While analyzing a variational problem, I came to the following question:

Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball.

Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega_f :\mathbb D^n \to \mathbb{R}^n$ be the unique harmonic map corresponding to the Dirichlet problem imposed by $f$, i.e. $\omega_f|_{\partial \mathbb D^n}=f|_{\partial \mathbb D^n}$.

$d\omega_f$ must be invertible outside a set of measure zero in $\mathbb D^n$. Indeed, $\omega_f$ is real-analytic, and so is $\det d\omega_f$, which is not identically zero, because $$ \int_{\mathbb D^n} \det d\omega_f = \int_{\mathbb D^n} \det df>0.$$

Now, the zero-set of a real-analytic function which is not identically zero has measure zero.

Question: Do there exist $f_k \in C^{\infty}(\mathbb D^n, \mathbb{R}^n)$ such that $d\omega_{f_k}$ are everywhere invertible and $f_k \to f$ in $W^{1,2}$?

($\omega_{f_k}$ is the harmonic map corresponding to the Dirichlet problem imposed by $f_k$.)

By the argument above, if we approximate $f$ by immersions, then $d\omega_{f_k}$ are invertible outside sets of measure zero in $\mathbb D^n$, but I want more than that.

Are there any tools which might help decide this problem one way or another?