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How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?

On page 3 of this preprint, after recalling the definition of flow generated by a vector field, the authors remark that "a necessary condition for a flow $\varphi_t(\cdot)$ generated by $a(t, \cdot)$ …

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it? Update. In an answer to this post, …

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$. Let $\tilde u = u$ a.e. Is it true th …

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 de …

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 $u …

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it? The analogous question for the Hau …

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 …

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 imp …

Consider the initial value problem $$\dot x(t) = F(t,x), \quad t \in (0,T)$$ with given initial datum $$x(0) = x_0 \in \mathbb R.$$ More precisely we consider the integral equation $$x(t)=x(0)+\int_0^ …

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 …

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 $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ 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 …

In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$. Heuristically, w …