Approximation of SBV functions with possibly infinite jump set

Abstract

We prove an approximation result for functions $u\in SBV(\Omega ;R^m)$ such that ∇u is p-integrable, $1\le p<\infty$, and $g_0\left(\right|\left[u\right]\left|\right)$ is integrable over the jump set (whose $H^{n-1}$ measure is possibly infinite), for some continuous, nondecreasing, subadditive function $g_0$, with $g_0^{-1}\left(0\right)=\left\{0\right\}$. 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 $g\left(\right[u_j],{\nu }_{u_j})$ 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 $H^{n-1}\left(J_u\right)<\infty$, we also have convergence of $H^{n-1}\left(J_{u_j}\right)$ to $H^{n-1}\left(J_u\right)$.