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We prove an $\varepsilon$-regularity theorem for $BV^\mathcal{B}$ minimizersof strongly $\mathcal{B}$-quasiconvex functionals with linear growth, where$\mathcal{B}$ is an elliptic operator of the first order. This generalises tothe $BV^\mathcal{B}$ setting the analogous result for $BV$ functions by F.Gmeineder and J. Kristensen [Arch. Rational Mech. Anal. 232 (2019)]. Theresults of this work cannot be directly derived from the $\mathcal{B} =\nabla$case essentially because of Ornstein's "non-inequality". This adaptationrequires an abstract local Poincar\'e inequality and a fine Fubini-typeproperty to avoid the use of trace theorems, which in general fail when$\mathcal{B}$ is elliptic.

Partial regularity for $BV^\mathcal{B}$ minimizers