*Published Paper*

**Inserted:** 7 jun 2001

**Last Updated:** 14 dec 2002

**Journal:** J. Differential Equations

**Volume:** 184

**Number:** 2

**Pages:** 475-525

**Year:** 2002

**Abstract:**

In this paper we study the minimizing total variation flow $u_t = {\rm div} (Du /\vert D u\vert)$ in $R^N$ for initial data $u_0$ in $L^1_{{\rm loc}}(R^N)$, proving an existence and uniqueness result. Then we characterize all bounded sets $\Omega$ of finite perimeter in $R^2$ which evolve without distortion of the boundary. In that case, $u_0 = \chi_{\Omega}$ evolves as $u(t,x) = (1-\lambda_\Omega t)^+ \chi_{\Omega}$, where $\chi_\Omega$ is the characteristic function of $\Omega$, $\lambda_\Omega := P(\Omega)/\vert \Omega\vert$, and $P(\Omega)$ denotes the perimeter of $\Omega$. We give examples of such sets. The solutions are such that $v := \lambda_\Omega \chi_\Omega$ solves the eigenvalue problem $- {\rm div} \left(\frac{Dv}{\vert Dv\vert}\right) = v$. We construct other explicit solutions of this problem. As an application, we construct explicit solutions of the denoising problem in image processing.

**Keywords:**
parabolic equations, Finite perimeter sets, Total variation flow