Bourgain-Brezis-Mironescu Domains

Abstract

Bourgain et al.(2001) proved that for $p>1$ and smooth bounded domain $\Omega\subseteq\mathbb{R}^N$, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{\Omega \times \Omega}\frac{\lvert f(x)-f(y) \rvert^p}{\lvert x-y \rvert^{N+sp}}dx dy=\kappa \int \limits_{\Omega}\lvert \nabla f(x) \rvert^p dx \end{equation*} for all $f\in L^p(\Omega)$. This gives a characterization of $W^{1,p}(\Omega)$ by means of $W^{s,p}(\Omega)$ seminorms only. For the case $p=1$, D\'avila(2002) proved that when $\Omega$ is a bounded domain with Lipschitz boundary, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{\Omega \times \Omega}\frac{\lvert f(x)-f(y) \rvert}{\lvert x-y \rvert^{N+s}}dx dy=\kappa [f]_{BV(\Omega)} \end{equation*} for all $f\in L^1(\Omega)$. This characterizes $BV(\Omega)$ in terms of $W^{s,1}(\Omega)$ seminorm. In this paper we extend the first result and partially extend the second result to extension domains.