Asymptotic form of solutions of the Tadjbakhsh-Odeh variational problem

Abstract

We consider a variational problem posed by Tadjbakhsh and Odeh to describe the shape of an elastic ring in the plane under uniform pressure. Regarding the ring as a smooth closed curve, the Euler-Lagrange equation reduces to a second order ordinary differential equation for the curvature with the periodic boundary condition. The asymptotic form of solutions is presented as the external pressure tends to infinity. This is done by studying a singular perturbation problem for the Euler-Lagrange equation.