Nonlinear Evolution Equation Associated with Hypergraph Laplacian

TL;DR

This paper introduces a nonlinear hypergraph p-Laplacian operator, establishes an inequality similar to PDEs, and analyzes the existence and long-term behavior of solutions to the associated heat equation on hypergraphs.

Contribution

It defines a hypergraph p-Laplacian, proves a Poincaré-Wirtinger type inequality, and studies the existence and asymptotic behavior of solutions to the hypergraph heat equation.

Findings

Established a Poincaré-Wirtinger type inequality for the hypergraph p-Laplacian.

Proved existence of solutions to the hypergraph heat equation.

Analyzed the large time behavior of solutions.

Abstract

Let $V$ be a finite set, $E \subset 2^{V} $ be a set of hyperedges, and $w : E \to (0, \infty)$ be an edge weight. On the (wighted) hypergraph $G = (V ,E ,w )$, we can define a multivalued nonlinear operator $L_{G,p}$ ($p \in [1 ,\infty )$) as the subdifferential of a convex function on $\mathbb{R} ^V $, which is called "hypergraph $p$-Laplacian." In this article, we first introduce an inequality for this operator $L_{G,p}$ which resembles the Poincar\'{e}-Wirtinger inequality in PDEs. Next we consider an ordinary differential equation on $\mathbb{R} ^V $ governed by $L_{G,p}$, which is referred as "heat" equation on the hypergraph and used to study the geometric structure of graph in recent researches. With the aid of the Poincar\'{e}-Wirtinger type inequality, we can discuss the existence and the large time behavior of solutions to the ODE by procedures similar to those for the standard heat equation in PDEs with the zero Neumann boundary condition.