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On the boundary conditions in estimating ∇ω by div ω and curl ω
- Source :
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 149:739-760
- Publication Year :
- 2018
- Publisher :
- Cambridge University Press (CUP), 2018.
-
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 ofωvanishes 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.
Details
- ISSN :
- 14737124 and 03082105
- Volume :
- 149
- Database :
- OpenAIRE
- Journal :
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Accession number :
- edsair.doi...........86ad8d016061c0c878288a653c62ad09