论文标题
Bourgain-Brezis-Mironescu域
Bourgain-Brezis-Mironescu Domains
论文作者
论文摘要
Bourgain等人（2001年）证明，对于$ p> 1 $和光滑的界面$ω\ subseteq \ subseteq \ mathbb {r}^n $，\ begin {equination*} \ lim \ limits_ {s \ s {s \ to1}（1-s）（1-s） f（x）-f（y）\ rvert^p} {\ lvert x-y \ rvert^{n+sp}} dx dy =κ\ int \ int \limits_Ω\ lvert \ lvert \ nabla f（x）\ rvert^p dx \ p dx \ end end end end eNd {queation*}这仅通过$ W^{s，p}（ω）$ seminorms来表征$ w^{1，p}（ω）$。对于情况，$ p = 1 $，dávila（2002）证明，当$ω$是具有Lipschitz边界的有界域时，\ begin {equination*} \ lim \ limits_ {s \ to1}（1-s）\ iint \ iint \ iint \ iint \ iint \ iint \ limits_ {ω\ timesω} \ frac frac \ frac \ frac \ lvert（） \ rvert} {\ lvert x-y \ rvert^{n+s}} dx dy =κ[f] _ {bv（ω）} \ end end {equation*} in l^1（ω）$ in l^1 in l^1（ω）$。这是$ w^{s，1}（ω）$ seminorm的$ bv（ω）$。在本文中，我们扩展了第一个结果，并部分将第二个结果扩展到扩展域。
Bourgain et al.(2001) proved that for $p>1$ and smooth bounded domain $Ω\subseteq\mathbb{R}^N$, \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 $f\in L^p(Ω)$. This gives a characterization of $W^{1,p}(Ω)$ by means of $W^{s,p}(Ω)$ seminorms only. For the case $p=1$, Dávila(2002) proved that when $Ω$ 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 $f\in L^1(Ω)$. This characterizes $BV(Ω)$ in terms of $W^{s,1}(Ω)$ seminorm. In this paper we extend the first result and partially extend the second result to extension domains.
