Non-Uniqueness of Minimizers for Strict Polyconvex Functionals

In this talk we consider a problem posed by J.M. Ball about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals, $$ \F(u)=\int_\Omega f(\nabla u(x))\,\dd x\quad\m{and}\quad u\vert_{\de\Omega}=u_0\,, $$ where $\Omega$ is homeomorphic to a ball.

We give several examples of non-uniqueness, the main of which is such a boundary value problem with at least two analytic different minimizers. All this examples are suggested by the theory of Minimal Surfaces.