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