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I think the obvious way is indeed the only way. First of all $n_j=1$ for all $j$, as boundaries do not have higher multiplicity. Secondly, the $l$ rays of your varifold split $\mathbb{R}^2$ into $l$ s …

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 …

There is the following interpretation coming from physics and continuum mechanics, which is a bit too long for a comment but might be helpful: If you think of $\mathcal{F}$ as an energy that you want …

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, …

This is not a full answer, since I do not know the counterexample Lin refers to, but I can offer some explanations and guesses which are too long for a comment: You can define a first variation for cu …

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 …

There is some interest in a related question in non-linear elasticity, specifically people there would consider a function "smoothable" if there is a close-by (in some norm applying both to function a …