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We show that the following generalized version of Korn's second inequalitywith nonconstant measurable matrix valued coefficients P  ||DuP+(DuP)^T||_q+||u||_q >= c ||Du||_q for u in W_0^{1,q}({\Omega};R^3),1<q<{\infty} is in general false, even if P is in SO(3), while theLegendre-Hadamard condition and ellipticity on C^n for the quadratic form |DuP+(DuP)^T|^2 is satisfied. Thus Garding's inequality may be violated forformally positive quadratic forms.

Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality