*Submitted Paper*

**Inserted:** 28 sep 2023

**Last Updated:** 28 sep 2023

**Year:** 2023

**Abstract:**

We prove an approximation result for functions $u\in SBV(\Omega;\mathbb R^m)$ such that $\nabla u$ is $p$-integrable, $1\leq p<\infty$, and $g_0(

[u]

)$ is integrable over the jump set (whose $\mathcal H^{n-1}$ measure is possibly infinite), for some continuous,
nondecreasing, subadditive function $g_0$, with $g_0^{-1}(0)=\{0\}$. The approximating functions $u_j$ are piecewise affine with piecewise affine jump set; the convergence is that of $L^1$ for $u_j$ and the convergence in energy for $

\nabla u_j

^p$ and $g([u_j],\nu_{u_j})$ for suitable functions $g$. In particular, $u_j$ converges to $u$ $BV$-strictly, area-strictly,
and strongly in $BV$ after composition with a bilipschitz map. If in addition $\mathcal H^{n-1}(J_u)<\infty$, we also have convergence of $\mathcal H^{n-1}(J_{u_j})$ to $\mathcal H^{n-1}(J_u)$.

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