Some properties of minimizers for the Chan-Esedoglu L1TV functional

Abstract

We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional $F(u)\equiv \int |\nabla u|dx+\lambda \int |u-f|dx$; $u,f:{\mathbb{R}}^n\to \mathbb{R}$. If we restrict to $u={\chi }_{\Sigma }$ and $f={\chi }_{\Omega }$, $\Sigma ,\Omega \in {\mathbb{R}}^n$, the $L^1$TV functional reduces to $E(Σ) = \Per(Σ) + λ|Σ\vartriangle Ω|$. We show that there is a minimizer $\Sigma$ such that its boundary $\partial \Sigma$ lies between the union of all balls of radius $\frac{n}{\lambda }$ contained in $\Omega$ and the corresponding union of $\frac{n}{\lambda }$-balls in ${\Omega }^c$. We also show that if a ball of radius $\frac{n}{\lambda }+\epsilon$ is almost contained in $\Omega$, a slightly smaller concentric ball can be added to $\Sigma$ 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.