Lower semicontinuity and relaxation of signed functionals with linear growth in the context of {\mathcal A}-quasiconvexity

A lower semicontinuity and relaxation result with respect to weak-* convergence of measures is derived for functionals of the form $$\mu \in \mathcal{M}(\Omega; \mathbb{R}^d) \to \int \limits_\Omega f(\mu^a(x))\,{\rm {d}}x +\int \limits_\Omega f^\infty \left( \frac{{\rm{d}}\mu^s}{d|\mu^s|}(x)\right) \, d| \mu^s|(x),$$ where admissible sequences {μ _n } are such that ${\{{\mathcal{A}}\mu_{n}\}}$ converges to zero strongly in ${W^{-1 q}_{\rm loc}(\Omega)}$ and ${\mathcal {A}}$ is a partial differential operator with constant rank. The integrand f has linear growth and L ^∞-bounds from below are not assumed.