I'm in the middle of a project concerning a Bernoulli-type free boundary problem in $\mathbb{R}^2$ and, as part of this project, I would like to understand the boundary behavior of conformal maps on domains that satisfy an exterior sphere condition.
Let $\Omega \subset \mathbb{R}^2$ be a bounded domain (open, connected) and assume that $\Omega$ satisfies a uniform exterior sphere condition. Let $f: \Omega \to D$ be a conformal map, where $D$ is the open unit disk. My first question is the following: what sort of regularity will $f$ have on $\overline{\Omega}$ and what is a good reference for this result? I know that, due to results of Lesley and others, $f$ is $C^{0,\alpha}$ for some $\alpha \in (0,1)$; I suspect this holds for any $\alpha \in (0,1)$, but could well be wrong. This follows from the fact that $\Omega$ will satisfy an exterior cone condition. Would anything change if I also assume the domain is Lipschitz?
I'm familiar with the Painlevé-type results that say if $\partial\Omega$ is $C^{k+\varepsilon}$, then $f$ will extend to be univalent and $C^k$ from $\overline{\Omega}$ into $\overline{D}$. However, I'm having some trouble finding results where the regularity of the boundary is between satisfying an exterior cone condition and $C^{k}$. I have Pommerenke's book and have been looking through it, but haven't yet found the right thing.
Edit: I have been thinking that I would impose a uniform exterior sphere condition: there exists $r > 0$ such that, for every $Q \in \partial\Omega$, there exists a ball $B \subset \mathbb{R}^2\setminus\Omega$ of radius $r$ such that $Q \in \partial B$.
However, if possible I would be interested in working with a (non-uniform) exterior sphere condition: for each $Q \in \partial\Omega$, there exists $r>0$ such that $Q \in \partial B$, where $B \subset \mathbb{R}^2\setminus\Omega$ is a ball of radius $r$.
So, if you have reference suggestions for (Lipschitz) domains satisfying either of these conditions, I would be very interested.
Edit 2: After doing some more research, I believe that $f$ will be in $C^{0,\alpha}(\overline{\Omega})$ for any $\alpha \in (0,1)$, however $f$ will not be Lipschitz up to the boundary. Indeed, as I understand it, a conformal map from the unit disk into a $C^1$ domain can fail to be Lipschitz up to the boundary. Hopefully this much is right? If anyone knows of any other, more subtle regularity properties of such an $f$ that might be of use, I would love to hear about them.
