Relaxation and optimization for linear-growth convex integral functionals under PDE constraints

We give necessary and sufficient conditions for minimality of generalized minimizers for linear-growth functionals of the form $$\mathcal F[u] := \int_\Omega f(x,u(x)) \, \text{d}x, \qquad u:\Omega \subset \mathbb R^N\to \mathbb R^d,$$ where $u$ is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures $\mathcal M(\Omega;\mathbb R^d)$, and the introduction of a set-valued pairing in $\mathcal M(\Omega;\mathbb R^N) \times {\rm L}^\infty(\Omega;\mathbb R^N)$. By these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle-point conditions. In particular, our results can be applied to relaxation and minimization problems in BV, BD.