preprint

Inserted: 12 nov 2021

Year: 2017

Abstract:

In this paper we study under what boundary conditions the inequality $$\
\nabla\omega\
_{L²(\Omega)}^2\leq C\left(\
{\rm curl}\omega\
_{L²(\Omega)}²⁺ \
{\rm div}\omega\
_{L²(\Omega)}^2+\
\omega\
_{L²(\Omega)}^2\right) $$ 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.