This is a follow-up question of this one.
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a smooth one-form.
Does there exist a sequence of smooth one-forms $\sigma_n \in \Omega^1(\mathbb{D}^2)$ such that
- $\sigma_n \to \sigma$ in $L^2$.
- $\sigma_n$ do not vanish on $\mathbb{D}^2$.
- $\sup_n \|\delta d \sigma_n \|_{L^p}< \infty, \sup_n \| d \delta \sigma_n \|_{L^p}< \infty$ for some $p>1$.
Note that I don't fix any boundary conditions on $\sigma_n$, so we don't have standard elliptic estimates. (In particular, $\sup_n \|\Delta \sigma_n \|_{L^p}< \infty$ does not imply $\sup_n \|\nabla^2 \sigma_n \|_{L^p}< \infty$).
Comment:
I think that one can make a perturbation to $\sigma$ which gives a form with only finitely many zeroes. After that, there is a procedure to push out the zeroes, one by one -- by composing with diffeomorphisms. I think that this procedure does not satisfy condition $3$.
If it matters, I am fine with assuming that $\delta \sigma=0, d\sigma=0$.
Note that we cannot approximate $\sigma$ by nowhere-vanishing $\sigma_n$ in $W^{1,1}$.
