# Is there any “extra regularity” to the solution to Poisson's equation posed on a 3-dimensional polyhedron?

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.

• Maz'ya and Rossman's Elliptic equations in polyhedral domains seems promising, and may be more digestible than the original papers. – Willie Wong Nov 21 '17 at 14:19
• Maybe this paper could also be useful. – Hannes Nov 21 '17 at 14:36
• @Hannes Your suggestion was exactly what I needed. I wouldn't have found it without you. What I asked for is pretty much exactly theorem 1 in that paper. If you want to write an answer to this I will accept your answer. Thank You! – fred Nov 22 '17 at 13:59
• @Hannes Are you familiar with the paper? Do you know if there is any reason why Theorem 1 couldn't be extended to the pure Nuemann case? – fred Nov 22 '17 at 14:09
• I think the pure Neumann case is excluded (although I couldn't find the precise point where it explicitly is) because the differential operator is not coercive on the Sobolev space if there is not a small Dirichlet part of the boundary: the operator lacks the classical "$+1$" or "$+u$", or the space lacks a condition which exludes constant functions other than the constant zero function, such as the mean over $\Omega$ as you posed it. I would expect this to be a more or less straightforward modification, though. If this is still helpful to you, I'll happily post the answer of course. – Hannes Nov 28 '17 at 14:39

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".