Approximation of $SBV$ functions with possibly infinite jump set

TL;DR

This paper establishes an approximation method for SBV functions with potentially infinite jump sets, using piecewise affine functions that converge in various senses, including energy and measure, under specific integrability conditions.

Contribution

It introduces a novel approximation scheme for SBV functions with possibly infinite jump sets, ensuring convergence in multiple senses and preserving measure properties.

Findings

Approximate SBV functions with infinite jump sets using piecewise affine functions.

Achieve convergence in L^1, energy, and measure of jump sets.

Ensure convergence in BV and area-strict senses, with measure preservation.

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 $\mathcal H^{n-1}(J_u)$.