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.
