Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality. Issue 9 (10th June 2013) - Neff, P.; Pompe, W.
Abstract We show that the following generalized version of Korn's second inequality with nonconstant measurable matrix valued coefficients P: Ω ⊂ ℝ3 → ℝ3× 3 <alternatives> <![CDATA[$$ \Vert DuP+(DuP)^T\Vert _q+\Vert u \Vert _q\geq c \Vert Du\Vert_q\quad \hbox{for $u\in W_0^{1, q}(\Omega;\mathbb{R}^3), \quad 1 \le q \le \infty$} $$]]> </alternatives> is in general false, even if P ∈ SO(3), while the Legendre‐Hadamard condition and ellipticity on ℂn for the quadratic form |Du P+(DuP)T|2 is satisfied. Thus Gårding's inequality may be violated for formally positive quadratic forms.