Non-uniqueness of minimizers for strictly polyconvex functionals

In this note we solve a problem posed by J. M. Ball in [3] about the uniqueness of smooth equilibrium solutions to boundary value prob-lems for strictly polyconvex functionals, F(u) = Ω f(∇u(x)) dx and u|∂Ω = u0, where Ω is homeomorphic to a ball. We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of Minimal Surfaces.