On the boundary conditions in estimating nabla ω by div ω and curl ω.

In this paper we study under what boundary conditions the inequality $$\|\nabla\omega\|_{L^2(\Omega)}^2\leq C\left(\|{\rm curl}\omega\|_{L^2(\Omega)}^2+ \|{\rm div}\omega\|_{L^2(\Omega)}^2+\|\omega\|_{L^2(\Omega)}^2\right) $$ holds true. It is known that such an estimate holds if either the tangential or normal component of $\omega$ vanishes on the boundary $\partial\omega.$ We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions we give an interpolation result between these two classical boundary conditions.