Dirichlet problem on locally finite graphs |
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Authors: | Mohammad Javaheri |
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Institution: | Department of Mathematics, University of Oregon, Eugene, OR 97403, USA |
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Abstract: | In this paper, we study the existence and uniqueness of solutions to the vertex-weighted Dirichlet problem on locally finite graphs. Let B be a subset of the vertices of a graph G. The Dirichlet problem is to find a function whose discrete Laplacian on G?B and its values on B are given. Each infinite connected component of G?B 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, we show that if G is a locally finite graph with l ends, then the set of solutions of any Dirichlet problem, if non-empty, is at least l-dimensional. |
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Keywords: | Vertex-weighted Laplacian Combinatorial Dirichlet problem Approximate solutions Discrete heat equation |
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