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The general prerequisites are almost the same as for currents, mainly a strong understanding of measure theory and a bit of geometrical intuition. There is an aspect of multilinear algebra and some fu …

If the $f_n$ are absolutely continuous, then $f'_n$ would also be the distributional derivative and $g=f'$ follows from the continuity of that. So if there is a counterexample it should probably invol …

Checking this in perfect detail might be a bit technical, but the following looks like it could work for any dimension up to n: For 1d: Take an S curve $\phi: [0,1] \to [0,1]$ such that $\phi(0) = 0$, …

In Evans & Gariepy's "Measure theory and fine properties of functions", Sec. 5.3., they construct the trace operator on a bounded Lipschitz domain $\Omega$ for BV-functions (and thus by inclusion for …

If by not agree you include that the derivative may not exist, you can get any dimension and measure that does not contradict the almost everywhere. Consider the worst case $d=1$: Take the constructio …

I assume that by maximal you mean with respect to inclusion. Then the answer is no. Consider the following counterexample on the real line: Let $\mathcal{B}_\epsilon := \epsilon\mathbb{Z}$ and $\widet …

Your intuition is right. The key is in the paper you cite, in that they consider uniqueness in the class $L^1_{\text{loc}}$, which does not allow for concentrations. If you add to this, that the Lagra …

This is a rather more fun problem than I thought. I must apologize though, as your question is a reference request and I have no references apart from pointing at any textbook on discrete optimization …

I think one of the classic counterexamples works here, to show that this is false: Let $\{q_i\}_{i\in\mathbb{N}}$ dense in $[0,1]^n$, $\delta >0$ and construct $$E = \bigcup_{i\in\mathbb{N}} B_{\delta …

One can use your infinite-density example, but replace the outer lines with very sparse dotted lines: $$S = (\{0\} \times \mathbb{R}) \cup \bigcup_{i=1}^\infty \{i^{-1},-i^{-1}\} \times \left[\bigcup_ …

I don't think your left hand side is well defined for the class of $u$ you are considering, I can change each $u(.,t)$ to a large value on the zero-set $S_{|t|}$, which will result in the supremum pic …

I think the following might be an example, though it will require a bit of work if you want to make it more precise: Take an immersion of a sphere, which is injective except for one cap at each pole, …

I am not the greatest expert on the details of this stuff, but since nobody else tried so far, let me have an attempt: Prelude: Measures Since you mention measures, I start with that, though this is m …

Building on Pietro Majer's answer to you previous question for a change, consider the following: Let $g: [0,\infty) \to \mathbb{R}$ be the unique continuous function such that $g(0)=0$, $g'(x) = 1$ fo …

Okay, let me try a writeup of the comment chain. For any reasonable subset $A\subset \Omega_2$ and $B := f^{-1}(A)$ you get $$\int_A |f^{-1}(y)| dy = \int_B \det df dx \leq \liminf_{n\to\infty} \int_B …