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On the boundary conditions in estimating ∇ω by div ω and curl ω

Part of: Inequalities Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  27 December 2018

Gyula Csató ,
Olivier Kneuss  and
Dhanya Rajendran
Gyula Csató
Affiliation:
Departamento de Matemática, Universidad de Concepción, Concepcion, Chile ([email protected])
Olivier Kneuss
Affiliation:
Departamento de Matemática, Universidade Federal do, Rio de Janeiro, Brasil ([email protected])
Dhanya Rajendran
Affiliation:
Departamento de Ingenería Matemática, Universidad de Concepción, Concepcion, Chile ([email protected])
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Abstract

In this paper, we study under what boundary conditions the inequality

$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$
holds true. It is known that such an estimate holds if either the tangential or normal component of ω vanishes on the boundary ∂Ω. 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.

Keywords

Gaffney inequalitydivergence and curl operatorstangential and normal componentsboundary conditions

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 35Q30: Navier-Stokes equations 35Q61: Maxwell equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 739 - 760
Copyright
Copyright © Royal Society of Edinburgh 2018 

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