Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $f \in C(\partial \Omega)$.
Consider the Dirichlet problem
$$\begin{cases}\Delta u=0&\text{ in } \Omega&\\u=f&\text{ on }\partial \Omega \end{cases}$$
I know that, by Perron's method, if $\Omega$ satisfies an exterior sphere condition at each boundary point that we can generate a solution $u\in C(\overline{\Omega})\cap C^{2,\alpha}(\Omega)$ solution of the above.
But according to the Wikipedia page on the Dirichlet Problem, it claims we can guarantee a solution if $\partial \Omega \in C^{1,\alpha}$.
Can anyone provide a resource or proof of this fact?
Additionally, what can we say about the regularity of $u$ in this situation?
