1
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Maz'ya and Rossman's Elliptic equations in polyhedral domains seems promising, and may be more digestible than the original papers. $\endgroup$ – Willie Wong Nov 21 '17 at 14:19
  • 1
    $\begingroup$ Maybe this paper could also be useful. $\endgroup$ – Hannes Nov 21 '17 at 14:36
  • $\begingroup$ @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! $\endgroup$ – fred Nov 22 '17 at 13:59
  • $\begingroup$ @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? $\endgroup$ – fred Nov 22 '17 at 14:09
  • $\begingroup$ 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. $\endgroup$ – Hannes Nov 28 '17 at 14:39
3
$\begingroup$

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".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.