Elastic Graphs with clamped boundary

In the Bernoulli model of an elastic rod described by a curve, the elastic energy is given by integral of the curvature squared with respect to arc-length. We study the minimization of this energy on curves given by the graph of a sufficiently smooth function satisfying Dirichlet boundary conditions. Using invariances of the problem, we are able to integrate the Euler-Lagrange equation once in two different ways. To illustrate the idea and the power of the method, we give also another application to unstable Willmore surfaces of revolution.