Higher differentiability of weak solutions to 2nd order elliptic PDEs with mixed boundary conditions 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.

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.
