All Questions

In this paper, it is written that Alberti’s rank says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...

Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...

Let $f: \mathbb R^N \to \mathbb R^N$ be a $BV$ function. Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $\operatorname{div} f \ll \mathcal L^N$. This ...

Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following: Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...

Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...