On the boundary conditions in estimating $\nabla \omega$ by div $\omega$ and curl $\omega .$

Abstract

In this paper we study under what boundary conditions the inequality $$\Vert \nabla \omega {\Vert }_{L^2(\Omega )}^2\le C\left(\Vert \mathrm{curl}\omega {\Vert }_{L^2(\Omega )}^2+\Vert \mathrm{div}\omega {\Vert }_{L^2(\Omega )}^2+\Vert \omega {\Vert }_{L^2(\Omega )}^2\right)$$ 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.