Continuum limit of $p$-biharmonic equations on graphs

TL;DR

This paper analyzes the asymptotic behavior of solutions to the $p$-biharmonic equation on graphs, showing that as data points increase, solutions converge to a weighted $p$-biharmonic PDE with Neumann boundary conditions.

Contribution

It establishes the continuum limit of the $p$-biharmonic equation on graphs, extending the understanding of hypergraph-based PDEs in the large data limit.

Findings

Solutions converge to a weighted $p$-biharmonic equation as data points grow.

Uniform $L^p$ and $L^ Infty$ estimates for solutions are derived.

The continuum limit involves homogeneous Neumann boundary conditions.

Abstract

This paper studies the $p$-biharmonic equation on graphs, which arises in point cloud processing and can be interpreted as a natural extension of the graph $p$-Laplacian from the perspective of hypergraph. The asymptotic behavior of the solution is investigated when the random geometric graph is considered and the number of data points goes to infinity. We show that the continuum limit is an appropriately weighted $p$-biharmonic equation with homogeneous Neumann boundary conditions. The result relies on the uniform $L^p$ estimates for solutions and gradients of nonlocal and graph Poisson equations. The $L^\infty$ estimates of solutions are also obtained as a byproduct.