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How can we show that a Dirichlet problem for Laplace's equation in a finite region has a unique solution.

Usually we can consider u2 - u1, a difference in values.

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Can you use the maximum principle? – littleO Nov 1 '12 at 9:21

If $u_1$ and $u_2$ solve Laplace's equation on the same domain with the same boundary conditions, then $u_2 - u_1$ solves Laplace's equation with $0$ boundary conditions. The maximum principle now implies that $u_2 - u_1 = 0$.

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Unfortunately, I'm not so sure if we can use the MaxPrinciple. Can you illustrate with another attempt? Thanks – Buddy Holly Nov 1 '12 at 10:12

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