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)?
[Math] Laplace equation with mixed boundary conditions
ap.analysis-of-pdesca.classical-analysis-and-odes
Best Answer
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.