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

## TL;DR

This paper establishes the asymptotic limit of the $W^{1/q,q}$-norm of mollified BV functions, revealing a connection to jump discontinuities, and applies this to analyze singular perturbation problems.

## Contribution

It proves a new limit formula for mollified BV functions' Sobolev norms and applies it to singular perturbation analysis, extending prior results by Figalli, Jerison, and Hernández.

## Key findings

- Limit of mollified BV functions' Sobolev norm characterized
- Connection between norm asymptotics and jump discontinuities
- 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.