Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity. Issue 1 (December 2013) - Lankeit, Johannes; Neff, Patrizio; Pauly, Dirk
Abstract Let <![CDATA[$\Omega\subset R^N$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cnj8" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> be bounded Lipschitz and <![CDATA[$\emptyset\neq\Gamma\subset \partial\Omega$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cnhq" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> relatively open. We show that the solution to the linear first order system 1: 1 <alternatives> <![CDATA[$\nabla \zeta = G \zeta, \quad \zeta \mid_{\Gamma} = 0$]]> </alternatives> vanishes if <![CDATA[$G \in {\rm L}^1(\Omega;{\rm I\!R}^{(N \times N)\times N})$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cnmc" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> and <![CDATA[$\zeta \in {\rm W}^{1, 1}(\Omega;{\rm I\!R}^N)$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cnkt" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, (e.g. <![CDATA[$\zeta \in L^2, G \in L^2$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cnpg" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />). We prove <alternatives> <![CDATA[$|\!|\!|\cdot |\!|\!| : {\rm C}^\infty_0(\Omega, \Gamma;{\rm I\!R}^3)\to[0, \infty), \quad u \mapsto \left\Vert {\rm sym}(\nabla uP^{-1})\right\Vert_{{\rm L}^2(\Omega)}$]]> </alternatives> to be a norm if <![CDATA[$P \in {\rm L}^\infty (\Omega;{\rm I\!R}^{3\times 3})$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn10" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> with <![CDATA[${\rm Curl}\; P \in {\rm L}^p (\Omega;{\rm I\!R}^{3\times 3})$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn2j" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, <![CDATA[${\rm Curl}\; P^{-1} \in {\rm L}^q (\Omega;{\rm I\!R}^{3\times 3})$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn4n" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> for some p, q > 1 with 1/p + 1/q = 1 and <![CDATA[${\rm det}P \geq c^{+} > 0$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn33" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. We give a new proof for the so called 'in‐finitesimal rigid displacement lemma' in curvilinear coordinates: Let <![CDATA[$\Phi \in {\rm H}^1(\Omega;{\rm I\!R}^{3}), \Omega \in {\rm I\!R}^{3}$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn0f" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, satisfy <![CDATA[${\rm sym} (\nabla\Phi^{\rm T} \nabla\Psi) = 0$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cnch" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> for some <![CDATA[$\Psi \in {\rm W}^{1, \infty} (\Omega;{\rm I\!R}^{3}) \cap {\rm H}^2 (\Omega;{\rm I\!R}^{3})$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cnbz" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> with <![CDATA[${\rm det}\nabla\Psi \geq c^{+} > 0$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn9d" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. Then there are <![CDATA[$a \in {\rm I\!R}^{3}$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn8v" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> and a constant skew‐symmetric matrix <![CDATA[$A \in {\rm so}(3)$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn79" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, such that <![CDATA[$\Phi = A\Psi +a$]]> <inline-graphic xlink:href="ark:/27927/pgg3z40cn6r" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)