A priori estimates of solutions to the motion of an inextensible hanging string

Abstract

We consider the initial boundary value problem to the motion of an inextensible hanging string of finite length under the action of the gravity. 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 of 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 solution is also proved.Comment: 40 pages, 3 figure