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 LegendreHadamard 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:

 Mathematics  Analysis of PDEs;
 74A35;
 74A30;
 74B20