Calculus of Variations and Geometric Measure Theory
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M. Gobbino

Entire solutions of the one-dimensional Perona-Malik equation

created on 07 May 2003
modified by gobbino on 13 Nov 2006

[BibTeX]

Accepted Paper

Inserted: 7 may 2003
Last Updated: 13 nov 2006

Journal: Comm. Partial Differential Equations
Year: 2003

Abstract:

We prove that every function $u:\re^{2}\to\re$ of class $C^{1}$, satisfying the Perona-Malik equation $$u{t}=\left(\frac{u{x}}{1+u{x}{2}}\right){x}$$ for every $(x,t)\in\re^{2}$, is a stationary affine solution, \ie\ of the form $u(x,t)=ax+b$, where $a$ and $b$ are suitable real constants.

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


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