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.

翻译：本文研究图上的$p$-双调和方程，该方程源于点云处理，从超图视角可视为图上$p$-拉普拉斯算子的自然推广。在随机几何图框架下，当数据点数量趋于无穷时，我们探究了解的渐近行为。结果表明，其连续极限是带有齐次Neumann边界条件的适当加权$p$-双调和方程。这一结论依赖于非局部方程及图泊松方程解与梯度的$L^p$一致估计，同时作为副产品还获得了解的$L^\infty$估计。