# Gradient estimates for the Perona-Malik equation

created by gobbino on 23 Mar 2005
modified on 13 Nov 2006

[BibTeX]

Accepted Paper

Inserted: 23 mar 2005
Last Updated: 13 nov 2006

Journal: Math. Ann.
Year: 2005

Abstract:

We consider the Cauchy problem for the Perona-Malik equation $$u{t}=\mathrm{div}\left(\frac{\nabla u}{1+ \nabla u {2}}\right)$$ in a bounded open set $\Omega\subseteq\re^{n}$, with Neumann boundary conditions.

If $n=1$, we prove some a priori estimates on $u$ and $u_{x}$. Then, extending such estimates to a discrete setting, we prove a compactness result for the semi-discrete scheme obtained by replacing the space derivatives by finite differences.

Finally, for $n>1$ we give examples to show that the corresponding estimates on $\nabla u$ are in general false.

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