Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a smooth and positive function on a neighbourhood of $\overline{\mathbb{D}}\subset \mathbb{C}$ such that the Riemannian surface $(\mathbb{D},\mu |dz|^2)$ is flat in a neighbourhood of $\partial \mathbb{D}.$ We consider the space $\mathcal{C}$ of continuous and $W^{1,2}$ maps from $\mathbb{D}\to\mathbb{D}$ that extend to the map $\varphi$ on the boundary, equipped with the $W^{1,2}$ norm.
I am interested in the following. Are there conditions on $\varphi$ that guarantee that $C^\infty(\mathbb{D},\mathbb{D})\cap \mathcal{C}$ is dense in $\mathcal{C}$?
This is presumably standard, but I can't find a source. Of course, I am also interested in generalizations (curvature, higher dimensions, etc.). I am aware that many density results for smooth maps fail for maps between higher dimensional manifolds (depends on homotopy, etc.).
