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For 1 <  p  <  ∞ , we prove an  L   p  ‐version of the generalized Korn inequality for incompatible tensor fields  P  inW    0    1  ,    p    (  Curl  ;  Ω  ,    ℝ    3  ×  3    )  . More precisely, let   Ω  ⊂    ℝ    3be a bounded Lipschitz domain. Then there exists a constant  c  =  c ( p , Ω) > 0 such that‖  P    ‖L    p    (  Ω  ,    ℝ    3  ×  3    )    ≤  c‖  sym    P    ‖L    p    (  Ω  ,    ℝ    3  ×  3    )    +  ‖  Curl    P    ‖L    p    (  Ω  ,    ℝ    3  ×  3    )holds for all tensor fields   P  ∈    W    0    1  ,    p    (  Curl  ;  Ω  ,    ℝ    3  ×  3    )  , that is, for all   P  ∈    W    1  ,    p    (  Curl  ;  Ω  ,    ℝ    3  ×  3    )   with vanishing tangential trace   P  ×  ν  =  0   on  ∂ Ω where  ν  denotes the outward unit normal vector field to  ∂ Ω. For compatible   P  =  D  u  , this recovers an  L   p  ‐version of the classical Korn's first inequality and for skew‐symmetric   P  =  A   an  L   p  ‐version of the Poincaré inequality.

Nečas–Lions lemma revisited: An L p ‐version of the generalized Korn inequality for incompatible tensor fields