Let $U\subset\mathbb{C}$ be a nonempty simply connected domain, which is not all of $\mathbb{C}$ and has locally connected boundary, then the Riemann map $\phi:\mathbb D\to U$ can be extended continuously to the boundary.
My question is whether we can extend $\phi$ to be a homeomorphism $\overline\phi:N(\overline{\mathbb D})\to N(\overline{U})$, where $N(\overline{\mathbb D})$ and $N(\overline{U})$ are some neighborhoods of $\overline{\mathbb D}$ and $\overline{U}$, respectively. Any suggestions?
Forget neighborhoods for the moment: to have a homeomorphism of $\overline{\mathbb{D}}$ onto $\overline{U}$, you need the boundaries of these sets to be homeomorphic. Thus, it is necessary to assume that $\partial U$ is a Jordan curve.
Thanks to Carathéodory's theorem this condition is also sufficient for the conformal map to extend to a homeomorphism of $\overline{\mathbb{D}}$ onto $\overline{U}$.
Once you have that, the Jordan-Schoenflies theorem assures an extension of this homeomorphism to a global homeomorphism $\mathbb{R}^2\to\mathbb{R}^2$.