• approximation
• free discontinuity problems
• SBV functions

The main objective of this master thesis is to compare different approximation results for SBV functions. Special functions of bounded variation were introduced by De Giorgi and Ambrosio to deal with “free discontinuity” variational problems, with the idea to replace the free discontinuity set by the set of “discontinuity points” of these functions, which turns out to be “sufficiently regular” so that weak notions of surface area, orientation and traces can be given.
The crucial tools to study integral functionals are compactness and approximation results. In this document, we will present a few of them, that have different characteristics. The main differences consist of the demanded regularity of the approximating sequences and the requested structure of the jump set.