1}(x), \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a regular domain, $u\in BV(\Omega)\cap L^\infty$, $q>1$ and $\eta_\epsilon(z)=\epsilon^{-N}\eta(z/\epsilon)$ is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.