Calculus of Variations and Geometric Measure Theory
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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

ArXiv: 1709.06117 PDF

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.

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