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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?

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  • $\begingroup$ Just for clarification: The main difference in your new problem compared to the first one is that you now allow the boundary parts to meet, right? (The boundary conditions are still of the same type.) $\endgroup$ – Hannes May 23 '18 at 9:41
  • $\begingroup$ In my new problem, I still have the boundary condition on $\Gamma_i$. So, I actually have three three boundary parts, two of which splits the original boundary $\Gamma_o$. $\endgroup$ – Julienne Franz May 23 '18 at 9:49
  • $\begingroup$ I edited my question above. I hope it is clear now. $\endgroup$ – Julienne Franz May 23 '18 at 9:54
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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.

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  • $\begingroup$ Thank you for your comment. Does this mean that, regardless of the regularity of the domain and the data, there is no hope of getting an $H^2$ regularity for the given problem. What happens when the Dirichlet data on $\Gamma_{o1}$ is replaced by a Neumann data? $\endgroup$ – Julienne Franz May 24 '18 at 4:42
  • $\begingroup$ Then you would have a mixed Neumann-Robin problem where Neumann- and Robin boundary conditions meet. I am, to be honest, not too sure how the Robin boundary data is affected by the counterexample I mentioned, but I suspect that some trickery is possible to relate the Robin boundary conditions to Neumann ones. (That's why I thought it might be helpful, but more of a comment.) In this sense, your then modified Neumann-Robin problem would be a bit more friendly, but please do not quote me on that :-) $\endgroup$ – Hannes May 24 '18 at 9:20
  • $\begingroup$ Your comment is of great help. I just found out from Grivard's book that an $H^2$ regularity cannot be expected in my problem, at least for the mixed Dirichlet-Neumann (or Dirichlet-Robin) case. $\endgroup$ – Julienne Franz May 24 '18 at 13:38

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