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Fractional Laplace operator on finite graphs

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arxiv 2403.19987 v3 pith:B3EJMQW3 submitted 2024-03-29 math.AP

Mengjie Zhang , Yong Lin , Yunyan Yang This is my paper · ORCID
classification math.AP
keywords deltafractionallaplaceoperatorconvergesfinitetendsarbitrary
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abstract

Nowadays a great attention has been focused on the discrete fractional Laplace operator as the natural counterpart of the continuous one. In this paper, we discretize the fractional Laplace operator $(-\Delta)^{s}$ for an arbitrary finite graph and any positive real number $s$. It is shown that $(-\Delta)^{s}$ can be explicitly represented by eigenvalues and eigenfunctions of the Laplace operator $-\Delta$. Moreover, we study its important properties, such as $(-\Delta)^{s}$ converges to $-\Delta$ as $s$ tends to $1$; while $(-\Delta)^{s}$ converges to the identity map as $s$ tends to $0$ on a specific function space. For related problems involving the fractional Laplace operator, we consider the fractional Kazdan-Warner equation and obtain several existence results via variational principles and the method of upper and lower solutions.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On finite-energy solutions of Kazan-Warner equations on the lattice graph

    math.AP 2025-09 unverdicted novelty 8.0 of 10

    Finite-energy solutions exist for Kazdan-Warner type equations on the square lattice for small κ, partially resolving an open problem for the lattice Liouville equation.

  2. Extension theorems for logarithmic Schr\"odinger and discrete Laplacian operators

    math.CA 2026-04 unverdicted novelty 6.0 of 10

    Logarithmic operators log L_V and log(−Δ_d) are realized as boundary values of solutions to suitable extension problems, in a more involved way than the fractional Laplacian case.

  3. The Logarithmic Laplacian on General Graphs

    math.AP 2025-07 conditional novelty 6.0 of 10

    The logarithmic Laplacian on weighted graphs is defined via a Bochner integral, given a kernel formula under stochastic completeness, and shown on Z^d to have sharp kernel bounds and exact diffusion asymptotics.

  4. Existence theory for elliptic equations of general exponential nonlinearity on finite graphs

    math.AP 2025-05 reject novelty 6.0 of 10

    A sign error in the graph-reduction step invalidates the claimed Brouwer degree formula and the existence theory built on it.

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