  We consider an overdetermined problem for Laplace equation on a disk with
partial boundary data where additional pointwise data inside the disk have to
be taken into account. After reformulation, this ill-posed problem reduces to a
bounded extremal problem of best norm-constrained approximation of partial L2
boundary data by traces of holomorphic functions which satisfy given pointwise
interpolation conditions. The problem of best norm-constrained approximation of
a given L2 function on a subset of the circle by the trace of a H2 function has
been considered in [Baratchart \& Leblond, 1998]. In the present work, we
extend such a formulation to the case where the additional interpolation
conditions are imposed. We also obtain some new results that can be applied to
the original problem: we carry out stability analysis and propose a novel
method of evaluation of the approximation and blow-up rates of the solution in
terms of a Lagrange parameter leading to a highly-efficient computational
algorithm for solving the problem.
