Maxwell meets Korn: A new coercive inequality for tensor fields in RN×N with square-integrable exterior derivative
✍ Scribed by Patrizio Neff; Dirk Pauly; Karl-Josef Witsch
- Publisher
- John Wiley and Sons
- Year
- 2011
- Tongue
- English
- Weight
- 131 KB
- Volume
- 35
- Category
- Article
- ISSN
- 0170-4214
- DOI
- 10.1002/mma.1534
No coin nor oath required. For personal study only.
✦ Synopsis
For a bounded domain with connected Lipschitz boundary, we prove the existence of some c > 0, such that
urn:x-wiley:1704214:media:mma1534:mma1534-math-0002
holds for all square‐integrable tensor fields , having square‐integrable generalized “rotation” tensor fields and vanishing tangential trace on __∂__Ω, where both operations are to be understood row‐wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in . For compatible tensor fields T, that is, T = ∇ v, the latter estimate reduces to a non‐standard variant of Korn's first inequality in , namely
urn:x-wiley:1704214:media:mma1534:mma1534-math-0007
for all vector fields , for which ∇ v~n~,n = 1, … ,N, are normal at __∂__Ω. Copyright © 2012 John Wiley & Sons, Ltd.