Continuum limit of $p$-Laplacian evolution problems on graphs:$L^q$ graphons and sparse graphs

TL;DR

This paper investigates the continuum limit of the $p$-Laplacian evolution on sparse graphs, extending previous results to more general kernels and graph sequences, and provides convergence rates for discretized models.

Contribution

It generalizes the continuum limit analysis to $L^q$-graphons and sparse graphs, offering new bounds and error estimates for the convergence of discrete to continuous solutions.

Findings

Derived bounds on trajectories for different kernels and initial data

Established error estimates for full discretization on sparse graphs

Provided convergence rates as graph size increases

Abstract

In this paper we study continuum limits of the discretized $p$-Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This extends the results of [24] to a far more general class of kernels, possibly singular, and graph sequences whose limit are the so-called $L^q$-graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of the $p$-Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.