Some properties of minimizers for the Chan-Esedoglu L1TV functional
Abstract
Description
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.