论文标题
一组有限周长的光滑内部近似
On smooth interior approximation of Sets of Finite Perimeter
论文作者
论文摘要
在本文中，我们证明，对于任何有限的周围$ω\ subset \ mathbb {r}^n $ \ label {moregeneralApproximation} \ limsup_ {i \ rightArrow \ infty} p（e_i）\ le p（ω）+c_1（n）\ Mathscr {h}^{h}^{n-1}（\ partialω\ partialω\ capcapω^1）。 \ end {align}在上述$ω^1 $中是$ω$，$ p（\ cdot）$的度量理论内部$表示周围的功能，而$ c_1（n）$是一个维常数。 相反，我们证明，对于任何集合$ e_k \ subsetω$满足$ e_k \ e_k \ rightarrowω$ in $ l^1 $，存在一个尺寸常数$ c_2（n）$，使以下不平等持有：\ begin {align} \ babel} \ label {gap} \ liminf_ {k \ rightArrow \ infty} p（e_k）\ ge p（ω）+ c_2（n）\ Mathscr {h}^^{n-1}（\ partialω\ capω^1）。 \ end {align}特别是，这些结果表明，对于有限周围的有限设置$ω$，\ begin {align} \ label {char*} \ Mathscr {h}^{n-1}（\ partialω\capΩ^1）= 0 \ end \ end {align}在且仅当存在一系列平滑集$ e_k $时，因此，$ e_k \ e_k \ e_k \ subsetω$，$ e_k $，$ e_k \ e_k \ rightarrow的$rightarrowΩ
In this paper, we prove that for any bounded set of finite perimeter $Ω\subset \mathbb{R}^n$, we can choose smooth sets $E_k \Subset Ω$ such that $E_k \rightarrow Ω$ in $L^1$ and \begin{align} \label{moregeneralapproximation} \limsup_{i \rightarrow \infty} P(E_i) \le P(Ω)+C_1(n) \mathscr{H}^{n-1}(\partial Ω\cap Ω^1). \end{align}In the above $Ω^1$ is the measure-theoretic interior of $Ω$, $P(\cdot)$ denotes the perimeter functional on sets, and $C_1(n)$ is a dimensional constant. Conversely, we prove that for any sets $E_k \Subset Ω$ satisfying $E_k \rightarrow Ω$ in $L^1$, there exists a dimensional constant $C_2(n)$ such that the following inequality holds: \begin{align} \label{gap} \liminf_{k \rightarrow \infty} P(E_k) \ge P(Ω)+ C_2(n) \mathscr{H}^{n-1}(\partial Ω\cap Ω^1). \end{align} In particular, these results imply that for a bounded set $Ω$ of finite perimeter,\begin{align} \label{char*} \mathscr{H}^{n-1}(\partial Ω\cap Ω^1)=0 \end{align} holds if and only if there exists a sequence of smooth sets $E_k$ such that $E_k \Subset Ω$, $E_k \rightarrow Ω$ in $L^1$ and $P(E_k) \rightarrow P(Ω)$.
