Function with divergence, curl and normal trace on boundary equals zero is zero Let $u\in H^1(\Omega)$ with $\nabla\times u=0$ in $\Omega\subset\mathbb{R}^3$ (open bounded domain), $u\times n=0$ on $\partial\Omega$ (where $n$ is a a normal vector to $\partial\Omega$), $\operatorname{div}(u)=0$ in $\Omega$ and $u\cdot n=0$ on $\partial\Omega$.
Prove that $u=0$.
Thanks.
 A: For $u\in H^1(\Omega)$, you can write :
\begin{equation}
||u||_{H^1(\Omega)}\leq C \{ ||u||_{L^2(\Omega)}+ ||div\ u||_{L^2(\Omega)} + ||curl\ u||_{L^2(\Omega)} + ||u\cdot n||_{H^{\frac{1}{2}}(\partial\Omega)}\}
\end{equation}
or,
\begin{equation}
||u||_{H^1(\Omega)}\leq C \{ ||u||_{L^2(\Omega)}+ ||div\ u||_{L^2(\Omega)} + ||curl\ u||_{L^2(\Omega)} + ||u\times n||_{H^{\frac{1}{2}}(\partial\Omega)}\}.
\end{equation}
Now if $u\cdot n =0$ on $\partial\Omega$, then you have stronger estimate to write 
\begin{equation}
||u||_{H^1(\Omega)}\leq C \{ ||div\ u||_{L^2(\Omega)} + ||curl\ u||_{L^2(\Omega)}\}
\end{equation}
or, if $u\times n=0$ on $\partial\Omega$ then also
\begin{equation}
||u||_{H^1(\Omega)}\leq C \{ ||div\ u||_{L^2(\Omega)} + ||curl\ u||_{L^2(\Omega)}\}.
\end{equation}
Now, if $div\ u=curl\ u =0$ in $\Omega$ then it is straightforward to conclude $u = 0$ in $\Omega$.
Remark: The above statement holds for $u\in W^{1,p}(\Omega)$ but not necessarily for $u\in L^p(\Omega)$. 
Source: (1) "On the Stokes equations with the Navier type boundary conditions": Cherif Amrouche and Nour Seloula. 
A: I could not comment in the above answer by tuhin, but strangely it assumes $u=0$ to prove $u=0$!!!!
This result is actually far from trivial. It is not necessarily true if you only assume $\Omega$ to be just open and bounded. However, if you assume $\partial\Omega$ to be $C^{2},$ the result is true. 
See Theorem 5.21 in the book "The Pullback Equation for Differential Forms" by Csato, Dacorogna, Kneuss,  Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, New York, 2012. 
The estimate, simplified to the case of the question is 
$$ \lVert u \rVert_{H^1(\Omega)} \leq C \left( \lVert \operatorname*{curl}u \rVert_{L^2(\Omega)} + \lVert \operatorname*{div}u \rVert_{L^2(\Omega)} + \lVert n \times u \rVert_{H^{1/2}(\partial\Omega)} 
+ \lVert n \cdot u \rVert_{L^1(\partial\Omega)} \right).$$
The estimate itself is a corollary of Theorem 2 in Bolik's paper--- Bolik, Jürgen, "H. Weyls Boundary Value Problems for Differential Forms",  Differential Integral Equations 14 (2001), no. 8, 937–952.   
