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 $\Omega \subset {\mathbb{R}}^N$ 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.

References in corpus (1)

• Properties of a Hilbertian Norm for Perimeter