Approximation of $SBV$ functions with possibly infinite jump set

Abstract

We prove an approximation result for functions $u\in SBV(\Omega ;{\mathbb{R}}^m)$ such that $\nabla u$ is $p$-integrable, $1\le p<\infty$, 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 ${\mathcal{H}}^{n-1}(J_u)$.