Manifold-constrained free discontinuity problems and Sobolev approximation

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano \cite{CFI-ARMA, CFI-AIHP} for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.