Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity. Issue 1 (29th November 2013)

• Record Type: Journal Article
• Authors: Lankeit, Johannes; Neff, Patrizio; Pauly, Dirk
• Is Part Of: Proceedings in applied mathematics and mechanics. Volume 13:Issue 1(2013)
• Page Start: 361
• Page End: 362
• Publication Date: 2013-11-29
• Subjects: Applied mathematics -- Periodicals; Engineering mathematics -- Periodicals; Mathematical physics -- Periodicals; 519
• DOI: 10.1002/pamm.201310176
• Languages: English
• ISSNs: 1617-7061

Abstract

Abstract: Let$\Omega\subset R^N$ be bounded Lipschitz and$\emptyset\neq\Gamma\subset \partial\Omega$ relatively open. We show that the solution to the linear first order system1 : 1 $\nabla \zeta = G \zeta, \quad \zeta \mid_{\Gamma} = 0$ vanishes if$G \in {\rm L}^1(\Omega;{\rm I\!R}^{(N \times N)\times N})$ and$\zeta \in {\rm W}^{1, 1}(\Omega;{\rm I\!R}^N)$, (e.g.$\zeta \in L^2, G \in L^2$ ). We prove $|\!|\!|\cdot |\!|\!| : {\rm C}^\infty_0(\Omega, \Gamma;{\rm I\!R}^3)\to[0, \infty), \quad u \mapsto \left\Vert {\rm sym}(\nabla uP^{-1})\right\Vert_{{\rm L}^2(\Omega)}$ to be a norm if$P \in {\rm L}^\infty (\Omega;{\rm I\!R}^{3\times 3})$ with${\rm Curl}\; P \in {\rm L}^p (\Omega;{\rm I\!R}^{3\times 3})$, ${\rm Curl}\; P^{-1} \in {\rm L}^q (\Omega;{\rm I\!R}^{3\times 3})$ for some p, q > 1 with 1/p + 1/q = 1 and${\rm det}P \geq c^{+} > 0$ . We give a new proof for the so called 'in‐finitesimal rigid displacement lemma' in curvilinear coordinates: Let$\Phi \in {\rm H}^1(\Omega;{\rm I\!R}^{3}), \Omega \in {\rm I\!R}^{3}$, satisfy${\rm sym} (\nabla\Phi^{\rm T} \nabla\Psi) = 0$ for some$\Psi \in {\rm W}^{1, \infty} (\Omega;{\rm I\!R}^{3}) \cap {\rm H}^2 (\Omega;{\rm I\!R}^{3})$ with${\rm det}\nabla\Psi \geq c^{+} > 0$ . Then there are$a \in {\rm I\!R}^{3}$ and a constant skew‐symmetric matrix$A \in {\rm so}(3)$, such that$\Phi = A\Psi +a$ . (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)