Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality
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.
- Publication:
-
Zeitschrift Angewandte Mathematik und Mechanik
- Pub Date:
- September 2014
- DOI:
- 10.1002/zamm.201300059
- arXiv:
- arXiv:1303.1387
- Bibcode:
- 2014ZaMM...94..784N
- Keywords:
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- Mathematics - Analysis of PDEs;
- 74A35;
- 74A30;
- 74B20