Asymptotic behavior of the W1∕q,q-norm of mollified BV functions and applications to singular perturbation problems
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:

\begin{equation*}
 %\label{hjhjjggjjjkhkhjjhjhhgghjhjhjljkjk}
 \lim_{\e\to
 0^+}\frac{1}{|\ln{\e}|}\big\|\eta_\e*u\big\|^q_{W^{1/q,q}(\Omega)}=
 C_0\int_{J_u}\Big|u^+(x)-u^-(x)\Big|^q{\rm d}\mathcal{H}^{N-1}(x),
\end{equation*}

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