Tukukkk

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Let $n,d in mathbb{N}$, and let $Omega subseteq R^n$ be open.
Let $f in W^{1,k}(Omega,R^d)$. Let $omega in Omega^k(R^d)$ be a smooth closed $k$-form; Is it true that $f^*omega$ is weakly closed? i.e. does
$$int_{Omega} IP{f^* omega}{de sig}=0$$
hold for every compactly-supported $k+1$-form $sigma in Omega^{k+1}(Omega)$?

I have read that this should be true, but I am only able to show this in two special cases:

1. The form $omega$ is constant.


2. 
   $f$ is continuous.

Does this hold for non-continuous Sobolev maps in general?

Here is the problem as I see it: We approximate $f$ via smooth functions; suppose that $f_n in C^{infty}(Omega,R^d)$ satisfy $f_n to f$ in $W^{1,k}(Omega,R^d)$. Then
$$
int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{d f_n^* omega}{ sig}=0.
$$
Now, we need to justify the passage to the limit $int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}$.

If $omega$ is constant, that is $omega_q= alpha$ independently of $q in R^d$, where $alpha$ is a fixed element in $bigwedge^k (R^d)^* $, then we have
$$
|f^* omega-f_n^* omega| le |alpha| , left|bigwedge^{k} df-bigwedge^{k} df_nright|_{op},
$$
thus
$$
left|int_{Omega} IP{f^* omega}{de sig}- IP{ f_n^* omega}{de sig}right| le |alpha| |de sig|_{sup} int_{Omega} left|bigwedge^{k} df-bigwedge^{k} df_nright|.
$$
and The RHS tends to zero since Sobolev approximation lifts to exterior powers.

When $omega$ is not constant, we have a problem that the point of evaluation "moves with the function" that pulls back, that is
$$
begin{split}
& |f^* omega-f_n^* omega|(p)= \
&left|omega_{f(p)} circ bigwedge^{k} df_p -omega_{f_n(p)} circ bigwedge^{k} (df_n)_p right| le \
&left|big(omega_{f(p)}-omega_{f_n(p)}) circ bigwedge^{k} df_p +omega_{f_n(p)} circ big( bigwedge^{k} df_p-bigwedge^{k} (df_n)_p) right| le \
&|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right|
end{split}
$$

So, we we need to estimate $omega_{f(p)}-omega_{f_n(p)}$. When $f$ is continuous, we can take $f_n$ which converges uniformly to $f$, and thus we can continue the estimate:

$$
begin{split}
&|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
&|Tomega|_{sup} , cdot , |f(p)-f_n(p)| , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
&|Tomega|_{sup} , cdot , |f-f_n|_{sup} , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| stackrel{L^1}{to} 0,
%&|alpha circ brk{bigwedge^{k} df_p-bigwedge^{k} (df_n)_p} | le |alpha| cdot |bigwedge^{k} df_p-bigwedge^{k} (df_n)_p|_{op}.
end{split}
$$

So, does the preservation of closedness hold in general (for non-continuous maps) or is there a counter-example?

This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 are close to be optimal. I will expand my answer when I have time. Let me know if Lemma 4.1 suits you.