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What can be said about the Hausdorff dimension of the image of a set by a $W^{1,1}$ map? In other words, what is the relationship between $\mathrm{dim}_H f(A)$ and $\mathrm{dim}_H A$, where $f \in …

Can the derivative of a BV function $f:\mathbb{R}^n\to\mathbb{R}^n$ be controlled by the symmetric part of the derivative $\frac{1}{2}(Df+(Df)^T)$?

Let $u\in BV(\mathbb{R}^N; \mathbb{R}^M)$. How does one prove that $$\operatorname{curl} Du = 0$$ holds in the sense of distributions?

Is there a "fractal" set $E \subset \mathbb R^2$ such that the indicator function $\mathbf{1}_E$ is in $BV(\mathbb R^2)$?

Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one? $$\lim_{r \to 0} \rlap{-}\!\!\int_ …

Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$. Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and t …

Where can I find a proof that the definition of total variation for functions of several variables is consistent with the definition of total variation for functions of one variable?

In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ s …

Let $u \in [BV(\mathbb R^N)]^N$. We have $$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$ where $a,b \in \mathbb S^{N-1}$. What is the projection of $D^{jump}u$ in the direction $a$? And how can w …

Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold? $$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \B …

Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation. Question 1. How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$? Question …

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 ( …

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 a …

Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Unfortunately, the formula $$f = \int_{\mathbb R} \chi_ …

Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent? For every $\tau >0$ and level surface $S$ of $u(\tau,\cdot)$, i …