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July/August 2015 Two cases of squares evolving by anisotropic diffusion
Piotr B. Mucha, Monika Muszkieta, Piotr Rybka
Adv. Differential Equations 20(7/8): 773-800 (July/August 2015). DOI: 10.57262/ade/1431115716

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.

Citation

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Piotr B. Mucha. Monika Muszkieta. Piotr Rybka. "Two cases of squares evolving by anisotropic diffusion." Adv. Differential Equations 20 (7/8) 773 - 800, July/August 2015. https://doi.org/10.57262/ade/1431115716

Information

Published: July/August 2015
First available in Project Euclid: 8 May 2015

zbMATH: 1330.35223
MathSciNet: MR3344618
Digital Object Identifier: 10.57262/ade/1431115716

Subjects:
Primary: 35K65 , 35K67

Rights: Copyright © 2015 Khayyam Publishing, Inc.

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Vol.20 • No. 7/8 • July/August 2015
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