Calculus of Variations and Geometric Measure Theory
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G. De Philippis - N. Fusco - A. Pratelli

On the approximation of SBV functions

created by pratelli on 16 Nov 2016
modified by dephilipp on 17 Nov 2016


Submitted Paper

Inserted: 16 nov 2016
Last Updated: 17 nov 2016

Year: 2016


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.


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