A canonical extension of Kornʼs first inequality to H ( Curl ) motivated by gradient plasticity with plastic spin
Patrizio Neff, Dirk Pauly, Karl‐Josef Witsch · Comptes Rendus Mathématique · 2011
We prove a Korn-type inequality in H ∘ ( Curl ; Ω , R 3 × 3 ) for tensor fields P mapping Ω to R 3 × 3 . More precisely, let Ω ⊂ R 3 be a bounded domain with connected Lipschitz boundary ∂ Ω . Then, there exists a constant c > 0 such that c ‖ P ‖ L 2 ( Ω , R 3 × 3 ) ⩽ ‖ sym P ‖ L 2 ( Ω , R 3 × 3 ) + ‖ Curl P ‖ L 2 ( Ω , R 3 × 3 ) holds for all tensor fields P ∈ H ∘ ( Curl ; Ω , R 3 × 3 ) , i.e., all P ∈ H ( Curl ; Ω , R 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 ∈ H 1 ( Ω , R 3