Chan-Esedoglu L1TV泛函极小元的一些性质
Some properties of minimizers for the Chan-Esedoglu L1TV functional
摘要：
我们提出了两个结果表征的Chan Esedoglu L1电视功能$F(u) \equiv \int |\nabla u | dx + λ\int |u - f| dx $极小; $u,f:\Bbb{R}^n \to \Bbb{R}$。如果我们限制为$u = χ_Σ$和$f = χ_Ω$、$Σ, Ω\in \Bbb{R}^n$，则$L^1$ TV功能降低为$E(Σ) = \Per(Σ) + λ|Σ\vartriangle Ω|$。证明了存在极小元$Σ$，使得它的边界$\partialΣ$位于$Ω$中包含的半径为$\frac{n}λ$的所有球的并集和$Ω^c$中的相应的$\frac{n}λ$ -球的并集之间.我们还表明，如果半径为$\frac{n}λ + ε$的球几乎包含在$Ω$中，则可以将稍小的同心球添加到$Σ$中以获得另一个极小值。最后，我们评论最近的结果阿拉德$L^1$电视极小化，以及如何这些与我们的结果。
英文摘要：
We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional $F(u) \equiv \int |\nabla u | dx + λ\int |u - f| dx $; $u,f:\Bbb{R}^n \to \Bbb{R}$. If we restrict to $u = χ_Σ$ and $f = χ_Ω$, $Σ, Ω\in \Bbb{R}^n$, the $L^1$TV functional reduces to $E(Σ) = \Per(Σ) + λ|Σ\vartriangle Ω|$. We show that there is a minimizer $Σ$ such that its boundary $\partialΣ$ lies between the union of all balls of radius $\frac{n}λ$ contained in $Ω$ and the corresponding union of $\frac{n}λ$-balls in $Ω^c$. We also show that if a ball of radius $\frac{n}λ + ε$ is almost contained in $Ω$, a slightly smaller concentric ball can be added to $Σ$ to get another minimizer. Finally, we comment on recent results Allard has obtained on $L^1$TV minimizers and how these relate to our results.
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