On an extension of the first Korn inequality to incompatible tensor fields on domains of arbitrary dimensions
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Contributors
Abstract
For a bounded domain Ω in RN with Lipschitz boundary Γ = ∂Ω and a relatively open and non-empty subset Γt of Γ, we prove the existence of a positive constant c such that inequality c||T ||L2(Ω,RN×N) ≤ ||sym T ||L2(Ω,RN×N) + || Curl T ||L2(Ω,RN×N(N−1)/2) holds for all tensor fields T ∈ H(Curl; Γt,Ω,RN×N), this is, for all T : Ω → RN×N which are square-integrable and possess a row-wise square-integrable rotation tensor field Curl T : Ω → RN×N(N−1)/2 and vanishing row-wise tangential trace on Γt. For compatible tensor fields T = ∇v with v ∈ H1(Ω,RN) having vanishing tangential Neumann trace on Γt the inequality reduces to a non-standard variant of the first Korn inequality since Curl T = 0, while for skew-symmetric tensor fields T the Poincaré inequality is recovered. If Γt = ∅, our estimate still holds at least for simply connected Ω and for all tensor fields T ∈ H(Curl; Ω,RN×N) which are L2(Ω,RN×N)-perpendicular to so(N), i.e., to all skew-symmetric constant tensors.
Details
Original language | English |
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Title of host publication | Computational Methods in Applied Sciences |
Publisher | Springer Netherland |
Pages | 139-159 |
Number of pages | 21 |
Publication status | Published - 2014 |
Peer-reviewed | Yes |
Publication series
Series | Computational Methods in Applied Sciences |
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Volume | 34 |
ISSN | 1871-3033 |
External IDs
ORCID | /0000-0003-4155-7297/work/145224254 |
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