Consider the problem

$$-\Delta u = 0 \mbox{ in }\Omega,\qquad \partial_\nu u+\tau u=g\mbox{ on }\partial\Omega,$$

where $\Omega\subset R^n$ is a bounded $C^2$-domain, $\tau>0$ is a constant, and $g\in L^p(\partial\Omega)$, $p>2$. What is the optimal regularity for the solution to this Laplace problem?

Of course, we cannot expect a strong solution $u\in W^{2,q}(\Omega)$, say, because in that case $g=\partial_\nu u+\tau u\in W^{1-1/q,q}(\partial\Omega)$, which is "too much", in general. So we will only find weak solutions. There are a bunch of results by Mitrea and Mitrea on problems like this for $p\in(1,2]$ in Lipschitz domains, but what about larger $p$?