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I am interested in regularity results for 2nd order elliptic PDEs with mixed boundary conditions like

$$\left\{\begin{array}{rl}-\text{div}(a\nabla u) =& f &\text{in }\Omega, \\ u=&\varphi &\text{on }\Gamma_D, \\ \frac{\partial u}{\partial\nu}=& g & \text{on }\Gamma_N, \end{array}\right.$$

where $\Omega\subset\mathbb{R}^N$ denotes a bounded domain and $\Gamma_D$ with positive surface measure and $\Gamma_N:=\partial\Omega/\Gamma_D$ the Dirichlet and Neumann part of the boundary $\partial\Omega$, respectively. So my question is the following:

What assumptions regarding regularity and compatibility do I have to make to ensure $u\in H^s(\Omega)$ holds for some given $s>1$?

I am aware that there are such results when dealing with a purely Dirichlet or Neumann boundary value problem. However, there are simple examples in a mixed boundary value setting, where smooth data and smooth boundary are not enough to ensure higher regularity.

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Unfortunately, there is no easy answer to your question. Mixed Dirichlet-Neumann problems have singular solutions even when the boundary conditions are regular. Take $f=\varphi=g=0$ then the function $u(r,\theta)=r^{1/2}\sin\frac\theta2$ is harmonic in the half-space $y>0$ and satisfies the Dirichlet-Neumann boundary conditions. This paper Costabel-Dauge has a few references you might want to look at. I don't know how to kill the singular part.

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  • $\begingroup$ Thank you very much for your answer. After looking a bit deeper into the matter, I also found articles stating the need of some sort of compatibility conditions, which are not known so far. Thank you again for your reference, I will check it, too. btw. This is the second very helpful comment of yours I came across in related topics to regularity. Thank you for that! $\endgroup$ – sgr Nov 1 '18 at 12:40

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