The Dirichlet problem for certain discrete structures
نویسندگان
چکیده
The paper reviews some of the early ideas behind the development of the theory of discrete harmonic functions. The connection to random walks is pointed out. Then a current and more general construction using weight functions is described. Discrete analogues of the Laplace operator are defined for Zn and a discrete planar hexagonal structure H. Discrete analogues of the Dirichlet problem and Poisson’s equation are formulated and existence and uniqueness of bounded solutions is proved for the finite case and also for a certain case of infinite sets in Zn. Explicit discrete analogues of the Green function are presented for Z3 and Z[i] equipped with a certain discrete calculus, which is also briefly presented.
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