*Submitted Paper*

**Inserted:** 7 feb 2017

**Last Updated:** 7 feb 2017

**Year:** 2016

**Abstract:**

We give necessary and sufficient conditions for minimality of generalized minimizers for linear-growth functionals of the form \[ \mathcal F[u] = \int_\Omega f(x,u(x)) \, \text{d}x, \quad u: \Omega \subset \mathbb R^d \to \mathbb R^N \] 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 $\mathcal M(\Omega;\mathbb R^N)$, and the introduction of a set-valued pairing in $\mathcal M(Ω;\mathbb R^N) \times {\rm L}^\infty(\Omega;\mathbb R^N)$. 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 ${\rm BV, BD}$.

**Keywords:**
relaxation, optimality condition, functional on measures, linear growth

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