# 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.