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" 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), implies what I need, but, like I said, I am out of my depth here and quickly get bogged down, and completely lost.

Elliptic equations in polyhedral domainsseems promising, and may be more digestible than the original papers. $\endgroup$ – Willie Wong Nov 21 '17 at 14:19