Approximation of SBV functions with possibly infinite jump set

We prove an approximation result for functions u∈SBV(Ω;Rm) such that ∇u is p-integrable, 1≤p<∞, and g0(|[u]|) is integrable over the jump set (whose Hn−1 measure is possibly infinite), for some continuous, nondecreasing, subadditive function g0, with g0−1(0)={0}. The approximating functions uj are piecewise affine with piecewise affine jump set; the convergence is that of L1 for uj and the convergence in energy for |∇uj|p and g([uj],νujavax.xml.bind.JAXBElement@4d3684be) for suitable functions g. In particular, uj converges to u BV-strictly, area-strictly, and strongly in BV after composition with a bilipschitz map. If in addition Hn−1(Ju)<∞, we also have convergence of Hn−1(Jujavax.xml.bind.JAXBElement@7ed7bf7f) to Hn−1(Ju).