Continuum limit of $p$-biharmonic equations on graphs

TL;DR

This work provides a rigorous continuum limit for the $p$-biharmonic equation on graphs, showing that as the data size grows on a random geometric graph the discrete $p$-biharmonic operator converges to a weighted $p$-biharmonic PDE with Neumann boundary. The authors recast the graph problem as a nonlocal equation via transportation maps and establish TL$^p$ convergence together with uniform $L^p$ and $L^\infty$ a priori bounds. Key contributions include consistency results for nonlocal and graph Laplacians, robust a priori estimates for nonlocal Poisson equations, and a complete discrete-to-continuum convergence to the weighted continuum model $-\Delta_\rho(|\Delta_\rho u|^{p-2}\Delta_\rho u)+\lambda(f-u)=0$ with $\frac{\partial u}{\partial n}=0$ on $\partial\Omega$. This provides theoretical justification for higher-order regularization on graphs and clarifies the link to hypergraph $p$-Laplacians, with implications for point-cloud processing and data analysis.

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.

Theorems & Definitions (23)

• Definition 1.1
• Definition 1.2
• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• Remark 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Lemma 3.1