Calculus of Variations and Geometric Measure Theory
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G. Bellettini - V. Caselles - M. Novaga

The Total Variation Flow in $R^N$

created on 07 Jun 2001
modified on 14 Dec 2002

[BibTeX]

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

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