The total variation flow in ℝN

Abstract

In this paper, we study the minimizing total variation flow ut = div(Du/Du) in ℝN for initial data u0 in Lloc1 (ℝN), proving an existence and uniqueness result. Then we characterize all bounded sets Ω of finite perimeter in ℝ2 which evolve without distortion of the boundary. In that case, uo = χΩ evolves as u(t, x) = (1 - λΩt)+ χΩ, where χΩ is the characteristic function of Ω, λΩ := P(Ω)/Ω, and P(Ω) denotes the perimeter of Ω. We give examples of such sets. The solutions are such that v := λΩχΩ solves the eigenvalue problem -div (Dr Dr) = v. We construct other explicit solutions of this problem. As an application, we construct explicit solutions of the denoising problem in image processing. © 2002 Elsevier Science (USA).