  We characterize lower semicontinuity of integral functionals with respect to
weak$^*$ convergence in $\mathrm{BV}$, including integrands whose negative part
has linear growth. In addition, we allow for sequences without a fixed trace at
the boundary. In this case, both the integrand and the shape of the boundary
play a key role. This is made precise in our newly found condition --
quasi-sublinear growth from below at points of the boundary -- which
compensates for possible concentration effects generated by the sequence. Our
work extends some recent results by J. Kristensen and F. Rindler (Arch. Rat.
Mech. Anal. 197 (2010), 539--598 and Calc. Var. 37 (2010), 29--62).
