The networks of this – primarily (but not exclusively) expository – compendium are strongly connected, finite directed graphs $X$, where each oriented edge $(x,y)$ is equipped with a positive weight (conductance) $a(x,y)$. We are not assuming symmetry of this function, and in general we do not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give rise to a difference operator, the normalised version of which we consider as our Laplace operator. It is associated with a Markov chain with state space $X$. A non-empty subset of $X$ is designated as the boundary. We provide a systematic exposition of the different types of Laplace equations, starting with the Poisson equation, Dirichlet problem and Neumann problem. For the latter, we discuss the definition of outer normal derivatives. We then pass to Laplace equations involving potentials, thereby also addressing the Robin boundary problem. Next, we study the bi-Laplacian and associated equations: the iterated Poisson equation, the bi-Laplace Neumann and Dirichlet problems, and the ”plate equation”. It turns out that the bi-Laplace Dirichlet to Neumann map is of non-trivial interest. The exposition concludes with two detailed examples.

这个——主要（但不完全）说明——纲要的网络是强连接的有限有向图 $X$，其中每个定向边 $(X,是)$ 配备正重（电导） $\mathrm{一种}(X,是)$. 我们不假设这个函数是对称的，通常我们不需要它与$(X,是)$， 还 $(是,X)$是一条边。权重产生差分算子，我们将其归一化版本视为我们的拉普拉斯算子。它与具有状态空间的马尔可夫链相关联$X$. 的非空子集$X$被指定为边界。我们从泊松方程、狄利克雷问题和诺依曼问题开始，系统地阐述了不同类型的拉普拉斯方程。对于后者，我们讨论外法线导数的定义。然后我们传递到涉及势能的拉普拉斯方程，从而也解决了罗宾边界问题。接下来，我们研究双拉普拉斯方程和相关方程：迭代泊松方程、双拉普拉斯诺依曼和狄利克雷问题，以及“板方程”。事实证明，双拉普拉斯狄利克雷到诺依曼映射具有重要意义。说明以两个详细的例子结束。