2007/10/22 by Vixie, Kevin R.
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.0710.3980
We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional F(u) ≡ ∫ |∇ u | dx + λ∫ |u - f| dx ; u,f:ℝn → ℝ. If we restrict to u = χΣ and f = χΩ, Σ, Ω∈ ℝn, the L1TV functional reduces to E(Σ) = \Per(Σ) + λ|Σ\vartriangle Ω|. We show that there is a minimizer Σ such that its boundary ∂Σ lies between the union of all balls of radius \fracnλ contained in Ω and the corresponding union of \fracnλ-balls in Ωc. We also show that if a ball of radius \fracnλ + 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 L1TV minimizers and how these relate to our results.