If $u\in H^1(\Omega)$ where $\Omega$ is a bounded domain, $u$ satisfies $u\cdot n=0$ and $\nabla\times u=0$ on the boundary, how to prove the following trace inequality:
$$||u||_{L^2(\partial\Omega)}^2\leq C||\nabla u||_{L^2(\Omega)} ||u||_{L^2(\Omega)}$$
It is an inequality in P.L.Lions' book. Any suggestions are welcome!