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Dirichlet problem on locally finite graphs

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Title: Dirichlet problem on locally finite graphs
Authors: Javaheri, Mohammad1 javaheri@uoregon.edu
Source: Discrete Applied Mathematics. Nov2007, Vol. 155 Issue 18, p2496-2506. 11p.
Subjects: Dirichlet problem, Boundary value problems, Parabolic differential equations, Mathematics
Abstract: 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 and its values on B are given. Each infinite connected component of 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. [Copyright &y& Elsevier]
Copyright of Discrete Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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PubType: Academic Journal
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  Data: 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 and its values on B are given. Each infinite connected component of 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. [Copyright &y& Elsevier]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Discrete Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.dam.2007.06.018
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      – Code: eng
        Text: English
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      – SubjectFull: Dirichlet problem
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      – SubjectFull: Boundary value problems
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      – SubjectFull: Parabolic differential equations
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              Text: Nov2007
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