Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality. Issue 9 (10th June 2013)

• Record Type: Journal Article
• Authors: Neff, P.; Pompe, W.
• Is Part Of: Zeitschrift für angewandte Mathematik und Mechanik. Volume 94:Issue 9(2014)
• Page Start: 784
• Page End: 790
• Publication Date: 2013-06-10
• Subjects: Mathematics -- Periodicals; Mechanics, Applied -- Periodicals; Engineering -- Periodicals; 519
• DOI: 10.1002/zamm.201300059
• Languages: English
• ISSNs: 0044-2267

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>We show that the following generalized version of Korn's second inequality with nonconstant measurable matrix valued coefficients P: Ω ⊂ ℝ³ → ℝ^{3× 3}<disp-formula content-type="mathematics" id="di-ueqn-1"><alternatives><graphic position="anchor" mimetype="image" xlink:href="ark:/27927/pgh12znknmg" orientation="portrait" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math notation="LaTeX"><![CDATA[$$ \Vert DuP+(DuP)^T\Vert _q+\Vert u \Vert _q\geq c \Vert Du\Vert_q\quad \hbox{for $u\in W_0^{1, q}(\Omega;\mathbb{R}^3), \quad 1 \le q \le \infty$} $$]]></tex-math></alternatives></disp-formula> 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.</p> </abstract>