Representation Formula for the Critical Points of the Tadjbakhsh--Odeh Functional and its Application

Abstract

In order to study the buckled states of an elastic ring under uniform pressure, Tadjbakhsh and Odeh [14] introduced an energy functional which is a linear combination of the total squared curvature (elastic energy) and the area enclosed by the ring. We prove that the minimizer of the functional is not a disk when the pressure is large, and its curvature can be expressed by Jacobian elliptic $\cn({}\cdot{})$ function. Moreover, the uniqueness of the minimizer is proven for certain range of the pressure.