论文标题
$ p $ -laplacian进化问题的连续限制图：$ l^q $图形和稀疏图
Continuum limit of $p$-Laplacian evolution problems on graphs:$L^q$ graphons and sparse graphs
论文作者
论文摘要
在本文中，我们研究了具有均匀的Neumann边界条件的稀疏图上的离散的$ P $ -Laplacian进化问题的连续限制。这将[24]的结果扩展到更一般的核类（可能是单数）和图形序列，其极限是所谓的$ l^q $ graphons。更确切地说，我们在两个不同的演化系统（即使用不同的内核，第二个成员和初始数据）定义的两个连续时轨迹之间的距离上得出了一个绑定。同样，我们在其中一个轨迹是离散时间而另一个是连续的情况下提供的。反过来，这些结果使我们确定了稀疏随机图上$ p $ laplacian问题的全部离散化的错误估计。特别是，随着顶点数量的增加，我们为离散模型提供解决方案的收敛速率。
In this paper we study continuum limits of the discretized $p$-Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This extends the results of [24] to a far more general class of kernels, possibly singular, and graph sequences whose limit are the so-called $L^q$-graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of the $p$-Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.
