Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.

Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ with $\det A > 0$ a.e.

Do there exist $u_n \in C^{\infty}\big(\Omega,\text{GL}(\mathbb{R}^n)\big)$ such that $u_n \to A$ in $W^{1,p}_{loc}$?

I am also interested in a weaker result: Are there $u_n \in C^{\infty}\big(\Omega,\text{End}( \mathbb{R}^n)\big)$, such that $u_n(x) \in \text{GL}(\mathbb{R}^n)$ a.e.

and$u_n \to A$ in $W^{1,p}_{loc}$?

I don't really need the $u_n$ to be defined on all $\Omega$. It suffices that for **every** arbitrarily small ball in $\Omega$, there would be a neighbourhood where such a sequence $u_n$ would be defined.

The problem is that *it is not always true* that $A_x \in \text{GL}( \mathbb{R}^n)$ for *every* $x \in \Omega$. The rank can fall on a subset of measure zero.

**If we knew $A(x) \in \text{GL}(\mathbb{R}^n)$ everywhere then the answer would be positive.** This follows from the facts that "being invertible" is an open condition, and that continuous Sobolev maps can be *approximated uniformly* by smooth maps over compact subsets.

In more detail, let $K \subseteq \Omega$ be compact. Since we assumed $A \in C\big(\Omega, \text{GL}(\mathbb{R}^n) \big)$, the map $\psi:x \to A_x$, considered as a map $K \to \text{GL}( \mathbb{R}^n)$, is continuous. Thus $\psi(K) $ is compact and $\text{dist}\big(\psi(K),\partial \text{GL}(\mathbb{R}^n)\big)>0$.

Now consider each component of $\psi(x)=A_x \in \text{End}(\mathbb{R}^n) $. We can approximate each component of $\psi$ using mollification on an open subset of $\Omega$ containing $K$. Since each component is a continuous function, the mollifications *converge uniformly* on $K$. This implies that from a certain point in the mollified sequence, $\text{dist}(u_n,A)<\text{dist}\big(\psi(K),\partial \text{GL}(\mathbb{R}^n)\big)$, so the $u_n$ are invertible.