Non-Uniqueness of Minimizers for Strictly Polyconvex Functionals

In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557–611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals, 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.