Fractional Laplace operator on finite graphs

2024/03/29 by Mengjie Zhang, Zhang, Mengjie, Yong Lin +3 · 3 citations
Mathematics · #Differential Equations and Boundary Problems #advanced mathematical theories #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2403.19987

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 (-Δ)^s for an arbitrary finite graph and any positive real number s. It is shown that (-Δ)^s can be explicitly represented by eigenvalues and eigenfunctions of the Laplace operator -Δ. Moreover, we study its important properties, such as (-Δ)^s converges to -Δ as s tends to 1; while (-Δ)^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.

Cited by

• Topological degree for negative fractional Kazdan--Warner equation on finite graphs
• On finite-energy solutions of Kazan-Warner equations on the lattice graph
• The Logarithmic Laplacian on General Graphs

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