Function with divergence, curl and normal trace on boundary equals zero is zero

Question

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.

Answer 1

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.

Answer 2

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.