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

TL;DR

This paper demonstrates that a generalized Korn's second inequality with variable coefficients can fail even under conditions that typically imply ellipticity, highlighting limitations in the theory of coerciveness for certain elliptic systems.

Contribution

It provides counterexamples showing the failure of a generalized Korn's inequality with measurable coefficients, challenging assumptions about coerciveness in elliptic systems.

Findings

Counterexamples show failure of generalized Korn's inequality

Ellipticity conditions do not guarantee coerciveness

Garding's inequality can be violated for positive quadratic forms

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.