Existence and uniqueness of equilibrium states of a rotating elastic rod (Q1326620)

An elastic rod, supported at one end, is being rotated around an axis with constant angular velocity. The author first sets up the mathematical model for equilibrium states, which is a certain boundary value problem, where one of the parameters is the angle \(\alpha\) the rod makes with the axis of rotation at the clamped end, and another crucial parameter \(\lambda\) is determined by the rotational velocity. Using a version of Schauder's fixed point theorem, he proves existence of a solution for each \(0<\alpha\leq {\pi\over 2}\) and uniqueness of the solution for \(\lambda\) less than the smallest eigenvalue \(\lambda_ 0\) of a related linearized problem. He then goes on to compute by standard perturbation techniques an approximate solution for small \(\alpha\), one for the case \(\lambda< \lambda_ 0\) and one for the case \(\lambda> \lambda_ 0\) assuming then that also \(\lambda- \lambda_ 0\) is small.