The Total variation flow in RN

Abstract

In this paper, we study the minimizing total variation flow ut=div(Du/∣Du∣) in N for initial data u0 in Lloc1(N), proving an existence and uniqueness result. Then we characterize all bounded sets Ω of finite perimeter in 2 which evolve without distortion of the boundary. In that case, u0=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. We construct other explicit solutions of this problem. As an application, we construct explicit solutions of the denoising problem in image processing.