A Note on Alberti’s Rank–one Theorem

The aim of these notes is to illustrate a proof of the following remarkable Theorem of Alberti (first proved in [1]). Here, when μ is a Radon measure on Ω ⊂ R, we denote by μ its absolutely continuous part (with respect to the Lebesgue measure L ), by μ := μ− μ its singular part, and by |μ| its total variation measure. Clearly, |μ|a = |μa| and |μ|s = μ. When μ = Du for some u ∈ BV (Ω,R), we will write Du and Du. If ν is a nonnegative measure, μ/ν will denote the Radon–Nykodim derivative of μ with respect to ν. Finally we recall the polar decomposition of Radon measures, namely the identity μ = μ |μ| |μ|, which implies that the vetor Borel map μ/|μ| has modulus 1 μ–a.e.. Theorem 1.1. Let u ∈ BV (Ω,R) for some open set Ω ⊂ R. Then rank (Du/|Du|(x)) = 1 for |Dsu|–a.e. x ∈ Ω.