Calculus of Variations and Geometric Measure Theory
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M. Ghisi - M. Gobbino

A class of local classical solutions for the one-dimensional Perona-Malik equation

created by gobbino on 13 Nov 2006

[BibTeX]

Preprint

Inserted: 13 nov 2006

Year: 2006

Abstract:

We consider the Cauchy problem for the one-dimensional Perona-Malik equation $$u{t}=\frac{1-u{x}{2}}{(1+u{x}{2}){2}}\,u{xx}$$ in the interval $[-1,1]$, with homogeneous Neumann boundary conditions.

We prove that the set of initial data for which this equation has a local-in-time classical solution $u:[-1,1]\times[0,T]\to\re$ is dense in $C^{1}([-1,1])$. Here ``classical solution'' means that $u$, $u_{t}$, $u_{x}$ and $u_{xx}$ are continuous functions in $[-1,1]\times[0,T]$.

Keywords: Perona-Malik equation, classical solution, forward-backward equation, anisotropic diffusion


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