The second Friedrichs' inequalities?

Question

In paper On the Validity of Friedrichs' Inequalities，$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le C\Big(\|\nabla\cdot\mathbf{u}\|_{0,\Omega} +\|\nabla\times\mathbf{u}\|_{0,\Omega}\Big) $$ for all $\mathbf{u}\in\mathbf{H}_0(\operatorname{div};\Omega) \cap \mathbf{H}(\operatorname{curl};\Omega)$ or $\mathbf{u}\in\mathbf{H}(\operatorname{div};\Omega) \cap \mathbf{H}_0(\operatorname{curl};\Omega)$.

If $\mathbf{u}\in\mathbf{H}(\operatorname{div};\Omega) \cap \mathbf{H}(\operatorname{curl};\Omega)$ with mixed boundary conditions: $$\mathbf{n}\cdot\mathbf{u} = 0 \text{ on }\Gamma_1, \quad \text{ and }\quad \mathbf{n} \times \mathbf{u} = \mathbf{0} \text{ on }\Gamma_2, $$ where $\Gamma_1,\Gamma_2\neq\emptyset$, $\Gamma_1\cap \Gamma_2 = \emptyset$, and $\overline{\Gamma}_1\cup \overline{\Gamma}_2 = \partial \Omega$. Does the inequality $(1)$ hold?

Answer 1

I know this is an old question, but I think it's worth to improve the previous answer with some remarks for the benefit of future readers.

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• The space $\{ \mathbf u \in \mathbf{H}(\operatorname{\textbf{curl}}; \Omega) \cap H(\operatorname{\textbf{div}};\Omega) : \mathbf u \cdot \mathbf n |_{\Gamma_1} = 0, \ \mathbf u \times \mathbf n |_{\Gamma_2} = \mathbf 0 \}$ is in fact compactly embedded into $\mathbf L^2(\Omega)$ if $\Omega$ is merely weak Lipschitz (see reference $[1]$). On the other hand, it is not embedded into $\mathbf{H}^1(\Omega)$.
• One can thus obtain a Poincaré-Friedrichs inequality similar to $(1)$ even with mixed boundary conditions, but with the $\| \cdot \|_{0,\Omega}$ norm of the LHS and imposing an orthogonality condition with respect to a space of so-called "Dirichlet-Neumann" harmonic fields - vectors with zero curl and divergence and suitable null normal/tangential traces -. You may look at [$1$, Thm. 5.1]. Interestingly enough, the additional requirement prevents the above counterexample.

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Reference:

$[1]$ Sebastian Bauer, Dirk Pauly and Michael Schomburg. "The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions." SIAM Journal on Mathematical Analysis 48.4 (2016): 2912-2943.