The annulus problem, using a von Karman membrane model

These notes supplement Robert Kohn’s Lecture 3 at the 2014 PCMI Graduate Summer School. For background and heuristic arguments, see the pdf of that lecture. For physicsoriented discussions of this topic see [4] and [3]. For upper and lower energy bounds similar to those presented here but using nonlinear elasticity and a rather general membrane model see [1]. We want to study deformations of a thin elastic sheet of annular shape, which is loaded radially at the inner and outer boundary by uniform forces Tin and Tout, respectively. To keep things simple we consider very simple form of the energy – Föppl-von Kárman energy1 with zero Poisson ratio. The deformation is then describe by the in-plane displacement w = (w1, w2) and out-of-plane displacement u3, and should minimize