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!
Update: I have found two relavent references:
[1] Bjorn E. J. Dahlberg and Carlos E. Kenig. Hardy Spaces and the Neumann Problem in Lp for Laplace's Equation in Lipschitz Domains. Annals of Mathematics , May, 1987, Second Series, Vol. 125, No. 3.
[2] E.B. Fabes, M. Jodeit and N.M. Rivière. Potential techniques for boundary value problems on $C^1$-domains. Acta Math. 141, 165–186 (1978).
The first paper proves that if $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$, then $u \in W^{1,p}(\Omega)$ for some $2<p<2+\epsilon$. Hence $u$ belongs to some Holder space when $n=2$. Of course, in the $W^{1,p}$ estimate, $2+\epsilon$ is optimal, since one can construct a counter example by considering a two-dimensional cone-domains suggested by @Math604. I would like to know whether Holder regularity is true for $n \ge 3$ and small $\alpha$. If not, can we prove continuity of solutions up to the boundary?
The second paper proves that if $\Omega$ is a bounded $C^1$ domain in $\mathbb{R}^n$, then $u$ must be in $W^{1,p}(\Omega)$ for any $1<p<\infty$. Hence $u$ must be in $C^{\alpha}(\overline{\Omega})$ for any $0<\alpha<1$.
