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Let $\Omega$ be a convex polyhedron in $\mathbf{R}^3$ with boundary $\partial\Omega$ consisting of $N$ polygons $\{\Gamma_j\}_{j=1}^N$. It's well known that in general

$$ H^s(\partial \Omega) \neq \prod_{j=1}^N H^{s}(\Gamma_j) $$

since the latter space says nothing about the behaviour of the functions at the edges and vertices of the polyhedron. I would like to know what additional compatibility conditions are required so that

$$ \left\{ u \in \prod_{j=1}^N H^{s}(\Gamma_j)\quad \mathrm{and} \quad u\,\,\mathrm{obeys}\,\, \mathrm{conditions}\,\, (...) \right\} = H^s(\partial\Omega) $$

I only care about the case $s=1$, but a reference for the more general case would be interesting to see. I know of a good reference for the necessary compatibility conditions for planar polygons, e.g. Grisvard's book Elliptic Problems in non-smooth domains (Th. 1.5.2.3). However, I can't find much in the case of polyhedrons in $\mathbf{R}^3$.

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  • $\begingroup$ How do you define $H^s(\partial\Omega)$? The boundary $\partial\Omega$ is not a smooth manifold so it can be hard to define it intrinsically over it, but the quotient $H^{s+1/2}(\Omega)/H^{s+1/2}_0(\Omega)$ gives a nice extrinsic definition. $\endgroup$
    – Joonas Ilmavirta
    Commented Jul 16, 2015 at 14:49

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I do not own the book but you might find something in Mazya's and Rossmann's Elliptic Equations in Polyhedral Domains. Hope this might help...

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