Partial regularity for $\mathcal{A}$-quasiconvex variational problems of linear growth

We prove that minimizers of variational integrals $$\mathcal{E}(v)={\int }_{\mathrm{\Omega }}f(v)\text{for }v\in \mathcal{M}(\mathrm{\Omega })\text{ such that }\mathcal{A}v=0,$$ are partially continuous provided that the integrands $f$ are strongly $\mathcal{A}$-quasiconvex in a suitable sense. We consider linear growth problems, linear PDE operators $\mathcal{A}$ of constant rank, and variations of the form $v+\phi$ with $\mathcal{A}$-free $\phi \in {\mathrm{C}}_{\mathrm{c}}^{\mathrm{\infty }}(\mathrm{\Omega })$. Our analysis also covers the ``potentials case’’ $$\mathcal{F}(u)={\int }_{\mathrm{\Omega }}f(\mathcal{B}u)\text{for }u\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })\text{ such that }\mathcal{B}u\in \mathcal{M}(\mathrm{\Omega }),$$ where $\mathcal{B}$ is a different linear pde operator of constant rank. Both our main results extend to $x$-dependent integrands.