On the boundary conditions in estimating $ abla \omega$ by div $\omega$ and curl $\omega.$
Gyula Csató, Olivier Kneuss, Rajendran Dhanya · arXiv (Cornell University) · 2017
In this paper we study under what boundary conditions the inequality $$\| abla\omega\|_{L^2(\Omega)}^2\leq C\left(\|{\rm curl}\omega\|_{L^2(\Omega)}^2+ \|{\rm div}\omega\|_{L^2(\Omega)}^2+\|\omega\|_{L^2(\Omega)}^2\right) $$ 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.