Non-uniqueness of minimizers for strictly polyconvex functionals

Abstract

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,
$$\mathcal {F}(u)=\int_\Omega f(\nabla u(x)) {\rm d}x\quad{\rm and}\quad u\vert_{\partial\Omega}=u_0,$$
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.