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arXiv · 1303.1387

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

Patrizio Neff·Waldemar Pompe
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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}({\Omega};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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Patrizio Neff, Waldemar Pompe. 2013-03-06. Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality. https://arxiv.org/abs/1303.1387

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