Advances in Differential Equations

Two cases of squares evolving by anisotropic diffusion

Piotr B. Mucha, Monika Muszkieta, and Piotr Rybka

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Abstract

We are interested in an anisotropic singular diffusion equation in the plane and in its regularization. We establish existence, uniqueness and the basic regularity of solutions to both equations. We construct explicit solutions showing the creation of facets, i.e., flat parts of graphs of solutions. Inspired by the formula for solutions, we rigorously prove that both equations create ruled surfaces out of convex initial data. We also notice that at each positive time, the solutions do not have strict (local) extrema either. We present results of numerical experiments suggesting that the two flows do not seem to differ much. Possible applications to the image reconstruction is pointed out, too.

Article information

Source
Adv. Differential Equations Volume 20, Number 7/8 (2015), 773-800.

Dates
First available in Project Euclid: 8 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1431115716

Mathematical Reviews number (MathSciNet)
MR3344618

Zentralblatt MATH identifier
1330.35223

Subjects
Primary: 35K67: Singular parabolic equations 35K65: Degenerate parabolic equations

Citation

Mucha, Piotr B.; Muszkieta, Monika; Rybka, Piotr. Two cases of squares evolving by anisotropic diffusion. Adv. Differential Equations 20 (2015), no. 7/8, 773--800. https://projecteuclid.org/euclid.ade/1431115716.


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