A property on the Green-St Venant strain tensor - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T17:52:41Z http://mathoverflow.net/feeds/question/95023 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95023/a-property-on-the-green-st-venant-strain-tensor A property on the Green-St Venant strain tensor pde_bk 2012-04-24T14:17:44Z 2012-04-24T14:17:44Z <p>Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient.</p> <p>Show that </p> <p>$u\in H^1(\Omega), E(u)\in L^r(\Omega), r\ge1\Rightarrow u\in W^{1,2r}(\Omega)$.</p> <p>Here $H^1, W^{1,k}$ are standard Sobolev spaces, $\Omega$ is bounded domain in $R^3$.</p> <p>It's an exercise(1.16) in P.G.Ciarlet's book "Mathematical Elasticity, Vol. I : Three-Dimensional Elasticity", cited as 'due to Luc Tartar'.</p> <p>I don't have a clue on it. Any idea and/or comment are very much appreciated.</p>