The second Friedrichs' inequalities? 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? 
 A: The answer is no. 
A pretty nice counter-example has been given by Stephen in this question: Friedrichs's inequality?

Backstory 1: 
$\mathbf{H}_0(\operatorname{div};\Omega) \cap \mathbf{H}(\mathbf{curl};\Omega)$ and $\mathbf{H}(\operatorname{div};\Omega) \cap \mathbf{H}_{0}(\mathbf{curl};\Omega)$ are compactly embedded in $\mathbf{L}^2(\Omega)$. However, if the homogeneous boundary condition is only on part of the boundary,  this compact embedding result is not true as of my knowledge. Rellich–Kondrachov compactness argument is used to prove the Poincaré-Friedrichs type inequality. 
For the estimate, please refer to Vector potentials in three-dimensional non-smooth domains by Cherif Amrouche, Christine Bernardi, Monique Dauge and Vivette Girault.
Backstory 2:
The function satisfies the boundary condition can be constructed implicitly using the following problem:
$$
\left\{
\begin{aligned}
-\Delta w = 0 & \text{ in }\Omega,
\\
w = 1 & \text{ on }\Gamma_2,
\\
\frac{\partial w}{\partial n} = 0& \text{ on }\Gamma_1.
\end{aligned}\right.
$$
Let the $\mathbf{u} = \nabla w$, we can see that this vector potential satisfies:
$$
\nabla\times \mathbf{u} = 0
\text{ and }
\nabla\cdot \mathbf{u} = 0,
\\
\mathbf{n}\times \mathbf{u}\big|_{\Gamma_2} = 0,\text{ and }
\mathbf{n}\cdot \mathbf{u}\big|_{\Gamma_1} = 0.
$$
Yet $\mathbf{u}$ itself is totally non-trivial.
A: 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.


*

*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.


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.
