Uniqueness of Integrable Solutions ∇ζ = G ζ, ζ∣ Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity

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 system 1: 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)