  We consider the initial boundary value problem to equations of motion of an
inextensible hanging string of finite length under the action of the gravity.
We also consider the problem in the case without any external forces. In this
problem, the tension of the string is also an unknown quantity. It is
determined as a unique solution to a two-point boundary value problem, which is
derived from the inextensibility of the string together with the equation of
motion, and degenerates linearly at the free end. We derive a priori estimates
for solutions to the initial boundary value problem in weighted Sobolev spaces
under a natural stability condition. The necessity for the weights results from
the degeneracy of the tension. Uniqueness of solutions is also proved.
