There are two different approaches to the Dirichlet minimization problem for variational integrals with linear growth. On the one hand, one commonly considers a generalized formulation in the space of functions of bounded variation. On the other hand, there is a closely related maximization problem in the space of divergence-free bounded vector fields, namely the dual problem in the sense of convex analysis.
In this paper, we extend previous results on the duality correspondence between the generalized and the dual problem to a full characterization of their extremals via pointwise extremality relations. Furthermore, we discuss related uniqueness issues for both kinds of solutions and their relevance in the regularity theory of generalized minimizers.
Our approach is sufficiently general to cover arbitrary dimensions, non-smooth integrands, and unbounded, irregular domains.