Continuum limit of $p$-biharmonic equations on graphs
This work addresses a theoretical problem in point cloud processing and hypergraph analysis, providing incremental insights by extending graph $p$-Laplacian methods to the $p$-biharmonic case with asymptotic convergence proofs.
The paper tackles the asymptotic behavior of solutions to the $p$-biharmonic equation on random geometric graphs as the number of data points increases, showing that the continuum limit converges to a weighted $p$-biharmonic equation with homogeneous Neumann boundary conditions, with results including uniform $L^p$ estimates for solutions and gradients and $L^\infty$ estimates as a byproduct.
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.