Laplace's equation on the rectangular region 0<x< a, 0<y<b, subject to the Dirichlet boundary conditions
is well posed. For the case of these example boundary conditions, one can show that the unique solution to this BVP is . If any one of the four boundary conditions is deleted, then the problem becomes ill-posed, because is would then admit multiple solutions. If a second, independent Dirichlet condition were added on any part of the boundary, the problem would again be ill-posed, in this case due to lack of existence of a solution. More generally, if two, independent boundary conditions are imposed on any part of the boundary of the region, then the problem will fail to have a solution.
Figure 1:Laplace's equation on the rectangular region 0 < x < a, 0 < y < b subject to the Dirichlet boundary conditions described in Example 8.