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** Let $a_1, a_2 >0$, $g_1 \in H^{1/2}(\Gamma_1)$ and $g_2\in H^{1/2}(\Gamma_2)$ where $\Gamma_1 \subset \Gamma$ and $\Gamma_2 = \Gamma\setminus \bar{\Gamma}_1$. Does the unique solution $u \in H^1(\Omega)$ of the BVP
\begin{align}
-\Delta u &= 0 & \text{in}\ \Omega\\
-\frac{\partial u}{\partial n} + a_1u &= g_1 & \text{on}\ \Gamma_1\\
-\frac{\partial u}{\partial n} + a_2u &= g_2 & \text{on}\ \Gamma_2
\end{align}
have higher regularity (in particular, is $u\in H^2(\Omega)$) for domain $\Omega$ of class $C^{1,1}$?

I know $u \in H^2(\Omega)$ if $\Gamma_1$ and $\Gamma_2$ are both closed, i.e., $\Omega$ is an annular domain. But, I do not know if I still have an $H^2$ regularity for the case given above.