Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to \begin{equation} \begin{cases} -\Delta u=0 \quad &\mbox{in $\Omega$}\\ \frac{\partial u}{\partial \nu}=g\quad &\mbox{on $\partial \Omega$}, \end{cases} \end{equation}where $g \in L^{\infty}(\partial \Omega)$ satisfies $$\int_{\partial \Omega}g \, d\sigma=0.$$ My question is that, how much regularity can we say on $u$? Does $u$ belong to $C^{\alpha}(\overline{\Omega})$ for some $\alpha \in (0,1)$? I cannot seem to find a suitable reference. Any help would be really appreciated!