A canonical extension of Korn's first inequality to H(Curl) motivated by gradient plasticity with plastic spin

Research output: Contribution to journalResearch articleContributedpeer-review

Contributors

  • Patrizio Neff - , University of Duisburg-Essen (Author)
  • Dirk Pauly - , Institute of Analysis, University of Duisburg-Essen (Author)
  • Karl Josef Witsch - , University of Duisburg-Essen (Author)

Abstract

We prove a Korn-type inequality in H̊(Curl;Ω,R{double-struck}3×3) for tensor fields P mapping Ω to R{double-struck}3×3. More precisely, let Ω⊂R{double-struck}3 be a bounded domain with connected Lipschitz boundary ∂ Ω. Then, there exists a constant c> 0 such that (1) c||P||L2(Ω,R{double-struck}3×3)≤||symP||L2(Ω,R{double-struck}3×3)+||CurlP||L2(Ω,R{double-struck}3×3) holds for all tensor fields P∈H̊(Curl;Ω,R{double-struck}3×3), i.e., all P∈H(Curl;Ω,R{double-struck}3×3) with vanishing tangential trace on ∂ Ω. Here, rotation and tangential traces are defined row-wise. For compatible P, i.e., P=∇;v and thus Curl. P=0, where v∈H1(Ω,R{double-struck}3) are vector fields having components vn, for which ∇;vn are normal at ∂ Ω, the presented estimate (1) reduces to a non-standard variant of Korn's first inequality, i.e., c||∇;v||L2(Ω,R{double-struck}3×3)≤||sym∇;v||L2(Ω,R{double-struck}3×3). On the other hand, for skew-symmetric P, i.e., sym. P=0, (1) reduces to a non-standard version of Poincaré's estimate. Therefore, since (1) admits the classical boundary conditions our result is a common generalization of these two classical estimates, namely Poincaré's resp. Korn's first inequality.

Details

Original languageEnglish
Pages (from-to)1251-1254
Number of pages4
JournalComptes Rendus Mathematique
Volume349
Issue number23-24
Publication statusPublished - Dec 2011
Peer-reviewedYes

External IDs

ORCID /0000-0003-4155-7297/work/145224235

Keywords

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