Bc. Li et S. Syngellakis, ON IMPROPERLY POSED CAUCHY-PROBLEMS AND THEIR APPROXIMATE SOLUTION, IMA journal of applied mathematics, 55(1), 1995, pp. 85-95
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.