On improperly posed Caucfay problems and their approximate solution

Abstract

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 for arbitrary boundary conditions. Firstly, it is shown that it is possible to construct an analytic function which assumes some prescribed value on part of the boundary of a simply-connected domain. This problem is then shown to be equivalent to the Cauchy problem under consideration, 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 functions over half the unit circle boundary. The uniqueness of the exact solution to the problem is also established. These results are actually true for any simply-connected domain which can be conformally mapped onto the unit circle so that the part of its boundary with prescribed potential and flux corresponds to one-half of the unit circle boundary. Finally, the feasibility of a boundary element formulation for a generic type of ill-posed boundary value problems is briefly discussed.