Calculus of Variations and Geometric Measure Theory

G. De Philippis - N. Fusco - A. Pratelli

On the approximation of SBV functions

created by pratelli on 16 Nov 2016
modified on 30 Nov 2017


Published Paper

Inserted: 16 nov 2016
Last Updated: 30 nov 2017

Journal: Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.
Year: 2017


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 Braides-Chiadò Piat,Cortesani-Toader. As we discuss, the first and the third result are sharp. We conclude with a simple application of our results.