If one of these conditions is not satisfied, the
PDE problem is said to be ill-posed. In practice, the
question of whether a PDE problem is well posed can be
difficult to settle. Roughly speaking the following
- The auxiliary conditions imposed must
not be too many or a solution will not exist.
- The auxiliary conditions imposed must not be too few or the
solution will not be unique.
- The kind of auxiliary conditions must be correctly
matched to the type of the PDE or the solution will not
depend continuously on the data.