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Does the Laplace equation on a rectangle with Dirichlet boundary conditions at two opposing sides and Neumann boundary conditions at the other two, always have a solution? If it does, is it unique? Is the same true for the discrete Laplace equation (with the standard five-point laplacian)?

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The solution to the Dirichlet problem is unique by the maximum principle. For the same reason, the solution to the Neumann problem is unique up to a constant.

At worst your problem would fail uniqueness by a constant, but those Dirichlet conditions you've got will prevent this.

This carries over to the discrete case. But note that the discrete case is an easier problem since you can just write out the matrix you need to invert and check if it's invertible.

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I am not familiar with the discrete case. As for continuous case, I think you can use the method of images to solve this problem. It should always have a solution and it should be unique, due to the fact that the underlying Green's function is unique. In particular, to construct the Green's function, after reflecting by a Neumann or Dirichlet boundary, you put a same (resp. opposite) source (Delta function). By this way, we have an infinitely many sources, then your Green's function is the sum of the effects by all the sources. Hope it can help. :-)

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I believe you can just use separation of variables.

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