ON IMPROPERLY POSED CAUCHY-PROBLEMS AND THEIR APPROXIMATE SOLUTION

It is well known that, in general, the Cauchy problem for the Laplace equation does not allow a solution and therefore is ill-posed in both the Hadamard and the Tikhonov senses. The present work focuses on the question whether the problem has any meaningful approximate solution f or arbitrary boundary conditions. Firstly, it is shown that it is poss ible to construct an analytic function which assumes some prescribed v alue on part of the boundary of a simply-connected domain. This proble m is then shown to be equivalent to the Cauchy problem under considera tion, the solution to which can thus be invariably approximated to any degree of accuracy on the unit circle centred at the origin when both the potential and the flux are specified as square-integrable functio ns over half the unit circle boundary. The uniqueness of the exact sol ution to the problem is also established. These results are actually t rue for any simply-connected domain which can be conformally mapped on to the unit circle so that the part of its boundary with prescribed po tential and flux corresponds to one-half of the unit circle boundary. Finally, the feasibility of a boundary element formulation for a gener ic type of ill-posed boundary value problems is briefly discussed.