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Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$.

Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\partial\Omega)$ we can define the duality pairing of $\vec{q}\cdot \vec{\eta}\in H^{-1/2}(\partial\Omega)$ and $u$ as

$$ \langle \vec{q}\cdot \vec{\eta}, u\rangle_{H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega)} = \inf_{U\in H^1(\Omega)} \int_\Omega\,(\vec{q}\cdot \vec{\nabla} U+ U \vec{\nabla}\cdot \vec{q}) \mathrm{d} x $$ where the infenum is taken over all $U$ such that $U\big|_{\partial\Omega}=u.$ We also get $$\big|\, \langle \vec{q}\cdot \vec{\eta}, u\rangle_{H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega)}\big| \le C \|\vec{q}\|_{H^{\mathrm{div}}(\Omega)} \|u\|_{H^{1/2}(\partial\Omega)}.$$

My question is, is there analogous way to define

$$ \langle \vec{q}\cdot \vec{n}, u \rangle_{H^{-1/2}(e_1),H^{1/2}(e_1)}? $$

And can we get $$ \big|\langle \vec{q}\cdot \vec{n}, u \rangle_{H^{-1/2}(e_1),H^{1/2}(e_1)}\big|\le C \| \vec{q}\|_{H^{\mathrm{div}}(\Omega)} \| u \|_{H^{1/2}(e_1)}? $$

I would especially appreciate a reference where $H^{\pm 1/2}$ defined on proper subsets of the boundary are discussed.

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  • $\begingroup$ The problem of properly defining Sobolev spaces on a subset of the boundary of a domain is briefly addressed in the appendix of H. Berninger's dissertation (248pp): page.mi.fu-berlin.de/haiko/dissertation_Berninger.pdf You might especially find the references valuable. $\endgroup$
    – anonymous
    Jun 25, 2016 at 12:15

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