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_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert^p}{\lvert x-y \rvert^{N+sp}}dx dy=κ\int \limits_Ω\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 $p=1$, Dávila(2002) proved that when $\Omega$ is a bounded domain with Lipschitz boundary, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert}{\lvert x-y \rvert^{N+s}}dx dy=κ[f]_{BV(Ω)} \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.

Cited by in corpus (4)

• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
• A Remake of Bourgain-Brezis-Mironescu Characterization of Sobolev Spaces
• Bourgain-Brezis-Mironescu formula for Riesz Potentials