A Formula for the Total Variation of Sbv Functions

The space SBV of special BV functions whose gradient measure has no Cantor part was singled out by De Giorgi and Ambrosio [4] as the natural setting to study variational problems where both volume and surface densities have to be taken into account. In fact, for a SBV function f the derivative Df is the sum of a measure Daf absolutely continuous with respect to the Lebesgue measure and a singular measure Dsf concentrated on the jump set Jf , which is a countable (n−1)rectifiable set. The density of Daf is equal to the approximate gradient ∇f . In this paper, following some ideas from [2] and [3], we give a characterization of a function f ∈ SBVloc(R) independent of the theory of distributions. Namely, we prove in Theorem 3.3 that if we define as in [2] for a function f ∈ Lloc(R) κε(f) := ε n−1 sup Gε ∑