An elementary proof of the rank-one theorem for BV functions

TL;DR

This paper presents a straightforward proof of a fundamental theorem by Alberti, demonstrating a rank-one property for the singular part of derivatives in BV functions, simplifying previous complex proofs.

Contribution

The paper offers an elementary proof of Alberti's rank-one theorem for BV functions, making the result more accessible and easier to understand.

Findings

Simplified proof of Alberti's rank-one theorem

Clarification of the structure of singular derivatives in BV functions

Enhanced understanding of BV function properties

Abstract

We provide a simple proof of a result, due to G. Alberti, concerning a rank-one property for the singular part of the derivative of vector-valued functions of bounded variation.