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arXiv · 1709.06117

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

Gyula Csató·Olivier Kneuss·Dhanya Rajendran
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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.

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Gyula Csató, Olivier Kneuss, Dhanya Rajendran. 2017-09-18. On the boundary conditions in estimating $\nabla \omega$ by div $\omega$ and curl $\omega.$. https://arxiv.org/abs/1709.06117

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