*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.

**Download:**