This paper is devoted to the study of integral functional denned on the space SBV(Ω ℝk) of vector-valued special functions with bounded variation on the open set Ω⊂ℝn, of the form
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0308210500028638/resource/name/S0308210500028638_eqnU1.gif?pub-status=live)
We suppose only that f is finite at one point, and that g is positively 1-homogeneous and locally bounded on the sets ℝk⊗vm, where {v1,…, vR} ⊂ Sn−1 is a basis of ℝn. We prove that the lower semicontinuous envelope of F in the L1(Ω;ℝk)-topology is finite and with linear growth on the whole BV(Ω;ℝk), and that it admits the integral representation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0308210500028638/resource/name/S0308210500028638_eqnU2.gif?pub-status=live)
A formula for ϕ is given, which takes into account the interaction between the bulk energy density f and the surface energy density g.