From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP \begin{align} -\Delta u &= 0 & \text{in}\ \Omega\\ -\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\ \end{align} where $a>0$ and $g \in H^{1/2}(\Gamma)$ has unique solution $u$ with $H^2$ regularity for open bounded domain $\Omega \subset \mathbb{R}^2$ with boundary $\Gamma$ of class $C^{1,1}$.
Question For $C^{2,1}$ boundary $\Gamma$ and $g \in H^{3/2}(\Gamma)$, is it true that $u \in H^3(\Omega)$? and, in general, is it true that $u \in H^{k+2}(\Omega)$ for $C^{k+1,1}$ boundary $\Gamma$ and $g \in H^{k+1/2}(\Gamma)$.