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 $; $u,f:\Bbb{R}^n \to \Bbb{R}$. If we restrict to $u = \chi_{\Sigma}$ and $f = \chi_{\Omega}$, $\Sigma, \Omega \in \Bbb{R}^n$, the $L^1$TV functional reduces to $E(\Sigma) = \Per(\Sigma) + \lambda |\Sigma\vartriangle \Omega |$. 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.