Let $\Omega \subset \mathbb{R}^2$ be an open bounded Lipschitz domain of class $C^{1,1}$ with boundary $\partial \Omega = \Gamma_i \cup \Gamma_o$, $\Gamma_i \cap \Gamma_o = \emptyset$ and dist$(\Gamma_i,\Gamma_o)>0$. Consider the PDE system \begin{align} -\Delta u &= 0 & \mbox{in}\ \Omega\\ u &= 1 & \mbox{on}\ \Gamma_o\\ -\partial_n u &= u+1 & \mbox{on} \ \Gamma_i. \end{align} I know that this BVP has a unique weak solution $u\in H^1(\Omega)$, and using the results in [Grisvard, Elliptic Problems in 9 Nonsmooth Domains, Chapter 2], I was able to show that $u$ has $H^2$ regularity.

Now, what if instead of just a Dirichlet condition on $\Gamma_o$, I have the following boundary conditions \begin{align} u &=1 &\mbox{on}\ \Gamma_{o1}\\ -\partial_n u &= au+b&\mbox{on}\ \Gamma_{o2} \end{align} where $a,b>0$ and $\Gamma_{o1}=\Gamma_o\setminus \overline{\Gamma}_{o2}$.

**Edit** So I have the new problem
\begin{align}
-\Delta u &= 0 & \mbox{in}\ \Omega\\
u &=1 &\mbox{on}\ \Gamma_{o1}\\
-\partial_n u &= au+b&\mbox{on}\ \Gamma_{o2}\\
-\partial_n u &= u+1 & \mbox{on} \ \Gamma_i.
\end{align}

**Question** Do I still have $H^2$ regularity for the weak solution $u$? If yes, how do I show this?

I know how to approach the problem when both $\Gamma_{o1}$ and $\Gamma_{o2}$ are only imposed with a pure Dirichlet (or Neumann) boundary condition (e.g., $u=1$ on $\Gamma_{o1}$ and $u=a$ on $\Gamma_{o2}$). However, I do not know how to proceed when I have the new BVP above.

Can someone please give me hint on how to deal with the problem and references that tackle such kind of problems?