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Math Help - prove these for this poisson equation

  1. #1
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    prove these for this poisson equation

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  2. #2
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    Because by the maximum principle the function u assumes it maximum and minimum on the boundary, S. Since it constatly assumes zero it must mean the harmonic function is identitically zero on V. As a corollary, we can show that the Dirichlet problem has at most one solution. Because if \xi and \mu are harmonic on an open set and solve the same Dirichlet problem then \xi - \mu is also harmonic, and furthermore \xi - \mu is identitcally zero on the boundary. Therefore, \xi = \mu on all points in open set.
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