## G. Csato - O. Kneuss - D. Rajendran

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

created by csato on 12 Nov 2021

[

BibTeX]

*preprint*

**Inserted:** 12 nov 2021

**Year:** 2017

**Abstract:**

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.