Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity. Issue 1 (December 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: 2013-12
• Subjects: Applied mathematics -- Periodicals; Engineering mathematics -- Periodicals; Mathematical physics -- Periodicals; 519
• DOI: 10.1002/pamm.201310176
• Languages: English
• ISSNs: 1617-7061

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>Let <tex-math notation="tex"><![CDATA[$\Omega\subset R^N$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cnj8" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> be bounded Lipschitz and <tex-math notation="tex"><![CDATA[$\emptyset\neq\Gamma\subset \partial\Omega$]]></tex-math><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 <xref ref-type="link" rid="eqn1">1</xref>:<disp-formula content-type="mathematics" id="eqn1"><label>1</label><alternatives><graphic position="anchor" mimetype="image" xlink:href="ark:/27927/pgg3z40cp45" orientation="portrait" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math notation="TeX"><![CDATA[$\nabla \zeta = G \zeta, \quad \zeta \mid_{\Gamma} = 0$]]></tex-math></alternatives></disp-formula> vanishes if <tex-math notation="tex"><![CDATA[$G \in {\rm L}^1(\Omega;{\rm I\!R}^{(N \times N)\times N})$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cnmc" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> and <tex-math notation="tex"><![CDATA[$\zeta \in {\rm W}^{1, 1}(\Omega;{\rm I\!R}^N)$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cnkt" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, (e.g. <tex-math notation="tex"><![CDATA[$\zeta \in L^2, G \in L^2$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cnpg" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />). We prove <disp-formula content-type="mathematics" id="di-ueqn-6"><alternatives><graphic position="anchor" mimetype="image" xlink:href="ark:/27927/pgg3z40cnnx" orientation="portrait" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math notation="TeX"><![CDATA[$|\!|\!|\cdot |\!|\!| : {\rm C}^\infty_0(\Omega, \Gamma;{\rm I\!R}^3)\to[0, \infty), \quad u \mapsto \left\Vert {\rm sym}(\nabla uP^{-1})\right\Vert_{{\rm L}^2(\Omega)}$]]></tex-math></alternatives></disp-formula> to be a norm if <tex-math notation="tex"><![CDATA[$P \in {\rm L}^\infty (\Omega;{\rm I\!R}^{3\times 3})$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cn10" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> with <tex-math notation="tex"><![CDATA[${\rm Curl}\; P \in {\rm L}^p (\Omega;{\rm I\!R}^{3\times 3})$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cn2j" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, <tex-math notation="tex"><![CDATA[${\rm Curl}\; P^{-1} \in {\rm L}^q (\Omega;{\rm I\!R}^{3\times 3})$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cn4n" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> for some <italic>p, q</italic> > 1 with 1/p + 1/q = 1 and <tex-math notation="tex"><![CDATA[${\rm det}P \geq c^{+} > 0$]]></tex-math><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 <tex-math notation="tex"><![CDATA[$\Phi \in {\rm H}^1(\Omega;{\rm I\!R}^{3}), \Omega \in {\rm I\!R}^{3}$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cn0f" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, satisfy <tex-math notation="tex"><![CDATA[${\rm sym} (\nabla\Phi^{\rm T} \nabla\Psi) = 0$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cnch" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> for some <tex-math notation="tex"><![CDATA[$\Psi \in {\rm W}^{1, \infty} (\Omega;{\rm I\!R}^{3}) \cap {\rm H}^2 (\Omega;{\rm I\!R}^{3})$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cnbz" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> with <tex-math notation="tex"><![CDATA[${\rm det}\nabla\Psi \geq c^{+} > 0$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cn9d" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. Then there are <tex-math notation="tex"><![CDATA[$a \in {\rm I\!R}^{3}$]]></tex-math><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 <tex-math notation="tex"><![CDATA[$A \in {\rm so}(3)$]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3z40cn79" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, such that <tex-math notation="tex"><![CDATA[$\Phi = A\Psi +a$]]></tex-math><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)</p> </abstract>