Dirichlet problem on locally finite graphs (Q2462387)

The existence and uniqueness of solutions to the vertex-weighted Dirichlet problem on locally finite graphs is studied. Let B be a subset of the vertices of a graph G. The Dirichlet problem is to find a function whose discrete Laplacian on the complement of B in G and its values on B are given. Each infinite connected component of the complement of B in G is called an end of G relative to B. If there are no ends, then there is a unique solution to the Dirichlet problem. Such a solution can be obtained as a limit of an averaging process or as a minimizer of a certain functional or as a limit-solution of the heat equation on the graph. On the other hand, it is shown that if G is a locally finite graph with n ends, then the set of solutions of any Dirichlet problem, if non-empty, is at least n-dimensional.