# A property on the Green-St Venant strain tensor

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.

Show that

$u\in H^1(\Omega), E(u)\in L^r(\Omega), r\ge1\Rightarrow u\in W^{1,2r}(\Omega)$.

Here $H^1, W^{1,k}$ are standard Sobolev spaces, $\Omega$ is bounded domain in $R^3$.

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'.

I don't have a clue on it. Any idea and/or comment are very much appreciated.

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Consider the trace of E. –  Michael Renardy Apr 24 '12 at 14:30
ok, let me try: consider the diagonal element, i.e., for each i=1,2,3, $E_{ii}={\partial u_i\over\partial x_i}+\sum_k |{\partial u_k\over\partial x_i }|^2$. 1. $r\le 2$ $u\in H^1\Rightarrow u_i\over\partial x_i \in L^2\Rightarrow u_i\over\partial x_i\in L^r$. $E_{ii}\in L^r\Rightarrow \sum_k |{\partial u_k\over\partial x_i }|^2\in L^r\Rightarrow \nabla u\in L^{2r}$. By the classical Poincare inequality here en.wikipedia.org/wiki/Poincar%C3%A9_inequality and note $u\in L^1$, we have $u\in L^{2r}$, done. –  pde_bk Apr 24 '12 at 19:07
2. for r>2 $E_{ii}\in L^r\Rightarrow E_{ii}\in L^2\Rightarrow |\nabla u|^2\in L^2$ since ${\partial u_i\over\partial x_i}\in L^2$. Hence $\nabla u\in L^4$ and by Poincare inequality again, we have $u\in L^4$. Hence if $r<\leq 4$, the proof is completed. Otherwise start with $u\in L^4$ and continue. –  pde_bk Apr 24 '12 at 19:15
haha, looks good, thanks M.R.! –  pde_bk Apr 24 '12 at 19:17