Higher regularity of solutions for Laplace equation with mixed boundary condition

Question

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?

Answer 1

This is more of an extended comment, but maybe it is helpful.

At least for the case of mixed boundary conditions involving Dirichlet and Neumann conditions where the corresponding boundary parts actually meet, there is an example by Shamir [1, Introduction] of an harmonic function (so, $\Delta u = 0$) in the positive halfspace $[\operatorname{Re} \lambda > 0]$ which satisfies homogeneous Dirichlet conditions on the positive imaginary axis and homogeneous Neumann conditions on the negative imaginary axis, but whose gradient is not in $L^4$.

This example can be modified to also work on a smooth domain $\Omega$ (also in more space dimensions) where the "new" boundary away from the imaginary axis is given a boundary condition from the already known function. Then you have a mixed boundary problem with quite regular data at hand whose solution is not in $W^{1,4}(\Omega)$ and thus in particular cannot be in $H^2(\Omega)$ for low space dimensions.

[1] Shamir, E., Regularization of mixed second-order elliptic problems, Isr. J. Math. 6, 150-168 (1968). ZBL0157.18202.