A lower semicontinuity result for a free discontinuity functional with a boundary term

We study the lower semicontinuity in $GSBV^{p}(Om;R^{m})$ of a free discontinuity functional~$F(u)$ that can be written as the sum of a crack term, depending only on the jump set~$S_{u}$, and of a boundary term, depending on the trace of~$u$ on~$partialOm$. We give sufficient conditions on the integrands for the lower semicontinuity of~$F$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of~$F$ can be represented by the sum of two integrals on~$S_{u}$ and~$partialOm$, respectively.