Confined elasticae and the buckling of cylindrical shells

For curves of prescribed length embedded into the unit disc in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds $2\pi$ and in the large length limit. In the small excess length case, we prove convergence to a fourth order obstacle type problem with integral constraint on the real line which we then solve. From the solution, we obtain the first order coefficient $\Theta\approx 37$ in the energy expansion $2\pi + \Theta \delta^{1/3} + o(\delta^{1/3})$ when a curve has length $2\pi + \delta$. We present an application of the scaling result to buckling in two-layer cylindrical shells where we can determine an explicit bifurcation point between compression and buckling in terms of universal constants and material parameters scaling with the thickness of the inner shell.