Boundary effects and weak${}^{\ast }$ lower semicontinuity for signed integral functionals on $\mathrm{B}\mathrm{V}$

We characterize lower semicontinuity of integral functionals with respect to weak${}^{\ast }$ convergence in $\mathrm{B}\mathrm{V}$, 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).