Parabolic theory of the discrete -Laplace operator

Mugnolo, Delio

doi:10.1016/j.na.2013.04.002

Cited by 32 publications

(24 citation statements)

References 49 publications

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“…The subject of diffusion in graphs is popular also owing to its applicative interest. We refer the reader to [23], [17] and to the references therein for more on this point.…”

Section: Definition 15mentioning

confidence: 99%

“…To the best of our knowledge such results are new in the framework of discrete nonlinear diffusion equations on graphs. We apply an alternative approach, more local than the one in [23], [19] where the global arguments of semigroup theory are extended to graphs, actually in a more general setting which is out of the scope of this paper. We comment below in the Introduction on the inherent difficulty and even partial unfeasibility of a local approach in graphs.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data

Andreucci

Tedeev

2019

Potential Anal

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We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex.Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is known, however, not all of this approach can have a direct counterpart in graphs. A basic tool here is a result connecting the supremum of the solution at a given positive time with the measure of its level sets at previous times.We also consider the case of slowly decaying initial data, where the total mass is infinite.

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“…The subject of diffusion in graphs is popular also owing to its applicative interest. We refer the reader to [23], [17] and to the references therein for more on this point.…”

Section: Definition 15mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data

Andreucci

Tedeev

2019

Potential Anal

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“…We refer to [BH09,Mug13] and the references therein for more information, including motivations to study the p-Laplacian. It follows from simple compactness arguments that there is a vector f (p) 1 achieving equality in (2.2).…”

Section: Combinatorial Graphs: Estimates Based On the Edge Connectivitymentioning

confidence: 99%

Edge connectivity and the spectral gap of combinatorial and quantum graphs

Berkolaiko

Kennedy

Kurasov

et al. 2017

J. Phys. A: Math. Theor.

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We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds is the well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Lévy. All proofs are general enough to yield corresponding estimates for the p-Laplacian and allow us to identify the minimizers.Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are "asymptotically correct", i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.

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“…[BHL + 15, BHY15, BKW15, DLM + 03, Fol14, Gol14, HJL15, HL17, HK14, KL12, KPP, Mün13, Woj, Web10], there is an upcoming interest in non-linear problems. On the one hand this concerns non-linear operators such as the p-Laplacian, [HM15, KM16,Mug13], and on the other hand non-linear equations such as the Kazdan-Warner equation [GLY16,Ge17].…”

Section: Introductionmentioning

confidence: 99%