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- W2963444786 abstract "For a bounded domain Ω⊂R3 with Lipschitz boundary Γ and some relatively open Lipschitz subset Γt≠∅ of Γ, we prove the existence of some c>0, such that(0.1)c‖T‖L2(Ω,R3×3)≤‖symT‖L2(Ω,R3×3)+‖CurlT‖L2(Ω,R3×3) holds for all tensor fields in H(Curl;Ω), i.e., for all square-integrable tensor fields T:Ω→R3×3 with square-integrable generalized rotation CurlT:Ω→R3×3, having vanishing restricted tangential trace on Γt. If Γt=∅, (0.1) still holds at least for simply connected Ω and for all tensor fields T∈H(Curl;Ω) which are L2(Ω)-perpendicular to so(3), i.e., to all skew-symmetric constant tensors. Here, both operations, Curl and tangential trace, are to be understood row-wise. For compatible tensor fields T=∇v, (0.1) reduces to a non-standard variant of the well known Korn's first inequality in R3, namelyc‖∇v‖L2(Ω,R3×3)≤‖sym∇v‖L2(Ω,R3×3) for all vector fields v∈H1(Ω,R3), for which ∇vn, n=1,…,3, are normal at Γt. On the other hand, identifying vector fields v∈H1(Ω,R3) (having the proper boundary conditions) with skew-symmetric tensor fields T, (0.1) turns to Poincaré's inequality since2c‖v‖L2(Ω,R3)=c‖T‖L2(Ω,R3×3)≤‖CurlT‖L2(Ω,R3×3)≤2‖∇v‖L2(Ω,R3). Therefore, (0.1) may be viewed as a natural common generalization of Korn's first and Poincaré's inequality. From another point of view, (0.1) states that one can omit compatibility of the tensor field T at the expense of measuring the deviation from compatibility through Curl T. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions and the Maxwell estimate." @default.
- W2963444786 created "2019-07-30" @default.
- W2963444786 creator A5041687955 @default.
- W2963444786 creator A5045667627 @default.
- W2963444786 creator A5046930021 @default.
- W2963444786 date "2015-02-01" @default.
- W2963444786 modified "2023-10-16" @default.
- W2963444786 title "Poincaré meets Korn via Maxwell: Extending Korn's first inequality to incompatible tensor fields" @default.
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- W2963444786 doi "https://doi.org/10.1016/j.jde.2014.10.019" @default.
- W2963444786 hasPublicationYear "2015" @default.
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