The first contribution of this thesis is a new proof of sequential weak* lower semicontinuity in $\mathrm{BV}(\Omega;\mathbb{R}^m)$ for integral functionals of the form \begin{align*} \mathcal{F}(u) &:= \int_\Omega f(x, \nabla u) \;\mathrm{d} x + \int_\Omega f^\infty \biggl(x, \frac{\mathrm{d} D^s u}{\mathrm{d} |D^s u|} \biggr) \;\mathrm{d} |D^s u| \\ &\qquad+ \int_{\partial\Omega} f^\infty \bigl( x, u|_{\partial \Omega} \otimes n_{\Omega} \bigr) \;\mathrm{d} \mathcal{H}^{d-1...