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 $\Gamma_{o1}=\Gamma_o\setminus \overline{\Gamma}_{o2}$.

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. However, I do not know how to proceed when the above boundary conditions are imposed.

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

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?