The Discrete and Classical Dirichlet Problem |
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Authors: | Nicolas Th. Varopoulos |
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Affiliation: | 1. 61, rue Raymond-Losserand, Paris, 75014, France
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Abstract: | Let W ì mathbbRd{Omega subset mathbb{R}^d} be some bounded domain with reasonable boundary and let f be a continuous function on the complement Ω c . We can construct an unique continuous function u that is harmonique on Ω and u = f on Ω c . Similarly, u d is the unique function on the lattice points such that for each lattice point of Ω satisfies the “average” property with respect to its nearest neighbours and u d = f on Ω c . In this paper when ∂Ω is Lipschitz I give a “best possible” estimate of ||u − u d ||∞. |
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