Parabolic theory of the discrete \(p\)-Laplace operator
type: Item
instance of: scholarly article
title: Parabolic theory of the discrete \(p\)-Laplace operator
zbMATH Open document ID: 1285.39004
author: Delio Mugnolo
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2014-01-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.1996
reviewed by: Zhenlai Han
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 35K92
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 6246145
zbMATH Keywords: discrete \(p\)-Laplace operator
zbMATH Keywords: porous medium equation
zbMATH Keywords: discrete symmetries
zbMATH Keywords: nonlinear \(C_0\)-semigroups generated by subdifferentials
zbMATH Keywords: operators on discrete graphs
zbMATH Keywords: discrete \(p\)-heat equation
zbMATH Keywords: discrete nonlinear diffusion-type problem
zbMATH Keywords: fractional discrete differential equation
MaRDI profile type: Publication
OpenAlex ID: W1991844959
arXiv ID: 1207.1996
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Parabolic theory of the discrete \(p\)-Laplace operator (cites work)
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Parabolic theory of the discrete \(p\)-Laplace operator (cites work)
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Parabolic theory of the discrete \(p\)-Laplace operator (cites work)
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Parabolic theory of the discrete \(p\)-Laplace operator (DOI)
DOI: 10.1016/J.NA.2013.04.002

Parabolic theory of the discrete \(p\)-Laplace operator (review text)
review text: The structure of this paper is clear, and the process of proofs is strict. The words of the paper are appropriate and normative. This paper is well written and contains some interesting and correct results.NEWLINENEWLINEThe author studies the elementary features of the discrete \(p\)-heat equation NEWLINE\[NEWLINE\frac{d}{dt}\sum\limits_{v\in V}\varphi(t,v)=-\sum\limits_{v\in V} \mathcal{I}(|\mathcal{I}^{T}\varphi|^{p-2}{I}^{T}\varphi)(t,v) +\sum\limits_{v\in V}f(t,v).NEWLINE\]NEWLINENEWLINENEWLINEBy introducing the relevant functional setting, the author formulates the main well-posedness result for the discrete \(p\)-heat equation. He shows that the nonlinear \(C_0\)-semigroup generated by the discrete \(p\)-Laplacian consists of an irreducible, sub-Markovian operator. Also, by energy methods, he discusses how these \(C_0\)-semigroups interact with the symmetries of the graph, based on the notion of an almost equitable partition of a graph. At last, he overviews a few popular generalizations of the discrete Laplacian, proposes some more and suggests how the variational structure of the \(p\)-heat equation can be generalized to discuss a broad class of discrete nonlinear diffusion-type problems, including a discretized porous medium equation.NEWLINENEWLINEThe author obtains that the results follow from the standard theory of ordinary differential equations whenever \(G\) is a finite graph. And the results are more general. The results are new in the case of fractional discrete differential equations, so one can apply these results rather than those presented in this paper.