On the approximation of SBV functions

Abstract

In this paper we deal with the approximation of SBV functions in the strong BV topology. In particular, we provide three approximation results. The first one, Theorem A, concerns general SBV functions; the second one, Theorem B, concerns SBV functions with absolutely continuous part of the gradient in $L^p$, $p>1$; and the third one, Theorem C, concerns SBV${}^p$ functions, that is, those SBV functions for which not only the absolutely continuous part of the gradient is in $L^p$, but also the jump set has finite ${\mathcal{H}}^{N-1}$-measure. The last result generalizes the previously known approximation theorems for\SBV${}^p$ functions, see [5, 7]. As we discuss, the first and the third result are sharp. We conclude with a simple application of our results.