This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be $C^1$ be bijective maps with $\det(df_n)>0$, and suppose that $f_n$ converges to a $C^1$ function $f: \bar\Omega_1 \to \bar\Omega_2$ strongly in $W^{1,2}$.
Question: Must $f$ be surjective?
Note that $f$ is surjective if and only if $|f^{-1}(y)| \le 1$ a.e. on $\Omega_2$:
By the area formula $$ \int_{\Omega_1} \det df_n = \int_{\Omega_2} |f_n^{-1}(y)|=\text{Vol}(\Omega_2), $$ so $$ \int_{\Omega_2} |f^{-1}(y)|= \int_{\Omega_1} \det df =\lim_{n \to \infty} \int_{\Omega_1} \det df_n=\text{Vol}(\Omega_2). $$ This implies the claim.
