Rank-one properties for BV functions

In this thesis we insvestigate some recent alternative proofs to the celebrated Alberti’s Rank-one Theorem for BV functions, in the standard Euclidean setting and in the context of Carnot groups. Alberti’s theorem is a rank-one property for the singular part of derivatives of functions of bounded variation, which was first conjectured by L. Ambrosio and E. De Giorgi, and is at the heart of many fundamental results in the theory of BV functions. The statement is as follows: Let Ω be an open subset of R^n and u : Ω → R^m a function of bounded variation. Denote DSu as the singular part of Du with respect to the Lebesgue measure. Then DSu is a rank one measure, or in other words for |DSu|-almost all x ∈ Ω the n × m matrix dDSu/d|DSu|(x) has rank one. In particular, we will first see two elementary proofs by Massaccesi-Vittone and De Philippis-Rindler: the former revolves around properties of the derivative of a BV functions and the perimeter of its subgraph, as well as a more geometric transversality lemma. Subsequently, Don, Massaccesi and Vittone were able to adapt their proof to a new result on a class of Carnot groups, and in particular to the Heisenberg groups H^n for n ≥ 2. Introducting a notion of functions of horizontal bounded variation, the result is akin to the Euclidean statement, but unfortunately this argument fails in three dimensions (i.e. for n = 1). In this regard, in the last chapter we explore a possible strategy for proving the Rank-one Theorem in H^1, which is still an open problem. The latter proof is a consequence of a more general and profound result concerning A-free measures: Let A be a linear partial differential operator defined on distributions in D(Ω; R^n). An R^m-valued Radon measure µ on some open Ω ⊆ R^d is called A-free if Aµ = 0 (in the sense of distributions in D′(Ω; R^n)). The main result is the following, which for A = curl recovers Alberti’s theorem: Denote by Λ_A the wave cone of A. Then, with the notation above, if µ is A-free and µ_s is its singular part, we have that dµ/d|µ|(x) ∈ Λ_A for |µ_s|-almost all x ∈ Ω. Apart from the direct implications for the Rank-one Theorem, the thesis also covers other interesting results in Geometric Measure theory, mainly concerning functions of bounded deformation, normal d-currents in R^d and the properties of measures satisfying Rademacher's differentiability theorem for Lipschitz functions.