Some properties of minimizers for the LTV functional

We present two results characterizing minimizers of the LTV functional F (u) ≡ ∫ |∇u|dx+λ ∫ |u−f |dx; u, f : R → R. If we restrict to u = χΣ and f = χΩ, Σ,Ω ∈ R, the LTV functional reduces to E(Σ) = Per(Σ) + λ|Σ M Ω|. We show that there is a minimizer Σ such that its boundary ∂Σ lies between the union of all balls of radius nλ contained in Ω and the corresponding union of nλ -balls in Ω . We also show that if a ball of radius nλ + 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 LTV minimizers and how these relate to our results.