Friedrichs's second inequality is stated as follows(see www.win.tue.nl/~drenth/Phd/friedrichs.ps): For all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n}\cdot\mathbf{u} = 0$ or $\mathbf{n} \times \mathbf{u} = \mathbf{0}$ on $\partial\Omega$ where $\Omega$ is a simply connected domain, then $$ \|\mathbf{u}\|_1 \le C_1 (\|\nabla\cdot\mathbf{u}\|_0 + \|\nabla\times\mathbf{u}\|_0). $$ My question is that if the boundary condition is satisfied only on the nonempty part of $\partial\Omega$, i.e., either $\mathbf{n}\cdot\mathbf{u} = 0$ or $\mathbf{n} \times \mathbf{u} = \mathbf{0}$ on $\emptyset \neq \Gamma \subset \partial\Omega$, does the inequality above hold?
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Thank Stephen Montgomery-Smith very much for your response. ^_^
In paper On the Validity of Friedrichs' Inequalities，$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ (1.4) \qquad \|\mathbf{u}\|_{1,\Omega} \le C(\|\nabla\cdot\mathbf{u}\|_{0,\Omega} +\|\nabla\times\mathbf{u}\|_{0,\Omega}) $$ for all $\mathbf{u}\in\mathbf{H}_0(div;\Omega) \cap \mathbf{H}(curl;\Omega)$ or $\mathbf{u}\in\mathbf{H}(div;\Omega) \cap \mathbf{H}_0(curl;\Omega)$.
If $\mathbf{u}\in\mathbf{H}(div;\Omega) \cap \mathbf{H}(curl;\Omega)$ with two types of boundary conditions: $\mathbf{n}\cdot\mathbf{u} = 0 \text{ on }\Gamma_1$ and $\mathbf{n} \times \mathbf{u} = \mathbf{0} \text{ on }\Gamma_2$, where $\Gamma_1,\Gamma_2\neq\emptyset$, does the inequality (1.4) hold?
Consider $\Omega = \{ x \in \mathbb R^2 : \frac12 < |x| < 1, x_1 > 0\}$, and let $u(x) = \frac x{|x|^2}$. Then $\nabla\cdot u = 0$ and $\nabla \times u = 0$. The condition $u \times n=0$ holds on the circular parts of the boundary, and the condition $u \cdot n=0$ holds on the straight parts of the boundary. But clearly $\|u\|_1 \ne 0$.