I am trying to write a proof and I am out of my depth. I need an elliptic regularity result of the form 
$$
\|u\|\_{H^{1+\epsilon}(\Omega)} \le C \|f\|\_{L^2(\Omega)}
$$


for some $\epsilon >0 $ where $u$ is the weak solution to either of the following PDEs.


\begin{align\*}
\nabla\cdot\nabla u &= f\quad x\in \Omega\\
u &= u\_D\quad x \in \partial \Omega\_D\\
\nabla u\cdot n& = 0\quad x\in \partial\Omega\_N 
\end{align\*}


or the pure Nuemann problem with the further restriction that $\int\_\Omega f \mathrm{d}x = 0$,
\begin{align\*}
 \nabla \cdot \nabla u &= f\quad x\in \Omega,\\
 \nabla u \cdot n &= 0 \quad x\in \partial\Omega,\\
 \int\_\Omega u\, \mathrm{d}x &= 0.
\end{align\*}


This result is known for the case of two dimensional polygons (I am interested in 3-dimensional polyhedra), and the largest $\epsilon$ depends on the measure of the interior angles.


I have looked into a few promising papers with ["Analytic Regularity for Linear Elliptic Systems in Polygons and 
Polyhedra](https://arxiv.org/pdf/1002.1772.pdf)" being among them. I suspect that Theorem 1.4, in that paper (which references theorem 2 in [On the Agmon-Miranda Maximum Principle for Solutions of Elliptic Equations in Polyhedral and Polygonal Domains](https://www.researchgate.net/publication/226589031_On_the_Agmon-Miranda_Maximum_Principle_for_Solutions_of_Elliptic_Equations_in_Polyhedral_and_Polygonal_Domains)), implies what I need, but, like I said, I am out of my depth here and quickly get bogged down, and completely lost.



Is there any "extra regularity" to the solution to Poisson's equation posed on a 3-dimensional polyhedron?
As discussed in the comments, such a result can be found in [Jochmann's "An $H^s$-Regularity Result for the Gradient of Solutions to Elliptic Equations with Mixed Boundary Conditions"](http://www.sciencedirect.com/science/article/pii/S0022247X99965186).


