A uniqueness theorem for the dual problem associated to a variational problem with linear growth

Uniqueness is proved for solutions of the dual problem which is associated to the minimum problem $\int_\Omega f(\nabla u)dx\to\min$ among mappings $u\colon\mathbb R^n\supset\Omega\to\mathbb R^N$ with prescribed Dirichlet boundary data and for smooth strictly convex integrands $f$ of linear growth. No further assumptions on $f$ or its conjugate function $f^*$ are imposed, in particular $f^*$ is not assumed to be strictly convex. One special solution of the dual problem is seen to be a mapping into the image of $\nabla f$ which immediately implies uniqueness.