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

TL;DR

This paper investigates the boundary conditions necessary for a key inequality relating the gradient, curl, and divergence of a vector field to hold, extending known conditions and providing new interpolation results in two dimensions.

Contribution

It generalizes the boundary conditions under which the inequality holds and introduces an interpolation result between classical boundary conditions in two dimensions.

Findings

The inequality holds under more general boundary conditions than previously known.

Vanishing tangential component is a special case of a broader boundary condition.

An interpolation result between boundary conditions is established in two dimensions.

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.