Continuum Limit of p-Biharmonic Equations on Graphs

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 hypergraphs. The asymptotic behavior of the solution is investigated for the random geometric graph when 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 𝐿𝑝 estimates for solutions and gradients of nonlocal and graph Poisson equations. The 𝐿∞ estimates of solutions are also obtained as a byproduct.