A priori estimates for solutions to equations of motion of an inextensible hanging string
Tatsuo Iguchi, Masahiro Takayama
TL;DR
The article analyzes the motion of an inextensible hanging string with unknown tension under gravity, formulating a hyperbolic-elliptic coupled system and a two-point boundary value problem for the tension. It develops a robust framework of weighted Sobolev spaces and Green's-function techniques to obtain a priori estimates, energy bounds for linearized dynamics, and uniqueness results under a stability condition that prevents degeneracy at the free end. A key contribution is the equivalence between the original and transformed formulations, enabling precise control of the tension and the string's geometry through detailed estimates. The results lay groundwork for local well-posedness in a rigorous functional-analytic setting, with the a priori estimates serving as a stepping stone toward an existence theory and providing quantitative insight into the stabilizing effects of the tension and gravity in inextensible string dynamics.
Abstract
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.